Practice Questions

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. 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 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. 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 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. 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. 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. 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

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. 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 [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. 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. 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. 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. 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. 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. 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

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 the equation of the circle, which touches x-axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y-axis be x2 + y2 −αx + βy + γ = 0. If the circle lies below x-axis, then the ordered pair (2a, b2) is equal to (1) (γ, β2 −4α) (2) (α, β2 + 4γ) (3) (γ, β2 + 4α) (4) (α, β2 −4γ) 2x

Q6. Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is : (1) 1 (2) 1 2 4 (3) 2 (4) 1 3 3 1 = a√3 + b, a, b ∈Z, then a2 + b2 is equal to : π π

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

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

Q6. Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is 4 of the area of the triangle OAB and AN : NB = λ : 1, then 9 the sum of all possible value(s) of is λ : (1) 2 (2) 5 2 (3) 1 (4) 13 2 6

Q6. x sin−1 x sin−1 x x 1 + If ∫ex + 1−x2 = g(x) + C, where C is the constant of integration, then g ( 2 ) equals (1−x2)3/2 ( √1−x2 )dx : (1) π (2) π 4 √e3 6 √e3 (3) π 4 √e2 (4) π6 √e2