Practice Questions

Q19.Consider the region R = {(x, y) : x ≤y ≤9 −113 x2, x ≥0}. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is: (1) 730 (2) 625 119 111 (3) 821 (4) 567 123 121

Q19.Let I(x) = ∫ 11 15 . If I(37) −I(24) = 4 1 − 1 b, c ∈N (x−11) 13 (x+15) 13 ( b 13 c 13 ), (1) 22 (2) 39 (3) 40 (4) 26

Q19.If in the expansion of (1 + x)p(1 −x)q , the coefficients of x and x2 are 1 and -2 , respectively, then p2 + q2 is equal to : (1) 18 (2) 13 (3) 8 (4) 20 a

Q19.Let the curve z(1 + i) + ¯z(1 −i) = 4, z ∈C, divide the region |z −3| ≤1 into two parts of areas α and β . Then |α −β| equals : (1) 1 + π2 (2) 1 + π3 (3) 1 + π6 (4) 1 + π4

Q20.Two equal sides of an isosceles triangle are along −x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is : (1) −2√10 (2) 12 (3) 6 (4) −6

Q20.Let the area of the region {(x, y) : 2y ≤x2 + 3, y + |x| ≤3, y ⩾|x −1|} be A. Then 6 A is equal to : (1) 16 (2) 12 (3) 14 (4) 18

Q20.Let z1, z2 and z3 be three complex numbers on the circle |z| = 1 with arg (z1) = −π4 , arg (z2) = 0 and arg (z3) = π4 . If |z1¯z2 + z2¯z3 + z3¯z1|2 = α + β√2, α, β ∈Z, then the value of α2 + β2 is : (1) 24 (2) 29 (3) 41 (4) 31

Q20.Let E : x2 + y2 = 1, a > b and H : x2 − y2 = 1. Let the distance between the foci of E and the foci of H a2 b2 A2 B2 be 2√3. If a −A = 2, and the ratio of the eccentricities of E and H is 13 , then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7 = α × 229 , then α is equal to ______

Q20.Let →a = ^i + 2^j + 3^k,→b = 3^i + ^j −^k and →c be three vectors such that →c is coplanar with →a and →b. If the vector →C is perpendicular to →b and →a ⋅→c = 5, then |→c| is equal to (1) √116 (2) 3√21 (3) 16 (4) 18

Q21.Let the circle C touch the line x −y + 1 = 0, have the centre on the positive x -axis, and cut off a chord of length 4 along the line −3x + 2y = 1. Let H be the hyperbola x2 −y2 = 1, whose one of the foci is the √13 α2 β2 centre of C and the length of the transverse axis is the diameter of C . Then 2α2 + 3β2 is equal to ______

Q21.Let P be the image of the point Q(7, −2, 5) in the line L : x−12 = y+13 = 4z and R(5, p, q) be a point on Then the square of the area of △PQR is ________. x + 1 + C, where C is the

Q21.Let A be a square matrix of order 3 such that det(A) = −2 and det(3 adj(−6 adj(3A))) = 2m+n ⋅3mn, m > n . Then 4 m + 2n is equal to _______ , then m −n is equal to _______

Q21.Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If d denotes the distance between A and B , and a denotes the area of △SAB, where S is the focus of the parabola y2 = 4x, then the value of (a + d) is -

Q22.Let M denote the set of all real matrices of order 3 × 3 and let S = {−3, −2, −1, 1, 2}. Let S1 = {A = [aij] ∈M : A = AT and aij ∈ S, ∀i, j}, S2 = {A = [aij] ∈M : A = −AT and aij ∈ S, ∀i, j}, S3 = {A = [aij] ∈M : a11 + a22 + a33 = 0 and aij ∈ S, ∀i, j}. If n ( S1 ∪2 US3) = 125α, then α equals _______

Q22.Let f : (0, ∞) →R be a twice differentiable function. If for some a ≠0, ∫10 f(λx)dλ = af(x), f(1) = 1 and f(16) = 18 , then 16 −f ′ ( 161 ) is equal to _______.

Q22.The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , is

Q22.If ∑5r=0 11C22r2r+2 = mn , gcd(m, n) = 1

Q23.Let A(6, 8), B(10 cos α, −10 sin α) and C(−10 sin α, 10 cos α), be the vertices of a triangle. If L(a, 9) and G(h, k) be its orthocenter and centroid respectively, then (5a −3h + 6k + 100 sin 2α) is equal to ______ -. , −1 < x < 1 such that

Q23.If y = y(x) is the solution of the differential equation, ( 2 ), −2 ≤x ≤2, y(2) = 4 , then y2(0) is equal to √4 −x2 dxdy = ((sin−1 x 2 x π2−8 ( 2 )) −y) sin−1

Q23.The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is _______.

Q24.Let the function, f(x) = {−3ax2a2 + bx,−2, xx <⩾11 be differentiable for all x ∈R, where a > 1, b ∈R. If the area of the region enclosed by y = f(x) and the line y = −20 is α + β√3, α, β ∈Z , then the value of α + β is ________

Q24.Let y = f(x) be the solution of the differential equation dydx + x2−1xy = √1−x2x6+4x f(0) = 0. If 6 ∫1/2−1/2 f(x)dx = 2π −α then α2 is equal to _______ .

Q24.Let f be a differentiable function such that 2(x + 2)2f(x) −3(x + 2)2 = 10 ∫x0 (t + 2)f(t)dt, f(2) is equal to ______.

Q24.Let y2 = 12x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ) = 1474 . Let C be the circle described taking PQ as a diameter. If the equation of a circle C is 64x2 + 64y2 −αx −64√3y = β , then β −α is equal to ________.

Q25.Let H1 : x2a2 −y2b2 A2 B2 and e2 respectively. If the product of the lengths of 12√5 respectively. Let their ecentricities be e1 = √52 their transverse axes is 100√10, then 25e22 is equal to ________.