Practice Questions

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

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. 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. 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 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 [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. 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. 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 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 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 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. 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 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. The product of all the rational roots of the equation (x2 −9x + 11)2 −(x −4)(x −5) = 3, is equal to (1) 14 (2) 21 (3) 28 (4) 7

Q7. If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at 440th position in this arrangement, is : (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAKU

Q7. Let the parabola y = x2 + px −3, meet the coordinate axes at the points P, Q and R . If the circle C with centre at (−1, −1) passes through the points P, Q and R, then the area of △PQR is : (1) 7 (2) 4 (3) 6 (4) 5

Q7. Let the line passing through the points (−1, 2, 1) and parallel to the line x−12 = y+13 = 4z intersect the line y−3 x+2 3 = 2 = z−41 at the point P . Then the distance of P from the point Q(4, −5, 1) is (1) 5 (2) 5√5 (3) 5√6 (4) 10

Q7. The area of the region enclosed by the curves y = x2 −4x + 4 and y2 = 16 −8x is : (1) 8 (2) 4 3 3 (3) 8 (4) 5 x ∈R. Then the numbers of local maximum and local minimum points of f ,

Q7. Let f : (0, ∞) →R be a function which is differentiable at all points of its domain and satisfies the condition x2f ′(x) = 2xf(x) + 3, with f(1) = 4. Then 2f(2) is equal to : (1) 39 (2) 19 (3) 29 (4) 23

Q7. If f(x) = , x ∈R, then ∑81k=1 f ( 82k ) is equal to 2x+√2 (1) 1.81√2 (2) 41 (3) 82 (4) 81 2

Q7. Let →a = ^i + 2^j + ^k and b = 2^i + 7^j + 3^k. Let L1 :→r= (−^i + 2^j + ^k) + λ→a, λ ∈R and → L2 :→r= (^j + ^k) + μb, μ ∈R be two lines. If the line L3 passes through the point of intersection of L1 and L2 , and is parallel to →a + →b, then L3 passes through the point : (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (−1, −1, 1) → →

Q8. If the set of all a ∈R, for which the equation 2x2 + (a −5)x + 15 = 3a has no real root, is the interval (α, β), and X = {x ∈Z : α < x < β}, then ∑x∈X x2 is equal to : (1) 2109 (2) 2129 (3) 2119 (4) 2139