Practice Questions

Q85.The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, center at the origin and passing through the point (0, 3) is (1) xyy′ −y2 + 9 = 0 (2) xyy′′ + x(y′)2 −yy′ = 0 (3) xyy′ + y2 −9 = 0 (4) x + yy′′ = 0 → → → →

Q88.If L1 is the line of intersection of the planes 2x −2y + 3z −2 = 0, x −y + z + 1 = 0 and L2 is the line of intersection of the planes x + 2y −z −3 = 0, 3x −y + 2z −1 = 0, then the distance of the origin from the plane, containing the lines L1 and L2 is (1) 1 (2) 1 √2 4√2 (3) 1 (4) 1 3√2 2√2

Q88.An angle between the lines whose direction cosines are given by the equations, l + 3m + 5n = 0 and 5lm −2mn + 6nl = 0, is (1) cos−1 ( 81 ) (2) cos−1 ( 61 ) (3) cos−1 ( 31 ) (4) cos−1 ( 41 )

Q89.An angle between the plane, x + y + z = 5 and the line of intersection of the planes, 3x + 4y + z −1 = 0 and 5x + 8y + 2z + 14 = 0 , is (1) cos−1 3 (2) cos−1 17 ( √17 ) (√3 ) 3 (4) (3) sin−1 sin−1 17 ( √17 ) (√3 )

Q90.A player X has a biased coin whose probability of showing heads is p and a player Y has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If X starts the game, and the probability of winning the game by both the players is equal, then the value of ' p ' is (1) 1 (2) 1 3 5 (3) 1 (4) 2 4 5 JEE Main 2018 (15 Apr Shift 2 Online) JEE Main Previous Year Paper

Q61.If, for a positive integer 𝑛, the quadratic equation, 𝑥𝑥+ 1 + 𝑥+ 1𝑥+ 2 + . .. + 𝑥+ 𝑛-¯ 1𝑥+ 𝑛= 10𝑛 has two consecutive integral solutions, then 𝑛 is equal to: (1) 12 (2) 9 (3) 10 (4) 11 JEE Main 2017 (02 Apr) JEE Main Previous Year Paper

Q64.For any three positive real numbers 𝑎, 𝑏 and 𝑐. If 925𝑎2 + 𝑏2 + 25𝑐2 - 3𝑎𝑐= 15𝑏3𝑎+ 𝑐. Then (1) 𝑏, 𝑐 and 𝑎 are in G.P. (2) 𝑏, 𝑐 and 𝑎 are in A.P. (3) 𝑎, 𝑏 and 𝑐 are in A.P. (4) 𝑎, 𝑏 and 𝑐 are in G.P.

Q66.The coefficient of x−5 in the binomial expansion of ( x 32 −x 31 +1 − x−x 21 ) where x ≠0,1 is (1) −1 (2) 4 (3) 1 (4) −4

Q66.If 5tan2⁡𝑥- cos2⁡𝑥= 2cos⁡ 2𝑥+ 9, then the value of cos⁡4𝑥 is 3 1 (1) - (2) 5 3 2 7 (3) (4) - 9 9

Q67.Let 𝑘 be an integer such that the triangle with vertices 𝑘, - 3𝑘, 5, 𝑘 and -𝑘, 2 has area 28 sq. units. Then the orthocenter of this triangle is at the point: (1) 2, - 1 (2) 1, 3 2 4 3 1 (3) 1, - (4) 2, 4 2

Q68.The radius of a circle, having minimum area, which touches the curve 𝑦= 4 - 𝑥2 and the lines, 𝑦= 𝑥 is: (1) 2√2 + 1 (2) 2√2 - 1 (3) 4√2 - 1 (4) 4√2 + 1 1

Q69.If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P , then the distance of P from the origin (units), is: + (1) 2(3 2√2) (2) 3 + 2√2 + (3) √2 + 1 (4) 2(√2 1) JEE Main 2017 (08 Apr Online) JEE Main Previous Year Paper

