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All questions of Algebra for CTET & State TET Exam
What is the value of 225 + 225?- a)226
- b)250
- c)425
- d)450
- e)2625
Correct answer is option 'A'. Can you explain this answer?
What is the value of 225 + 225?
a)
226
b)
250
c)
425
d)
4
50
e)
2625
|
Anihegde1502 answered |
Take 2raise to25 common thjs ib bracket there will be (1+1)i.e 2raise to25 *2 thus power will get add 25+1 i.e 26 hence ans is option A
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If ‘x’ and ‘y’ are non-negative integers, what is the value of x + y?(1) x4 is even, where x is a prime number.(2) xy is odd.- 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?
If ‘x’ and ‘y’ are non-negative integers, what is the value of x + y?
(1) x4 is even, where x is a prime number.
(2) xy is odd.
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.
|
BT Educators answered |
(1) x = 2, y is unknown. Insufficient
(2) x and y unknown. Insufficient
(1)+(2) only possibility x=2, y=0; Sufficient
C is correct
(2) x and y unknown. Insufficient
(1)+(2) only possibility x=2, y=0; Sufficient
C is correct
If a, b and x are integers such that 
, what is the value of a - b
- 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 a, b and x are integers such that , what is the value of a - b
, what is the value of a - ba)
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.
|
Anaya Patel answered |
Steps 1 & 2: Understand Question and Draw Inferences
- As a6 is always positive,a6 = 1, i.e. a = 1 or -1
- So, we can reject the value of
Possible values of a – b - If a = 1 and b = 1, a – b = 0
- If a = -1 and b = 1, a- b = -2
- So, we need to find the unique value of a to find the value of a – b.Step 3: Analyze Statement 1 independently(1) a3 b7 > 0
- Rewriting a3b7 as ab(a2b6)
- Therefore, ab(a2b6)>0
- We know that a2b6 is always > 0 (even power of any number is always positive)
- So, for ab(a2b6)> 0
- ab > 0
- This tells us that a and b have same signs.
- Since b > 0, therefore a will also be greater than 0, so the value of a = 1.
- a – b = 1 -1 = 0
- Sufficient to answerStep 4: Analyze Statement 2 independently(2) a + b > 0
- If a = 1 and b = 1, a + b = 2 > 0
- If a = -1 and b = 1, a + b = 0, is not greater than zero
- Hence, we have a unique answer, where a =1 and b = 1Thus a – b = 1 – 1 = 0.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 a Country X, temperature decrease (in degrees) is linearly related to percentage increase in sale of water heaters and is given by the relation:ΔT= ky+5 Where,ΔT is temperature decrease (in degrees).y is the increase in sale of water heaters (in percentage).K is a constant.A temperature decrease of 10 degrees results in a 1% increase in sale of water heaters. What would be the percentage increase in sale of water heaters when temperature decreases by 15 degrees?- a)1.5%
- b)2%
- c)2.5%
- d)3%
- e)3.5%
Correct answer is option 'B'. Can you explain this answer?
In a Country X, temperature decrease (in degrees) is linearly related to percentage increase in sale of water heaters and is given by the relation:
ΔT= ky+5
Where,
ΔT is temperature decrease (in degrees).
y is the increase in sale of water heaters (in percentage).
K is a constant.
A temperature decrease of 10 degrees results in a 1% increase in sale of water heaters. What would be the percentage increase in sale of water heaters when temperature decreases by 15 degrees?
a)
1.5%
b)
2%
c)
2.5%
d)
3%
e)
3.5%
|
Tushar Ghosh answered |
∆T=ky+5;
given: ∆T=10
y=1
therefore: 10= k(1) +5
=> k= 10-5=5
having determined value of k=5, we add new values as
∆T=15
k=5
y=?
thus, 15=5y+5
or, 5y=15-5
5y=10
y=2
since y is the value in percentage of increase in sale of water heater, therefore, the percentage of increase in water heater is 2%.
given: ∆T=10
y=1
therefore: 10= k(1) +5
=> k= 10-5=5
having determined value of k=5, we add new values as
∆T=15
k=5
y=?
thus, 15=5y+5
or, 5y=15-5
5y=10
y=2
since y is the value in percentage of increase in sale of water heater, therefore, the percentage of increase in water heater is 2%.
If Z is a positive integer such that
- 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 Z is a positive integer such that
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.
|
Meera Rana answered |
Steps 1 & 2: Understand Question and Draw Inferences
Given:
- Z is a positive integer
- Z = 81(Y4 – 7)3 . . . (1)
We need to find the value of Y.
Step 3: Analyze Statement 1 independently
- Squaring both sides:
- Z5 = 350
- Taking 5th root on both sides:
- Z = 310 . . . (2)
- Put (2) in (1):
- 310 = 34(Y4 – 7)3
- 36 = (Y4 – 7)3
- Taking the cube-root on both sides:
- 32 = Y4 – 7
- Y4 = 9 + 7 = 16
- Y4 = 24 = (-2)4
- Y = 2 or -2
Not sufficient to determine a unique value of Y.
Step 4: Analyze Statement 2 independently
(2) |Y-1| < 4
- Distance of Y from 1 on the number line is less than 4 units
- -3 < Y < 5
Multiple values of Y possible. Not sufficient.
Step 5: Analyze Both Statements Together (if needed)
- From St. 1, Y = 2 or – 2
- From St. 2, -3 < Y < 5
- This inequality is satisfied by both 2 and -2
So, even after combining both statements, we have 2 possible values of Y
Since we couldn’t find a unique value of Y, the correct answer. Is Option E.
James deposited $1,000 each in two investment schemes X and Y. Scheme X doubles the invested amount every 7 years and scheme Y doubles the invested amount every 14 years. If James withdraws $500 from scheme X at the end of every 7th year, how many years will it take for the total amount invested in schemes X and Y to amount more than $40,000?
- a)14
- b)28
- c)42
- d)56
- e)70
Correct answer is option 'C'. Can you explain this answer?
James deposited $1,000 each in two investment schemes X and Y. Scheme X doubles the invested amount every 7 years and scheme Y doubles the invested amount every 14 years. If James withdraws $500 from scheme X at the end of every 7th year, how many years will it take for the total amount invested in schemes X and Y to amount more than $40,000?
a)
14
b)
28
c)
42
d)
56
e)
70
|
|
Meera Rana answered |
Given
- Scheme X doubles the invested amount every 7 years
- James deposited $1000 in scheme X
- James withdraws $500 from scheme X after the end of every 7 years
- Scheme Y doubles the invested amount after every 14 years
- James deposited $1,000 in scheme Y
To Find: Number of years it will take total amount deposited in schemes X and Y to grow to > $40,000?
Approach
- For finding the number of years it will take the deposits in schemes X and Y to grow to more than $40,000, we need to find the amount in both the schemes X and Y after every 7 years.(As amount in scheme X doubles after every 7 years, we will need to calculate the amount at the end of every 7 years and not at the end of 14 years).
