Problem Solving Select Many Practice Test 1


Question 1

Which of the following is a correct representation of the set X = {1, 1/4, 1/9, 1/16, 1/25,...} in the set builder form? Indicate all correct options.

[n2 = n*n

sqrt(n) = square root of n]

  1. X = {x: x is an element of N}
  2. X = {n: 1/n2 is an element of N}
  3. X = {1/n2: n is an element of N}
  4. X = {1/x2: x is a natural number}
  5. X = {1/sqrt(x): x is a natural number}

Correct Answer: C and D

Explanation:

For X = {x: x is an element of N}, X is the set of natural numbers and hence option (A) is not correct.
For X = {n: 1/n2 is an element of N}, X = {-1, 1, 1/sqrt(2), 1/sqrt(3),...} and hence option (B) is not correct.
For X = {1/n2: n is an element of N}, X = {1, 1/4, 1/9, 1/16, 1/25,...} and hence option (C) is correct.
For X = {1/x2: x is a natural number}, X = {1, 1/4, 1/9, 1/16, 1/25,...} and hence option (D) is correct.
For X = {1/sqrt(x): x is a natural number}, X = {-1, 1, 1/sqrt(2), 1/sqrt(3),...} and hence option (E) is not correct.

Question 2

Which of the following statements is true for two points A(6,3) and B(-4,5)? Indicate all correct options.

  1. They lie in the same quadrant
  2. The mid point of AB is (1, 4)
  3. The point P(0, 21/5) divides AB in the ratio 3:2
  4. The point P(0, 21/5) divides AB in the ratio 2:3
  5. The two points are in different quadrants

Correct Answer: B, C and E

Explanation:

A lies in the first quadrant and B lies in the second quadrant.
Option A is false and E is true.
The mid point of AB is given by ((6-4)/2, (3+5)/2)
= (2/2, 8/2)
= (1,4)
Option B is true.

The co-ordinates of a point that divides AB in ratio 3:2 are given by
((3*-4+2*6)/(3+2), (3*5+2*3)/(3+2))
= ((-12+12)/5, (15+6)/5)
= (0, 21/5)
This is the same point as P. Hence, option C is true and D is false.

Question 3

How many diagonals are there in a polygon with 9 sides? Indicate all correct options.

  1. C(9,2)
  2. P(9,2)
  3. 36
  4. 72
  5. C(9,4)

Correct Answer: A and C

Explanation:

Number of diagonals = C(9,2) = 9!/[(9-2)!2!]
= 9!/(7!2!) = 9*8/2 = 9*4 = 36
Options (A) and (C) are correct.

Question 4

The following table shows the percentage expenditure on different items in constructing a flat. If the cost of the flat is Rs.5,40,000, then which of the following statements is true? Indicate all correct options.

Item Labour Timber Cement Bricks Steel
Percentage Expenditure 250/9% 25% 125/6% 125/9% 25/2%

  1. The amount spent on timber is twice that spent on steel
  2. Labour is the costliest commodity
  3. Steel is the cheapest commodity
  4. The amount spent on steel was Rs. 11250
  5. The amount spent on timber was Rs. 67500

Correct Answer: A, B and C

Explanation:

It is clear from the table that the amount spent on timber is twice the amount spent on steelsince the percentage expenditure on timber is 25% and the percentage expenditure on steel is 25/2%.
Option A is true.

250/9% = 27.78%
125/6% = 20.83%
125/9% = 13.89%
25/2% = 12.5%

Hence, the maximum amount was spent on Labour. Option B is true.
Steel is the cheapest commodity. Option C is true.

The amount spent on steel = 25/2*540000/100 = Rs.67500
The amount spent on timber = 25*540000/100 = Rs. 135000
Options D and E are false.

Question 5

A bag contains 5 red balls and 5 blue balls. In how many ways can 5 balls be drawn at random such that there are at least two balls of each colour? Indicate all correct options.

