# Problem Solving Select Many Practice Test 2

Question 1

Consider the following sets: A = {1, 2}, B = {1, 4, 6} and C = { 1, 2, 4, 6, 8, 10}.

Which of the following statements is true? Indicate all correct statements.

[ A U B = A union B]

1. A is a subset of B
2. C is a superset of A
3. A U B is a subset of C
4. B and C are disjoint sets
5. A U B U C = {1, 2, 4, 6}

Explanation:

All elements of A do not exist in B and hence option (A) is not true.
All elements of A exist in C and hence option (B) is true.
A U B = {1, 2, 4, 6}, all elements of AUB exist in C and hence option (C) is true.
All the elements of B exist in C and hence they are not disjoint. Option (D) is not true.
A U B U C = { 1, 2, 4, 6, 8, 10} which is not equal to {1, 2, 4, 6}. Option (E) is not true.

Question 2

Find f(x)-1/f(x), when f(x) = (x+1)/(x-1)? Indicate all correct options.

[n2 = n*n]

1. 2
2. 2/(x2-1)
3. 4x/(x2-1)
4. 4x/[(x+1)(x-1)]
5. 2(x2+2x-1)/(x2-1)

Explanation:

f(x) - 1/f(x) = (x+1)/(x-1) - (x-1)/(x+1)
= [(x+1)(x+1)-(x-1)(x-1)]/[(x+1)(x-1)]
= (x2+2x+1-x2+2x-1)/[(x+1)(x-1)]
= 4x/[(x+1)(x-1)]
= 4x/(x2-1)

Question 3

Which of the following numbers is not prime? Indicate all correct options.

1. 431
2. 432
3. 433
4. 434
5. 435

Correct Answer: B, D and E

Explanation:

431 and 433 are prime numbers. The other numbers are not prime.

Question 4

If the product of three numbers in GP is 27 and the sum of their product in pairs is 39, then which of the following are the numbers?

Indicate all correct options.

1. 1
2. 2
3. 4
4. 3
5. 9

Correct Answer: A, D and E

Explanation:

Let the three numbers be a/r, a and ar.
(a/r)*a*ar = a3 = 27
a = 3
a/r*a + a*ar + ar*a/r = 39
a2(1/r + r + 1) = 39
a2(1+r+r2) = 39*r
32(1+r+r2) = 39r
9 + 9r + 9r2 - 39r = 0
9r2 - 30r + 9 = 0
3r2 - 10r + 3 = 0
3r2 -9r - r + 3 = 0
3r(r-3) -1(r-3) = 0
(3r-1)(r-3) = 0
r = 1/3, 3
The three numbers are 1, 3 and 9

Question 5

Which of the following numbers is not divisible by 6? Indicate all correct options.

1. 234
2. 27789
3. 9876
4. 3521
5. 19872

Explanation:

A number is divisible by 6 if the number is even and the sum of its digits is divisible by 3.
234 is even and 2+3+4 = 9, which is divisible by 3. 234 is divisible by 6.
27789 is odd and hence not divisible by 6.
9876 is even and 9+8+7+6 = 30 is divisible by 3. 9876 is divisible by 6.
3521 is odd and hence not divisible by 6.
19872 is even and 1+9+8+7+2 = 27 is divisible by 3 and hence 19872 is divisible by 6.
Options B and D are not divisible by 6.

Question 6

Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, A = { 1, 2, 3}, B = { 4, 5, 7} and C = { 8, 9, 11}.
Which of the following is equal to (A U B U C)' ? Indicate all correct options.

[ (AUB)' = compliment of A union B]

1. { 6, 10}
2. {12, 10, 11, 5}
3. {10, 14, 3, 6}
4. {6, 10, 12}
5. {10, 12, 6}

Explanation:

(A U B U C) = {1, 2, 3, 4, 5, 7, 8, 9, 11}
(A U B U C)' = {6, 10, 12}
Options (D) and (E) are true.

Question 7

The ratio between two numbers is 3:5. When they are increased by 10 each, the ratio becomes 5:7. Which of the following are the two numbers? Indicate all correct options.

1. 5
2. 10
3. 15
4. 20
5. 25

Explanation:

Let the numbers be x and y
x:y = 3:5
x = 3y/5

x+10:y+10 =5:7
(x+10)*7 = (y+10)*5
7x + 70 = 5y + 50
7x + 20 = 5y
7(3y/5) + 20 = 5y
21y + 100 = 25y
25y-21y = 100
4y = 100
y = 25
The other number x = 3*25/5 = 75/5 = 15
The two numbers are 15 and 25.

Question 8

Which of the following is equal to sqrt(8750)/sqrt(7)? Indicate all correct options.

[sqrt(n) = square root of n]

1. sqrt(625)
2. sqrt(61250)
3. 25
4. 5
5. -5

Explanation:

sqrt(8750)/sqrt(7)
= sqrt(8750/7)
= sqrt(625)
= 25

Question 9

A can complete a work in 20 days. B is 25% faster than A and C is 60% faster than B. Which of the following statements is true? Indicate all correct options.