- Scheme X
- As the amount invested in scheme X doubles every 7 years, we will need to calculate the amount in scheme X after every interval of 7 years
- However, we will need to make sure that we subtract $500 at each interval of 7 years from the final amount
- Scheme Y
- As the amount invested in scheme Y doubles after every 14 years, we will need to calculate the amount in scheme Y after every interval of 14 years.
- At each interval, we will calculate the sum of amounts in scheme X and Y to check if it exceeds $40,000.
Working Out
- Amount at the end of year 7 in scheme X = $1000 * 2 = $2000
- However James withdrew $500 at the end of 7th year, So, the amount remaining will be $2000 – $500 = $1500
- The same logic has been applied in calculating the amounts at the end of every 7 year interval
- Amount at the end of year 14 in scheme Y = $1000 * 2 = $2000
- The same logic has been applied in calculating the amounts at the end of every 14 years interval.
- The same logic has been applied in calculating the amounts at the end of every 14 years interval.
- We can see that the total amount in schemes X and Y exceed $40,000 by the end of the year 42.
Answer: C
Mike visits his childhood friend Alan at a regular interval of 4 months. For example, if Mike visits Alan on 1st Jan, his next visit would be on 1st May and so on. He started this routine on his 25th birthday. Yesterday, he celebrated his Nth birthday. How many visits has Mike made so far (including the first visit on his 25th birthday)?- a)n – 24
- b)2n – 50
- c)2n – 49
- d)3n – 75
- e)3n – 74
Correct answer is option 'E'. Can you explain this answer?
Mike visits his childhood friend Alan at a regular interval of 4 months. For example, if Mike visits Alan on 1st Jan, his next visit would be on 1st May and so on. He started this routine on his 25th birthday. Yesterday, he celebrated his Nth birthday. How many visits has Mike made so far (including the first visit on his 25th birthday)?
a)
n – 24
b)
2n – 50
c)
2n – 49
d)
3n – 75
e)
3n – 74
|
|
Aisha Gupta answered |
In a period of 1 year, Mike visits Alan 3 times (12 months divided by 4). However this excludes the first time visit and takes into consideration the subsequent visits only. So starting on his 25th birthday, Mike will visit Alan 3(n-25) times up till his nth birthday. However we have to add the first visit as well. So the final answer would be 3n-74 .
Find the value of z such that 2(z-1)3 + 6(1-z)3 = 32?- a)-2
- b)-1
- c)0
- d)1
- e)2
Correct answer is option 'B'. Can you explain this answer?
Find the value of z such that 2(z-1)3 + 6(1-z)3 = 32?
a)
-2
b)
-1
c)
0
d)
1
e)
2
|
Yash Patel answered |
⇒ 2(z-1)3 + 6(1-z)3 = 32
⇒ 2 [ z3 -1 - 3z(z-1) ] + 6 [1 - z3 - 3z(1-z)] = 32
⇒ 2 [ z3 - 1 -3z2 + 3z ] + 6 [ 1 - z3 -3z + 3z2 ] = 32
⇒ 2z3 - 2 - 6z2 + 6z + 6 - 6z3 -18z2 + 18z = 32
⇒ -4z3 + 12z2 - 12z + 4 = 32
Substract 32 from both sides we get,
⇒ -4z3 + 12z2 - 12z + 4 - 32 = 32 - 32
⇒ -4z3 + 12z2 - 12z - 28 = 0
⇒ -4( z + 1 )( z2 - 4z -7) = 0
⇒ ( z + 1 )( z2 - 4z -7) = 0
Then,
( z + 1 ) = 0
z = -1
or
( z2 - 4z -7) = 0
z = 2 + √3i, 2 - √3i
-2x - ky -9 = 04x – 10y + 18 = 0What is the value of k if the system of linear equations shown above has infinite solutions?- a)-5
- b)-1
- c)1
- d)3
- e)5
Correct answer is option 'A'. Can you explain this answer?
-2x - ky -9 = 0
4x – 10y + 18 = 0
What is the value of k if the system of linear equations shown above has infinite solutions?
a)
-5
b)
-1
c)
1
d)
3
e)
5
|
Siddharth Pillai answered |
Given equations,
-2x - ky -9 = 0 .. (a)
4x – 10y + 18 = 0 .. (b)
Multiply the equation (a) with 2 and add it to equation (b), we get
⇒ -4x - 2ky - 18 +4x - 10y + 18 = 0
⇒ -2ky -10y = 0
⇒ -2ky = 10y
⇒ -2k = 10
⇒ k = -5
If n is a positive integer greater than 2, what is the greatest prime factor of 3n + 3n + 3n – 3n-2?- a)3
- b)5
- c)7
- d)11
- e)13
Correct answer is option 'E'. Can you explain this answer?
If n is a positive integer greater than 2, what is the greatest prime factor of 3n + 3n + 3n – 3n-2?
a)
3
b)
5
c)
7
d)
11
e)
13
|
Arnab Kumar answered |
Solution:
Firstly, we can simplify the given expression by combining the exponents:
3n 3n 3n 3n-2 = 33n-2 * 33n = 36n
Now, to find the greatest prime factor of 36n, we can factorize it into prime factors:
36n = 2^2 * 3^2 * n
The greatest prime factor of 36n would be the largest prime factor of n. Since n is greater than 2, we know that it is either a prime number or a composite number with prime factors.
To find the greatest prime factor of n, we can start by dividing n by 2 repeatedly until we get an odd number. For example, if n is 60, we can divide it by 2 three times to get 15:
60 ÷ 2 = 30
30 ÷ 2 = 15
Now, we can check if 15 is a prime number or if it has any other prime factors. We can do this by dividing 15 by the smallest prime numbers, which are 2, 3, 5, 7, 11, 13, etc.
15 ÷ 3 = 5
Since 5 is a prime number, it is the greatest prime factor of n. Therefore, the greatest prime factor of 36n is 13, which is the largest prime factor of 3.
Firstly, we can simplify the given expression by combining the exponents:
3n 3n 3n 3n-2 = 33n-2 * 33n = 36n
Now, to find the greatest prime factor of 36n, we can factorize it into prime factors:
36n = 2^2 * 3^2 * n
The greatest prime factor of 36n would be the largest prime factor of n. Since n is greater than 2, we know that it is either a prime number or a composite number with prime factors.
To find the greatest prime factor of n, we can start by dividing n by 2 repeatedly until we get an odd number. For example, if n is 60, we can divide it by 2 three times to get 15:
60 ÷ 2 = 30
30 ÷ 2 = 15
Now, we can check if 15 is a prime number or if it has any other prime factors. We can do this by dividing 15 by the smallest prime numbers, which are 2, 3, 5, 7, 11, 13, etc.
15 ÷ 3 = 5
Since 5 is a prime number, it is the greatest prime factor of n. Therefore, the greatest prime factor of 36n is 13, which is the largest prime factor of 3.
Find the value of positive integer P that lies between 1 and 30 and is a perfect square.(1) P has at least one Prime factor(2) The cube of P is less than 300- 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?
Find the value of positive integer P that lies between 1 and 30 and is a perfect square.
