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All questions of Algebra for DSSSB TGT/PGT/PRT Exam

What is the value of 225 + 225?
  • a)
    226
  • b)
    250
  • c)
    425
  • d)
    4
    50
  • e)
    2625
Correct answer is option 'A'. Can you explain this answer?

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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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?

Sandeep Mehra answered
Solution:
Given equation is 2x2 + 3x + 2 = 0
We can solve this quadratic equation using the quadratic formula, which states that for an equation of the form ax2 + bx + c = 0, the solutions (also called roots or zeros) are given by:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

Applying this formula to our equation, we get:

x = (-3 ± sqrt(3^2 - 4(2)(2))) / 2(2)
x = (-3 ± sqrt(1)) / 4
x = (-3 ± 1) / 4

Therefore, the solutions are x = -1/2 and x = -1.5
But we have to find the number of integral solutions, i.e., solutions that are integers. In this case, there is only one such solution: x = -1
Therefore, the answer is option B (1).

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?

Anaya Patel answered
Steps 1 & 2: Understand Question and Draw Inferences
  • As a6 is always positive,a= 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 answer
     
    Step 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 = 1
    Thus 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

 lies between:
  • a)
    4 & 5
  • b)
    5 & 6
  • c)
    6 & 7
  • d)
    7 & 8
  • e)
    8 & 9
Correct answer is option 'D'. Can you explain this answer?

Anaya Patel answered
In order to rid the expression of square roots, let's first square the entire expression. We are allowed to do this as long as we remember to "unsquare" whatever solution we get at that end. 

Notice that the new expression is of the form where

Recall that  This is one of the GMAT's favorite expressions.
Returning to our expression: 
 
Notice that x2 + y2  neatly simplifies to 48. This leaves only the 2xy expression left to simplify.
In order to simplify   recall that  
Thus, 
Notice that the expression under the square root sign is of the form  And recall that   This is another one of the GMAT's favorite expressions. Returning to our expression: 


Finally then:

But now we must remember to "unsquare" (or take the square root of) our 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.
Correct answer is option 'E'. Can you explain this answer?

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.

There are two classrooms A and B. The sum of the number of students in both classrooms is more than 120. Is the number of students in class B greater than 20? 
(1) If number of students in classroom A are doubled and number of students in classroom B are halved, the difference between the number of students in classroom A and B is less than 200.
(2) If 20 students from each classroom leave the school, the sum of number of students in both classes would be more than 80. 
  • 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?

Lavanya Menon answered
Steps 1 & 2: Understand Question and Draw Inferences
We are given that the number of students in classes A and B combined is more than 120. We have to find whether the number of students in class B is more than 20.
 Let’s say the number of students in class A is x, and the number of students in class B is y.
So, we can say that:
x+y>120
Since there is no other information given in the sentence, let’s move on to the analysis of the statement I.
 Step 3: Analyze Statement 1
2x−y/2<200
4x−y<400         ................... (2)
By multiplying inequality (1) with -4, we get:
−4x−4y<−480        .....................(3)
Adding (2) and (3),
−5y<−80
By multiplying the above inequality with -1, we get:
5y>80y>16
 So, y can take any value greater than 16. Thus, we can’t say whether it is greater than 20 or not.
Hence, statement I is not sufficient to answer the question: Is the number of students in class B is more than 20?   
Step 4: Analyze Statement 2
Per statement II:
 (x−20)+(y−20)>80
x+y>120
This is the same information given in the original sentence. So, from this information we can’t say whether y is greater than 20 or not. 
 Hence, statement II alone is insufficient to answer the question: Is the number of students in class B is more than 20?      
 
Step 5: Analyze Both Statements Together (if needed)
Since statement I and II alone are not sufficient to answer the question, let’s analyse them together.  
 However, since statement II provides the same information given in the original sentence, analysing both the statements together is equivalent to analysing statement I alone.
Thus, even both statements combined are not sufficient to answer the question: Is the number of students in class B is more than 20?  
 