1. B takes 10 days to complete the work
2. C takes 13 days to complete the work
3. B takes 15 days to complete the work
4. C takes 10 days to complete the work
5. B takes 16 days to complete the work

Explanation:

Time taken by A to complete the work = 20 days
Work done by A in one day = 1/20
Work done by B in one day = 1/20+(25/100)*(1/20)
= 1/20 + 1/80 = (4+1)/80
= 5/80 = 1/16
B takes 16 days to complete the work.

Work done by C in one day = 1/16 + (60/100)*(1/16)
= 1/16 + (3/5)*(1/16)
= (1+3/5)*(1/16)
= (5+3)/5*1/16
= 8/5*1/16 = 1/10
C takes 10 days to complete the work.
Options D and E are true.

Question 10

If ad and d>e, then which of the following is true? Indicate all correct options.

1. b
2. b
3. a
4. c>e
5. e>a

Explanation:

a < b, b < c, c > d and d > e
or a < b, b < c, d < c, e < d

Though b < c and d < c, we cannot say that b < d. Hence, option A is not true.
Though b < c, e < d < c, we cannot establish a relation between b and e. Hence, option B is not true.
a < b < c, hence a < c. Option C is true.
e < d < c, hence e < c or c > e.
Option D is true.

We cannot establish a relation between a and e. Hence, option E is not true.

Question 11

Let A = { -2, -1, 0, 1} and B = {0, 1, 2, 3, 4, 5}. g is a function from A to B defined by g(x) = x2.
Which of the following is true? Indicate all correct statements.

1. g(2) = 4
2. g(-2) = 4
3. g(-1) = g(1)
4. g(0) = 2
5. g(-1) + g(1) = g(0)

Explanation:

g(2) is not defined as 2 is not an element of A. Option (A) is false.
g(-2) = (-2)2 = 4. Option (B) is true.
g(-1) = 1 = g(1). Option (C) is true.
g(0) = 02 = 0. Option (D) is not true.
g(-1) + g(1) = (-1)2 + (1)2 = 1 + 1 = 2. Option (E) is not true.

Question 12

A man bought pens at the rate of 6 for Rs.20 and sold them at 4 for Rs.16. Which of the following statements is true? Indicate all correct options.

1. He gained 10%
2. He lost 10%
3. He gained 20%
4. If he sold 2 for Rs.16, his profit would be more than double
5. If he sold 8 for Rs.16, his profit would be half

Explanation:

Cost price (CP) for one pen = Rs.20/6 = Rs.10/3
Selling price (SP) for one pen = Rs.16/4 = Rs.4

Profit% = (SP-CP)/CP*100
= (4-10/3)/(10/3)*100
=2/3*3/10*100
= 20%
Options A and B are false and C is true.

If he sold them at Rs.16/2 = Rs.8 his profit would be
Profit% = (SP-CP)/CP*100
= (8-10/3)/(10/3)*100
=14/3*3/10*100
= 140%
140%>2*20% = 40%
Option D is true.

If he sold them at Rs.16/8 = Rs.2, he would suffer a loss.
Hence, option E is false.

Question 13

An article was bought for Rs.390. Which of the following statements is true? Indicate all correct options.

1. The transaction earned a loss of 20% if it was sold for Rs.355
2. The transaction led to profit of 20% if it was sold for Rs.425
3. The profit after selling it for Rs.425 was equal to the loss after selling it for Rs.355
4. The profit percent after selling it for Rs.425 was equal to the loss percent after selling it for Rs.355
5. The transaction earned a profit if it was sold for Rs.395.

Explanation:

Let the cost price and the selling price be CP and SP respectively.
SP = (100-loss%)/100*CP
SP = (100+gain%)/100*CP
When loss = 20%
SP = (100-20)/100*390
= 80/100*390 = 312
Hence, option A is false.

When profit = 20%
SP = (100+20)/100*390
= 120/100*390
= 468
Hence, option B is false.
Loss = CP-SP
= 390-355 = 35
Profit = SP-CP
= 425-390 = 35
Option C is true.

Profit% = gain/CP*100
= 35/425*100 = 3500/425
Loss% = Loss/CP*100
= 35/390*100 = 3500/390
Option D is false.

395 > 390. Option E is true.

Question 14

Two numbers are in ratio 3:4.Which of the following statements is true? Indicate all correct options.

1. Their duplicate ratio is 6/8
2. Their duplicate ratio is 9/16
3. Their triplicate ratio is 27/64
4. Their triplicate ratio is 9/12
5. Their triplicate ratio is (3/4)3

Correct Answer: B, D and E

Explanation:

The given ratio is 3/4
The duplicate ratio is 32/42 = 9/16
The triplicate ratio is 33/43 = 27/64
Options B, D and E are true.

Question 15

The ratio between two numbers is 4:5. Which of the following is their inverse ratio? Indicate all correct options.

1. 16:25
2. 5:4
3. 1/4:1/5
4. 1/5:1/4
5. 25:16

Explanation:

The inverse ratio of 4:5 is given by
1/4:1/5, which is equal to 5:4
Options B and C are true.