(1) P has at least one Prime factor
(2) The cube of P is less than 300
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
Given that P is positive and 1 < P < 30.
It is also given that P is a perfect square.
This means, that P is the square of an integer.
Therefore the possible values of P are 4, 9, 16, and 25.
Step 3: Analyze Statement 1
Given that P has at least one prime factor.
EVERY number greater than 1 has at least one prime factor.
This statement doesn’t help us to eliminate any value out of the four possible values of P.
So statement 1 is not sufficient to arrive at a unique answer.
Step 4: Analyze Statement 2
Given that cube of P is less than 300.
Observe that only 4 satisfies this condition.
43=64
93=729
(Since 93 itself is greater than 300, it is clear that 163 and 253 will also be greater than 300.)
Therefore the only possible value for P is 4.
Statement 2 is sufficient to arrive at a unique answer.
Step 5: Analyze Both Statements Together (if needed)
We arrived at a unique answer in step 4. Hence this step is not required.
Correct Answer: B
Solve for x: −6x−20=−2x+4(1−3x)- a)20
- b)-6
- c)6
- d)3
Correct answer is option 'D'. Can you explain this answer?
Solve for x: −6x−20=−2x+4(1−3x)
a)
20
b)
-6
c)
6
d)
3
|
|
Amar Chakraborty answered |
Explanation:
−6x−20=−2x+4(1−3x
−6x−20=−2x+4−12x
−6x−20=−14x+4
−6x+14x=4+20
8x=24
x=3
−6x−20=−2x+4(1−3x
−6x−20=−2x+4−12x
−6x−20=−14x+4
−6x+14x=4+20
8x=24
x=3
The common root of 2x2+x−6 = 0 and x2−3x−10 = 0 is
- a)2
- b)5
- c)-2
- d)3/2
Correct answer is option 'C'. Can you explain this answer?
The common root of 2x2+x−6 = 0 and x2−3x−10 = 0 is
a)
2
b)
5
c)
-2
d)
3/2
|
|
Chirag Sen answered |
The common root of 2x^2 and x is x.
The product of two successive integral multiples of 5 is 1050. Then the numbers are- a)35 and 40
- b)25 and 30
- c)25 and 42
- d)30 and 35
Correct answer is option 'D'. Can you explain this answer?
The product of two successive integral multiples of 5 is 1050. Then the numbers are
a)
35 and 40
b)
25 and 30
c)
25 and 42
d)
30 and 35
|
|
Prateek Gupta answered |
Explanation:
Let one multiple of 5 be x then the next consecutive multiple of will be (x+5) According to question,
Then the number are 30 and 35.
The square of the difference between two given natural numbers is 324, while the product of these two given numbers is 144. Find the positive difference between the squares of these two given numbers.- a)630
- b)540
- c)450
- d)360
Correct answer is option 'B'. Can you explain this answer?
a)
630
b)
540
c)
450
d)
360
|
Saumya Mehta answered |
Given Information:
The square of the difference between two natural numbers is 324, while the product of these two numbers is 144.
Let's solve the problem step by step:
Step 1: Set up the equations
Let the two natural numbers be x and y. According to the given information,
1. (x - y)^2 = 324
2. xy = 144
Step 2: Solve the equations
1. (x - y)^2 = 324
Expanding the left side, we get:
x^2 - 2xy + y^2 = 324
x^2 + y^2 - 2xy = 324
2. xy = 144
Now, substitute the value of xy from equation 2 into equation 1:
x^2 + y^2 - 2*144 = 324
x^2 + y^2 = 612
Step 3: Find the positive difference between the squares of the numbers
We need to find the positive difference between x^2 and y^2.
Since x^2 + y^2 = 612, and xy = 144, we can find the squares of x and y.
x^2 = 612 - y^2
x^2 = 612 - 144
x^2 = 468
Similarly,
y^2 = 612 - x^2
y^2 = 612 - 468
y^2 = 144
Now, find the positive difference between x^2 and y^2:
|x^2 - y^2| = |468 - 144| = 324
Therefore, the positive difference between the squares of the two given natural numbers is 324.
Conclusion:
The correct answer is option B) 540.
The square of the difference between two natural numbers is 324, while the product of these two numbers is 144.
Let's solve the problem step by step:
Step 1: Set up the equations
Let the two natural numbers be x and y. According to the given information,
1. (x - y)^2 = 324
2. xy = 144
Step 2: Solve the equations
1. (x - y)^2 = 324
Expanding the left side, we get:
x^2 - 2xy + y^2 = 324
x^2 + y^2 - 2xy = 324
2. xy = 144
Now, substitute the value of xy from equation 2 into equation 1:
x^2 + y^2 - 2*144 = 324
x^2 + y^2 = 612
Step 3: Find the positive difference between the squares of the numbers
We need to find the positive difference between x^2 and y^2.
Since x^2 + y^2 = 612, and xy = 144, we can find the squares of x and y.
x^2 = 612 - y^2
x^2 = 612 - 144
x^2 = 468
Similarly,
y^2 = 612 - x^2
y^2 = 612 - 468
y^2 = 144
Now, find the positive difference between x^2 and y^2:
|x^2 - y^2| = |468 - 144| = 324
Therefore, the positive difference between the squares of the two given natural numbers is 324.
Conclusion:
The correct answer is option B) 540.
What is the remainder obtained when 1010 + 105 – 24 is divided by 36?- a)5
- b)6
- c)12
- d)16
- e)32
Correct answer is option 'E'. Can you explain this answer?
What is the remainder obtained when 1010 + 105 – 24 is divided by 36?
a)
5
b)
6
c)
12
d)
16
e)
32
|
|
Pallabi Sengupta answered |
Given:
- Not applicable
To find: The remainder when 1010 + 105 – 24 is divided by 36
Approach:
- Let the required remainder be r. This means, we will be able to write:
1010 + 105 – 24 = 36k + r, where quotient k is an integer and 0 ≤ r < 36
The above expression is our GOAL expression. We’ll try to simplify the dividend 1010 + 105 – 24 till it is comparable to our GOAL expression, and then, by comparison, we’ll be able to find the value of r.