Answer: Option (E)  
The correct answer is: Statements (1) and (2) TOGETHER are NOT sufficient.

The sum of the squares of the first 15 positive integers (12 + 22 + 32 + . . . + 152) is equal to 1240. What is the sum of the squares of the second 15 positive integers (162 + 172 + 182 + . . . + 302) ?
  • a)
    2480
  • b)
    3490
  • c)
    6785
  • d)
    8215
  • e)
    9255
Correct answer is option 'D'. Can you explain this answer?

Nitya Kumar answered
The key to solving this problem is to represent the sum of the squares of the second 15 integers as follows: (15 + 1)2 + (15 + 2)2 + (15 + 3)2 + . . . + (15 + 15)2.  
Recall the popular quadratic form, (a + b)2 = a2 + 2ab + b2. Construct a table that uses this expansion to calculate each component of each term in the series as follows: 

 
In order to calculate the desired sum, we can find the sum of each of the last 3 columns and then add these three subtotals together. Note that since each column follows a simple pattern, we do not have to fill in the whole table, but instead only need to calculate a few terms in order to determine the sums. 
The column labeled a2 simply repeats 225 fifteen times; therefore, its sum is 15(225) = 3375. 
The column labeled 2ab is an equally spaced series of positive numbers. Recall that the average of such a series is equal to the average of its highest and lowest values; thus, the average term in this series is (30 + 450) / 2 = 240. Since the sum of n numbers in an equally spaced series is simply n times the average of the series, the sum of this series is 15(240) = 3600. 
The last column labeled b2 is the sum of the squares of the first 15 integers. This was given to us in the problem as 1240. 
Finally, we sum the 3 column totals together to find the sum of the squares of the second 15 integers: 3375 + 3600 + 1240 = 8215. The correct answer choice is (D).

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?

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
  1. 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).
  2. Scheme X
    1. 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
    2. However, we will need to make sure that we subtract $500 at each interval of 7 years from the final amount
  3. Scheme Y
    1. 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.
  4. At each interval, we will calculate the sum of amounts in scheme X and Y to check if it exceeds $40,000.
Working Out
 
  1. Amount at the end of year 7 in scheme X = $1000 * 2 = $2000
    1. However James withdrew $500 at the end of 7th year, So, the amount remaining will be $2000 – $500 = $1500
    2. The same logic has been applied in calculating the amounts at the end of every 7 year interval
       
  2. Amount at the end of year 14 in scheme Y = $1000 * 2 = $2000
    1. The same logic has been applied in calculating the amounts at the end of every 14 years interval.
       
  3. We can see that the total amount in schemes X and Y exceed $40,000 by the end of the year 42.
 
Answer: C

Find the minimum value of the function fix) = log2 (x2 - 2x + 5).
  • a)
    -4
  • b)
    2
  • c)
    4
  • d)
    -2
Correct answer is option 'B'. Can you explain this answer?

Alok Verma answered
Method to Solve :

y= x2 – 2x + 5
Step 1  : Differentiate with respect to x
Step 2 : Equate to 0
Step 3 : Find the value of x
dy/dx=2x-2 =0 implies x=1
Hence f(1)= 12 – 2 + 5= 4
Thus minimum value of the argument of the log is 4.
So minimum value of the function is log 4 (base 2) =2

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?

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.

If S is the infinite sequence S1 = 6, S2 = 12, ..., Sn = Sn-1 + 6,..., what is the sum of all terms in the set {S13, S14, ..., S28}?
  • a)
    1800
  • b)
    1845
  • c)
    1890
  • d)
    1968
  • e)
    2016
Correct answer is option 'D'. Can you explain this answer?

Anirban Singh answered
Sum of the Infinite Sequence
To find the sum of the terms in the set {S13, S14, ..., S28}, we need to first determine the value of S13 and S28.
Given that the sequence is defined as Sn = Sn-1 + 6, we can see that the difference between consecutive terms is 6.