  1. 200
  2. 2C(5,2)*C(5,3)
  3. 100
  4. C(5,2)*C(5,3)
  5. C(10,5)

Correct Answer: A and B

Explanation:

We either select 2 red and 3 blue balls or 3 red and 2 blue balls.

Total number of ways = C(5,2)*C(5,3) + C(5,3)*C(5,2)
= 2[5!/(2!3!) * 5!/(2!3!)]
= 2[(5*4)/2*(5*4)/2]
= 2(100) = 200

Options (A) and (B) are correct.

Question 6

Which of the following is an element of the set A = {x: x is a positive integer and x is a divisor of 12}? Indicate all correct options.

  1. 2
  2. 1
  3. 0
  4. 9
  5. 24

Correct Answer: A and B

Explanation:

The set A = {x: x is a positive integer and x is a divisor of 12}
= {1, 2, 3, 4, 6, 12}
1 and 2 are elements of A.

Question 7

A bag contains 5 red balls, 8 white balls and 7 green balls. Which of the following is true? Indicate all correct options.

  1. The probability of drawing a blue ball is 7/25
  2. The probability of drawing a red ball is 1/4
  3. The probability of drawing a ball which is not green is 13/20
  4. The probability of drawing two red balls is 1/4
  5. The probabilty of drawing three green balls is 3/7

Correct Answer: B and C

Explanation:

The probability of drawing a blue ball is 0 since there are no blue balls in the bag.
Option A is false.

The probability of drawing a red ball = 5/(5+8+7)= 5/20 = 1/4|
Option B is true.

The probability of drawing a ball which is not green = (5+8)/(5+8+7)= 13/20
Option C is true.

The probability of drawing two red balls is = C(5,2)/C(20,2)
= 5!/(2!3!)/20!/(18!2!)= (5*4)/2/(20*19)/2= 20/(20*19) = 1/19
Option D is false.

The probability of drawing three green balls is = C(7,3)/C(20,3)
= 7!/(4!3!)/20!/(17!3!)
=(7*6*5)/(3*2)/(20*19*18)/(3*2)= 35/(20*19*3)= 7/(4*19*3)= 7/228
Option E is false.

Question 8

The given table shows the weight of the students of a class in kilograms. Which of the given statements is true? Indicate all correct options. Weight (in kg): 67 70 72 73 75 Number of students: 4 3 2 2 1

Weight (in kg): 67 70 72 73 75
Number of students: 4 3 2 2 1

  1. There are in all 12 students
  2. The average weight is 70 kgs
  3. Most of the students weigh less than 72 kgs
  4. The weight of the students is proportional to their heights
  5. The average weight is 70.25 kgs

Correct Answer: A, C and E

Explanation:

Total students in the class = 4+3+2+2+1 = 12
Option A is true.

Average weight = (67*4+70*3+72*2+73*2+75*1)/12
= (268 + 210 + 144 + 146 + 75)/12
= 843/12 = 70.25 kgs
Option B is false and E is true.

7 students weigh less than 72 kgs. Option C is true.
Since we do not know the heights of the students, we cannot say that option D is true.

Question 9

A cubical block of iron of 88 cm edge is recast into small spheres of 4 cm diameter. Which of the following statemetns is true? Indicate all correct options.

  1. 2541 spheres can be made
  2. number of spheres = volume of iron cube/volume of sphere
  3. number of spheres = surface area of iron cube/surface area of sphere
  4. 20328 spheres can be made
  5. 5082 spheres can be made

Correct Answer: B and D

Explanation:

Volume of a block of iron = 88*88*88
= 681472

Volume of sphere = (4/3)*pi*r3, where r is the radius of the sphere
= (4/3)*22/7*23

Number of spheres = volume of cube/volume of sphere
= [88*88*88]/[(4/3)*22/7*23]
= (88*88*88*3*7)/(4*22*8)
= 20328
Options B and D are true.
[pi=22/7, r3=r*r*r]

Question 10

Which of the following statements is true for the quadratic equation 2x2+x-1=0? Indicate all correct options.