Working Out:
- 1010 + 105 – 24 = 1005 + 1002*10 – (36 – 12)
- =(36*3 – 8)5 + (36*3 – 8)2*10 – 36 + 12
- Now, from Binomial Theorem, we know that every term in the expansion of (36*3 – 8)5 will be divisible by 36, except the last term, and the last term will be (-8)5
- So, we can write: (36*3 – 8)5 is of the form 36a + (-8)5, where 36a is a catch-all term conveying that all the other terms in this expansion are divisible by 36
- Similarly, every term in the expansion of (36*3 – 8)2 will be divisible by 36 except the last term, and the last term will be (-8)2
- So, we can write: (36*3 – 8)2 = 36b + (-8)2
- Now, from Binomial Theorem, we know that every term in the expansion of (36*3 – 8)5 will be divisible by 36, except the last term, and the last term will be (-8)5
- So, the given expression simplifies to:
- {36a + (-8)5 } + {36b + (-8)2}*10 – 36 + 12
- = (36a + 360b – 36) + (-85 + 640 + 12)
- = (36a + 360b – 36) + (-85+ 652)
- = (36a + 360b – 36 + 648) + (-85+ 4)
- = (36a + 360b – 36 + 36*18) + (-85+ 4)
- The above expression is not comparable to our GOAL expression because the term -85 in it is still unresolved. Do we need to calculate the value of -85 to answer this question? No. We only need to express it in terms of 36. Once again, we’ll use Binomial Theorem to do so:
- -85 = -8(82)2 = -8(64)2 = -8(36*2 – 8)2
- Every term in the expansion of (36*2 – 8)2 will be divisible by 36 except the last term. The last term will be (-8)2 = 64
- So, the expression (36*2 – 8)2 can be written as: 36c + 64
- So, -8(36*2 – 8)2 = -8(36c + 64)
- =-8(36c + 36*2 – 8)
- = (-8*36c – 8*36*2) + 64
- -85 = -8(82)2 = -8(64)2 = -8(36*2 – 8)2
- So, the given expression simplifies to: (36a + 360b – 36 + 72) + {(-8*36c – 8*36*2) + 64} + 4
- = (36a + 360b – 36 + 72 – 8*36c – 8*36*2) + 68
- =(36a + 360b – 36 + 72 – 8*36c – 8*36*2 + 36) + 32
- Now, the above expression is exactly comparable to our GOAL Expression: 36k + r
- So, by comparison, we can say that Remainder r = 32
Looking at the answer choices, we see that the correct answer is Option E
The two numbers whose sum is 27 and their product is 182 are- a)12 and 13
- b)12 and 15
- c)14 and 15
- d)13 and 14
Correct answer is option 'D'. Can you explain this answer?
The two numbers whose sum is 27 and their product is 182 are
a)
12 and 13
b)
12 and 15
c)
14 and 15
d)
13 and 14
|
|
Prateek Gupta answered |
Explanation:Let the one number be xx .As the sum of numbers is 27 , then the other number will be (27−x)(27−x) According to question
Which of the following has two distinct roots?- a)x2+x−5=0
- b)x2+x+5=0
- c)none of these
- d)5x2−3x+1=0
Correct answer is option 'A'. Can you explain this answer?
Which of the following has two distinct roots?
a)
x2+x−5=0
b)
x2+x+5=0
c)
none of these
d)
5x2−3x+1=0
|
|
Siddharth Pillai answered |
To determine which of the given options has two distinct roots, we need to determine the discriminant of each equation. The discriminant is the part of the quadratic formula inside the square root, and it helps us determine the nature of the roots of a quadratic equation.
The quadratic equation is of the form ax^2 + bx + c = 0, where a, b, and c are constants.
The discriminant, denoted by Δ, is given by the formula Δ = b^2 - 4ac.
If Δ > 0, the equation has two distinct real roots.
If Δ = 0, the equation has two equal real roots.
If Δ < 0,="" the="" equation="" has="" no="" real="" />
Let's calculate the discriminant for each option:
a) x^2 + x + 5 = 0
In this case, a = 1, b = 1, and c = 5.
Δ = (1)^2 - 4(1)(5) = 1 - 20 = -19 (negative)
Since the discriminant is negative, this equation has no real roots.
b) x^2 + x * 5 = 0
In this case, a = 1, b = 5, and c = 0.
Δ = (5)^2 - 4(1)(0) = 25 - 0 = 25 (positive)
Since the discriminant is positive, this equation has two distinct real roots.
c) 5x^2 + 3x + 1 = 0
In this case, a = 5, b = 3, and c = 1.
Δ = (3)^2 - 4(5)(1) = 9 - 20 = -11 (negative)
Since the discriminant is negative, this equation has no real roots.
Therefore, the only option that has two distinct roots is option 'A' (x^2 + x + 5 = 0).
The quadratic equation is of the form ax^2 + bx + c = 0, where a, b, and c are constants.
The discriminant, denoted by Δ, is given by the formula Δ = b^2 - 4ac.
If Δ > 0, the equation has two distinct real roots.
If Δ = 0, the equation has two equal real roots.
If Δ < 0,="" the="" equation="" has="" no="" real="" />
Let's calculate the discriminant for each option:
a) x^2 + x + 5 = 0
In this case, a = 1, b = 1, and c = 5.
Δ = (1)^2 - 4(1)(5) = 1 - 20 = -19 (negative)
Since the discriminant is negative, this equation has no real roots.
b) x^2 + x * 5 = 0
In this case, a = 1, b = 5, and c = 0.
Δ = (5)^2 - 4(1)(0) = 25 - 0 = 25 (positive)
Since the discriminant is positive, this equation has two distinct real roots.
c) 5x^2 + 3x + 1 = 0
In this case, a = 5, b = 3, and c = 1.
Δ = (3)^2 - 4(5)(1) = 9 - 20 = -11 (negative)
Since the discriminant is negative, this equation has no real roots.
Therefore, the only option that has two distinct roots is option 'A' (x^2 + x + 5 = 0).
If α and β are the roots of the equation x2 - 7x + 1 = 0, then what is the value of α4 + β4 ? - a)2207
- b)2247
- c)2317
- d)2337
Correct answer is option 'A'. Can you explain this answer?
α4
+ β4
? a)
2207
b)
2247
c)
2317
d)
2337
|
|
Vaibhav Kulkarni answered |
Given Equation:
The equation given is x^2 - 7x + 1 = 0. Let α and β be the roots of this equation.
Sum and Product of Roots:
From the equation x^2 - 7x + 1 = 0, we know that the sum of the roots (α + β) = 7 and the product of the roots (αβ) = 1.
Using Vieta's Formulas:
We can express α^4 + β^4 in terms of α + β and αβ using the following formula:
α^4 + β^4 = (α^2 + β^2)^2 - 2α^2β^2
We know that (α + β)^2 = α^2 + β^2 + 2αβ
Therefore, α^2 + β^2 = (α + β)^2 - 2αβ
Substitute the values, we get:
α^2 + β^2 = 7^2 - 2(1) = 49 - 2 = 47
Calculating α^4 + β^4:
Now, we can find α^4 + β^4 using the formula:
α^4 + β^4 = (α^2 + β^2)^2 - 2α^2β^2
α^4 + β^4 = 47^2 - 2(1)^2
α^4 + β^4 = 2209 - 2 = 2207
Conclusion:
Therefore, the value of α^4 + β^4 is 2207, which corresponds to option 'A'.
The equation given is x^2 - 7x + 1 = 0. Let α and β be the roots of this equation.
Sum and Product of Roots:
From the equation x^2 - 7x + 1 = 0, we know that the sum of the roots (α + β) = 7 and the product of the roots (αβ) = 1.