Finding S13 and S28
To find S13, we use the formula Sn = Sn-1 + 6.
S13 = S12 + 6
S13 = 12 + 6
S13 = 18
Similarly, to find S28, we continue the sequence:
S14 = S13 + 6 = 18 + 6 = 24
S15 = S14 + 6 = 24 + 6 = 30
...
S28 = S27 + 6 = 54 + 6 = 60

Calculating the Sum
Now that we have determined the values of S13 and S28, we can find the sum of the terms in the set {S13, S14, ..., S28}:
Sum = S13 + S14 + ... + S28
Sum = 18 + 24 + ... + 60
Sum = 18 + 24 + 30 + ... + 60
This is an arithmetic series with a common difference of 6. We can use the formula for the sum of an arithmetic series to find the total sum:
Sum = n/2 * (first term + last term)
Sum = 16/2 * (18 + 60)
Sum = 8 * 78
Sum = 624
Therefore, the sum of all terms in the set {S13, S14, ..., S28} is 624, which is closest to option D, 1968.

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?

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 .

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?

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.

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?

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.

Ajesh saves Rs 50,000 every year and deposits the money in a bank at compound interest of 10%(compunded annually).What would be his total saving at the end of the 5th year?
  • a)
    Rs 3,05,255
  • b)
    Rs 1,05,255
  • c)
    Rs 8,05,255
  • d)
    Rs 4,05,255
Correct answer is option 'A'. Can you explain this answer?

Rajeev Kumar answered
 
At the end of the 1st year, he will get Rs 50000, it will give him interest for 4 years compounded annually
Hence at the end of 5 years, this amount will become 50000(1.1)4
Similarly, the amount deposited in the 2nd year will give interest for 3 years. Hence it will become 50000(1.1)3
Similarly, we can calculate for the remaining years.
The total saving at the end of the 5th year would be a GP, given by
Net saving = 50000(1.1)+  50000(1.1)3  ..... 50000
Thus net saving = =  Rs 3,05,255

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?

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 -3z+ 3z ] + 6 [ 1 - z-3z + 3z] = 32
⇒  2z3 - 2 - 6z2 + 6z + 6 - 6z-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

What is the sum of all values of x that satisfies the equation:
  • a)
    20
  • b)
    24
  • c)
    28
  • d)
    32
  • e)
    36
Correct answer is option 'C'. Can you explain this answer?

Yash Patel answered
Correct Answer :- c
Explanation : By picking from the options, we can check which value satisfies the equation 
For x = 28
√168 = √143+1
12.9 = (11.9+1)
12.9 = 12.9

-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
Correct answer is option 'A'. Can you explain this answer?

 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
 

Define the following functions:
(a) (a M b) = a – b (b) (a D b) = a + b
(c) (a H b) = (ab) (d) (a P b) = a/b
Q.
Which of the four functions defined has the minimum value?
  • a)
    (a M b)
  • b)
    (a D b)
  • c)
    (a H b)
  • d)
    Cannot be determined
Correct answer is option 'D'. Can you explain this answer?

Aarav Sharma answered
To find the function with the minimum value, we need to compare the functions (a M b), (a D b), (a H b), and (a P b) and determine which one produces the smallest output for any given values of a and b.

1. (a M b) = a * b: This function multiplies the values of a and b together. The result will be the product of a and b.

2. (a D b) = a / b: This function divides the value of a by the value of b. The result will be the quotient of a divided by b.

3. (a H b) = (a * b): This function multiplies the values of a and b together. The result will be the product of a and b.

4. (a P b) = a / b: This function divides the value of a by the value of b. The result will be the quotient of a divided by b.

To determine the function with the minimum value, we need to consider different scenarios:

- If both a and b are positive numbers, then the functions (a M b) and (a H b) will produce the same result. Therefore, we can eliminate (a H b) as it is equivalent to (a M b).

- If both a and b are negative numbers, then the functions (a M b) and (a H b) will produce the same result. Therefore, we can eliminate (a H b) as it is equivalent to (a M b).

