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All questions of Algebra for Civil Engineering (CE) Exam
If f(x) = -2x + 8 & f(p) = 16, find the value of p?- a)-12
- b)-8
- c)-4
- d)4
- e)12
Correct answer is option 'C'. Can you explain this answer?
If f(x) = -2x + 8 & f(p) = 16, find the value of p?
a)
-12
b)
-8
c)
-4
d)
4
e)
12
|
Palak Saha answered |
Given, f(x) = -2x + 8 and f(p) = 16
To find: The value of p
Solution:
Substitute f(p) = 16 in the equation f(x) = -2x + 8
f(p) = -2p + 8 = 16
-2p = 16 - 8
-2p = 8
Divide both sides by -2
p = 8/-2
p = -4
Therefore, the value of p is -4.
Hence, the correct option is (c) -4.
To find: The value of p
Solution:
Substitute f(p) = 16 in the equation f(x) = -2x + 8
f(p) = -2p + 8 = 16
-2p = 16 - 8
-2p = 8
Divide both sides by -2
p = 8/-2
p = -4
Therefore, the value of p is -4.
Hence, the correct option is (c) -4.
A polynomial function P(x) is defined as,
P(x) = 4x3 – 2x2
If P (z -2) =0 & z ≠ 2, find the value of z?
- a)-3/2
- b)+1/2
- c)1
- d)+5/2
- e)+7/2
Correct answer is option 'D'. Can you explain this answer?
A polynomial function P(x) is defined as,
P(x) = 4x3 – 2x2
If P (z -2) =0 & z ≠ 2, find the value of z?
a)
-3/2
b)
+1/2
c)
1
d)
+5/2
e)
+7/2
|
Siddharth Pillai answered |
Solution:
Given, P(x) = 4x3 - 2x2
Let z-2 = k
P(z-2) = P(k) = 4k3 - 2k2
We know that P(k) = 0
Therefore, 4k3 - 2k2 = 0
2k2(2k - 1) = 0
k = 0 or k = 1/2
Now, z-2 = k
So, z = k+2
For k = 0, z = 2
For k = 1/2, z = 5/2
Therefore, the possible values of z are 2 and 5/2, but since z ≤ 2, the only possible value of z is 5/2.
Hence, the correct answer is option (D) 5/2.
Given, P(x) = 4x3 - 2x2
Let z-2 = k
P(z-2) = P(k) = 4k3 - 2k2
We know that P(k) = 0
Therefore, 4k3 - 2k2 = 0
2k2(2k - 1) = 0
k = 0 or k = 1/2
Now, z-2 = k
So, z = k+2
For k = 0, z = 2
For k = 1/2, z = 5/2
Therefore, the possible values of z are 2 and 5/2, but since z ≤ 2, the only possible value of z is 5/2.
Hence, the correct answer is option (D) 5/2.
If f(x) = 3x + 6, then what is the value of f (2) + f(7)? - a)f(8)
- b)f(9)
- c)f(10)
- d)f(11)
- e)f(12)
Correct answer is option 'D'. Can you explain this answer?
If f(x) = 3x + 6, then what is the value of f (2) + f(7)?
a)
f(8)
b)
f(9)
c)
f(10)
d)
f(11)
e)
f(12)
|
|
Arjun Iyer answered |
Solution:
Given: f(x) = 3x - 6
We need to find f(2) + f(7)
Substituting x = 2 and x = 7 in the given equation, we get:
f(2) = 3(2) - 6 = 0
f(7) = 3(7) - 6 = 15
Therefore, f(2) + f(7) = 0 + 15 = 15
Hence, the correct answer is option D.
Given: f(x) = 3x - 6
We need to find f(2) + f(7)
Substituting x = 2 and x = 7 in the given equation, we get:
f(2) = 3(2) - 6 = 0
f(7) = 3(7) - 6 = 15
Therefore, f(2) + f(7) = 0 + 15 = 15
Hence, the correct answer is option D.
If g(x) = -2x2 + 8 and g (-q) = -24, which of the following could be the value of q?- a)-4
- b)-2
- c)-1
- d)1
- e)2
Correct answer is option 'A'. Can you explain this answer?
If g(x) = -2x2 + 8 and g (-q) = -24, which of the following could be the value of q?
a)
-4
b)
-2
c)
-1
d)
1
e)
2
|
|
Parth Singh answered |
Given:
- g(x) = -2x^2 + 8
- g(-q) = -24
To find:
- Possible values of q
Solution:
Substitute -q in place of x in g(x) to get g(-q)
g(-q) = -2(-q)^2 + 8
g(-q) = -2q^2 + 8
Given that g(-q) = -24, we can set up the equation:
-2q^2 + 8 = -24
Simplifying, we get:
-2q^2 = -32
Dividing by -2, we get:
q^2 = 16
Taking the square root of both sides, we get:
q = ±4
Therefore, the possible values of q are -4 and 4.
Option A (-4) is the correct answer.
- g(x) = -2x^2 + 8
- g(-q) = -24
To find:
- Possible values of q
Solution:
Substitute -q in place of x in g(x) to get g(-q)
g(-q) = -2(-q)^2 + 8
g(-q) = -2q^2 + 8
Given that g(-q) = -24, we can set up the equation:
-2q^2 + 8 = -24
Simplifying, we get:
-2q^2 = -32
Dividing by -2, we get:
q^2 = 16
Taking the square root of both sides, we get:
q = ±4
Therefore, the possible values of q are -4 and 4.
Option A (-4) is the correct answer.
If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?- a)29
- b)34
- c)81
- d)86
- e)91
Correct answer is option 'A'. Can you explain this answer?
If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?
a)
29
b)
34
c)
81
d)
86
e)
91
|
Lavanya Menon answered |
Given:
- Sum of the first 5 terms of an arithmetic sequence = 120
- Sum of the next 5 terms of the same arithmetic sequence = 245
- Let the first term of this arithmetic sequence be x1 and let the common difference be d.
To Find:
- 4th term of the arithmetic sequence.
- So the 4th term of the sequence will become x1+3d
- So we need to find the value of x1 and d or the value of x1+3d to find the 4th term of the sequence.
Approach:
- We know that the sum of first n terms of the Arithmetic Sequence is given as
where n is the number of terms in the arithmetic sequence.- Using the formula above for the sum of first 5 terms of the sequence, we will get an equation in terms of and common difference d, as we are given the sum of first 5 terms of the sequence.
- We are also given the sum of next 5 terms of the sequence. So, we will be able to calculate the sum of first 10 terms of the sequence.
→ Sum of first 10 terms of sequence = Sum of first 5 terms + sum of next 5 terms. - Using the formula above for the sum of first 10 terms of sequence, we will get another equation in terms of x1 and common difference d.
- Using these two equations in x1 and d, we will be able to calculate the value of x1 and d.
- Knowing the values of x1 and d, we will be able to calculate the fourth term of the sequence, which is equal to x1+3d
Working out:
- Sum of first 5 terms of the arithmetic sequence = 120
- Putting this in formula of sum of first n terms, where n=5 and z=120, we get
- Sum of the next 5 terms of the sequence = 245
- Sum of the first 10 terms of the sequence = Sum of the first five terms + Sum of the next five terms.
- Sum of the first 10 terms of the sequence = 120+245 = 365
- Sum of the first 10 terms of the sequence = 120+245 = 365
- Now, using the formula of the sum of first n terms of an arithmetic sequence, we get
- Solving Equations 1 and 2.
- Multiplying ‘equation 1’ by 2, we have 10x1+20d =240 ...(Equation 3)
Now that we have values of x1 and d. The value of 4th term of the sequence will be
⇒ x1+3(d)=14+3(5)=29
Answer:
- The value of 4th term of the sequence is 29.
- Hence the correct answer is option A
Alternate method
- Let the first term be 'a' and common dfference between any two cosecutive terms be 'd'
Therefore,
- 1st term = a
- 5th term = a + 4d
- 6th term = a + 5d
- 10th term = a + 9d
- Average of first five terms of an arithemetic sequence = (First term + Last term)/2 = (a + a +4d) / 2 = a + 2d
- Sum of first five terms = Average of first five terms * 5 = (a + 2d) * 5 = 120
- a + 2d = 120/5 = 24 ---------------- Eq(1)
- Average of next five terms of the arithemetic sequence = (First term + Last term)/2 = (a+ 5d + a +9d) / 2 = a + 7d
- Sum of five terms = Average of five terms * 5 = (a + 7d) * 5 = 245
- a + 7d = 245/5 = 49---------------- Eq(2)
Solving Eq(1) and (2) we get
- d = 5
- 4th term =
- a + 3d = (a+2d)+ d = 24 + 5 = 29
Correct Answer: Option A
$x = 6x +4 and £x = 8x – 2Find the value of x for which $x = £x?- a)-3
- b)-2
- c)1
- d)2
- e)3
Correct answer is option 'E'. Can you explain this answer?
$x = 6x +4 and £x = 8x – 2
Find the value of x for which $x = £x?
a)
-3
b)
-2
c)
1
d)
2
e)
3
|
|
Bhavana Kulkarni answered |
Solution:
Given, $x = 6x + 4$ and $x = 8x - 2$
Simplifying the above equations, we get
$5x = -4$ and $7x = 2$
Solving for x, we get
$x = -\frac{4}{5}$ and $x = \frac{2}{7}$
Since both values of x are not equal, we cannot find the value of x for which $x = x$ from the given equations.
Therefore, the answer is none of the given options.
Given, $x = 6x + 4$ and $x = 8x - 2$
Simplifying the above equations, we get
$5x = -4$ and $7x = 2$
Solving for x, we get
$x = -\frac{4}{5}$ and $x = \frac{2}{7}$
Since both values of x are not equal, we cannot find the value of x for which $x = x$ from the given equations.
Therefore, the answer is none of the given options.
Given the equation x2 + bx + c = 0, where b and c are constants, what is the value of c?(1) The sum of the roots of the equation is zero.(2) The sum of the squares of the roots of the equation is equal to 18.- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- d)EACH statement ALONE is sufficient.
- e)Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'C'. Can you explain this answer?
Given the equation x2 + bx + c = 0, where b and c are constants, what is the value of c?
(1) The sum of the roots of the equation is zero.
(2) The sum of the squares of the roots of the equation is equal to 18.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
c)
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
d)
EACH statement ALONE is sufficient.
e)
Statements (1) and (2) TOGETHER are NOT sufficient.
|
|
Kiran Nambiar answered |
Solution:
Given equation is x2 bx c = 0, where b and c are constants
To find: value of c
Statement 1: The sum of the roots of the equation is zero.
Let the roots of the equation be α and β. Then, we know that:
α + β = −b
αβ = c
The sum of the roots is zero, so:
α + β = 0
−b = 0
b = 0
From the equation, we can conclude that c = αβ = 0. This statement alone is sufficient to answer the question.
Statement 2: The sum of the squares of the roots of the equation is equal to 18.
Let the roots of the equation be α and β. Then, we know that:
α + β = −b
αβ = c
We are given that:
α2 + β2 = 18
Squaring the equation α + β = −b, we get:
α2 + 2αβ + β2 = b2
Substituting the values of αβ and b2, we get:
α2 + 2c + β2 = b2
α2 + β2 = b2 − 2c
Substituting the given value of α2 + β2, we get:
18 = b2 − 2c
We can solve for c in terms of b:
c = (b2 − 18)/2
However, we do not know the value of b, so we cannot determine the value of c. This statement alone is not sufficient to answer the question.
Together, statements 1 and 2 give us:
b = 0
α2 + β2 = b2 − 2c = 0 − 2c = −2c
Substituting the given value of α2 + β2, we get:
18 = −2c
c = −9
Therefore, both statements together are sufficient to answer the question. The answer is (C).
