Practice Questions

Q1. Let x1, x2, … , x10 be ten observations such that ∑10i=1 (xi −2) = 30, ∑10i=1 (xi −β)2 = 98, β > 2, and their variance is 4 . If μ and σ2 are respectively the mean and the variance of 2 (x1 −1) + 4β , 5 2 (x2 −1) + 4β, … . , 2 (x10 −1) + 4β , then βμσ2 is equal to : (1) 100 (2) 120 (3) 110 (4) 90

Q1. For a 3 × 3 matrix M , let trace (M) denote the sum of all the diagonal elements of M . Let A be a 3 × 3 matrix such that |A| = 12 and trace (A) = 3. If B = adj(adj(2A)), then the value of |B|+ trace (B) equals : (1) 56 (2) 132 (3) 174 (4) 280

Q1. Let O be the origin, the point A be z1 = √3 + 2√2i , the point B (z2) be such that √3 |z2| = |z1| and arg (z2) = arg (z1) + π6 . Then (1) area of triangle ABO is 11 (2) ABO is an obtuse angled isosceles triangle √3 (3) area of triangle ABO is 11 (4) ABO is a scalene triangle 4

Q1. The distance of the line x−2 2 = y−63 = z−34 from the point (1, 4, 0) along the line x1 = y−22 = z+33 is : (1) √17 (2) √15 (3) √14 (4) √13

Q1. Let a1, a2, a3, … be a G.P. of increasing positive terms. If a1a5 = 28 and a2 + a4 = 29, then a6 is equal to: (1) 628 (2) 812 (3) 526 (4) 784 = 0. If x(1) = 1, then x ( 12 ) is :

Q1. Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to : (1) 8750 (2) 9100 (3) 8925 (4) 8575

Q2. Let ^a be a unit vector perpendicular to the vectors b = ^i −2^j + 3^k and→c= 2^i + 3^j −^k, and makes an angle of α cos−1 (−13 ) with the vector ^i + ^j + ^k. If ^a makes an angle of π3 with the vector ^i + α^j + ^k, then the value of is : (1) √6 (2) −√6 (3) −√3 (4) √3

Q2. In a group of 3 girls and 4 boys, there are two boys B1 and B2 . The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but B1 and B2 are not adjacent to each other, is : (1) 96 (2) 144 (3) 120 (4) 72

Q2. Let x = x(y) be the solution of the differential equation y2 dx + (x −1y )dy (1) 1 2 + e (2) 3 + e (3) 3 −e (4) 32 + e

Q2. Let A = {(x, y) ∈R × R : |x + y| ⩾3} and B = {(x, y) ∈R × R : |x| + |y| ≤3}. If C = {(x, y) ∈A ∩B : x = 0 or y = 0}, then ∑(x,y)∈C |x + y| is : (1) 15 (2) 24 (3) 18 (4) 12

Q2. Let in a △ABC , the length of the side AC be 6 , the vertex B be (1, 2, 3) and the vertices A, C lie on the line x−6 3 = y−72 = z−7−2 . Then the area (in sq. units) of △ABC is: (1) 17 (2) 21 (3) 56 (4) 42 y2 on the ellipse x2 + = 1, (a > b), be 74 . Then the a2 b2

Q2. Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is : (1) 90 (2) 84 (3) 122 (4) 108 − − = 0 is: √x √x

Q2. x + 2y −3z = 2 If the system of equations 2x + λy + 5z = 5 has infinitely many solutions, then λ + μ is equal to : 14x + 3y + μz = 33 (1) 13 (2) 10 (3) 12 (4) 11 and

Q2. If the components of →a = α^i + β^j + γ^k along and perpendicular to b = 3^i + ^j −^k respectively, are 16 γ 2 is equal to : 11 (3^i + ^j −^k) and 111 (−4^i −5^j −17^k), then α2 + β2 + (1) 26 (2) 18 (3) 23 (4) 16

Q2. One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is (1) 2 (2) 1 3 2 (3) 4 (4) 3 9 5

Q3. If for the solution curve y = f(x) of the differential equation dydx + (tan x)y = (1+22+secsec xx)2 , x ∈( −π2 , π2 ), f ( π3 ) = √310 , then f ( π4 ) is equal to : (1) √3+1 (2) 5−√3 10(4+√3) 2√2 (3) 9√3+3 (4) 4−√2 10(4+√3) 14

Q3. Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is m , n where gcd(m, n) = 1, then m + n is equal to : (1) 4 (2) 14 (3) 13 (4) 11

Q3. Let the product of the focal distances of the point (√3, 12 ) absolute difference of the eccentricities of two such ellipses is (1) 1−√3 (2) 3−2√2 √2 2√3 (3) 3−2√2 (4) 1−2√2 3√2 √3 Q4. 2x −y + z = 4 If the system of equations 5x + λy + 3z = 12 has infinitely many solutions, then μ −2λ is equal to 100x −47y + μz = 212 (1) 57 (2) 59 (3) 55 (4) 56

Q4. The sum of all local minimum values of the function ⎧ 1 −2x, x < −1 f(x) = 3 (7 + 2|x|), −1 ≤x ≤2 ⎨ 1 11 ⎩ 18 (x −4)(x −5), x > 2 is (1) 157 (2) 131 72 72 (3) 171 (4) 167 72 72

Q4. The area of the region enclosed by the curves y = ex, y = |ex −1| and y-axis is: (1) 1 −loge 2 (2) loge 2 (3) 1 + loge 2 (4) 2 loge 2 −1 y2

Q4. Let a line pass through two distinct points P(−2, −1, 3) and Q , and be parallel to the vector 3^i + 2^j + 2^k. If the distance of the point Q from the point R(1, 3, 3) is 5 , then the square of the area of △PQR is equal to : (1) 148 (2) 136 (3) 144 (4) 140

Q4. Let P be the foot of the perpendicular from the point (1, 2, 2) on the line L : x−11 = y+1−1 = z−22 . Let the line →r = (−^i + ^j −2^k) + λ(^i −^j + ^k), λ ∈R, intersect the line L at Q . Then 2(PQ)2 is equal to : (1) 25 (2) 19 (3) 29 (4) 27

Q4. Let ∫x3 sin x dx = g(x) + C , where C is the constant of integration. If 8 (g ( π2 ) + g′ ( π2 )) = απ3 + βπ2 + γ, α, β, γ ∈Z , then α + β −γ equals : (1) 48 (2) 55 (3) 62 (4) 47

Q4. Define a relation R on the interval [0, π2 ) by xRy if and only if sec2 x −tan2 y = 1. Then R is : (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric not transitive (4) both reflexive and symmetric but not transitive

Q4. The product of all solutions of the equation e5(loge x)2+3 = x8, x > 0, is : (1) e8/5 (2) e6/5 (3) e2 (4) e