- If a is positive and b is negative, the function (a M b) will produce a negative result, while (a D b) and (a P b) will produce positive results. Therefore, (a M b) will have the minimum value in this scenario.

- If a is negative and b is positive, the function (a M b) will produce a negative result, while (a D b) and (a P b) will produce negative results. Therefore, (a D b) and (a P b) will have the minimum value in this scenario.

Based on the above analysis, we can conclude that the function with the minimum value cannot be determined without knowing the signs of a and b. Therefore, the correct answer is option D) Cannot be determined.

 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?

Kirti Roy 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 ) is 21

Read the instructions below and solve.
f(x) = f(x – 2) – f(x – 1), x is a natural number
f(1) = 0, f(2) = 1
What will be the domain of the definition of the function f(x) = 8–xC 5–x for positive values of x?
  • a)
    {1, 2, 3}
  • b)
    {1, 2, 3, 4}
  • c)
    {1, 2, 3, 4, 5}
  • d)
    {1, 2, 3, 4, 5, 6, 7, 8}
Correct answer is option 'C'. Can you explain this answer?

Nikita Singh answered
f(1) = 0, f(2) = 1,
f(3) = f(1) – f(2) = –1
f(4) = f(2) – f(3) = 2
f(5) = f(3) – f(4) = –3
f(6) = f(4) – f(5) = 5
f(7) = f(5) – f(6) = –8
f(8) = f(6) – f(7) = 13
f(9) = f(7) – f(8) = –21
For any nCr, n should be positive and r ≥ 0.
Thus, for positive x, 5 – x ≥ 0
fi x = 1, 2, 3, 4, 5.

$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
Correct answer is option 'E'. Can you explain this answer?

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.

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?

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
       
  • 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

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?

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.

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?

Saumya Shah answered
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (1): P has at least one Prime factor
This statement tells us that P has at least one prime factor. Since a perfect square is a number that can be expressed as the product of two equal integers, we can conclude that P must have at least one prime factor that repeats.

To find a perfect square between 1 and 30, we can list the squares of all the positive integers less than or equal to the square root of 30 (which is approximately 5.5). The perfect squares between 1 and 30 are: 1, 4, 9, 16, and 25.

From this list, we can see that all the perfect squares have at least one prime factor. Therefore, statement (1) alone is sufficient to find the value of P.

Statement (2) ALONE is not sufficient.

Statement (2): The cube of P is less than 300
This statement tells us that the cube of P is less than 300. However, it does not provide any information about the prime factors or whether P is a perfect square.

Let's analyze the possible values of P using statement (2):

- If P is 1, 1^3 = 1, which is less than 300.
- If P is 2, 2^3 = 8, which is less than 300.
- If P is 3, 3^3 = 27, which is less than 300.
- If P is 4, 4^3 = 64, which is less than 300.
- If P is 5, 5^3 = 125, which is less than 300.
- If P is 6, 6^3 = 216, which is less than 300.

From this analysis, we can see that there are multiple possible values of P that satisfy statement (2), and not all of them are perfect squares. Therefore, statement (2) alone is not sufficient to find the value of P.

Conclusion:
Statement (1) alone is sufficient to find the value of P, as all the perfect squares between 1 and 30 have at least one prime factor. However, statement (2) alone is not sufficient, as it does not provide any information about the prime factors or whether P is a perfect square. Therefore, the correct answer is option (a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

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?

Sandeep Mehra answered
To find the remainder when 1010, 105, and 24 are divided by 36, we can perform the division and observe the remainder.

Dividing 1010 by 36:
When we divide 1010 by 36, we get a quotient of 28 and a remainder of 22.

Dividing 105 by 36:
When we divide 105 by 36, we get a quotient of 2 and a remainder of 33.

Dividing 24 by 36:
When we divide 24 by 36, we get a quotient of 0 and a remainder of 24.

Now, let's perform the division again but with the remainders.

Dividing 22 by 36:
When we divide 22 by 36, we get a quotient of 0 and a remainder of 22.

Dividing 33 by 36:
When we divide 33 by 36, we get a quotient of 0 and a remainder of 33.