Given equation is x2 bx c = 0, where b and c are constants
To find: value of c
Statement 1: The sum of the roots of the equation is zero.
Let the roots of the equation be α and β. Then, we know that:
α + β = −b
αβ = c
The sum of the roots is zero, so:
α + β = 0
−b = 0
b = 0
From the equation, we can conclude that c = αβ = 0. This statement alone is sufficient to answer the question.
Statement 2: The sum of the squares of the roots of the equation is equal to 18.
Let the roots of the equation be α and β. Then, we know that:
α + β = −b
αβ = c
We are given that:
α2 + β2 = 18
Squaring the equation α + β = −b, we get:
α2 + 2αβ + β2 = b2
Substituting the values of αβ and b2, we get:
α2 + 2c + β2 = b2
α2 + β2 = b2 − 2c
Substituting the given value of α2 + β2, we get:
18 = b2 − 2c
We can solve for c in terms of b:
c = (b2 − 18)/2
However, we do not know the value of b, so we cannot determine the value of c. This statement alone is not sufficient to answer the question.
Together, statements 1 and 2 give us:
b = 0
α2 + β2 = b2 − 2c = 0 − 2c = −2c
Substituting the given value of α2 + β2, we get:
18 = −2c
c = −9
Therefore, both statements together are sufficient to answer the question. The answer is (C).
If f(x) = 3x2 – 5x + 9 and g(x) = 4x – 5, then find the value of g( f(x)) at x = 3. - a)7
- b)51
- c)56
- d)79
- e)121
Correct answer is option 'D'. Can you explain this answer?
If f(x) = 3x2 – 5x + 9 and g(x) = 4x – 5, then find the value of g( f(x)) at x = 3.
a)
7
b)
51
c)
56
d)
79
e)
121
|
|
Kiran Nambiar answered |
The information provided in the question is:
f(x) = 3x2 – 5x + 9
g(x) = 4x – 5
We have to find out the value of g( f(x)) at x = 3.
f(x) = 3x2 – 5x + 9
f(3) = 3*(3)2 – 5*3 + 9
= 27 – 15 + 9
= 21
g(x) = 4x – 5
g(f(x)) = 4f(x) -5
g( f(3)) = 4f(3) – 5
= 4*21 – 5
= 84 – 5
= 79
Answer: Option (D)
List A consists of 10 distinct integers arranged in ascending order. Is the difference between the sixth term and the fifth term of list A greater than 5?(1) The difference between any two integers in list A is a multiple of 5.(2) The median of the list is an integer.- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'C'. Can you explain this answer?
List A consists of 10 distinct integers arranged in ascending order. Is the difference between the sixth term and the fifth term of list A greater than 5?
(1) The difference between any two integers in list A is a multiple of 5.
(2) The median of the list is an integer.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
|
|
Rhea Gupta answered |
Given:
- List A = {a1, a2, a3, a4, a5, a6, a7, a8, a9, a10}
- consists of all integers,
- where a10 > a9> a8…> a2> a1
To Find: Is a6 – a5 > 5?
Step 4: Analyse Statement 2 independently
The median of the list is an integer.
- Median =?
- List contains 10 elements
- a5 & a6 are integers at the centre of the list (in ascending order)
- Median of the list = a5+a62
-
- = m, where m is an integer
- Rearranging, we get a5+ a6 = 2m = even
- If sum of a6 & a5 is even, then
- Difference of a6 & a5 is also even
- No information about values of the terms , so cannot tell if (a6 – a5) > 5
- Hence, statement 2 is insufficient to answer the question.
Step 5: Analyse Both Statements Together (if needed)
- From statement 1, a6 – a5 = {5, 10, 15}
- From statement 2, a6 – a5 = even
- Combining the two,
- a6 – a5 = even multiples of 5 = {10, 20, 30…}
- a6 – a5 > 5
- So, we can answer the question.
- Hence statement 1 and statement 2 together are sufficient to arrive at a definite answer.
Answer: C
Step 3: Analyse Statement 1 independently
The difference between any two integers in list A is a multiple of 5.
- Difference between any two terms of A can be = {5,10,15…}
- a6 – a5 = {5, 10, 15…}
- Since the difference can also = 5, we cannot be certain that a6 – a5 > 5.
- Hence, Statement 1 is insufficient to answer the question.
Ted and Robin start from the same point at 7 AM and drive in opposite directions. Ted doubles his speed after every 90 minutes whereas Robin reduces her speed by half after every 120 minutes. If Ted starts driving at a speed of 10 kilometers/hour and Robin starts driving at a speed of 120 kilometers/hour, how far in kilometers will they be from one another at 1 PM?- a)195
- b)485
- c)525
- d)645
- e)675
Correct answer is option 'D'. Can you explain this answer?
Ted and Robin start from the same point at 7 AM and drive in opposite directions. Ted doubles his speed after every 90 minutes whereas Robin reduces her speed by half after every 120 minutes. If Ted starts driving at a speed of 10 kilometers/hour and Robin starts driving at a speed of 120 kilometers/hour, how far in kilometers will they be from one another at 1 PM?
a)
195
b)
485
c)
525
d)
645
e)
675
|
|
Sounak Iyer answered |
To solve this problem, we need to calculate the distances covered by Ted and Robin separately and then find the sum of their distances.
Ted's journey:
Ted doubles his speed after every 90 minutes. So, he will have different speeds during different time intervals.
From 7 AM to 8:30 AM (90 minutes):
Ted's speed = 10 km/hour
Distance covered = Speed × Time = 10 km/hour × 1.5 hours = 15 km
From 8:30 AM to 10 AM (90 minutes):
Ted's speed = 2 × 10 km/hour = 20 km/hour
Distance covered = Speed × Time = 20 km/hour × 1.5 hours = 30 km
From 10 AM to 11:30 AM (90 minutes):
Ted's speed = 2 × 20 km/hour = 40 km/hour
Distance covered = Speed × Time = 40 km/hour × 1.5 hours = 60 km
From 11:30 AM to 1 PM (90 minutes):
Ted's speed = 2 × 40 km/hour = 80 km/hour
Distance covered = Speed × Time = 80 km/hour × 1.5 hours = 120 km
Total distance covered by Ted = 15 km + 30 km + 60 km + 120 km = 225 km
Robin's journey:
Robin reduces her speed by half after every 120 minutes. So, she will have different speeds during different time intervals.
From 7 AM to 9 AM (120 minutes):
Robin's speed = 120 km/hour
Distance covered = Speed × Time = 120 km/hour × 2 hours = 240 km
From 9 AM to 11 AM (120 minutes):
Robin's speed = 120 km/hour ÷ 2 = 60 km/hour
Distance covered = Speed × Time = 60 km/hour × 2 hours = 120 km
From 11 AM to 1 PM (120 minutes):
Robin's speed = 60 km/hour ÷ 2 = 30 km/hour
Distance covered = Speed × Time = 30 km/hour × 2 hours = 60 km
Total distance covered by Robin = 240 km + 120 km + 60 km = 420 km
Total distance between Ted and Robin at 1 PM = Distance covered by Ted + Distance covered by Robin = 225 km + 420 km = 645 km
Therefore, the correct answer is option D) 645 km.
Ted's journey:
Ted doubles his speed after every 90 minutes. So, he will have different speeds during different time intervals.
From 7 AM to 8:30 AM (90 minutes):
Ted's speed = 10 km/hour
Distance covered = Speed × Time = 10 km/hour × 1.5 hours = 15 km
From 8:30 AM to 10 AM (90 minutes):
Ted's speed = 2 × 10 km/hour = 20 km/hour
Distance covered = Speed × Time = 20 km/hour × 1.5 hours = 30 km
From 10 AM to 11:30 AM (90 minutes):
Ted's speed = 2 × 20 km/hour = 40 km/hour
Distance covered = Speed × Time = 40 km/hour × 1.5 hours = 60 km
From 11:30 AM to 1 PM (90 minutes):
Ted's speed = 2 × 40 km/hour = 80 km/hour
Distance covered = Speed × Time = 80 km/hour × 1.5 hours = 120 km
Total distance covered by Ted = 15 km + 30 km + 60 km + 120 km = 225 km
Robin's journey:
Robin reduces her speed by half after every 120 minutes. So, she will have different speeds during different time intervals.
From 7 AM to 9 AM (120 minutes):
Robin's speed = 120 km/hour
Distance covered = Speed × Time = 120 km/hour × 2 hours = 240 km
From 9 AM to 11 AM (120 minutes):
Robin's speed = 120 km/hour ÷ 2 = 60 km/hour
Distance covered = Speed × Time = 60 km/hour × 2 hours = 120 km
From 11 AM to 1 PM (120 minutes):
Robin's speed = 60 km/hour ÷ 2 = 30 km/hour
Distance covered = Speed × Time = 30 km/hour × 2 hours = 60 km
Total distance covered by Robin = 240 km + 120 km + 60 km = 420 km
Total distance between Ted and Robin at 1 PM = Distance covered by Ted + Distance covered by Robin = 225 km + 420 km = 645 km
Therefore, the correct answer is option D) 645 km.
The sequence a1, a2,…an is such that an = an-1 +n*d for all n > 1, where d is a positive integer. If a3 = 20 and a5 = 47, what is the value of a7?- a)53
- b)65
- c)75
- d)80
- e)86
Correct answer is option 'E'. Can you explain this answer?
The sequence a1, a2,…an is such that an = an-1 +n*d for all n > 1, where d is a positive integer. If a3 = 20 and a5 = 47, what is the value of a7?
a)
53
b)
65
c)
75
d)
80
e)
86
|
|
Kavya Chakraborty answered |
Given
- A sequence a1, a2,…an
- an = an-1 +n*d for all n > 1, where d is an integer > 0
- a3 = 20
- a5 = 47
To Find: a7?
Approach
- As an = an-1 +n*d, we can express a7 in terms of a1 and d
- So, we need to find the value of a1 and d.
- As we are given the values of a3 and a5, we will express them in terms of a1 and d to get 2 equations in a1 and d.
- We will then solve these two equations to find out the value of a1 and d.
Working Out
Solving (1) and (2), we have a1 = 5 and d = 3 
Answer: E
If one of the roots of the quadratic equation x2 + bx + 98 = 0 is the average (arithmetic mean) of the roots of the equation x2 + 28x – 588 = 0, what is the other root of the equation x2 + bx + 98 = 0?- a)-7
- b)−5/2
- c)5/2
- d)7
- e)21
Correct answer is option 'A'. Can you explain this answer?
If one of the roots of the quadratic equation x2 + bx + 98 = 0 is the average (arithmetic mean) of the roots of the equation x2 + 28x – 588 = 0, what is the other root of the equation x2 + bx + 98 = 0?
a)
-7
b)
−5/2
c)
5/2
d)
7
e)
21
|
|
Sounak Iyer answered |
Problem Analysis:
We are given two quadratic equations: x^2 + bx + 98 = 0 and x^2 + 28x + 588 = 0. We need to find the other root of the equation x^2 + bx + 98 = 0, given that one of the roots is the average of the roots of the equation x^2 + 28x + 588 = 0.
Key Observations:
Let's denote the roots of x^2 + 28x + 588 = 0 as α and β.
The sum of the roots, α + β, can be found using the formula: α + β = -b/a.
The product of the roots, α * β, can be found using the formula: α * β = c/a.
Solution:
1. Find the sum and product of the roots of x^2 + 28x + 588 = 0:
- Using the formula α + β = -b/a, we find: α + β = -28/1 = -28.
- Using the formula α * β = c/a, we find: α * β = 588/1 = 588.
2. Find the average of the roots of x^2 + 28x + 588 = 0:
- The average of the roots is (α + β)/2 = (-28)/2 = -14.
3. Set up the equation for the other root of x^2 + bx + 98 = 0:
- We know that one of the roots is -14.
