Practice Questions

Q1. Let f(x) = ∫t0 (1) 253 (2) 154 (3) 125 (4) 157 →

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 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. Let circle C be the image of x2 + y2 −2x + 4y −4 = 0 in the line 2x −3y + 5 = 0 and A be the point on C such that OA is parallel to x-axis and A lies on the right hand side of the centre O of C . If B(α, β), with β < 4, lies on C such that the length of the are AB is (1/6)th of the perimeter of C , then β −√3α is equal to (1) 3 + √3 (2) 4 (3) 4 −√3 (4) 3

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

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. 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. If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to (1) −1080 (2) −1020 (3) −1200 (4) −120

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

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. 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. 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 ^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. 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. 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. 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 f : R →R be a function defined by f(x) = (2 + 3a)x2 + ( a+2a−1 )x + b, a ≠1. If f(x + y) = f(x) + f(y) + 1 −27 xy , then the value of 28 ∑5i=1 |f(i)| is (1) 545 (2) 715 (3) 735 (4) 675

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. 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. Let A, B, C be three points in xy-plane, whose position vector are given by √3^i + ^j,^i + √3^j and a^i + (1 −a)^j respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between −−→ → the vectors OA and OB is 9 , then the sum of all the possible values of a is : √2 (1) 2 (2) 9/2 (3) 1 (4) 0

Q3. Let ABCD be a trapezium whose vertices lie on the parabola y2 = 4x. Let the sides AD and BC of the trapezium be parallel to y -axis. If the diagonal AC is of length 25 and it passes through the point (1, 0), then 4 the area of ABCD is (1) 75 4 (2) = 252 (3) 125 (4) 75 8 8

Q3. Let X = R × R. Define a relation R on X as : (a1, b1)R (a2, b2) ⇔b1 = b2 Statement I : R is an equivalence relation. Statement II : For some (a, b) ∈X , the set S = {(x, y) ∈X : (x, y)R(a, b)} represents a line parallel to y = x. In the light of the above statements, choose the correct answer from the options given below : (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true

Q3. Let A = {x ∈(0, π) −{ π2 } : log(2/π) | sin x| + log(2/π) | cos x| = 2} B = {x ⩾0 : √x(√x −4) −3|√x −2| + 6 = 0}. Then n(A ∪B) is equal to : (1) 4 (2) 8 (3) 6 (4) 2

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. The number of solutions of the equation ( x9 9 + 2) ( x2 7 + 3) (1) 2 (2) 3 (3) 1 (4) 4