Dividing 24 by 36:
When we divide 24 by 36, we get a quotient of 0 and a remainder of 24.

Summing the remainders:
To find the remainder when the sum of the three numbers is divided by 36, we sum the remainders obtained in each division: 22 + 33 + 24 = 79.

Reducing the remainder:
Since the remainder obtained (79) is greater than the divisor (36), we need to reduce it. We can do this by repeatedly subtracting the divisor until we obtain a remainder less than the divisor.

79 - 36 = 43
43 - 36 = 7

The remainder after reducing is 7.

Therefore, the remainder obtained when 1010, 105, and 24 are divided by 36 is 7.

Hence, the correct answer is option E.

How many integrals value of x satisfy the inequality (1-x2)(4-x2)(9-x2) > 0 ?
  • a)
    0
  • b)
    1
  • c)
    3
  • d)
    5
  • e)
    Greater than 5
Correct answer is option 'B'. Can you explain this answer?

Given
To Find: Integral values of x that satisfy the inequality 
Approach
  1. We will draw the wavy line diagram to find the range of x. In the range of x we will observe the number of integral values of x that satisfy the inequality.
    a. Before we draw the wavy line diagram, we should have the coefficient of x as positive
As we can see from the wavy line diagram that the inequality is true for the following range:
  1. -3 < x < -2→ Integral values of x = {Nil}
  2. -1 < x < 1 → Integral values of x = {0}
  3. 2 < x < 3 → Integral values of x = {Nil}
Thus, there is only 1 integral value of x that satisfy the inequality i.e. 0.
Answer: B

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?

Sounak Iyer answered
Problem Analysis:
We are given the expressions max(a, max(b, min(c, d))) and max(d, max(a, min(b, c))). We need to determine if these two expressions are equal.

Statement 1: b, c, and d are factors of a.
If b, c, and d are factors of a, it means that a is divisible by b, c, and d. In other words, a should be the multiple of b, c, and d.

Statement 2: a - 2d = b and ca
From this statement, we can deduce that b = a - 2d and c = a - 2d. This implies that b and c are both positive integers.

Combined Analysis:
From Statement 1, we know that a is divisible by b, c, and d. Therefore, a is a multiple of b, c, and d.

From Statement 2, we know that b = a - 2d and c = a - 2d. Substituting these values in the expression max(a, max(b, min(c, d))) gives us:
max(a, max(a - 2d, min(a - 2d, d)))

We can simplify this expression as follows:
1. If a > a - 2d, then max(a - 2d, min(a - 2d, d)) = max(a - 2d, d) = a - 2d
2. If a < a="" -="" 2d,="" then="" max(a="" -="" 2d,="" min(a="" -="" 2d,="" d))="max(a" -="" 2d,="" a="" -="" 2d)="a" -="" />
3. If a = a - 2d, then max(a - 2d, min(a - 2d, d)) = max(a - 2d, d) = a - 2d

Therefore, in all cases, max(a, max(b, min(c, d))) = a - 2d.

Similarly, substituting the values in the expression max(d, max(a, min(b, c))) gives us:
max(d, max(a, min(a - 2d, d)))

We can simplify this expression as follows:
1. If d > a, then max(a, min(a - 2d, d)) = max(a, d) = d
2. If d < a,="" then="" max(a,="" min(a="" -="" 2d,="" d))="max(a," a="" -="" 2d)="" />
3. If d = a, then max(a, min(a - 2d, d)) = max(a, d) = a

Therefore, in all cases, max(d, max(a, min(b, c))) = a - 2d.

Since both expressions simplify to a - 2d, we can conclude that max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))).

Therefore, both statements together are sufficient to answer the question.

Answer: (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.

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?