- Let the other root be r.
- The sum of the roots is -14 + r = -b/a.
- The product of the roots is -14 * r = 98/a.
4. Solve the equation to find r:
- Rearrange the equation -14 + r = -b/a to get r = -14 - b/a.
- Substitute the value of α * β = 98/a from step 3: -14 * r = 98/a.
- Simplify: -14r = 98/a.
- Multiply both sides by -a/14: r = -98/14 = -7.
5. Answer:
- The other root of the equation x^2 + bx + 98 = 0 is -7.
- Therefore, the correct answer is option A.
We are given two quadratic equations: x^2 + bx + 98 = 0 and x^2 + 28x + 588 = 0. We need to find the other root of the equation x^2 + bx + 98 = 0, given that one of the roots is the average of the roots of the equation x^2 + 28x + 588 = 0.
Key Observations:
Let's denote the roots of x^2 + 28x + 588 = 0 as α and β.
The sum of the roots, α + β, can be found using the formula: α + β = -b/a.
The product of the roots, α * β, can be found using the formula: α * β = c/a.
Solution:
1. Find the sum and product of the roots of x^2 + 28x + 588 = 0:
- Using the formula α + β = -b/a, we find: α + β = -28/1 = -28.
- Using the formula α * β = c/a, we find: α * β = 588/1 = 588.
2. Find the average of the roots of x^2 + 28x + 588 = 0:
- The average of the roots is (α + β)/2 = (-28)/2 = -14.
3. Set up the equation for the other root of x^2 + bx + 98 = 0:
- We know that one of the roots is -14.
- Let the other root be r.
- The sum of the roots is -14 + r = -b/a.
- The product of the roots is -14 * r = 98/a.
4. Solve the equation to find r:
- Rearrange the equation -14 + r = -b/a to get r = -14 - b/a.
- Substitute the value of α * β = 98/a from step 3: -14 * r = 98/a.
- Simplify: -14r = 98/a.
- Multiply both sides by -a/14: r = -98/14 = -7.
5. Answer:
- The other root of the equation x^2 + bx + 98 = 0 is -7.
- Therefore, the correct answer is option A.
If x2+4x+p=13 , where p is a constant, what is the product of the roots of this quadratic equation?(1) -2 is one of the roots of the quadratic equation(2) x2+4x+p=13 has equal roots- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?
If x2+4x+p=13 , where p is a constant, what is the product of the roots of this quadratic equation?
(1) -2 is one of the roots of the quadratic equation
(2) x2+4x+p=13 has equal roots
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
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Devansh Shah answered |
Statement (1): -2 is one of the roots of the quadratic equation
Let's assume the quadratic equation is in the form ax^2 + bx + c = 0.
From the given information, we know that one of the roots is -2. This means that when x = -2, the quadratic equation is satisfied. Plugging this value into the equation, we get:
(-2)^2 + 4(-2) + p = 13
4 - 8 + p = 13
-4 + p = 13
p = 17
Statement (2): x^2 + 4x + p = 13 has equal roots
This means that the discriminant of the quadratic equation is zero. The discriminant is given by b^2 - 4ac.
In this case, a = 1, b = 4, and c = p - 13.
So, the discriminant is:
(4)^2 - 4(1)(p - 13) = 0
16 - 4p + 52 = 0
-4p + 68 = 0
-4p = -68
p = 17
Combined solution:
From both statements, we have determined that p = 17. Therefore, we can find the roots of the quadratic equation:
x^2 + 4x + 17 = 0
Using the quadratic formula, the roots can be calculated as follows:
x = (-b ± √(b^2 - 4ac)) / (2a)
x = (-4 ± √(4^2 - 4(1)(17))) / (2(1))
x = (-4 ± √(16 - 68)) / 2
x = (-4 ± √(-52)) / 2
Since the discriminant is negative, the roots are imaginary and the product of the roots will also be imaginary. Therefore, the product of the roots cannot be determined based on the given information.
Answer:
Neither statement alone nor both statements together are sufficient to answer the question asked.
Let's assume the quadratic equation is in the form ax^2 + bx + c = 0.
From the given information, we know that one of the roots is -2. This means that when x = -2, the quadratic equation is satisfied. Plugging this value into the equation, we get:
(-2)^2 + 4(-2) + p = 13
4 - 8 + p = 13
-4 + p = 13
p = 17
Statement (2): x^2 + 4x + p = 13 has equal roots
This means that the discriminant of the quadratic equation is zero. The discriminant is given by b^2 - 4ac.
In this case, a = 1, b = 4, and c = p - 13.
So, the discriminant is:
(4)^2 - 4(1)(p - 13) = 0
16 - 4p + 52 = 0
-4p + 68 = 0
-4p = -68
p = 17
Combined solution:
From both statements, we have determined that p = 17. Therefore, we can find the roots of the quadratic equation:
x^2 + 4x + 17 = 0
Using the quadratic formula, the roots can be calculated as follows:
x = (-b ± √(b^2 - 4ac)) / (2a)
x = (-4 ± √(4^2 - 4(1)(17))) / (2(1))
x = (-4 ± √(16 - 68)) / 2
x = (-4 ± √(-52)) / 2
Since the discriminant is negative, the roots are imaginary and the product of the roots will also be imaginary. Therefore, the product of the roots cannot be determined based on the given information.
Answer:
Neither statement alone nor both statements together are sufficient to answer the question asked.
A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following could be the value of the ordered set (a, b, c)?I. (-1, 4, -3)II. (1, 4, 3)III. (3, -10√3, 9) - a)I Only
- b)II Only
- c)III Only
- d)I and II Only
- e)I, II and III Only
Correct answer is option 'D'. Can you explain this answer?
A quadratic equation ax2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following could be the value of the ordered set (a, b, c)?
I. (-1, 4, -3)
II. (1, 4, 3)
III. (3, -10√3, 9)
a)
I Only
b)
II Only
c)
III Only
d)
I and II Only
e)
I, II and III Only
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Srestha Basu answered |
To solve this question, we need to use the quadratic formula and the given information about the sum and squares of the roots.
The quadratic formula states that for a quadratic equation ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Let's first analyze the given information about the sum and squares of the roots. The square of the sum of the roots is given by:
(x1 + x2)^2 = (x1^2 + 2x1x2 + x2^2)
The sum of the squares of the roots is given by:
x1^2 + x2^2
According to the given information, the square of the sum of the roots is 6 greater than the sum of the squares of the roots, so we can write the equation:
(x1^2 + 2x1x2 + x2^2) = (x1^2 + x2^2) + 6
Simplifying this equation, we get:
2x1x2 = 6
Now let's analyze each option given and check if they satisfy this condition:
I. (-1, 4, -3)
Using the quadratic formula, we find the roots of the equation -x^2 + 4x - 3 = 0 as x1 = 1 and x2 = 3.
The sum of the roots is 1 + 3 = 4, and the sum of the squares of the roots is 1^2 + 3^2 = 10.
The square of the sum of the roots is (1 + 3)^2 = 16, which is not 6 greater than the sum of the squares of the roots.
Therefore, option I does not satisfy the given condition.
II. (1, 4, 3)
Using the quadratic formula, we find the roots of the equation x^2 + 4x + 3 = 0 as x1 = -1 and x2 = -3.
The sum of the roots is -1 + (-3) = -4, and the sum of the squares of the roots is (-1)^2 + (-3)^2 = 10.
The square of the sum of the roots is (-1 + (-3))^2 = 16, which is 6 greater than the sum of the squares of the roots.
Therefore, option II satisfies the given condition.
III. (3, -103, 9)
Using the quadratic formula, we find the roots of the equation 3x^2 - 103x + 9 = 0 as x1 = 1 and x2 = 9/3 = 3.
The sum of the roots is 1 + 3 = 4, and the sum of the squares of the roots is 1^2 + 3^2 = 10.
The square of the sum of the roots is (1 + 3)^2 = 16, which is 6 greater than the sum of the squares of the roots.
Therefore, option III satisfies the given condition.
Based on the above analysis, options II and III satisfy the given condition, so the correct answer is option D, "I and II Only."
The quadratic formula states that for a quadratic equation ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Let's first analyze the given information about the sum and squares of the roots. The square of the sum of the roots is given by:
(x1 + x2)^2 = (x1^2 + 2x1x2 + x2^2)
The sum of the squares of the roots is given by:
x1^2 + x2^2
According to the given information, the square of the sum of the roots is 6 greater than the sum of the squares of the roots, so we can write the equation:
(x1^2 + 2x1x2 + x2^2) = (x1^2 + x2^2) + 6
Simplifying this equation, we get:
2x1x2 = 6
Now let's analyze each option given and check if they satisfy this condition:
I. (-1, 4, -3)
Using the quadratic formula, we find the roots of the equation -x^2 + 4x - 3 = 0 as x1 = 1 and x2 = 3.
The sum of the roots is 1 + 3 = 4, and the sum of the squares of the roots is 1^2 + 3^2 = 10.
The square of the sum of the roots is (1 + 3)^2 = 16, which is not 6 greater than the sum of the squares of the roots.
Therefore, option I does not satisfy the given condition.
II. (1, 4, 3)
Using the quadratic formula, we find the roots of the equation x^2 + 4x + 3 = 0 as x1 = -1 and x2 = -3.
The sum of the roots is -1 + (-3) = -4, and the sum of the squares of the roots is (-1)^2 + (-3)^2 = 10.
The square of the sum of the roots is (-1 + (-3))^2 = 16, which is 6 greater than the sum of the squares of the roots.
Therefore, option II satisfies the given condition.
III. (3, -103, 9)
Using the quadratic formula, we find the roots of the equation 3x^2 - 103x + 9 = 0 as x1 = 1 and x2 = 9/3 = 3.
The sum of the roots is 1 + 3 = 4, and the sum of the squares of the roots is 1^2 + 3^2 = 10.
The square of the sum of the roots is (1 + 3)^2 = 16, which is 6 greater than the sum of the squares of the roots.
Therefore, option III satisfies the given condition.
Based on the above analysis, options II and III satisfy the given condition, so the correct answer is option D, "I and II Only."
For any integers x and y, min(x, y) and max(x, y) denote the minimum and the maximum of x and y, respectively. For example, min(2, 1) = 1 and max(2,1) = 2. If a, b, c and d are distinct positive integers, is max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))) ?(1) b, c and d are factors of a(2) a – 2d = b + c- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?
For any integers x and y, min(x, y) and max(x, y) denote the minimum and the maximum of x and y, respectively. For example, min(2, 1) = 1 and max(2,1) = 2. If a, b, c and d are distinct positive integers, is max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))) ?
(1) b, c and d are factors of a
(2) a – 2d = b + c
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
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Snehal Banerjee answered |
Steps 1 & 2: Understand Question and Draw Inferences
- min(x, y) → minimum of x and y, where x and y are integers
- max(x, y) → maximum of x and y, where x and y are integers
- a, b, c, d are distinct integers > 0
To Find: Is max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))) ?
Step 3: Analyze Statement 1 independently
(1) b, c and d are factors of a
- So, we can write a = bx = cy = dz, where x, y and z are distinct integers > 1
- For example, since 2, 3 and 6 are factors of 36, we can write: 36 = 2x =3y = 6z (in this example, x, y and z will be equal to 18, 12 and 6 respectively)
- Hence, a > {b, c, d}
- Like, in the above example, 36 > {2, 3, 6}
- For solving the expression, we need to start from the innermost bracket. Since we do not know which of c or d is smaller, max(a, max(b, min(c, d))) = max(a, max(b, (c or d)))
- Now, we do not know, which of b, c or d is the greatest, max(a, max(b, (c or d))) = max (a, (b, c or d))
- However, we do know that a > b, c, d. So, we can write max (a, (b, c or d)) = a
- Similarly, we do not know which b or c is smaller, so, max(d, max(a, min(b, c))) = max (d, max (a, (b or c)))
- However, we do know that a > b , c. So, we can write max (d, max (a, (b or c)))= max(d, a)
- Also, as a > d, we can write max (d, a) = a
So, max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))).
