Practice Questions

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 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. 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

Q7. If ∑13r=1 { sin( 4 +(r−1) 6 ) sin( π4 + rπ6 ) } (1) 10 (2) 4 (3) 2 (4) 8

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. (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. 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) → →

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. 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 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

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 )

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. 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 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 →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. 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. 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. 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 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 )

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. 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

Q9. Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f(x) = [x] + |x −2|, −2 < x < 3, is not continuous and not differentiable. Then m + n is equal to : (1) 6 (2) 8 (3) 9 (4) 7