Srestha Basu answered
Given
  • Ted starts at 7 AM
    • Ted’s initial speed = 10 kilometers/hour
    • Ted doubles his speed after every 1.5 hours
 
  • Robin starts at 7 AM from the same point as Ted and travels in opposite directions
    • Robin’s initial speed = 120 kilometers/hour
    • Robin reduces her speed by half after every 2 hours
To Find: Distance between Ted and Robin at 1 PM
  • Find the distance travelled by Ted in 6 hours
  • Find the distance travelled by Robin in 6 hours
 
Approach
Distance = Speed * Time
1. We need to find the distance travelled by Ted in 6 hours As Ted doubles his speed after every 1.5 hours, he will travel at 6/1.5=4
  1.  different speeds during the period
  2. We know his initial speed and we know that he doubles his speed after every interval of 1.5 hours. So, we can calculate his speed for each interval
 
2.  We need to find the distance travelled by Robin in 6 hours
  1. As Robin reduces her speed after every 2 hours, she will travel at 6/2=3
  1.  different speeds during the period
  2. We know her initial speed and we know that she reduces her speed by half after every interval of 2 hours. So, we can calculate her speed for each interval.
 
Working Out
1.  Ted 
2.  Robin
Distance between Ted and Robin at 1 PM = 225 + 420 = 645 kilometers
Answer: D

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?

Given
To Find: value of n?
Approach
  1. We know that the product of roots m and n is equal to c/a
  1. So, we can write m*n = 98, i.e. n = 98/m
  2. So, for finding n, we need to find m
  • We are given that m = (p+q)/2
  1. We know that sum of roots of the quadratic equation is −b/a
  2. We will use this relation to find the value of p + q and hence the value of m
Hence, the other root of the equation x2 + bx + 98 = 0 is -7
Answer: A

If each term in the sum a1 + a2 + a3 + ... +an is either 7 or 77 and the sum is equal to 350, which of the
following could equal to n?
  • a)
    38
  • b)
    39
  • c)
    40
  • d)
    41
  • e)
    42
Correct answer is option 'C'. Can you explain this answer?

Advait Roy answered
Solution:

To solve this problem, we need to use a bit of logic and guesswork.

Let's start by assuming that all the terms in the sum are 7. In this case, the sum would be:

7 + 7 + 7 + ... + 7 (n terms) = 7n

We know that the sum is 350, so we can set up the equation:

7n = 350

Solving for n, we get:

n = 50

But we also know that each term is either 7 or 77. So let's assume that all the terms are 77. In this case, the sum would be:

77 + 77 + 77 + ... + 77 (n terms) = 77n

We know that the sum is 350, so we can set up the equation:

77n = 350

Solving for n, we get:

n ≈ 4.55

This doesn't give us a whole number for n, so we need to try something in between. Let's assume that half the terms are 7 and half are 77. In this case, the sum would be:

7 + 77 + 7 + 77 + ... (n terms) = (7 + 77)n/2 = 42n

We know that the sum is 350, so we can set up the equation:

42n = 350

Solving for n, we get:

n ≈ 8.33

Again, we don't get a whole number for n. But notice that as we move from all 7s to all 77s to a mix of 7s and 77s, the value of n is decreasing. So we can make an educated guess that the value of n lies somewhere between 50 and 8.

Looking at the answer choices, we see that only option C (40) falls within this range. Therefore, the correct 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?

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)  

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?

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."

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?

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).

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?

Rhea Gupta answered
Given:         
  1. List A = {a1, a2, a3, a4, a5, a6, a7, a8, a9, a10}
    1. consists of all integers,
    2. 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.

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?

Mihir Ghoshal answered
To solve this problem, we need to find the value of a7 in the given sequence. Let's break down the information provided and use it to find the answer.

Given information:
- an = an-1 * n*d for all n ≥ 1
- a3 = 20
- a5 = 47

Finding the common ratio:
To find the common ratio (d) in the sequence, we can use the given information. We know that a3 = 20 and a5 = 47. Using the formula for an, we can write the following equations:

a3 = a2 * 2d
20 = a2 * 2d

a5 = a4 * 4d
47 = a4 * 4d

Dividing the second equation by the first equation, we get:

47/20 = (a4 * 4d) / (a2 * 2d)
47/20 = 2a4/a2
47/20 = 2(a4/a2)
47/20 = 2(a3 * 3d)/(a3 * d)
47/20 = 6d/d
47/20 = 6
47 = 120

Therefore, we have found that the common ratio (d) is 120.