Sufficient to answer.
Sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) a – 2d = b + c
- Rearranging the terms: a = b + c + 2d. So, a > b, c and d
- We know that a > b, c, d and for this case we have already evaluated in statement-1 that max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))).
So, max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))).
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
If p and q are the roots of the quadratic equation ax2 + bx + c = 0 where a ≠ 0, what are the roots of the equation ayx2 + byx + cy = 0 where 0 < y ≤ 1?- a)p and q
- b)py and qy
- c)p/yand q/y
- d)y and -y
- e)Cannot be determined
Correct answer is option 'A'. Can you explain this answer?
If p and q are the roots of the quadratic equation ax2 + bx + c = 0 where a ≠ 0, what are the roots of the equation ayx2 + byx + cy = 0 where 0 < y ≤ 1?
a)
p and q
b)
py and qy
c)
p/yand q/y
d)
y and -y
e)
Cannot be determined
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Rajdeep Nair answered |
Given
- ax2+bx+c=0
- has two roots p and q a ≠ 0, 0 < y ≤ 1
To Find
- Roots of ayx2+byx+cy=0
Approach
- To find the roots of the equation ayx2+byx+cy=0 , we will do the following steps:
- Take y common and the get the equation in terms of ax2+bx+c=0
- As we know the roots of ax2+bx+c=0, we will use this knowledge to find the roots of the equation ayx2+byx+cy=0
- Working OutSolving the equation ayx2+byx+cy=0
- ayx2+byx+cy=0
- y(ax2+bx+c)=0
So we have either y = 0 or ax2+bx+c=0
Since we are given that y > 0, y ≠ 0.
Hence ax2+bx+c=0
.As the roots of the equation ax2+bx+c=0 are p and q, the roots of the equation ayx2+byx+cy=0 will also be p and q.
Answer: A
Given the three quadratic equations above, which pair of equations has at least one common root?- a)I and II
- b)II and III
- c)I and III
- d)I, II and III
- e)None of the above
Correct answer is option 'E'. Can you explain this answer?
Given the three quadratic equations above, which pair of equations has at least one common root?
a)
I and II
b)
II and III
c)
I and III
d)
I, II and III
e)
None of the above
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Nandita Chauhan answered |
Given
- Three quadratic equations:
- x2 – 16x + 55 = 0
- 2x2 – 4x - 70 = 0
- x (x +7) = 44
To Find: Which pair of equations has at least one common root?
Approach
- As we need to find the common roots among the equation, we will solve each quadratic equation to find their roots and then find the pair of equations that have at least one common root.
Working Out
So, the roots of Equations I, II and III are (5, 11), (-5, 7) and (-11, 4) respectively.
Hence, none of the equations have even one root in common.
Answer: E
r and s are the roots of the quadratic equation ax2 + bx + c = 0 where a ≠ 0 & s >0, such that r is 50 percent greater than s. If the product of the roots of the equation is 150, what is the sum of the roots of the equation?- a)-25
- b)-15
- c)15
- d)25
- e)Cannot be determined
Correct answer is option 'D'. Can you explain this answer?
r and s are the roots of the quadratic equation ax2 + bx + c = 0 where a ≠ 0 & s >0, such that r is 50 percent greater than s. If the product of the roots of the equation is 150, what is the sum of the roots of the equation?
a)
-25
b)
-15
c)
15
d)
25
e)
Cannot be determined
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Hridoy Gupta answered |
To solve this problem, let's start by assigning variables to the roots of the quadratic equation. Let's say that r is one root and s is the other root.
Given that r is 50% greater than s, we can express this relationship as:
r = 1.5s
We are also given that the product of the roots is 150. This means that:
rs = 150
Now, let's express the quadratic equation in terms of the roots:
ax^2 + bx + c = 0
Since r and s are the roots, we can write the equation as:
a(x - r)(x - s) = 0
Expanding this equation, we get:
ax^2 - (ar + as)x + rs = 0
Now, let's substitute the values of r and s into the equation:
ax^2 - (1.5as + as)x + 150 = 0
Simplifying further, we get:
ax^2 - 2.5asx + 150 = 0
Since the quadratic equation is in the form ax^2 + bx + c = 0, we can equate the coefficients to find the sum of the roots:
b/a = -2.5as/a
b = -2.5as
Since the sum of the roots is equal to -b/a, we can substitute the value of b:
Sum of roots = -(-2.5as)/a
Sum of roots = 2.5s
Now, we can substitute the value of rs = 150 into the equation rs = 150:
s(1.5s) = 150
1.5s^2 = 150
Dividing both sides by 1.5, we get:
s^2 = 100
Taking the square root of both sides, we get:
s = 10
Now, substituting the value of s into the equation for the sum of the roots:
Sum of roots = 2.5s
Sum of roots = 2.5(10)
Sum of roots = 25
Therefore, the sum of the roots of the quadratic equation is 25. So, the correct answer is option D) 25.
Given that r is 50% greater than s, we can express this relationship as:
r = 1.5s
We are also given that the product of the roots is 150. This means that:
rs = 150
Now, let's express the quadratic equation in terms of the roots:
ax^2 + bx + c = 0
Since r and s are the roots, we can write the equation as:
a(x - r)(x - s) = 0
Expanding this equation, we get:
ax^2 - (ar + as)x + rs = 0
Now, let's substitute the values of r and s into the equation:
ax^2 - (1.5as + as)x + 150 = 0
Simplifying further, we get:
ax^2 - 2.5asx + 150 = 0
Since the quadratic equation is in the form ax^2 + bx + c = 0, we can equate the coefficients to find the sum of the roots:
b/a = -2.5as/a
b = -2.5as
Since the sum of the roots is equal to -b/a, we can substitute the value of b:
Sum of roots = -(-2.5as)/a
Sum of roots = 2.5s
Now, we can substitute the value of rs = 150 into the equation rs = 150:
s(1.5s) = 150
1.5s^2 = 150
Dividing both sides by 1.5, we get:
s^2 = 100
Taking the square root of both sides, we get:
s = 10
Now, substituting the value of s into the equation for the sum of the roots:
Sum of roots = 2.5s
Sum of roots = 2.5(10)
Sum of roots = 25
Therefore, the sum of the roots of the quadratic equation is 25. So, the correct answer is option D) 25.
For the positive integers k, m and n, k(m)n means that the remainder when m is divided by n is k. If k(13766)9 and p(137)k, where p is a positive integer, then p is equal to.- a)1
- b)2
- c)5
- d)7
- e)8
Correct answer is option 'B'. Can you explain this answer?
For the positive integers k, m and n, k(m)n means that the remainder when m is divided by n is k. If k(13766)9 and p(137)k, where p is a positive integer, then p is equal to.
a)
1
b)
2
c)
5
d)
7
e)
8
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Snehal Banerjee answered |
Given:
- For positive integers k, m and n, k(m)n implies k is the remainder when m is divided by n
- k(13766)9 → k is the remainder when 13766 is divided by 9
- p(137)k → p is the remainder when 137 is divided by k
- p is a positive integer
To find: p = ?
Approach:
- To find the value of p, we need to evaluate the expression p(137)k
- However, this expression also contains k
- So, we’ll be able to find the value of p only when we first know the value of k
- We can find the value of k by evaluating the expression k(13766)9
Working Out:
- Finding the value of k
- k(13766)9 means that the remainder when 13766 is divided by 9 is k.
- Sum of digits of 13766 = 1+3+7+6+6 = 23
- So, remainder when 13766 is divided with 9 = Remainder that 23 leaves with 9 = 5
- Thus, k = 5
- Finding the value of p
- p(137)k means that the remainder when 137 is divided by k is p
- We’ve inferred above that k = 5
- So, the above expression means that when 137 is divided by 5, the remainder is p
- Now, 137 leaves a remainder of 2 with 5
- So, p= 2
- p(137)k means that the remainder when 137 is divided by k is p
Looking at the answer choices, we see that the correct answer is Option B
The function SCI is defined as SCI(x, y) = z, where z is the sum of y consecutive positive integers starting from positive integer x. If a and n are positive integers, is SCI(a,n) divisible by n?(1) 3n +2a is not divisible by 2(2) 3a +2n is divisible by 2- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'A'. Can you explain this answer?
The function SCI is defined as SCI(x, y) = z, where z is the sum of y consecutive positive integers starting from positive integer x. If a and n are positive integers, is SCI(a,n) divisible by n?
(1) 3n +2a is not divisible by 2
(2) 3a +2n is divisible by 2
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
|
|
Sankar Desai answered |
Statement (1): 3n - 2a is not divisible by 2
- This means that 3n - 2a is an odd number, as it is not divisible by 2.
- We can rewrite 3n - 2a as 2n + n - 2a, where 2n is divisible by 2 and n - 2a is an odd number.
- Therefore, 2n + n - 2a can be written as an even number + an odd number, which is always an odd number.
- Since SCI(a, n) is the sum of y consecutive positive integers starting from a, it will be an odd number if y is odd and an even number if y is even.
- Therefore, SCI(a, n) cannot be divisible by n if y is odd.
- This statement alone is sufficient to determine that SCI(a, n) is not divisible by n.
Statement (2): 3a + 2n is divisible by 2
- This means that 3a + 2n is an even number, as it is divisible by 2.
- We can rewrite 3a + 2n as 2a + a + 2n, where 2a is divisible by 2 and a + 2n is an even number.
- Therefore, 2a + a + 2n can be written as an even number + an even number, which is always an even number.
- Since SCI(a, n) is the sum of y consecutive positive integers starting from a, it will be an odd number if y is odd and an even number if y is even.
- Therefore, SCI(a, n) can be divisible by n if y is even.
- This statement alone is sufficient to determine that SCI(a, n) is divisible by n when y is even.
Combined Statements:
- From statement (1), we know that SCI(a, n) is not divisible by n if y is odd.
- From statement (2), we know that SCI(a, n) is divisible by n if y is even.
- Therefore, when we combine the statements, we have sufficient information to determine whether SCI(a, n) is divisible by n for any value of y.
- The answer is option (A) - Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question.
- This means that 3n - 2a is an odd number, as it is not divisible by 2.
- We can rewrite 3n - 2a as 2n + n - 2a, where 2n is divisible by 2 and n - 2a is an odd number.
- Therefore, 2n + n - 2a can be written as an even number + an odd number, which is always an odd number.
- Since SCI(a, n) is the sum of y consecutive positive integers starting from a, it will be an odd number if y is odd and an even number if y is even.
- Therefore, SCI(a, n) cannot be divisible by n if y is odd.
- This statement alone is sufficient to determine that SCI(a, n) is not divisible by n.
Statement (2): 3a + 2n is divisible by 2
- This means that 3a + 2n is an even number, as it is divisible by 2.
- We can rewrite 3a + 2n as 2a + a + 2n, where 2a is divisible by 2 and a + 2n is an even number.
- Therefore, 2a + a + 2n can be written as an even number + an even number, which is always an even number.
- Since SCI(a, n) is the sum of y consecutive positive integers starting from a, it will be an odd number if y is odd and an even number if y is even.
- Therefore, SCI(a, n) can be divisible by n if y is even.
- This statement alone is sufficient to determine that SCI(a, n) is divisible by n when y is even.
Combined Statements:
- From statement (1), we know that SCI(a, n) is not divisible by n if y is odd.
- From statement (2), we know that SCI(a, n) is divisible by n if y is even.
- Therefore, when we combine the statements, we have sufficient information to determine whether SCI(a, n) is divisible by n for any value of y.