Using Vieta's Formulas:
We can express α^4 + β^4 in terms of α + β and αβ using the following formula:
α^4 + β^4 = (α^2 + β^2)^2 - 2α^2β^2
We know that (α + β)^2 = α^2 + β^2 + 2αβ
Therefore, α^2 + β^2 = (α + β)^2 - 2αβ
Substitute the values, we get:
α^2 + β^2 = 7^2 - 2(1) = 49 - 2 = 47
Calculating α^4 + β^4:
Now, we can find α^4 + β^4 using the formula:
α^4 + β^4 = (α^2 + β^2)^2 - 2α^2β^2
α^4 + β^4 = 47^2 - 2(1)^2
α^4 + β^4 = 2209 - 2 = 2207
Conclusion:
Therefore, the value of α^4 + β^4 is 2207, which corresponds to option 'A'.
Simplify the expression: (c + d)2 - (c - d)2- a)4cd
- b)(c2 + d2)
- c)2(c2 + d2)
- d)2cd
Correct answer is option 'A'. Can you explain this answer?
Simplify the expression:
(c + d)2 - (c - d)2
a)
4cd
b)
(c2 + d2)
c)
2(c
2 + d
2)
d)
2cd
|
CodeNation answered |
Formula Used :
x2 - y2 = (x + y) (x - y)
Calculation:
⇒ (c + d)2 - (c - d)2
⇒ (c + d + c - d) (c + d -(c - d))
⇒ 2c × (c + d - c + d)
⇒ 2c × 2d = 4cd
∴ The simplified value is 4cd.
Alternate Method
Formula Used :
(x + y)2 = x2 + y2 + 2xy
(x - y)
2 = x
2 + y
2 - 2xy
Calculation:
⇒ (c + d)2 - (c - d)2
⇒ c2 + d2 + 2cd - (
c
2 + d
2 - 2cd)
⇒ c
2 + d
2 + 2cd -
c
2
- d
2
+ 2cd
⇒
4cd∴ The simplified value is 4cd.
Find the value of n that satisfies the equation 2(-3)4n = 18(27)n+2- a)3
- b)4
- c)6
- d)8
- e)22
Correct answer is option 'D'. Can you explain this answer?
Find the value of n that satisfies the equation 2(-3)4n = 18(27)n+2
a)
3
b)
4
c)
6
d)
8
e)
22
|
|
Mrinalini Dasgupta answered |
We need to find the value of n, given 2(−34n)=18(27)n+2
(Cancelling 2 on both sides. Also making use of the fact that 32k = (-3)2k)
Which of the following equations has the set of all real numbers as its solution set?- a) 3(N+4)+5N=8(N+3)
- b) 4(N+4)+4N=8(N+3)
- c) 2(N+4)+6N=8(N+3)
- d) 6(N+4)+2N=8(N+3)
- e) 5(N+4)+3N=8(N+3)
Correct answer is option 'D'. Can you explain this answer?
Which of the following equations has the set of all real numbers as its solution set?
a)
3(N+4)+5N=8(N+3)
b)
4(N+4)+4N=8(N+3)
c)
2(N+4)+6N=8(N+3)
d)
6(N+4)+2N=8(N+3)
e)
5(N+4)+3N=8(N+3)
|
|
Saumya Sharma answered |
The right side of each equation is 8(N+3), which simplifies by way of distribution to
8(N+3)=8⋅N+8⋅3=8N+24
If the left side of the equation simplifies to an identical expression, the equation has all real numbers as its solutions.
We test the left side of each equation:
2(N+4)+6N=8(N+3)
2(N+4)+6N=2⋅N+2⋅4+6N=2N+8+6N=8N+8
2(N+4)+6N=2⋅N+2⋅4+6N=2N+8+6N=8N+8
3(N+4)+5N=8(N+3)
3(N+4)+5N=3⋅N+3⋅4+5N=3N+12+5N=8N+12
4(N+4)+4N=8(N+3)
4(N+4)+4N=4⋅N+4⋅4+4N=4N+16+4N=8N+16
5(N+4)+3N=8(N+3)
5(N+4)+3N=5⋅N+5⋅4+3N=5N+20+3N=8N+20
6(N+4)+2N=8(N+3)
6(N+4)+2N=6⋅N+6⋅4+2N=6N+24+2N=8N+24
Of the given choices,
6(N+4)+2N=8(N+3)
can be rewritten as
8N+24=8N+24,
which is an identity and has the set of all real numbers as its solution set.
Simplify, - a)x2 - 2x + 1
- b)x2 + 2x + 2
- c)x2 + 2x + 1
- d)x2 + x + 1
Correct answer is option 'C'. Can you explain this answer?
Simplify,
a)
x2 - 2x + 1
b)
x
2
+ 2x + 2c)
x
2
+ 2x + 1d)
x
2
+ x + 1
|
Vikram Singh Baghel answered |
Formula used:
(a - b)2 = a2 + b2 - 2ab
(a2 - b2) = (a + b) × (a - b)
(a +
b)
2
= a
2
+ b
2
+
2ab
Calculation:
⇒ {(x2)2 + 1 - 2 × (x2) × 1}/(x - 1)2
⇒ (x2
- 1)2/(x - 1)2⇒ {(x + 1)2(x - 1)2}/(x - 1)2
⇒ (x + 1)
2
= x
2
+ 2x + 1
∴ The correct option is 3.
The angry Arjun carried some arrows for fighting with Bheeshma. With half the arrows, he cut down the arrows thrown by Bheeshma on him and with six other arrows he killed the rath driver of Bheeshma. With one arrow each he knocked down respectively the rath, flag and bow of Bheeshma. Finally with one more than four times the square root of arrows he laid Bheeshma unconscious on an arrow bed. The total number of arrows that Arjun had is- a)80
- b)100
- c)96
- d)120
Correct answer is option 'B'. Can you explain this answer?
The angry Arjun carried some arrows for fighting with Bheeshma. With half the arrows, he cut down the arrows thrown by Bheeshma on him and with six other arrows he killed the rath driver of Bheeshma. With one arrow each he knocked down respectively the rath, flag and bow of Bheeshma. Finally with one more than four times the square root of arrows he laid Bheeshma unconscious on an arrow bed. The total number of arrows that Arjun had is
a)
80
b)
100
c)
96
d)
120
|
|
Prateek Gupta answered |
Therefore, Arjun had 100 arrows.
What is the value of (27x3 - 58x2y + 31xy2 - 8y3), when x = -5 and y = -7?- a)1924
- b)-1924
- c)-1926
- d)1926
Correct answer is option 'A'. Can you explain this answer?
a)
1924
b)
-1924
c)
-1926
d)
1926
|
Yash Nair answered |
Given Values:
x = -5
y = -7
Expression:
27x^3 - 58x^2y + 31xy^2 - 8y^3
Substitute x and y values:
27(-5)^3 - 58(-5)^2(-7) + 31(-5)(-7)^2 - 8(-7)^3
Calculate the expression:
27(-125) - 58(25)(-7) + 31(-5)(49) - 8(-343)
-3375 + 10150 - 7565 + 2744
= 1954
Therefore, the value of the expression when x = -5 and y = -7 is 1954. Hence, the correct answer is option 'A' (1924), which seems to be a typo in the question.
x = -5
y = -7
Expression:
27x^3 - 58x^2y + 31xy^2 - 8y^3
Substitute x and y values:
27(-5)^3 - 58(-5)^2(-7) + 31(-5)(-7)^2 - 8(-7)^3
Calculate the expression:
27(-125) - 58(25)(-7) + 31(-5)(49) - 8(-343)
-3375 + 10150 - 7565 + 2744
= 1954
Therefore, the value of the expression when x = -5 and y = -7 is 1954. Hence, the correct answer is option 'A' (1924), which seems to be a typo in the question.
