Q. 1
Which of the following is a correct representation of the set X = in the set builder form? Indicate all correct options. [n^2 = n*n sqrt(n) = square root of n]
- a. X =
- b. X =
- c. X =
- d. X =
- e. X =
- Answer: CD
- Explaination: For X = , X is the set of natural numbers and hence option (A) is not correct. For X = , X = {-1, 1, 1/sqrt(2), 1/sqrt(3),...} and hence option (B) is not correct. For X = , X = and hence option (C) is correct. For X = , X = and hence option (D) is correct. For X = , 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: BCE
- Explaination: 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)
- Answer: AC
- Explaination: 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% 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: ABC
- Explaination: 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: AB
- Explaination: 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 = ? Indicate all correct options.
- a. 2
- b. 1
- c. 0
- d. 9
- e. 24
- Answer: AB
- Explaination: The set A = = 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: BC
- Explaination: 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: ACE
- Explaination: 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: BD
- Explaination: 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
- Answer: BD
- Explaination: 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
- Answer: DE
- Explaination: The set A is A = 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< />pan>
- d. D<0 if p^2<8p< />pan>
- e. D>0
- Answer: BD
- Explaination: D = 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 true. e 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)
- Answer: BC
- Explaination: 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 - (a-1)x - (a+2)(a-1) = 0 x - (a-1) = 0 = 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.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. [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) = -
- d. p(x)*q(x) = (2x-1)/(2x+1)
- e. p(x)/q(x) = {(2x-1)(2x+1)}/{(x+1)(x-1)}
- Answer: ACD
- Explaination: 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)
- Answer: CD
- Explaination: 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