- The answer is option (A) - Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question.
Mike took 5 mock tests before appearing for the GMAT. In each mock test he scored 10 points more than the previous mock test. If he scored 760 on the GMAT and his average score for the mocks was 720, what is the difference between his last mock score and his GMAT score?- a)10
- b)20
- c)30
- d)40
- e)50
Correct answer is option 'B'. Can you explain this answer?
Mike took 5 mock tests before appearing for the GMAT. In each mock test he scored 10 points more than the previous mock test. If he scored 760 on the GMAT and his average score for the mocks was 720, what is the difference between his last mock score and his GMAT score?
a)
10
b)
20
c)
30
d)
40
e)
50
|
|
Rhea Gupta answered |
Given
Mike took 5 mock tests
- Let his score in the 1st mock be x.
- So, his scores in the other mocks = x+ 10, x+20, x+30, x + 40
- Mike’s score on GMAT = 760
- Average score of mocks = 720
To Find:
- 760 – (x +40)
Working Out
Average score of mocks = (x + x + 10 + … x + 40) /5 = (5x + 100)/5 = x + 20
(The other way to think about this is, as Mike’s scores in the mocks are in arithmetic sequence, average will be the middle term)
(The other way to think about this is, as Mike’s scores in the mocks are in arithmetic sequence, average will be the middle term)
- Mike’s average score in the mocks = x + 20 = 720
- x = 700
Therefore
- x + 40 = 740
- 760 – (x+40) = 760 – 740 = 20
Correct Answer: Option B
A function f(x) is defined as f(x)=3x2−20x+c, where c is a constant. Also given f(1) = -16. What is the value of f(c) + f(-c) ?- a)6
- b)8
- c)10
- d)12
- e)30
Correct answer is option 'B'. Can you explain this answer?
A function f(x) is defined as f(x)=3x2−20x+c, where c is a constant. Also given f(1) = -16. What is the value of f(c) + f(-c) ?
a)
6
b)
8
c)
10
d)
12
e)
30
|
|
Snehal Banerjee answered |
Given Info:
- Function f(x) is given as f(x)=3x2−20x+c
- , where c is a constant
- The above function is a quadratic function in x
- Also given f(1)=−16
To Find:
- Value of f(c)+f(−c)
⇒f(c)+f(−c)=6c2+2c
Approach:
Working out:
- Now, f(c)=3c2−20c+c
- (Putting value of c in the given function → f(x)=3x2−20x+c
- And, f(−c)=3c2+20c+c
- Adding both functions, we get
⇒f(c)+f(−c)=3c2−20c+c+3c2+20c+c- Now in order to calculate the above value, we need to determine the value of c.
- To determine the value of c, we will work on the quadratic function in x. We know the value of function at x=1 as’ -16’ as given in the question, so we will calculate the value of c by putting the value of f(x) and the value of x.
- After knowing the value of c from the given value of the function at x=1, we will find the value of f(c)+f(−c)
- f(x)=3x2−20x+c
- Given f(1)=−16
⇒ f(1)=3(12)−20(1)+c
⇒ f(1)=c−17
- Now f(1)=−16
⇒ c−17=−16
⇒ c=1
- Now we have already established above, f(c)+f(−c)=6c2+2c
- Putting value of c from above, we get
⇒ f(c)+f(−c)=6(1)2+2(1)
⇒ f(c)+f(−c)=8
Answer
- So the value of f(c)+f(−c) is 8
- Hence answer option B is correct.
For which of the following functions is f(ab) = f(a) * f(b)? - f(x) = x2
- f (x) = √x
- f (x) = 2x
- a)I only
- b)I, II and III
- c)III only
- d)I and II
Correct answer is 'D'. Can you explain this answer?
For which of the following functions is f(ab) = f(a) * f(b)?
- f(x) = x2
- f (x) = √x
- f (x) = 2x
a)
I only
b)
I, II and III
c)
III only
d)
I and II
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|
Srestha Basu answered |
Ex
The function f(x) = ex satisfies the property f(ab) = f(a) * f(b) because:
f(ab) = eab = ea * eb = f(a) * f(b)
Using the laws of exponents, we can see that eab = ea * eb, which means that f(ab) = f(a) * f(b) for this function.
The function f(x) = ex satisfies the property f(ab) = f(a) * f(b) because:
f(ab) = eab = ea * eb = f(a) * f(b)
Using the laws of exponents, we can see that eab = ea * eb, which means that f(ab) = f(a) * f(b) for this function.
In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?(1) 3 and 4 are roots of the equation.(2) The product of the roots of the equation is 12.- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- d)EACH statement ALONE is sufficient.
- e)Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'E'. Can you explain this answer?
In the equation ax2 + bx + c = 0, where a, b and c are constants and a ≠ 0, what is the value of b?
(1) 3 and 4 are roots of the equation.
(2) The product of the roots of the equation is 12.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
c)
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
d)
EACH statement ALONE is sufficient.
e)
Statements (1) and (2) TOGETHER are NOT sufficient.
|
|
Ameya Yadav answered |
Understanding the Problem
To find the value of \( b \) in the equation \( ax^2 + bx + c = 0 \), we can use the relationships between the roots of the quadratic equation. The roots of the equation can be denoted as \( r_1 \) and \( r_2 \).
Key Relationships
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)
Evaluating Statement (1)
- Statement (1): 3 and 4 are roots of the equation.
- From this, we can calculate:
- Sum of the roots: \( 3 + 4 = 7 \)
- Therefore, \( -\frac{b}{a} = 7 \) implies \( b = -7a \).
- However, without knowing the value of \( a \), we cannot determine a unique value for \( b \). Thus, statement (1) alone is insufficient.
Evaluating Statement (2)
- Statement (2): The product of the roots of the equation is 12.
- This gives us: \( r_1 \cdot r_2 = 12 \) implies \( \frac{c}{a} = 12 \).
- We cannot determine \( b \) as we still lack information about the sum of the roots. Thus, statement (2) alone is also insufficient.
Combining Statements (1) and (2)
- Combining both statements:
- From statement (1), we have \( b = -7a \).
- From statement (2), we know the product of the roots is 12, but we still lack the actual values of \( a \) and \( c \).
- Therefore, even together, the statements do not provide enough information to uniquely determine \( b \).
Conclusion
Thus, the correct answer is option E: Statements (1) and (2) together are NOT sufficient.
To find the value of \( b \) in the equation \( ax^2 + bx + c = 0 \), we can use the relationships between the roots of the quadratic equation. The roots of the equation can be denoted as \( r_1 \) and \( r_2 \).
Key Relationships
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)
Evaluating Statement (1)
- Statement (1): 3 and 4 are roots of the equation.
- From this, we can calculate:
- Sum of the roots: \( 3 + 4 = 7 \)
- Therefore, \( -\frac{b}{a} = 7 \) implies \( b = -7a \).
- However, without knowing the value of \( a \), we cannot determine a unique value for \( b \). Thus, statement (1) alone is insufficient.
Evaluating Statement (2)
- Statement (2): The product of the roots of the equation is 12.
- This gives us: \( r_1 \cdot r_2 = 12 \) implies \( \frac{c}{a} = 12 \).
- We cannot determine \( b \) as we still lack information about the sum of the roots. Thus, statement (2) alone is also insufficient.
Combining Statements (1) and (2)
- Combining both statements:
- From statement (1), we have \( b = -7a \).
- From statement (2), we know the product of the roots is 12, but we still lack the actual values of \( a \) and \( c \).
- Therefore, even together, the statements do not provide enough information to uniquely determine \( b \).
Conclusion
Thus, the correct answer is option E: Statements (1) and (2) together are NOT sufficient.
The nth term of an increasing sequence S is given by Sn = Sn-1 + Sn-2 for n > 2 and the nth term of a sequence S’ is given by S’n = S’n-1 - S’n-2 for n > 2. If S5 = S’5, what is the average (arithmetic mean) of S2 and S’2?(1) The difference between the fourth term and the second term of sequence S is 14.(2) The sum of the fourth term and the second term of sequence S’ is 14.- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'A'. Can you explain this answer?
The nth term of an increasing sequence S is given by Sn = Sn-1 + Sn-2 for n > 2 and the nth term of a sequence S’ is given by S’n = S’n-1 - S’n-2 for n > 2. If S5 = S’5, what is the average (arithmetic mean) of S2 and S’2?
(1) The difference between the fourth term and the second term of sequence S is 14.
(2) The sum of the fourth term and the second term of sequence S’ is 14.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
|
Aarav Sharma answered |
Steps 1 & 2: Understand Question and Draw Inferences
- Increasing sequence S whose nth term is represented as Sn = Sn-1 + Sn-2 for n > 2
- For a sequence S’, the nth term of which is represented as S’n = S’n-1 - S’n-2 for n > 2
- S5 = S’5
Thus we need to find the value of S3 to find the average of S2 andS′2
Step 3: Analyze Statement 1 independently
(1) The difference between the fourth term and the second term of sequence S is 14
- S4−S2=14
- Substituting the expression of S4=S3+S2
- (S3+S2)−S2=14,i.e.S3=14
As we know the value of S3, the statement is sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The sum of the fourth term and the second term of sequence S’ is 14.
- S′2+S′4=14
- Substituting the expression of S′4=S′3−S′2
- S′2+(S′3−S′2)=14
- S′3=14
Does not tell us anything about the value of S3, the statement is insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step -3, this step is not required.
Answer: A
If v* = (1/v), find the value ofv* + (v*)* + ((v*)*)* if v = 1/2.- a)3
- b)7/2
- c)4
- d)9/2
- e)5
Correct answer is option 'D'. Can you explain this answer?
If v* = (1/v), find the value of
v* + (v*)* + ((v*)*)* if v = 1/2.
a)
3
b)
7/2
c)
4
d)
9/2
e)
5
|
|
Abhishek Choudhury answered |
Step 1: Question statement and Inferences
A function has been defined called v* such that v* = 1v
Step 2 & 3: Finding required values and calculating the final answer
v = 1/2
=> v*=1v=2
=> (v*)* = (2)* = 1/2
=> ((v*)*)* = (1/2)* = 2
=> v* + (v*)* + ((v*)*)* = 2 + 1/2 + 2 = 9/2.
Answer: Option (D)
Edward invested five-ninths of his money at an annual rate of 2r% compounded semi-annually, and the remaining money at an annual rate of r% compounded annually. If after one year, Edward’s money had grown by one-thirds, the value of r is equal to which of the following?- a)10%
- b)15%
- c)20%
- d)25%
- e)33%
Correct answer is option 'C'. Can you explain this answer?
Edward invested five-ninths of his money at an annual rate of 2r% compounded semi-annually, and the remaining money at an annual rate of r% compounded annually. If after one year, Edward’s money had grown by one-thirds, the value of r is equal to which of the following?
a)
10%
b)
15%
c)
20%
d)
25%
e)
33%
|
|
Tanishq Choudhury answered |
Given: Let the total money be y
- First Investment:
-
-
- Rate of interest = 2r% p.a. compounded every 6 months = r% per 6-months
- Time = 1 year
-
- Second Investment:
-
-
- Rate of interest = r% p.a.
- Time = 1 year
-
- Total Interest earned in 1 year
To find: The value of r
Approach
1. Total interest earned in 1 year = (Interest earned from 1st investment) + (Interest earned from 2nd investment)
i. In the given time frame of 1 year, the 1st investment will pay interest twice (since this investment pays interest every 6 months). So, the formula for compound interest will be applicable for the 1st investment
ii. The 2nd investment pays interest after 1 year. Since the given time frame is also 1 year, this investment will yield simple interest
2. The only unknown in the above equation will be r. So, using this equation, we can find the value of r
Working Out
- Amount of the first investment after 1 year
- Compound Interest earned from 1st investment
- Calculating the interest earned from the 2nd investment:
- Simple interest earned from 2nd investment =
- So, Total interest earned in 1 year =
- Multiplying both sides of the above equation with 9/y :
- Simple interest earned from 2nd investment =
- Rejecting the negative value since the money is given to have grown.