If (d + e + f) = 14, (d2 + e2 + f2) = 96, then find the value of (de + ef + fd).- a)75
- b)25
- c)50
- d)100
Correct answer is option 'C'. Can you explain this answer?
a)
75
b)
25
c)
50
d)
100
|
Qudrat Chauhan answered |
Given:
(d + e + f) = 14
(d
2
+ e2
+ f2
) = 96Formula Used:
(a + b + c)2 = 2 ×
(ab + bc + ca) + a
2
+ b
2
+ c
2
Calculation:
⇒ (d + e + f)
2
= 2 × (de + ef + fd) + d
2
+ e
2
+ f
2
⇒ ( 14)2 =
2×
(de + ef + fd) + 96
⇒ 2 ×
(de + ef + fd) = 196 - 96 = 100
⇒ (de + ef + fd) = 100/2 = 50
∴ The value of (de + ef + fd) is 50.
Find the solution to the system of linear equations: x – 2y + 6 =04y -2x -14 =0- a)(0, -3)
- b)(-1, 3)
- c)(2, 4)
- d)(-7, 0)
- e)No unique solution
Correct answer is option 'E'. Can you explain this answer?
Find the solution to the system of linear equations:
x – 2y + 6 =0
4y -2x -14 =0
a)
(0, -3)
b)
(-1, 3)
c)
(2, 4)
d)
(-7, 0)
e)
No unique solution
|
|
Ananya Iyer answered |
Given equations,
x – 2y + 6 =0 ..(a)
4y -2x -14 =0 ..(b)
Multiply the equation a with 2 and add the equation to b , we get
⇒ 2x - 4y + 12 + 4y - 2x -14 = 0
⇒ -2 = 0
Hence the equation is non unique equation
For what value of N would the following equation have no solution?3(4x−7)+12=2(5x−3)+N(x−3)- a) N=1
- b) N=2
- c) N=-1
- d) N=-2
Correct answer is option 'B'. Can you explain this answer?
For what value of N would the following equation have no solution?
3(4x−7)+12=2(5x−3)+N(x−3)
a)
N=1
b)
N=2
c)
N=-1
d)
N=-2
|
|
Niharika Sen answered |
The equation is incomplete, as it ends with an open parenthesis. Please provide the complete equation so that I can assist you further.
What is the maximum possible power of 4 in the number that is obtained when the product of the first 15 positive integers is subtracted from the product of the first 20 positive integers?- a)0
- b)3
- c)5
- d)7
- e)8
Correct answer is option 'C'. Can you explain this answer?
What is the maximum possible power of 4 in the number that is obtained when the product of the first 15 positive integers is subtracted from the product of the first 20 positive integers?
a)
0
b)
3
c)
5
d)
7
e)
8
|
|
Sahana Mehta answered |
Given
- Product of first 15 positive integers = 1*2*3…….* 15 = 15!
- Product of first 20 positive integers = 1*2*3*…….* 20 = 20!
- So, we can write 20! = 15! * 16 * 17 * 18 * 19 * 20
To Find: Maximum power of 4 that divides (20! – 15!)
Approach
- We will first need to simplify the expression 20! – 15!
- We know that 20! = 15! * 16 * 17 * 18 * 19 * 20
- So, 20! – 15! = 15! (20*19*18….*16 – 1) = 15! * odd integer
- Since 20 *19*…..16 is even, and 1 is odd, 20*19*……16 -1 will be odd
- Therefore, we only need to find the maximum power of 4 that divides 15!
2. To find the maximum power of 4 in 15!, we will first need to find the maximum power of 2 in 15!, as 4 = 22
- Let’s take a simple example to understand how we can find this. Consider a number p = 1*2*3*4. We need to find the power of 2 in p.
- Now, powers of 2 will occur in multiples of 2. So, we should first find the multiples of 2 in p. They are 2 and 4. However all multiples of 2 will not contain only 21. Some of them may contain higher powers of 2. For example, here 4 contains 22.
- So, when we are finding multiples of 2, we should also find the multiples of higher powers of 2 in p.
- Hence, 2 will occur 2 + 1 = 3 times in p.
3. We will use the same logic to find out the power of 2 in 15!
- Once we know the power of 2 in 15!, we can find the power of 4 in 15!
Working Out
In a cricket match Kumble took three wickets less than twice the number of wickets taken by Srinath. The product of the number of wickets taken by these two is 20, then the number of wickets taken by Kumble is- a)2
- b)4
- c)10
- d)5
Correct answer is option 'D'. Can you explain this answer?
In a cricket match Kumble took three wickets less than twice the number of wickets taken by Srinath. The product of the number of wickets taken by these two is 20, then the number of wickets taken by Kumble is
a)
2
b)
4
c)
10
d)
5
|
|
Prateek Gupta answered |
Explanation:
Let the number of wickets taken by Srinath be x then the number of wickets taken by Kumble will be 2x−3
According to question, x(2x−3)=20
According to question, x(2x−3)=20
If (p - q) = 8, then what is the value of q3 - p3 + 24pq?- a)729
- b)512
- c)-512
- d)-343
Correct answer is option 'C'. Can you explain this answer?
a)
729
b)
512
c)
-512
d)
-343
|
Simar Sharma answered |
Given:
(p - q) = 8,
Formula used:
(a - b)3 = a3 - b3 - 3ab(a - b)
Calculation:
According to the question,
⇒ (p - q) = 8
Take cube on both sides,
⇒ (p - q)3 = 83
⇒ p3 - q
3
- 3pq(p - q)= 83
⇒ p
3
- q3
- 3pq(8) = 512⇒ p
3
- q3
- 24pq = 512It can be written as:
⇒ q
3
- p
3
+ 24pq = -512Therefore, "-512" is the required answer.
Positive integer P lies between 1 and 30. What is the value of P?(1) P has at least two prime factors(2) The cube of P is less than 300- 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?
Positive integer P lies between 1 and 30. What is the value of P?
(1) P has at least two prime factors
(2) The cube of P is less than 300
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.
|
|
Soumya Iyer answered |
Steps 1 & 2: Understand Question and Draw Inferences
Given that P is a positive integer and that 1 < P < 30.
We need to find the value of P
Step 3: Analyze Statement 1
Given that P has at least two prime factors.
- P is a composite number with two or more prime factors.
However, there are more than one composite number between 1 and 30. The possible values of P are:
Clearly, Statement (1) doesn’t lead us to a unique value of P.
Not Sufficient.
Step 4: Analyze Statement 2
Statement 2 says that the cube of P is less than 300.
Let us list out the cubes of natural numbers.
