Practice Questions

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. 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

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 →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. 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 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. 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 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. 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. 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

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 )

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. 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. 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. 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 f be a real valued continuous function defined on the positive real axis such that g(x) = ∫x0 tf(t)dt. If g (x3) = x6 + x7 , then value of ∑15r=1 f (r3) is : (1) 270 (2) 340 (3) 320 (4) 310

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. Two number k1 and k2 are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1 + ik2, (i = √−1) is non-zero, equals ⎪ ⎪ 2025 (28 Jan Shift 1) JEE Main Previous Year Paper (1) 1 (2) 3 2 4 (3) 1 (4) 2 4 3

Q8. If the line 3x −2y + 12 = 0 intersects the parabola 4y = 3x2 at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to ⎪ ⎪ 2025 (23 Jan Shift 1) JEE Main Previous Year Paper (1) tan−1 ( 45 ) (2) tan−1 ( 97 ) (3) tan−1 ( 119 ) (4) π2 −tan−1 ( 32 )

Q9. Let f(x) be a real differentiable function such that f(0) = 1 and f(x + y) = f(x)f ′(y) + f ′(x)f(y) for all x, y ∈R. Then ∑100n=1 loge f(n) is equal to : (1) 2525 (2) 5220 (3) 2384 (4) 2406

Q9. The length of the chord of the ellipse x2 4 + 2 = 1, whose mid-point is (1, 12 ), is : (1) 5 3 √15 (2) 13 √15 (3) 2 3 √15 (4) √15

Q9. Let A = [aij] be a 2 × 2 matrix such that aij ∈{0, 1} for all i and j. Let the random variable X denote the possible values of the determinant of the matrix A . Then, the variance of X is : 2025 (29 Jan Shift 2) JEE Main Previous Year Paper (1) 3 (2) 5 4 8 (3) 3 (4) 1 8 4