Looking at the answer choices, we see that Option C is correct
(Note: You could also have solved this question by framing the first equation in terms of the amount that each investment grows to, as under:
(Total Amount after 1 year) = (Amount that the 1st investment grows to) + (Amount that the 2nd investment grows to)
This equation leads to a similar calculation and the same result as in the solution above /End of Note)
If p and q are the roots of the quadratic equation ax2 + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?(1) |p + q| = |p| + |q|(2) ac > 0- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?
If p and q are the roots of the quadratic equation ax2 + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?
(1) |p + q| = |p| + |q|
(2) ac > 0
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
|
|
Pallavi Sharma answered |
Understanding the Problem
We need to determine if the product of the roots p and q of the quadratic equation ax² + bx + c = 0 is greater than 0, given that a, b, and c are non-zero.
Key Concept
The product of the roots of a quadratic equation is given by the formula p*q = c/a. To determine if p*q > 0, we need to analyze the signs of c and a.
Statement (1): |p + q| = |p| + |q|
This statement implies that both p and q have the same sign. When both roots share the same sign, their product p*q will be positive.
Thus, statement (1) alone is sufficient to conclude that the product of the roots is greater than 0.
Statement (2): ac > 0
This statement means that both a and c are either positive or negative.
- If both a and c are positive, then p*q = c/a > 0.
- If both a and c are negative, then p*q = c/a > 0.
In both cases, the product p*q is greater than 0, making statement (2) alone sufficient as well.
Conclusion
Since both statements independently confirm the product of the roots p and q is greater than 0, we can conclude:
- Statement (1) alone is sufficient.
- Statement (2) alone is sufficient.
Thus, the correct answer is option 'd': EACH statement ALONE is sufficient to answer the question asked.
We need to determine if the product of the roots p and q of the quadratic equation ax² + bx + c = 0 is greater than 0, given that a, b, and c are non-zero.
Key Concept
The product of the roots of a quadratic equation is given by the formula p*q = c/a. To determine if p*q > 0, we need to analyze the signs of c and a.
Statement (1): |p + q| = |p| + |q|
This statement implies that both p and q have the same sign. When both roots share the same sign, their product p*q will be positive.
Thus, statement (1) alone is sufficient to conclude that the product of the roots is greater than 0.
Statement (2): ac > 0
This statement means that both a and c are either positive or negative.
- If both a and c are positive, then p*q = c/a > 0.
- If both a and c are negative, then p*q = c/a > 0.
In both cases, the product p*q is greater than 0, making statement (2) alone sufficient as well.
Conclusion
Since both statements independently confirm the product of the roots p and q is greater than 0, we can conclude:
- Statement (1) alone is sufficient.
- Statement (2) alone is sufficient.
Thus, the correct answer is option 'd': EACH statement ALONE is sufficient to answer the question asked.
If f(x) = x + x2, is f(a+1) – f(a) divisible by 4, where a is an integer > 0(1) f(a) is divisible by 4(2) (-1)a < (-1)a+1- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- d)EACH statement ALONE is sufficient.
- e)Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'B'. Can you explain this answer?
If f(x) = x + x2, is f(a+1) – f(a) divisible by 4, where a is an integer > 0
(1) f(a) is divisible by 4
(2) (-1)a < (-1)a+1
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
c)
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
d)
EACH statement ALONE is sufficient.
e)
Statements (1) and (2) TOGETHER are NOT sufficient.
|
|
Disha Mehta answered |
Steps 1 & 2: Understand Question and Draw Inferences
The question wants us to know whether f(a+1) – f(a) is divisible by 4. Let’s simplify the expression given to us.
f(a+1) – f(a) = (a+1) + (a+1)2 – (a + a2)
Simplifying we get f(a+1) – f(a) = (a + 1 –a) + ((a+1)2 -a2))
- 1 + (a + 1 –a)(a+1+a) = 2 + 2a = 2(1+a) … (using a2 – b2 = (a-b)(a+b))
- From the statement above we can conclude that the given expression is always divisible by 2.
Hence, for f(a+1) – f(a) to be divisible by 4, (1+a) must be divisible by 2, which means that a must be odd.
Step 3: Analyze Statement 1
Statement 1 says that f(a) is divisible by 4.
f(a) = a(1+a)
a(1+a) is the product of two consecutive integers. Therefore, one term out of a and 1+a will be even and the other will be odd. The product of these two terms will be even and will always be divisible by 2.
But, we are given that a(1+a) is divisible by 4 also.
This can happen only if
a) a is divisible by 4 or
b) 1+a is divisible by 4 or
c) Both a and 1+a are divisible by 2
Case c) is ruled out since a and 1+a are consecutive terms. Therefore, they cannot be both even.
If a is divisible by 4, then a is even.
If 1+a is divisible by 4, then a is odd.
Thus, we cannot determine with confidence whether a is odd or not.
Since Statement 1 does not give us a unique answer, it is not sufficient.
Step 4: Analyze Statement 2
Statement 2 says that (-1)a < (-1)a+1
This is only possible if a is odd, implying that a+1 is even.
Thus, a is indeed odd.
Since statement 2 gives us a unique answer, it is sufficient to arrive at the conclusion.
Step 5: Analyze Both Statements Together (if needed)
Since statement 2 gave us a unique answer, this step is not needed.
Correct Answer: B
The function {x} is defined as the lowest odd integer greater than x. What is the value of {x}?(1) -3.1 < x < -2.5(2) x2 < 9- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- d)EACH statement ALONE is sufficient.
- e)Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'C'. Can you explain this answer?
The function {x} is defined as the lowest odd integer greater than x. What is the value of {x}?
(1) -3.1 < x < -2.5
(2) x2 < 9
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
c)
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
d)
EACH statement ALONE is sufficient.
e)
Statements (1) and (2) TOGETHER are NOT sufficient.
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|
Sreemoyee Nambiar answered |
Steps 1 & 2: Understand Question and Draw Inferences
Let’s understand the meaning of {x}. If x = 2, then the odd integers greater than 2 are 3, 5, etc. The lowest among them is 3. Therefore {2} = 3.
Step 3: Analyze Statement 1
If x is between -3.1 and -3, then {x} = -3. If x is between -3 and -2.5, then {x} = -1.
INSUFFICIENT.
Step 4: Analyze Statement 2
x2 < 9 only for values of x between -3 and 3.
If x = is between -3 and – 1, {x} = -1, if x is between -1 and 1, {x} = 1 and if x is between 1 and 3, {x} = 3.
Therefore, {x} can be -1, 1 or 3.
INSUFFICIENT.
Step 5: Analyze Both Statements Together (if needed)
When we combine statements (1) and (2),
We know that
-3.1 < x < -2.5
AND
-3 < x < 3
The common solution is: -3 < x < -2.5
For this set of values, {x} = -1.
SUFFICIENT.
Answer: Option (C)
What is the value of s+t if the equation 3x2 + tx + s has 2 and 3 as its roots?- a)1
- b)2
- c)3
- d)4
- e)5
Correct answer is option 'C'. Can you explain this answer?
What is the value of s+t if the equation 3x2 + tx + s has 2 and 3 as its roots?
a)
1
b)
2
c)
3
d)
4
e)
5
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EduRev GATE answered |
- Given the equation 3x² + tx + s = 0 with roots 2 and 3:
- Sum of roots: 2 + 3 = 5 ⇒ -t /3 = 5 ⇒ t = -15
- Product of roots: 2 × 3 = 6 ⇒ s /3 = 6 ⇒ s = 18
- s + t = 18 + (-15) = 3
- Answer: 3
If x ≠ 0, for which of following functions is 
, for all values of x?- a)
- b)
- c)
- d)
- e)
Correct answer is 'D'. Can you explain this answer?
If x ≠ 0, for which of following functions is , for all values of x?
, for all values of x?a)
b)
c)
d)
e)
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KS Coaching Center answered |
Step 1: Question statement and Inferences
We have to pick the only option for which 
, Since the option will have to be true for all the values of x, substituting numbers is not the best approach. We shall try to replace x with 1/x in each option.
, Since the option will have to be true for all the values of x, substituting numbers is not the best approach. We shall try to replace x with 1/x in each option.Step 2 & 3: Finding required values and finding the final answer
Let’s check each option one by one.
A. 
.
.Not equal to f(x)
B. 
Not equal to f(x)
C. 
. Not equal to f(x)
. Not equal to f(x)D. 
Not equal to f(x)
Not equal to f(x)In the exam, you need not evaluate option (E) – since we already know that Option D is the answer- but here for practice, let’s evaluate it.
Not equal to f(x).
If the sum of the first 30 positive odd integers is k, what is the sum of first 30 non-negative even integers?- a)k-29
- b)k-30
- c)k
- d)k+29
- e)k+30
Correct answer is option 'B'. Can you explain this answer?
If the sum of the first 30 positive odd integers is k, what is the sum of first 30 non-negative even integers?
a)
k-29
b)
k-30
c)
k
d)
k+29
e)
k+30
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Moumita Sen answered |
Given
- 1 + 3 + 5………..30*2 -1 = k.
- Let’s call this sequence O
To Find: 0 + 2+ 4………30 *2 -2 = ?
- Let’s call this sequence E
Approach
- To express the sum of sequence E in terms of k, we need to express the terms of sequence E in terms of sequence O
- Now, we see that we can write 0 = 1 – 1
- Similarly, we can write 2 = 3 -1
- Continuing the same pattern, we can write 58 = 59 -1
- Observe that 1, 3…..59 are terms of sequence O. So, using the above process we have captured all the terms of sequence O in sequence E
- We will use the above logic to represent the sum of sequence E in terms of k
Working Out
- 0 + 2+ 4……58 = (1-1) + (3-1) +…….(59- 1) = 1+ 3+ 5…….59 – (1 + 1 + 1……….30 times)
- 0 + 2 + 4 …….+ 58 = k – 30
Answer: B
What is the number of integral solutions of the equation 2x2 – 3x – 2 = 0?- a)0
- b)1
- c)2
- d)3
- e)4
Correct answer is option 'B'. Can you explain this answer?
What is the number of integral solutions of the equation 2x2 – 3x – 2 = 0?
a)
0
b)
1
c)
2
d)
3
e)
4
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EduRev GATE answered |
- Using the quadratic formula:
- (3 ± √25) /4
- The two roots are:
- 8 /4 = 2
- -2 /4 = -1⁄2
- Since only x = 2 is an integer, the number of integral solutions is: 1
If x is an integer and 
what is the value of x?- a)-3
- b)-1
- c)0
- d)2
- e)3
Correct answer is option 'E'. Can you explain this answer?
If x is an integer and what is the value of x?
what is the value of x?a)
-3
b)
-1
c)
0
d)
2
e)
3
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Devika Yadav answered |
Given equation is
And, we need to find the value of x, given that x is an integer.
Step 2 & 3: Finding required values and calculating the final answer
Let us rewriting the last term of (I) to get x-1 in the denominator:
Let us now bring the terms with the same denominator together:
- (x + 1)2 = {2(x – 1)}2
- [(x+1)2-{2(x-1)}2 ]=0 ………….. (II)
- Observe that this is of the form a2 – b2 = (a+b)(a-b)So let us say:a = x + 1b = 2(x – 1) = 2x - 2Therefore:a + b = x + 1 + 2x – 2 = 3x -1a – b = x + 1 - 2x + 2 = -x + 3Therefore we can rewrite (II) as:(3x-1)(3-x)=0
- Putting each factor to zero:
Since it is given that x is an integer,
Thus, x = 3 is the required answer.