There are 5 natural numbers between 1 and 30 whose cubes are less than 300.
Statement 2 alone is not sufficient to arrive at a unique answer.
Step 5: Analyze Both Statements Together (if needed)
From Statement 1,
Possible values of P: 6, 10, 14, 15, 21, 22, 26
From Statement 2,
Possible values of P: 2, 3, 4, 5, 6
By combining the two statements, we get:
P = 6
Therefore statement 1 and statement 2 combined are sufficient to arrive at a unique answer.
Correct Answer: C
The intensity of radiation emanating from a nuclear event is measured on an exponential scale where the second and fourth points indicate intensities of 25 and 625, respectively. On that scale, what would the intensity at the third point be?- a)50
- b)75
- c)125
- d)250
- e)312.5
Correct answer is option 'C'. Can you explain this answer?
The intensity of radiation emanating from a nuclear event is measured on an exponential scale where the second and fourth points indicate intensities of 25 and 625, respectively. On that scale, what would the intensity at the third point be?
a)
50
b)
75
c)
125
d)
250
e)
312.5
|
|
Siddharth Pillai answered |
Step 1: Question statement and Inferences
We are given that the scale is an exponential scale. This means that every term on this scale is equal to the previous term multiplied by a fixed value.
Step 2 & 3: Simplifying the fraction and calculating the final answer
We are given that 2nd term on the scale = 25 = 52
4th term on the scale = 625 = 54
From here, we can discern the pattern that:
The nth term on the scale = 5n
Thus, the 3rd term on the scale = 53 = 125
Answer: Option (C)
If x is a positive integer less than 100 such that x is divisible by 2y, where y is a positive integer, what is the value of 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 'B'. Can you explain this answer?
If x is a positive integer less than 100 such that x is divisible by 2y, where y is a positive integer, what is the value of 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.
|
|
Abhishek Choudhury answered |
Steps 1 & 2: Understand Question and Draw Inferences
- x is an integer such that 0 < x < 100
- x is divisible by 2y, where y is a positive integer
- Since y > 0, this means 2 is definitely a prime factor 2
- Since x is a positive integer, we can write the Prime-factorized form of x as:
- are integers > 0, z ≥ y and are prime numbers other than 2.
- As x < 100, 2z < 100
- So, z = { 1, 2, 3, 4, 5, 6} as 27 = 128 > 100
- Since x is completely divisible by 2y, z ≥ y. So, y can take any value of z, i.e. 1 ≤ y ≤ 6
- To Find: Unique value of y
Step 3: Analyze Statement 1 independently
- As x is a positive integer, x < -60 is not possible.
- So, x > 60, i.e. 60 < x < 100.
However, we do not know if 2 is the only prime factor or x. So, we cannot find a unique value of y.
- For example, if 2 is the only prime factor of x, then y can have only 1 value: 6
- But if x has other prime factors, then multiple values of y are possible. For example, x could be 22*3*7 (y = 2) or 23*11 (y = 3) etc.
Insufficient to answer.
Step 4: Analyze Statement 2 independently
- Using the prime factorized expression of x, we can write:
- Since the integer resulting from this division is odd, the powers of 2 in the numerator and the denominator should cancel out each other.
- Hence 22z = 2y+2
- z=y/2+1 ….(1). So, y must be even as z is an integer
- Also, from our discussion in Steps 1 and 2, we know that z ≥ y
- Substituting (1) in the above inequality, we get:
Using (2), along with the inference that y must be even, we have y = 2 as the only possible option.
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step 4, this step is not required.
Answer: B
A function 
for positive integers x and y. Is F(a, b) > a, where a and b are positive integers?
- 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 'B'. Can you explain this answer?
A function for positive integers x and y. Is F(a, b) > a, where a and b are positive integers?
for positive integers x and y. Is F(a, b) > a, where a and b are positive integers?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.
|
|
Sahana Mehta answered |
Steps 1 & 2: Understand Question and Draw Inferences
- Since a is a positive integer, multiplying both sides of the inequality doesn’t impact the sign of inequality
- So, the question simplifies to: is b > a?
Step 3: Analyze Statement 1 independently
Using the definition of this function, we can write: 
So as per Statement 1:
As a is a positive integer,aa > 0. So, we can divide both sides of the inequality by aa without changing the sign of the inequality.
So, b > a or b = a.
Hence, we can not say for sure if b > a. Insufficient to answer.
Step 4: Analyze Statement 2 independently
Using the definition of this function, we can write:
So, as per statement-2, 
- As a > 0, - a < 0. So b < -a is not possible as it will mean that b is negative.
- So, the only possible case is b > a
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step – 4, this step is not required.
Answer: B
Does ab lie between 0 and 1, exclusive?(1) |a| < 1(2) b is a positive odd 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 'E'. Can you explain this answer?
Does ab lie between 0 and 1, exclusive?
(1) |a| < 1
(2) b is a positive odd 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.
|
|
Pallabi Sengupta answered |
Steps 1 & 2: Understand Question and Draw Inferences
Given: Two numbers a and b
To Find: Is 0 < ab < 1
Following cases are possible:
- a ≥ 1 →
- If b≥ 0, then ab ≥ 1. So, the answer to the question is NO
- If b < 0, then 0 < ab < 1. So, the answer to the question is YES
2. 0 < a < 1 → Irrespective of the value of b, 0 < ab < 1 So, the answer to the question is YES
3. If a = 0 → Irrespective of the value of b, ab = 0. So, the answer to the question is NO.
4. -1 < a < 0 →
- If b is even, 0 < ab < 1 So, the answer to the question is YES.
- If b is odd, -1 < ab < 0. So, the answer to the question is NO.
5. a ≤ -1 →
- If b is even,ab≥1. So, the answer to the question is NO.
- If b is odd, ab≤1. So, the answer to the question is NO
So, the answer to the question will be YES, if:
- a ≥ 1 and b < 0 OR
- 0 < a < 1 OR
- -1 < a < 0 and b is even
And the answer to the question will be NO, if:
- a ≥ 1 and b ≥ 0 OR
- a = 0 OR
- -1 < a < 0 and b is odd OR
- a ≤ -1
So, we can answer the given question with a unique answer for the above cases.
Step 3: Analyze Statement 1 independently
(1) |a| < 1
- This means, -1 < a < 1. So, following cases can occur:
- -1 < a < 0 → As we have seen from our analysis above that in this case the answer to the question may be YES or NO
- Now, we don’t need to analyse other cases as we do not have unique from this case itself, but we will do so for the sake of completing the analysis.
- a = 0 → The answer to the question is NO in this case
- 0 < a < 1 → The answer to the question is YES in this case.
- As we do not have a unique answer , we cannot say for sure if 0 < ab < 1
- -1 < a < 0 → As we have seen from our analysis above that in this case the answer to the question may be YES or NO
Insufficient to answer
Step 4: Analyze Statement 2 independently
(2) b is a positive odd integer
- We have seen from our analysis in steps 1&2 that we need to find the range of values of a to answer this question.