Q78.Let 𝑎, 𝑏, 𝑐∈𝑅 . If 𝑓𝑥= 𝑎𝑥2 + 𝑏𝑥+ 𝑐 is such that 𝑎+ 𝑏+ 𝑐= 3 and 𝑓𝑥+ 𝑦= 𝑓𝑥+ 𝑓𝑦+ 𝑥𝑦, ∀ 𝑥, 𝑦∈𝑅 , 10 then ∑ 𝑓(𝑛) is equal to: 𝑛= 1 (1) 330 (2) 165 (3) 190 (4) 255 1 6𝑥√𝑥

Q79.Let f(x) = 210x + 1 and g(x) = 310x −1. If (fog)(x) = x, then x is equal to: JEE Main 2017 (08 Apr Online) JEE Main Previous Year Paper (1) 210−1 (2) 1−2−10 210−3−10 310−2−10 (3) 310−1 (4) 1−3−10 310−2−10 210−3−10 15 15 dy is equal to + + x dx , then (x2 −1) dx2d2y

Q83.If n→∞( (1) 17 (2) 15 2 2 (3) 7 (4) 8

Q84.The area (in sq. units) of the smaller portion enclosed between the curves, x2 + y2 = 4 and y2 = 3x, is: (1) √3 1 + 4π3 (2) √31 + 2π3 (3) 2√3 1 + π3 (4) 2√31 + 2π3

Q84.The area (in sq. units) of the region 𝑥, 𝑦: 𝑥≥0, 𝑥+ 𝑦≤3, 𝑥2 ≤4𝑦 and 𝑦≤1 + √𝑥 is 59 3 (1) sq . units (2) sq . units 12 2 (3) 7 sq . units (4) 5 sq . units 3 2

Q85.A tangent to the curve, y = f(x) at P(x, y) meets x -axis at A and y -axis at B . If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the point (1) ( 13 , 24) (2) ( 21 , 4) (3) (2, 18 ) (4) (3, 281 ) → → →

Q86.Given, →𝑎= 2 ^𝑖+ ^𝑗- 2 ^𝑘 and 𝑏= ^𝑖+ ^𝑗. Let →𝑐 be a vector such that →𝑐- →𝑎= 3, →𝑎× 𝑏× →𝑐= 3 and the angle between →𝑐 and →𝑎× →𝑏 be 30° . Then →𝑎⋅ →𝑐 is equal to: 25 (1) (2) 2 8 (3) 5 (4) 1 8

Q87.If the image of the point 𝑃1, - 2, 3 in the plane, 2𝑥+ 3𝑦- 4𝑧+ 22 = 0 measured parallel to the line, 𝑥 𝑦 𝑧 = = is 𝑄, then 𝑃𝑄 is equal to: 1 4 5 (1) 3√5 (2) 2√42 (3) √42 (4) 6√5

Q87.The coordinates of the foot of the perpendicular from the point (1, −2, 1) on the plane containing the lines x+1 6 = y−17 = z−38 and x−13 = y−25 = z−37 , is: (1) (2, −4, 2) (2) (1, 1, 1) (3) (0, 0, 0) (4) (−1, 2, −1) = 2, is,

Q87.If the line, x−3 1 = y+2−1 = z+λ−2 lies in the plane, 2x −4y + 3z = 2 , then the shortest distance between this line and the line, x−1 12 = 9y = 4z is (1) 1 (2) 2 (3) 3 (4) 0

Q89.For three events, 𝐴, 𝐵 and 𝐶, 𝑃(Exactly one of 𝐴 or 𝐵 occurs) = 𝑃(Exactly one of 𝐵 or 𝐶 occurs) 1 1 = 𝑃(Exactly one of 𝐶 or 𝐴 occurs) = and 𝑃(All the three events occur simultaneously) = . 4 16 Then the probability that at least one of the events occurs, is: (1) 7 (2) 7 32 16 7 3 (3) (4) 64 16

Q89. From a group of 10 men and 5 women, four member committees are to be formed each of which must contain at least one women. Then the probability for these committees to have more women than men, is : (1) 3 (2) 2 11 23 (3) 1 (4) 21 11 220

Q67.If A > 0, B > 0 and A + B = π6 , then the minimum positive value of (tan A + tan B) is : (1) √3 −√2 (2) 4 −2√3 (3) 2 (4) 2 −√3 √3 be two sets. Then and Q = : sin θ −cos θ = √2 cos θ} {θ : sin θ + cos θ = √2 sin θ},