Answer: Option (E)
If integers p, q and r are the roots of the equation x3-7x2+12x = 0, and p < q < r, what is the value of 
? - a)0
- b)1/3
- c)1
- d)3
- e)4
Correct answer is option 'C'. Can you explain this answer?
If integers p, q and r are the roots of the equation x3-7x2+12x = 0, and p < q < r, what is the value of ?
?a)
0
b)
1/3
c)
1
d)
3
e)
4
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Telecom Tuners answered |
- Step 1: Question statement and Inferences
- Given equation is x3−7x2+12x=0
- and p, q, r are the roots of this equation.
- We need to find the value of the expression
- . For this we need to find the values of p, q and r by solving the given equation.
- Step 2 & 3: Finding required values and calculating the final answer
- x3−7x2+12x=0
- ⇒x(x2−7x+12)=0 ....(I)
- Note that x2−7x+12
- is a standard form quadratic equation. It can be factorized as follows:
- x2−4x−3x+12=(x−3)(x−4)
- Therefore (I) can be rewritten as:
- x(x-3)(x-4) = 0
- By putting each factor equal to zero, we can find the roots of (I):
- x = 0(Root 1)
- x – 3 = 0
- ⇒ x = 3 (Root 2)
- x – 4 = 0
- ⇒ x = 4 (Root 3)
- Therefore possible values for p, q, r: {0, 3, 4}
- Since it is given that p < q < r, we have:
- p = 0, q = 3 and r =4
- Therefore
- =
- When a number is raised to the power zero, we get 1.
- So,=1
- ⇒=1
- Answer: Option (C)
240 students are to be arranged in n rows in the assembly hall of a school. If the first row is the closest to the stage and each subsequent row has 10 more students than the row ahead of it, what is the value of n?(1) There are 45 students in the 4th row from the stage.(2) The number of students in the nth row is 10 less than 5 times the number of students in the first row.- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?
240 students are to be arranged in n rows in the assembly hall of a school. If the first row is the closest to the stage and each subsequent row has 10 more students than the row ahead of it, what is the value of n?
(1) There are 45 students in the 4th row from the stage.
(2) The number of students in the nth row is 10 less than 5 times the number of students in the first row.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
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|
Rhea Gupta answered |
Steps 1 & 2: Understand Question and Draw Inferences
- As the number of students in a row is 10 more than the number of students in the previous row, the number of students in the rows form an arithmetic sequence with common difference, d = 10
- Number of rows = n
- Let the number of students in the first row be a1
and that in the last row be an
(average of the 1st and the nth term is the average of all the terms of the sequence)
To Find: Unique value of n
- Need to find the value of a1 or express a1 in terms of n to find the value of n.
Step 3: Analyze Statement 1 independently
(1) There are 45 students in the 4th row from the stage.
Substituting the value of a1 in (1), we have
As n cannot be negative, n = 6.
Sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The number of students in the nth row is 10 less than 5 times the number of students in the first row.
Putting the value of in (1), we have
As n has to be an integer, n = 6.
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
In the sequence S, the difference between any two consecutive terms is equal. If the sum of the fourth term and the fifth term of the sequence is equal to the seventh term of the sequence, what is the value of the second term of the sequence?- a)-4
- b)0
- c)4
- d)8
- e)Can't be determined
Correct answer is option 'B'. Can you explain this answer?
In the sequence S, the difference between any two consecutive terms is equal. If the sum of the fourth term and the fifth term of the sequence is equal to the seventh term of the sequence, what is the value of the second term of the sequence?
a)
-4
b)
0
c)
4
d)
8
e)
Can't be determined
|
|
Sharmila Singh answered |
Given
- In a sequence S, difference between any two terms is equal
- S is an arithmetic sequence
- Let the first term of the sequence be a and the common difference be d
- 4th term + 5th term = 7th term
- 4th term = a + (4-1)*d = a+ 3d
- 5th term = a + (5-1)*d = a +4d
- 7th term = a + (7-1)d = a + 6d
To Find: a + d = ?
Approach
- To find the value of a + d, we need to either find the values of a and d or a + d
- We are given a relation
- (a + 3d) + (a + 4d) = a + 6d
- As we are given only one equation between a and d, we will be unable to find the values of both a and b. So, we will try to see if we can find the value of a + d
Working Out
- (a + 3d) + (a+4d) = a+ 6d
- 2a + 7d = a + 6d
- a + d = 0
Hence, the value of the 2nd term of the sequence = a + d = 0
Answer: B
If x and y are non-zero numbers, what is the value of x/y?
- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'E'. Can you explain this answer?
If x and y are non-zero numbers, what is the value of x/y?
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
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|
Manasa Gupta answered |
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
Not sufficient to get a unique value of the ratio x/y.
Step 4: Analyze Statement 2 independently
Multiplying both sides of the inequality with will not impact the sign of inequality:
From the Wavy Line Method:
Not sufficient to determine the exact value.
Step 5: Analyze Both Statements Together (if needed)
From Statement 1:
From Statement 2:
Both the values of 
obtained from Statement 1 is less than −1/5
obtained from Statement 1 is less than −1/5Therefore, both these values satisfy Statement 2
So, we are still not able to determine a unique value of the ratio x/y
Answer: Option E
For any non-zero numbers p and q, 
What is the value of p$q?
- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'C'. Can you explain this answer?
For any non-zero numbers p and q, What is the value of p$q?
What is the value of p$q?a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
|
|
Nitya Kumar answered |
Steps 1 & 2: Understand Question and Draw Inferences
Given:
In order to find the required value, we need to know the value of 
Step 3: Analyze Statement 1 independently 
=> q= p2 ( Since |p|2 is always positive, we can remove the modulus sign)
Since we do not know the value of |p|, we cannot find a unique value of
Statement 1 is not sufficient.
Step 4: Analyze Statement 2 independently
- However, q may be positive or negative
- This equation conveys that p is positive. So, |p| = p
- =13(if q is positive) or − 13 (if q is negative)
Thus, Statement 2 is not sufficient to find a unique value of the ratio
Step 5: Analyze Both Statements Together (if needed)
- From Statement 1: q = p2
- From Statement 2: =13 (if q is positive ) or − 13 ( if q is negative)
- Since q is the square of a non-zero number, q must be positive
- Combining both statements: =13 (since q is positive )
Thus, the two statements together are sufficient to obtain a unique value of
Answer: Option C
The function f(x) is defined as 
where x is not equal to -1. If a is not equal to -1 or 0, which of the following expressions must be true for all values of a? 
- a)I only
- b)II only
- c)III only
- d)I, II and III
- e)None of the above
Correct answer is option 'B'. Can you explain this answer?
The function f(x) is defined as where x is not equal to -1. If a is not equal to -1 or 0, which of the following expressions must be true for all values of a?
where x is not equal to -1. If a is not equal to -1 or 0, which of the following expressions must be true for all values of a? a)
I only
b)
II only
c)
III only
d)
I, II and III
e)
None of the above
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|
Nandita Chauhan answered |
Given:
- Function f(x) is defined as
- To find: Which of the 3 expressions is/are true for all values of a?Approach:
- Since the three expressions contain f(a – 1), f(a) and f(1), we’ll first find the expressions for these 3 functions
- Then, we’ll evaluate the given expressions one by one.
- Working Out:
- Finding the expressions for f(a – 1), f(a) and f(1)
- Evaluating Expression I
- f(a−1)=f(a)–f(1)
- We will assume that f(a-1) = f(a) – f(1) is true and then see if it leads to a real value of a or not.
- Equating the above expression with the expression for f(a-1):
- For no real value of a can the square of a+1/2 be negative
- Therefore, no real value of a satisfies the equation given in Expression I.
- So, Expression I is not true.
- Since Left hand side of Expression II has the same value as the right hand side, this expression will hold true for all values of a.
Evaluating Expression III
- f(a−1)=f(a)∗f(1)
- The value of the left hand side of Expression III = 1/a
- The value of the right hand side of Expression III = 1/2(a+1)
- Equating the 2 values, we get:
- Thus, Expression III is true for only 1 value of a (-2) – not for other values.
- So, Expression III is not a must be true expression.
- Getting to the answer
- Thus, we have seen that out of the 3 expressions, only Expression II is a must be true expression.
Looking at the answer choices, we see that the correct answer is Option B
If x + y = -10 and xy = 16, which of the following statements must be true?
(I) x – y = -6
(II) x2 + y2 = 68
(III) (x-y)2 leaves a remainder of 1 when divided by 7
- a)I only
- b)II only
- c)III only
- d)I and II
- e)II and III
Correct answer is option 'E'. Can you explain this answer?
If x + y = -10 and xy = 16, which of the following statements must be true?
(I) x – y = -6
(II) x2 + y2 = 68
(III) (x-y)2 leaves a remainder of 1 when divided by 7
a)
I only
b)
II only
c)
III only
d)
I and II
e)
II and III
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|
Pranav Das answered |
Step 1: Question statement and Inferences
It is given that
x+y=−10 ....(i)
xy=16 ....(ii)
We need to find which of the given statements must be true.
Step 2 & 3: Finding required values and calculating the final answer
Squaring (i) on both sides, we get:
(x+y)2=100
x2+y2+2xy=100
....(iii)
We know that
(x−y)2=x2+y2−2xy
(x−y)2=x2+y2+2xy−4xy
(x−y)2=(x+y)2−4xy
Using (ii) and (iii) we get:
(x−y)2=100−64
(x−y)2=36
⇒(x−y)=±6
Therefore (I) is not a must be true statement.
Now let us look at (II).
(iii) – 2*(ii) gives us:
(x2+y2+2xy)−2xy=100−32
⇒x2+y2=68
Therefore statement II is true.
We already observed above that
(x−y)2=36
We know that
36=35+136=7∗5+1
Therefore
(x−y)2
leaves a remainder of 1 when divided by 7.
Answer: Option (E)
If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is - a)13
- b)9
- c)21
- d)17
Correct answer is option 'C'. Can you explain this answer?
If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is
a)
13
b)
9
c)
21
d)
17
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|
EduRev GATE answered |
- Let 3 consecutive terms A.P is a –d, a , a + d. and the sum is 51
- so, (a –d) + a + (a + d) = 51
- ⇒ 3a –d + d = 51
- ⇒ 3a = 51
- ⇒ a = 17
- The product of first and third terms is 273
- So it stand for ( a –d) (a + d) = 273
- ⇒ a2 –d2 = 273
- ⇒ 172 –d 2 = 273
- ⇒ 289 –d 2 = 273
- ⇒ d 2 = 289 –273
- ⇒ d 2 = 16
- ⇒ d = 4
- Hence the 3rd terms ( a+d )= 17 + 4 = 21
The area of a rectangle is 28 square centimeter. What is the perimeter of the rectangle?(1) If the length of the rectangle is increased by 10 centimeter and the breadth is decreased by 5 centimeter, the perimeter of the rectangle is eight times the original length of the rectangle.(2) If the length of the rectangle is increased by 350% and the breadth is increased to 350% of the original breadth, the perimeter of the rectangle is 63 centimeters more than the original perimeter of the rectangle.- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'A'. Can you explain this answer?
The area of a rectangle is 28 square centimeter. What is the perimeter of the rectangle?
(1) If the length of the rectangle is increased by 10 centimeter and the breadth is decreased by 5 centimeter, the perimeter of the rectangle is eight times the original length of the rectangle.
(2) If the length of the rectangle is increased by 350% and the breadth is increased to 350% of the original breadth, the perimeter of the rectangle is 63 centimeters more than the original perimeter of the rectangle.
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
|
|
Prashanth Chawla answered |
Steps 1 & 2: Understand Question and Draw Inferences
Given:
Let the length and breadth of rectangle be L and B respectively
- LB = 28…..............................................(1)
To find: The value of 2(L+B)
- To find perimeter, we need to know the value of L and B.
