Practice Questions

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 α, β, γ and δ be the coefficients of x7, x5, x3 and x respectively in the expansion of 5 5 αu + βv = 18 + , x > 1. If u and v satisfy the equations then u + v equals : (x + √x3 −1) (x −√x3 −1) γu + δv = 20 (1) 5 (2) 3 (3) 4 (4) 8

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. Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be ^i + 2^j + ^k,^i + 3^j −2^k and 2^i + ^j −^k respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E . If the length of AD is √110 and the volume of the 3 tetrahedron is √805 , then the position vector of E is 6√2 (1) 12 1 (7^i + 4^j + 3^k) (2) 12 (^i + 4^j + 7^k) (3) 1 6 (12^i + 12^j + ^k) (4) 16 (7^i + 12^j + ^k)

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. 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. If A, B , and (adj (A−1) + adj (B−1)) are non-singular matrices of same order, then the inverse of A(adj (A−1) + adj (B−1))−1 B , is equal to (1) AB−1 + A−1 B (2) adj (B−1) + adj (A−1) BA−1 (3) AB−1 (4) 1 (adj(B) + adj(A)) |A| + |B| |AB|

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. 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. Let the coefficients of three consecutive terms Tr, Tr+1 and Tr+2 in the binomial expansion of (a + b)12 be in a G.P. and let p be the number of all possible values of r. Let q be the sum of all rational terms in the binomial expansion of (4√3 + 3√4)12 . Then p + q is equal to : (1) 283 (2) 287 (3) 295 (4) 299

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. 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

Q5. Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line x + 2y = 2. If the centroid of △PQR is the point (α, β), then 15(α −β) is equal to : (1) 19 (2) 24 (3) 21 (4) 22

Q5. The equation of the chord, of the ellipse x2 = 1, whose mid-point is (3, 1) is : 25 + 16 (1) 48x + 25y = 169 (2) 5x + 16y = 31 (3) 25x + 101y = 176 (4) 4x + 122y = 134

Q5. A rod of length eight units moves such that its ends A and B always lie on the lines x −y + 2 = 0 and y + 2 = 0, respectively. If the locus of the point P , that divides the rod AB internally in the ratio 2 : 1 is 9 (x2 + αy2 + βxy + γx + 28y) −76 = 0, then α −β −γ is equal to : (1) 22 (2) 21 (3) 23 (4) 24

Q5. If A and B are two events such that P(A ∩B) = 0.1, and P(A ∣B) and P(B ∣A) are the roots of the equation – 12x2 −7x + 1 = 0, then the value of P(A∪B) is : P(A∩B) (1) 4 (2) 7 3 4 (3) 5 (4) 9 3 4

Q5. For some n ≠10, let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of (1 + x)n+4 be in A.P. Then the largest coefficient in the expansion of (1 + x)n+4 is: (1) 20 (2) 10 (3) 35 (4) 70

Q5. Let nCr−1 = 28, nCr = 56 and nCr+1 = 70. Let A(4 cos t, 4 sin t), B(2 sin t, −2 cos t) and C (3r −n, r2 −n −1) be the vertices of a triangle ABC , where t is a parameter. If (3x −1)2 + (3y)2 = α, is the locus of the centroid of triangle ABC , then α equals (1) 6 (2) 18 (3) 8 (4) 20

Q5. Marks obtains by all the students of class 12 are presented in a freqency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is (1) 52 (2) 48 (3) 44 (4) 40

Q5. Let A = [aij] be a matrix of order 3 × 3, with aij = (√2)i+j . If the sum of all the elements in the third row of A2 is α + β√2, α, β ∈Z, then α + β is equal to : (1) 280 (2) 224 (3) 210 (4) 168

Q5. Let [x] denote the greatest integer less than or equal to x. Then the domain of f(x) = sec−1(2[x] + 1) is : (1) (−∞, −1] ∪[0, ∞) (2) (−∞, −1] ∪[1, ∞) (3) (−∞, ∞) (4) (−∞, ∞) −{0}

Q5. Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersects at the points A and B, then (AB)2 is equal to : (1) 392 (2) 384 (3) 192 (4) 96

Q6. Let the points ( 112 , α) lie on or inside the triangle with sides x + y = 11, x + 2y = 16 and 2x + 3y = 29. Then the product of the smallest and the largest values of α is equal to : (1) 44 (2) 22 (3) 33 (4) 55

Q6. Let a curve y = f(x) pass through the points (0, 5) and (loge 2, k). If the curve satisfies the differential equation 2(3 + y)e2xdx −(7 + e2x)dy = 0, then k is equal to (1) 4 (2) 32 (3) 8 (4) 16