Problem Solving Select Many Practice Test 1
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}
 Answer: C and D

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

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 ((64)/2, (3+5)/2) = (2/2, 8/2) = (1,4) Option B is true. The coordinates 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)
 Answer: A and C

Number of diagonals = C(9,2) = 9!/[(92)!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% 25/9% 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

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)
 Answer: A and B

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
 Answer: A and B

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
 Answer: B and C

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

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
 Answer: B and D

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+x1=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
 Answer: B and D

2x^2+x1=0 2x^2 +2xx1=0 2x(x+1)1(x+1)=0 (2x1)(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
 Answer: D and E

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
 Answer: B and D

D = p^2  4*1*2q = p^2  8q D=0 if p^28q = 0 p^2 = 8q Option A is wrong and B is true. D<0 if p^28q<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+a2) = 0? Indicate all correct options. [n^2 = n*n]
 A. (a+2)
 B. (a1)
 C. (a+2)
 D. (1a)
 E. (2a)
 Answer: B and C

x^2 + 3x (a^2+a2) = 0 x^2 + 3x  (a+2)(a1) = 0 x^2 +{(a+2)(a1)}x  (a+2)(a+1) = 0 {x^2 +(a+2)x}  (a1)x  (a+2)(a1) = 0 x{x + (a+2)}  (a1){x + (a+2)} = 0 {x + (a+2)}{x  (a1)} = 0 x = (a+2), (a1) Hence, options B and C are true.
Q.14
For two rational expressions p(x) = (2x1)/(x+1) and q(x) = (x+1)/(2x+1), which of the following is true? Indicate all correct options.For two rational expressions p(x) = (2x1)/(x+1) and q(x) = (x+1)/(2x+1), which of the following is true? Indicate all correct options. [n^2 = n*n]
 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) = (2x1)/(2x+1)
 E. p(x)/q(x) = {(2x1)(2x+1)}/{(x+1)(x1)}
 Answer: A, C and D

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) = [(2x1)/(x+1)]*[(x+1)/(2x+1)] = (2x1)/(2x+1) Option D is true. p(x)/q(x) = [(2x1)/(x+1)]/[(x+1)/(2x+1)] = [(2x1)/(x+1)]*[(2x+1)/(x+1)] = [(2x1)(2x+1)]/[(x+1)(x+1)] Option E is false.
Q.15
If p(x) = (x+2)/(x2) and q(x) = x/(x^24), then which of the following represents p(x)*q(x)? Indicate all correct options. [n^2 = n*n]
 A. x/(x2)
 B. x(x2)^2
 C. x/[(x2)(x2)]
 D. x/(x2)^2
 E. x(x+2)/(x2)
 Answer: C and D

p(x)*q(x) = [(x+2)/(x2)]*[x/(x^24)] = [(x+2)*x]/[(x2)(x+2)(x2)] = x/[(x2)(x2)] = x/(x2)^2