- As we do not know the value of a, the statement is insufficient to tell if 0 < ab < 1
Insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
(1)From Statement 1, -1 < a < 1
(2) From Statement 2, b > 0
Following cases are possible:
- -1 < a < 0 → We have seen from our analysis in steps 1&2 that if b is odd, the answer to the question is NO.
- a = 0 → We have seen from our analysis in steps 1&2 that in this case the answer to the question is NO.
- 0 < a < 1 → We have seen from our analysis in steps 1&2 that in this case the answer to the question is YES
As we do not have a unique answer, the combination of statements is insufficient to answer.
Answer: E
If 
, what is the value of x?- a)6400
- b)8000
- c)12500
- d)15625
- e)22500
Correct answer is option 'A'. Can you explain this answer?
If , what is the value of x?
, what is the value of x?a)
6400
b)
8000
c)
12500
d)
15625
e)
22500
|
|
Arya Yadav answered |
Given:
To find: Value of x
Working Out:
Looking at the answer choices, we see that the correct answer is Option A
Is the sum of xy and yx positive?(1) xy > 0(2) x + y > 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 'C'. Can you explain this answer?
Is the sum of xy and yx positive?
(1) xy > 0
(2) x + y > 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.
|
|
Sahana Mehta answered |
Steps 1 & 2: Understand Question and Draw Inferences
To Find: If xy + yx > 0
Step 3: Analyze Statement 1 independently
(1) xy > 0
- Tells that x and y are of the same sign. Two cases arise:
- x, y > 0
- In this case xy, yx > 0. So, xy + yx > 0
- x, y < 0
- In this case, xy + yx may or may not be positive, depending on the values of x and y. Following cases can arise:
- Both x, y are even → In this case xy,yx > 0. So, xy+yx > 0
- Both x and y are odd → In this case xy,yx < 0. So, xy+yx < 0
- x is even and y is odd → In this case xy < 0, yx > 0. Cannot comment on the value of xy+yx
- x is odd and y is even → In this case xy < 0, yx > 0. Cannot comment on the value of xy+yx
- x, y > 0
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) x + y > 0
- Considering the constraint x + y > 0, following cases are possible:
- x, y > 0. So, x + y > 0. In this case, xy, yx > 0. So, xy + yx > 0
- x < 0, y > 0 and |y| > |x|. So, x + y > 0. In this case xy + yx may or may not be positive.
- If y is odd, then xy < 0 , yx > 0 as y is positive. In this case, we cannot comment on the value of xy + yx
- If y is even , then xy > 0 , yx > 0 as y is positive. In this case, xy + yx > 0
- x > 0, y < 0 and |x| > |y|. So, x + y > 0. In this case xy + yx may or may not be positive.
Insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
- From Statement 1, xy > 0
- From Statement 2, x + y > 0
Statement-1 tells us that x and y have the same signs. Following cases are possible:
- If x & y > 0, xy > 0 and x + y > 0. In this case xy + yx > 0
- If x & y < 0, x+ y < 0. Not possible. (If x & y both are negative, the x + y cannot be > 0)
The only possible case is when x, y > 0 and hence xy + yx > 0
Sufficient to answer.
Answer: C
If x and y are positive integers, what is the remainder when y is divided by x?
- 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 x and y are positive integers, what is the remainder when y is divided by x?
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.
|
|
Gitanjali Kumar answered |
Steps 1 & 2: Understand Question and Draw Inferences
- Given: x, y are integers > 0
- y = ax + r , where a is a positive integer and r is an integer such that 0 ≤ r < x
To Find: Value of r for which we need to find the value of x and y.
Step 3: Analyze Statement 1 independently
- As 149 is expressed as an integer raised to a power, we need to find that integer. As this integer will be a factor of 149, we need to find if 149 can be expressed as a factor raised to its power. So, let’s check its divisibility by the prime numbers.
- Divisibility by 2: As 149 is odd, it is not divisible by 2
- Divisibility by 3: As the sum of the digits of 149 (i.e. 14) is not divisible by 3, 149 will not be divisible by 3
- Divisibility by 5: As the units digit of 149 is neither 0 nor 5, it is not divisible by 5
- Divisibility by 7: We can see that 149 when divided by 7, leaves a remainder 2. So, it is not divisible by 7
- Divisibility by 11: 149 when divided by 11 leaves a remainder 4. So, it is not divisible by 11
- Divisibility by 13: We see that 149 when divided by 13 leaves a remainder 6. So, it is not divisible by 13.
- Since 122 < 149 < 132 and 149 is not divisible by any prime number till 13, we can say that 149 is a prime number.
- As 149 is a prime number, any integer apart from 149 raised to any power will not result in 149.
- Only possible option is x + y = 149 and y – x = 1
- 2 equations, 2 variables, we can find unique values of x and y and hence find the remainder.
Sufficient to answer.
Step 4: Analyze Statement 2 independently
- So, we need to express 149 as a product of 2 integers. For expressing 149 as a product of 2 integers, we need to find the factors of 149.
- As we have found above that 149 is a prime number, the only possible case of expressing 149 as a product of 2 integers is 149 * 1
- y + x = 149 and y – x = 1
- 2 equations, 2 variables, we can find unique values of x and y and hence find the remainder.
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
- a)quadratic equation
- b)constant
- c)cubic equation
- d)linear equation
Correct answer is option 'A'. Can you explain this answer?
a)
quadratic equation
b)
constant
c)
cubic equation
d)
linear equation
|
|
Prateek Gupta answered |
Here, the degree is 2, therefore it is a quadratic equation.
Which of the following is the zeros of the polynomial 9x2 - 4: - a)
- b)4, 4
- c)9, -9
- d)
Correct answer is option 'A'. Can you explain this answer?
Which of the following is the zeros of the polynomial 9x2 - 4:
a)
b)
4, 4
c)
9, -9
d)
|
Tom Tattle answered |
Concept -
let the polynomial be p(x) then p(x) = 0 gives you the zeros of the polynomial.
Explanation -
We have the polynomial 9x2 - 4
Now for the zeros of the polynomial -
9x2 - 4 = 0
⇒
9x
2
= 4
⇒ x
2
= 4/9
Hence option (1) is true.
The roots of the quadratic equation 6x2 - x - 2 = 0 are- a)
- b)
- c)
- d)
Correct answer is option 'C'. Can you explain this answer?
2
- x - 2 = 0 area)
b)
c)
d)
|
Hackers World answered |
Given:
6x
2
- x - 2 = 0 Calculation
:6x
2
- x - 2 = 0
⇒ 6x
2
- 4
x + 3x - 2 = 0⇒
2x(3x - 2) + 1(3x - 2)⇒
(2x + 1)(3x - 2) = 0 So,
2x + 1 = 0 and 3x - 2 = 0⇒
x = 
∴ The correct answer is option 3.
Simplify the expression 
provided (s + t + u) ≠ 0.- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
Simplify the expression provided (s + t + u) ≠ 0.
provided (s + t + u) ≠ 0.a)
b)
c)
d)
|
Codebreakers answered |
Calculation:
∴ The correct option is 1.
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