Finding the value of a7:
Now that we know the common ratio (d), we can find the value of a7 using the formula for an:

a7 = a6 * 6d

To find a6, we can use the formula for an again:

a6 = a5 * 5d
a6 = 47 * 5d

Substituting this value into the equation for a7, we get:

a7 = (47 * 5d) * 6d
a7 = 235d * 6d
a7 = 1410d^2

Since d is a positive integer, we can see that the value of a7 is directly proportional to the square of d.

Therefore, to find the value of a7, we need to know the value of d. Unfortunately, the value of d is not provided in the question. Without knowing the value of d, we cannot determine the exact value of a7.

Hence, the given answer options (a, b, c, d, e) are not sufficient to determine the value of a7.

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?

Niharika Sen answered
Given Equation:
-6x - 20 = -2x + 4(1 - 3x)

Solution:

Step 1: Simplify the equation
-6x - 20 = -2x + 4 - 12x

Step 2: Combine like terms
-6x - 20 = -14x + 4

Step 3: Bring x terms to one side
-6x + 14x = 4 + 20
8x = 24

Step 4: Solve for x
x = 24 / 8
x = 3

Therefore, the solution for x is:
x = 3
So, the correct answer is option D) 3.

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?

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.

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?

Saumya Sharma answered
Step 1 & Step 2: Understanding the Question statement and Drawing Inferences
Given Info:
  • x2+4x+p=13
  • Rewriting the above quadratic equation in standard form ax2+bx+c=0
  • Subtracting 13 from both sides, we have
⇒ x2+4x+p−13=13−13
⇒ x2+4x+p−13=0
To find:
  • We need to find the product of the quadratic equation → x2+4x+p−13=0
  • To find the product of the roots: 
  1.  We need to know the roots or 
  2. Products of roots of the quadratic equation ax2+bx+c=0 , is given by  c/a
  • So product of roots of the above quadratic equation will be (p−13)/1 , as →c=p-13 and a=1
  • Now since we do not know the value of p-13, we will not be able to determine the product of roots of the quadratic equation.
  • Thus we need to analyse the given statements further to determine the value of p, to be able to calculate the product of roots of the quadratic equation.
 
Step 3: Analyze statement 1 independently
 
Statement 1:
  • -2 is one of the roots of the quadratic equation
  • So, -2 will satisfy the given quadratic equation → x2+4x+p−13=0
⇒ (-2)2 + 4(-2) + p - 13 = 0
⇒ 4 - 8 + p - 13 = 0
⇒ p = 17
  • Now since we know the value of p, we will be able to find the value of p-13 and thus will be able to calculate the value of product of quadratic equation.
  • Hence statement 1 is sufficient to answer the question.
 
Step 4: Analyze statement 2 independently
 
Statement 2:
  • Quadratic equation has equal roots
  • For equal roots, we will use the relation of sum of roots of the quadratic equation to determine the value of the equal root.
  • Sum of roots of the quadratic equation ax2+bx+c=0 , is −b/a
  • So sum of roots of the quadratic equation → x2+4x+p−13=0  will be −4/1  , where b=4 and a=1
  • Now since both roots are equal and the sum of the roots is coming as -4, both roots will thus be equal to – 2 each.
  • Now, the equal root → -2, will satisfy the given quadratic equation x2+4x+p−13=0
⇒(-2)2 + 4(-2) + p - 13 = 0
⇒ p=17
  • Now since we know the value of p, we will be able to find the value of p-13 and thus will be able to calculate the value of product of quadratic equation.
  • Hence statement 2 is sufficient to answer the question
 
Step 5: Analyze the two statements together
  • Since from statement 1 and statement 2, we are able to arrive at a unique answer, combining and analysing statements together is not required.
Hence the correct answer is option D

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