[n2 = n*n]

  1. The equation has equal roots
  2. The equation has real and distinct roots
  3. x = 1
  4. x = -1
  5. x = -1/2

Correct Answer: B and D

Explanation:

2x2+x-1=0
2x2 +2x-x-1=0
2x(x+1)-1(x+1)=0
(2x-1)(x+1)=0
x=1/2, -1
Clearly, the equation has real and distinct roots.
Options B and D are correct.

Question 11

A is the set of vowels. How many elements does the power set of A have? Indicate all correct options.

[24 = 2*2*2*2]

  1. 24
  2. 0
  3. 5
  4. 32
  5. 25

Correct Answer: D and E

Explanation:

The set A is A = {a, e, i, o, u}
The set has five elements and the number of elements in the power set of A is given by 25 = 32

Question 12

Which of the following is true for the equation x2+px+2q=0, where D is the discriminant? Indicate all correct options.

[n2 = n*n]

  1. D = 0
  2. D >0 if p2 = 8q
  3. D<0
  4. D<0 if p2<8p
  5. D>0

Correct Answer: B and D

Explanation:

D = p2 - 4*1*2q
= p2 - 8q
D=0 if p2-8q = 0
p2 = 8q
Option A is wrong and B is true.
D<0 if p2-8q<0
p2<8q

Option C is false and D is true.
Option E is false.

Question 13

Which of the following is a root of the equation x2 + 3x -(a2+a-2) = 0? Indicate all correct options.

[n2 = n*n]

  1. (a+2)
  2. (a-1)
  3. -(a+2)
  4. (1-a)
  5. (2-a)

Correct Answer: B and C

Explanation:

x2 + 3x -(a2+a-2) = 0
x2 + 3x - (a+2)(a-1) = 0
x2 +{(a+2)-(a-1)}x - (a+2)(a+1) = 0
{x2 +(a+2)x} - (a-1)x - (a+2)(a-1) = 0
x{x + (a+2)} - (a-1){x + (a+2)} = 0
{x + (a+2)}{x - (a-1)} = 0
x = -(a+2), (a-1)
Hence, options B and C are true.

Question 14

For two rational expressions p(x) = (2x-1)/(x+1) and q(x) = (x+1)/(2x+1), which of the following is true? Indicate all correct options.

  1. p(x) + q(x) = q(x) + p(x)
  2. p(x) - q(x) = q(x) - p(x)
  3. p(x) - q(x) = -{q(x) - p(x)}
  4. p(x)*q(x) = (2x-1)/(2x+1)
  5. p(x)/q(x) = {(2x-1)(2x+1)}/{(x+1)(x-1)}

Correct Answer: A, C and D

Explanation:

p(x) + q(x) = q(x) + p(x) since addition of rational expressions is commutative.

Option A is true.

Option B is false and C is true since subtraction of rational expressions is not commutative.
p(x)*q(x) = [(2x-1)/(x+1)]*[(x+1)/(2x+1)]
= (2x-1)/(2x+1)
Option D is true.

p(x)/q(x) = [(2x-1)/(x+1)]/[(x+1)/(2x+1)]
= [(2x-1)/(x+1)]*[(2x+1)/(x+1)]
= [(2x-1)(2x+1)]/[(x+1)(x+1)]
Option E is false.

Question 15

If p(x) = (x+2)/(x-2) and q(x) = x/(x2-4), then which of the following represents p(x)*q(x)? Indicate all correct options.

[n2 = n*n]

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

Correct Answer: C and D

Explanation:

p(x)*q(x) = [(x+2)/(x-2)]*[x/(x2-4)]
= [(x+2)*x]/[(x-2)(x+2)(x-2)]
= x/[(x-2)(x-2)]
= x/(x-2)2