Step 3: Analyze Statement 1 independently
(1) If the length of the rectangle is increased by 10 centimeter and the breadth is decreased by 5 centimeter, the perimeter of the rectangle is eight times the original length of the rectangle.
- New L = L +10
- New B = B - 5
So,
- New Perimeter = 2(L+10 + B-5)
- 2(L+10 + B-5) = 8L
- 2L + 2B + 10 = 8L
- 6L – 2B – 10 = 0
- 3L – B – 5 = 0
- B = 3L - 5 . . . . . . . . . . . . . . .. . . . .(2)
Substituting (2) in (1):
- 2 values of L (roots) will be obtained from this quadratic equation
Comparing this with the standard quadratic form : ax2 + bx + c = 0, we get : a = 1 ; b = -5/3 ; c = -28/3
- Product of these 2 values = (c/a) −283
-
- Since the product is negative, one root is positive and the other is negative
- The negative root will be rejected
- L, being the length of a rectangle, cannot be negative
- Thus, a unique value of L is obtained
- Using Equation (2), a unique value of B is also obtained
Hence, Statement 1 alone is sufficient.
Step 4: Analyze Statement 2 independently
- If the length of the rectangle is increased by 350% and the breadth is increased to 350% of the original breadth, the perimeter of the rectangle is 63 centimeters more than the original perimeter of the rectangle
- New L = 4.5L
- New B = 3.5B
So,
Substituting (3) in (1):
- 2 values of L (roots) will be obtained from this quadratic equation
Comparing this with the standard quadratic form : ax2 + bx + c = 0, we get :
a = 1 ; b = - 9 ; c = 20
- Product of these 2 values = (c/a) = 20
- Since the product is positive, the two roots are either both positive or both negative
- Sum of roots = (-b/a) = -(-9) = 9
- Since the sum is positive, it means both roots are positive
- Thus, St. 2 leads to 2 values of L
- From Equation (3), 2 values of B will be obtained
- So, 2 values of Perimeter will be obtained.
St. 2 is not sufficient to obtain a unique value of the perimeter.
Step 5: Analyze Both Statements Together (if needed)
Since we get a unique answer in Step 3, this step is not required
Answer: Option A
If 
where the given expression extends till infinity, which of the following statements must be true ?I. Two values are possible for zII. 4 - z2 = 2III. z8 = 16 - a)I only
- b)II only
- c)III only
- d)I, II and III
- e)None of the above
Correct answer is option 'D'. Can you explain this answer?
If
where the given expression extends till infinity, which of the following statements must be true ?
I. Two values are possible for z
II. 4 - z2 = 2
III. z8 = 16
a)
I only
b)
II only
c)
III only
d)
I, II and III
e)
None of the above
|
|
Kirti Roy answered |
Given:
.To find: Which of the given 3 statements must be true about z?
Approach:
- Looking at the 3 statements, we realize that in order to validate them, we’ll need to know the value of z
- We’re given an expression for z2. Using that expression, we’ll find the value of z
Working Out:
- Finding the value of z
-
- Notice that the series of within within
is never-ending. So, we can write:
-
To make our calculations easy, let’s replace z2 with another variable, say 'y'
- So, we get:
- Now, squaring both sides:
- y2 = 2 + y
- y2 – y – 2 = 0
- (y-2)(y+1) = 0
- So, y = 2 or y = -1
- Since y = z2 and perfect squares are never negative, y cannot be -1
- So, y = z2 = 2
- This means, either z√2 or z-√2
- Checking the validity of the 3 Statements
- Statement I: Two values are possible for z
-
- As we’ve found above, indeed only 2 values are possible for z
- Therefore, this statement is true
- Statement II: 4 - z2 = 2
- As we calculated above, z2 = 2
- So, 4 – z2 = 4 – 2 = 2
- So, this statement too is true
- Statement III: z8 = 16
- Since z2 = 2, (z2)4 = 24
- So, z8 = 16
- Therefore, this statement is true as well
- Looking at the answer choices, we see that the correct answer is Option D
A number x is multiplied with itself and then added to the product of 4 and x. If the result of these two operations is -4, what is the value of x?- a)-4
- b)-2
- c)2
- d)4
- e)Cannot be determined.
Correct answer is option 'B'. Can you explain this answer?
A number x is multiplied with itself and then added to the product of 4 and x. If the result of these two operations is -4, what is the value of x?
a)
-4
b)
-2
c)
2
d)
4
e)
Cannot be determined.
|
|
Rajdeep Nair answered |
Given
- x∗x+4∗x=−4
- x2 + 4x + 4 = 0
To Find: value of x?
Approach
- As we have a quadratic equation x2+4x+4=0
- , we will solve this equation to find out the value of x
Working Out
x2+4x+4=0
⇒(x+2)2=0
⇒x=−2
Answer: B
If x is a negative number, what is the value of x?(1) 16x2 – 16x – 5 = 0(2) ||4x+3| - 5| = 3- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'A'. Can you explain this answer?
If x is a negative number, what is the value of x?
(1) 16x2 – 16x – 5 = 0
(2) ||4x+3| - 5| = 3
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
|
|
Pranab Dasgupta answered |
Steps 1 & 2: Understand Question and Draw Inferences
Given: x < 0
To find: x = ?
Step 3: Analyze Statement 1 independently
(1) 16x2 – 16x – 5 = 0
x2–x−5/16=0
- 2 values of x (roots) will be obtained from this quadratic equation
- Product of these 2 values = −5/16
-
-
- Since the product is negative, one root is positive and the other is negative
-
- The positive root will be rejected (given: x < 0)
Thus, a unique value of x will be obtained from Statement 1
Statement 1 is sufficient to answer the question.
Step 4: Analyze Statement 2 independently
x= -5/4 We can reject this value as we know that x < 0
Thus, we get 3 possible negative values of x from Statement 2:
Not sufficient to find a unique value of x
Step 5: Analyze Both Statements Together (if needed)
Since we get a unique answer in Step 3, this step is not required
Answer: Option A
If x is equal to 
, where the given expressions extend to an infinite number of roots, then what is the value of x?- a)-3
- b)-4
- c)3
- d)4
- e)12
Correct answer is option 'D'. Can you explain this answer?
If x is equal to , where the given expressions extend to an infinite number of roots, then what is the value of x?
, where the given expressions extend to an infinite number of roots, then what is the value of x?a)
-3
b)
-4
c)
3
d)
4
e)
12
|
|
Saumya Sharma answered |
Given
- to an infinite terms
- As the value of x is equal to the square root of something, x ≥ 0
To Find:
The value of x
Approach
x= to an infinite terms.
to an infinite terms.- Since the expression under the root extends infinitely, the expression under the root will be equal to x itself.
- So,
0………….(1) - By solving this equation and applying the constraint that x ≥0, we will find the value of x
Working Out
a.
b. Squaring both sides, we have
As we know that x≥ 0, x = 4
Answer: D
Alternate solution
- Once we have deduced that x will be positive the options -3 and -4 are eliminated
- We know √9 is 3. So the value of √(12+ ……) will be more than 3. Now we are left with 4 and 12 in the options
- Now √(12 + …..) cannot give us 12..
- So the answer has be 4.
Which one of the following is not a document related to fulfill the customs formalities- a)Letter of insurance
- b)Shipping bill
- c)Export licence
- d)Proforma invoice
Correct answer is option 'D'. Can you explain this answer?
Which one of the following is not a document related to fulfill the customs formalities
a)
Letter of insurance
b)
Shipping bill
c)
Export licence
d)
Proforma invoice
|
Notes Wala answered |
Document Related to Fulfilling Customs Formalities:
There are several documents that are typically required to fulfill customs formalities. However, one of the following is not a document related to fulfilling customs formalities:
A: Letter of insurance
- The letter of insurance is not directly related to customs formalities but rather pertains to insurance coverage for the shipment.
B: Shipping bill
- The shipping bill is a document that contains details about the exported goods and is required by the customs authorities to process the shipment.
C: Export license
- An export license is a document issued by the appropriate government authority that grants permission to export specific goods. It is an essential document for customs clearance.
D: Proforma invoice
- The proforma invoice is a preliminary invoice that provides a detailed description of the goods to be exported, including their value and other relevant information. It is a crucial document for customs purposes.
Answer: D
- The proforma invoice is indeed a document related to fulfilling customs formalities. The correct answer is D, as stated in the question.
There are several documents that are typically required to fulfill customs formalities. However, one of the following is not a document related to fulfilling customs formalities:
A: Letter of insurance
- The letter of insurance is not directly related to customs formalities but rather pertains to insurance coverage for the shipment.
B: Shipping bill
- The shipping bill is a document that contains details about the exported goods and is required by the customs authorities to process the shipment.
C: Export license
- An export license is a document issued by the appropriate government authority that grants permission to export specific goods. It is an essential document for customs clearance.
D: Proforma invoice
- The proforma invoice is a preliminary invoice that provides a detailed description of the goods to be exported, including their value and other relevant information. It is a crucial document for customs purposes.
Answer: D
- The proforma invoice is indeed a document related to fulfilling customs formalities. The correct answer is D, as stated in the question.
For distinct positive integers x and y, where x < y, the function FP(x, y) returns the smallest prime number between x and y, exclusive, or the text string ‘NULL’ if no such number is found. If FP(a, b) +FP(c, d) = FP(e, f), where a, b, c, d, e and f are distinct positive integers, what is the value of ca ?(1) FP(g, h) = a, where g and h are distinct positive integers(2) c is less than the minimum possible value of the function FP(x,y).- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?
For distinct positive integers x and y, where x < y, the function FP(x, y) returns the smallest prime number between x and y, exclusive, or the text string ‘NULL’ if no such number is found. If FP(a, b) +FP(c, d) = FP(e, f), where a, b, c, d, e and f are distinct positive integers, what is the value of ca ?
(1) FP(g, h) = a, where g and h are distinct positive integers
(2) c is less than the minimum possible value of the function FP(x,y).
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
|
|
Pallavi Sharma answered |
Steps 1 & 2: Understand Question and Draw Inferences
- For positive integers x and y where x < y, FP(x, y) returns the smallest prime number between x and y, exclusive
- FP(a, b) +FP(c, d) = FP(e, f)
- FP(a, b), FP(c, d) and FP(e, f) each return a prime number
- So, the above equation conveys that the Sum of two prime numbers is equal to another prime number
- As all prime numbers except 2 are odd, FP(e, f) will be odd and either of FP(a, b) or FP(c, d) will be even i.e. 2
- For example, if both FP(a,b) and FP(c, d) are odd, then their sum i.e. FP(e, f) will be even, i.e. 2, which is not possible as there are no prime numbers less than 2
- So, the only case possible is that either of FP(a, b) or FP(c, d) is 2 and hence the sum of FP(a, b) and FP(c, d), which is FP (e, f) is odd
- If FP(a, b) = 2
- a = 1, ca = c
- If FP(c, d) = 2
- If c = 1, ca = 1
To Find: Value of c
Step 3: Analyze Statement 1 independently
(1) FP(g, h) = a, where g and h are distinct positive integers
- Since we know that FP(g, h) will return a prime number, we can infer that a is a prime number
- As a is a prime number, a > 1. So, FP(a, b) ≠ 2
- Hence FP(c, d) = 2
- For this to be possible, c = 1
If c = 1, ca = 1
Sufficient to answer
Step 4: Analyze Statement 2 independently
(2) c is less than the minimum possible value of the function FP(x,y).
- Minimum possible value of the function FP(x, y) = smallest prime number = 2
- So, c < 2,
- As c is given to be a positive integer and now we know that c < 2, the only possible value of c = 1
Hence ca = 1
Sufficient to answer
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
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