Quantitative Aptitude (QA) Test

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Quantitative Aptitude (QA) Test - 32

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Question 1
3 / -1

The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is

A
15

B
52

C
25

D
51

Solution
Question 2
3 / -1

In a company, 20% of the employees work in the manufacturing department. If the total salary obtained by all the manufacturing employees is one-sixth of the total salary obtained by all the employees in the company, then the ratio of the average salary obtained by the manufacturing employees to the average salary obtained by the non-manufacturing employees is

A
5:4

B
4:5

C
5:6

D
6:5

Solution
Question 3
3 / -1

The price of a precious stone is directly proportional to the square of its weight. Sita has a precious stone weighing 18 units. If she breaks it into four pieces with each piece having distinct integer weight, then the difference between the highest and lowest possible values of the total price of the four pieces will be 288000. Then, the price of the original precious stone is

A
972000

B
1296000

C
1620000

D
1944000

Solution

Let price and weight of stone be P and w

So, P is directly proportional to w²

=> P = kw² where k is a constant

Also weight of stone = 18 units and the stone is broken into four parts of distinct weights

Now, maximum possible total price of all 4 pieces would be when one of the piece has maximum possible weight So, the weights would be 1, 2, 3 and 12

And similarly, minimum possible total price of all 4 pieces would be when the weights are as closest as possible So, the weights would be 3, 4, 5 and 6
Question 4
3 / -1

Jayant bought a certain number of white shirts at the rate of Rs.1000 per piece and a certain number of blue shirts at the rate of Rs.1125 per piece. For each shirt, he then set a fixed market price which was 25% higher than the average cost of all the shirts. He sold all the shirts at a discount of 10% and made a total profit of Rs.51000. If he bought both colors of shirts, then the maximum possible total number of shirts that he could have bought is (type in numerical)

A
407

B
854

C
964

D
367

Solution

Let the number of white shirts = x and blue shirts = y
Question 5
3 / -1

Let n and m be two positive integers such that there are exactly 41 integers greater than 8m and less than 8n, which can be 16expressed as powers of 2. Then, the smallest possible value of n + m is

A
16

B
44

C
14

D
42

Solution
Question 6
3 / -1

A quadratic equation x²+ bx + c = 0 has two real roots. If the difference between the reciprocals of the roots is 1/3 and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b + c) is (in numerical value).

A
9

B
8

C
5

D
6

Solution
Question 7
3 / -1

Anil mixes cocoa with sugar in the ratio 3:2 to prepare mixture A, and coffee with sugar in the ratio 7:3 to prepare mixture B. He combines mixtures A and B in the ratio 2:3 to make a new mixture C. If he mixes C with an equal amount of milk to make a drink, then the percentage of sugar in this drink will be

A
17

B
24

C
16

D
21

Solution

Ratio of cocoa: sugar is 3:2 and ratio of coffee:sugar is 7:3.

Total 5 litres of A has 3 litres of cocoa and 2 litres of sugar

Total 10 litres of B has 7 litres of coffee and 3 litres of sugar

Since A and B are mixed in the ratio 2:3, so total 15 litres will be divided in this ratio. Hence we take 6 litres of A and 9 litres of B

Total Mixture = 15 + 15 = 30 litres. Mixture C will have 3.6 litres cocoa, 6.3 litres coffee and 2.4 + 2.7 = 5.1 litres of sugar.
Question 8
3 / -1

If f(x) = |x| + |x + 5| + |x + 10| + ……………………………..+ |x + 5t|, where x is an integer and t is a positive integer, if t = 7, for how many values of x, f(x) will be minimum?

A
8

B
7

C
6

D
5

Solution

When t = 7,

f(x) = |x| + |x + 5| + |x + 10| + ……………………………..+ |x + 35|

This expression has two middle terms: |x + 15|, |x + 20|

The value of f(x) will be minimized when the sum of the two middle terms are minimized

Thus, |x + 15| + |x + 20| should be minimum

The minimum happens when -20 ≤ x ≤ -15, Note that for x = -20, -19, -18, -17, -16, -15 the value of the sum of the middle terms = 5.

Therefore, for all 6 values of x, f(x) will have a minimum value.
Question 9
3 / -1

What is the product of all the real solutions of the equation x² -13|x| -30 = 0? ( where |.| represents modulus function)

A
225

B
300

C
-300

D
-225

Solution

The equation can be written as |x|² -13 |x| – 30 = 0.

Or, (|x| – 15) (|x| + 2) = 0

|x| = 15 or –2. |x| cannot be –2.

|x| = 15, x = 15 or –15.

Thus, the product of all the real solutions of the given equation is (15) × (-15) = – 225

Therefore, Option (D) is the answer.
Question 10
3 / -1

In a race A, B and C are climbing on a moving escalator in the same direction. A beats B by 429 steps and B beats C by 352 steps. Find the sum of digits of the sum of number of steps and speed of escalator, if the speed of A, B and C is 10 steps/sec, 7 steps/sec and 5 steps/sec respectively.

A
17

B
20

C
15

D
19

Solution

Let number of steps be N and speed of escalator be e steps/sec, as per question

Steps covered by A is N, then B covers N - 429 steps and similarly if steps covered by B is N then steps covered by C is N - 352.

If A takes t time to finish the race then N = 10t + et and N - 429 = 7t + et

from both equations we get, t = 143sec.

similarly if B covered N steps in t1 time then

N = 7t1 + et1 and N - 352 = 5t1 + et1

we get t1 = 176sec. from t1 and t, it is clear B can cover 429 steps in (176 - 143) = 33sec

so steps covered by B in 33sec = 33 × 7 = 231 rest steps are covered by escalator = 198 steps in 33secs so speed of escalator is 198/33 = 6 steps/sec.

so N = 10t + et (for A) = 10 × 143 + 6 × 143 = 2288 steps

so e + N = 2288 + 6 = 2294

so sum of digits is= 2 + 2 + 9 + 4 = 17.