Practice Questions

Q3. The number of solutions of the equation ( x9 9 + 2) ( x2 7 + 3) (1) 2 (2) 3 (3) 1 (4) 4

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 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

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

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. 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. 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

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 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. 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

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. 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. 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. 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. 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. 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

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. 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. 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 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

Q6. If the square of the shortest distance between the lines x−2 1 = y−12 = z+3−3 and x+12 = y+34 = z+5−5 is mn , where m, n are coprime numbers, then m + n is equal to : (1) 21 (2) 9 (3) 14 (4) 6 x

Q6. Let for f(x) = 7 tan8 x + 7 tan6 x −3 tan4 x −3 tan2 x, I1 = ∫π/40 f(x)dx and I2 = ∫π/40 xf(x)dx. Then 7I1 + 12I2 is equal to : (1) 2 (2) 1 (3) 2π (4) π