Q. 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. [n^2 = n*n sqrt(n) = square root of n]

• A. X = {x: x is an element of N}
• B. X = {n: 1/n^2 is an element of N}
• C. X = {1/n^2: n is an element of N}
• D. X = {1/x^2: x is a natural number}
• E. X = {1/sqrt(x): x is a natural number}
• 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/n^2
is an element of N}, X = {-1, 1, 1/sqrt(2), 1/sqrt(3),…} and hence option (B) is not correct. For X = {1/n^2:
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/x^2: 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.

Q. 2

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

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

Q. 3

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

• A. C(9,2)
• B. P(9,2)
• C. 36
• D. 72
• E. C(9,4)
• 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.

Q. 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%

• A. The amount spent on timber is twice that spent on steel
• B. Labour is the costliest commodity
• C. Steel is the cheapest commodity
• D. The amount spent on steel was Rs. 11250
• E. The amount spent on timber was Rs. 67500
• Answer: A, B and C
• Explanation:

It is clear from the table that the amount spent on timber is twice the amount spent on steel
since 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.

Q. 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.

• A. 200
• B. 2C(5,2)*C(5,3)
• C. 100
• D. C(5,2)*C(5,3)
• E. C(10,5)
• 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.

Q. 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.

• A. 2
• B. 1
• C. 0
• D. 9
• E. 24
• 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.

Q. 7

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

• A. The probability of drawing a blue ball is 7/25
• B. The probability of drawing a red ball is 1/4
• C. The probability of drawing a ball which is not green is 13/20
• D. The probability of drawing two red balls is 1/4
• E. The probabilty of drawing three green balls is 3/7
• 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.

Q. 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

• A. There are in all 12 students
• B. The average weight is 70 kgs
• C. Most of the students weigh less than 72 kgs
• D. The weight of the students is proportional to their heights
• E. The average weight is 70.25 kgs
• 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.

Q. 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.

• A. 2541 spheres can be made
• B. number of spheres = volume of iron cube/volume of sphere
• C. number of spheres = surface area of iron cube/surface area of sphere
• D. 20328 spheres can be made
• E. 5082 spheres can be made
• Explanation:

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

Volume of sphere = (4/3)*pi*r^3, where r is the radius of the sphere
= (4/3)*22/7*2^3

Number of spheres = volume of cube/volume of sphere
= [88*88*88]/[(4/3)*22/7*2^3] = (88*88*88*3*7)/(4*22*8)
= 20328
Options B and D are true.

[pi=22/7, r^3=r*r*r]

Q. 10

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

[n^2 = n*n]

• A. The equation has equal roots
• B. The equation has real and distinct roots
• C. x = 1
• D. x = -1
• E. x = -1/2
• Explanation:

2x^2+x-1=0
2x^2 +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.

Q. 11

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

[2^4 = 2*2*2*2]

• A. 2^4
• B. 0
• C. 5
• D. 32
• E. 2^5
• 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 2^5 = 32

Q. 12

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

[n^2 = n*n]

• A. D = 0
• B. D >0 if p^2 = 8q
• C. D<0
• D. D<0 if p^2<8p
• E. D>0
• Explanation:

p^2 – 4*1*2q
= p^2 – 8q

D=0 if p^2-8q = 0
p^2 = 8q
Option A is wrong and B is true.

D<0 if p^2-8q<0 p^2<8q Option C is false and D is true. Option E is false.

Q. 13

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

[n^2 = n*n]

• A. (a+2)
• B. (a-1)
• C. -(a+2)
• D. (1-a)
• E. (2-a)
• Explanation:

x^2 + 3x -(a^2+a-2) = 0
x^2 + 3x – (a+2)(a-1) = 0
x^2 +{(a+2)-(a-1)}x – (a+2)(a+1) = 0
{x^2 +(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.

Q. 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.

• A. p(x) + q(x) = q(x) + p(x)
• B. p(x) – q(x) = q(x) – p(x)
• C. p(x) – q(x) = -{q(x) – p(x)}
• D. p(x)*q(x) = (2x-1)/(2x+1)
• E. p(x)/q(x) = {(2x-1)(2x+1)}/{(x+1)(x-1)}
• 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.

Q. 15

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

[n^2 = n*n]

• A. x/(x-2)
• B. x(x-2)^2
• C. x/[(x-2)(x-2)]
• D. x/(x-2)^2
• E. x(x+2)/(x-2)
• Explanation:

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

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