Practice Questions

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. 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

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. 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

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. 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. 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

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. If ∑13r=1 { sin( 4 +(r−1) 6 ) sin( π4 + rπ6 ) } (1) 10 (2) 4 (3) 2 (4) 8

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) → →

Q7. x2 {sin (k1 + 1)x + sin (k2 −1)x}, x < 0 ⎧ If the function f(x) = 4, x = 0 is continuous at x = 0, then k21 + k22 is ⎨ 2 2+k1x x > 0 x loge ( 2+k2x ), ⎩ equal to (1) 20 (2) 5 (3) 8 (4) 10

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. 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. 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. (2x2−3x+5)(3x−1) 2 limx→∞ is equal to : (3x2+5x+4)√(3x+2)x (1) 2 (2) 2e √3e √3 (3) 2 (4) 2e 3√e 3

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

Q8. Let the point A divide the line segment joining the points P(−1, −1, 2) and Q(5, 5, 10) internally in the ratio −−−−→ → → → r : 1(r > 0). If O is the origin and (OQ ⋅OA) −15 |OP × OA|2 = 10, then the value of r is : (1) √7 (2) 14 (3) 3 (4) 7 2025 (23 Jan Shift 2) JEE Main Previous Year Paper y2

Q8. If 7 = 5 + 17 (5 + α) + 721 (5 + 2α) + 731 (5 + 3α)+ ∞, then the value of α is : (1) 6 (2) 6 7 (3) 1 (4) 1 7

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

Q8. Let →a = 2^i −^j + 3^k, b = 3^i −5^j + ^k and→cbe a vector such that →a×→c=→c× b and (→a + →c) ⋅(→b + →c) = 168. Then the maximum value of |→c|2 is : (1) 462 (2) 77 (3) 154 (4) 308 π

Q8. Let the lines 3x −4y −α = 0, 8x −11y −33 = 0, and 2x −3y + λ = 0 be concurrent. If the image of the point (1, 2) in the line 2x −3y + λ = 0 is ( 5713 , −4013 ), then |αλ| is equal to (1) 84 (2) 113 (3) 91 (4) 101

Q8. Let f(x) = ∫x20 t2−8t+15et dt, respectively, are : (1) 2 and 3 (2) 2 and 2 (3) 3 and 2 (4) 1 and 3

Q8. Let L1 : x−12 = y−23 = z−34 and L2 : x−23 = y−44 = z−55 be two lines. Then which of the following points lies on the line of the shortest distance between L1 and L2 ? (1) ( 143 , −3, 223 ) (2) (−53 , −7, 1) (3) (2, 3, 13 ) (4) ( 83 , −1, 13 )