Practice Questions

Q76.A value of x satisfying the equation sin[cot−1(1 + x)] = cos[tan−1x], is: JEE Main 2017 (09 Apr Online) JEE Main Previous Year Paper (1) −12 (2) 0 (3) −1 (4) 21

Q77.The function f : N →I defined by f(x) = x −5[ x5 ] , where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is: (1) one-one but not onto (2) one-one and onto (3) neither one-one nor onto (4) onto but not one-one Q78. 4 tantan 5x4x π 5 ) , 0 < x < 2 π The value of k which the function f(x) = is continuous at x = 2 , is 2 π {( k + 5 , x = 2 (1) 2 5 (2) −25 (3) 17 (4) 3 20 5 , then λ + k is equal to

Q77.The function 𝑓 : 𝑅→-1 1 defined as 𝑓𝑥= 𝑥 is: 2, 2 1 + 𝑥2, (1) Invertible (2) Injective but not surjective (3) Surjective but not injective (4) Neither injective nor surjective

Q78.The value of tan−1[ √1+x2−√1+x2+ √1−x2√1−x2 ], (1) π 4 + 21 cos−1x2 (2) π4 −cos−1x2 (3) π 4 −12 cos−1x2 (4) π4 + cos−1x2

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.If 2x = y 15 + y−15 and (x2 −1) dx2d2y + λx dxdy + ky = 0 (1) 26 (2) −24 (3) −23 (4) −26

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

Q79.If for 𝑥∈0, 4, the derivative of tan-1⁡1 - 9𝑥3 is √𝑥 ⋅𝑔𝑥 , then 𝑔𝑥 equals: JEE Main 2017 (02 Apr) JEE Main Previous Year Paper 9 3𝑥√𝑥 (1) (2) 1 + 9𝑥3 1 - 9𝑥3 3𝑥 3 (3) (4) 1 - 9𝑥3 1 + 9𝑥3

Q80.Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is: (1) 12 . 5 (2) 10 (3) 25 (4) 30

Q80.If y = [x + √x2 −1] [x −√x2 −1] (1) 224 y2 (2) 125 y (3) 225 y (4) 225 y2

Q80.The function f defined by f(x) = x3 −3x2 + 5x + 7 is: (1) Decreasing in R (2) Increasing in R (3) Increasing in (0, ∞) and decreasing in (−∞, 0) (4) Decreasing in (0, ∞) and increasing in (−∞, 0)

Q81.The normal to the curve 𝑦𝑥- 2 𝑥- 3 = 𝑥+ 6 at the point where the curve intersects the 𝑦-axis passes through the point: (1) -1 - 1 (2) 1 1 2, 2 2, 2 (3) 1 - 1 (4) 1 1 2, 3 2, 3

Q81.If f( 3x−43x+4 ) = x + 2, x ≠−43 , and ∫f(x)dx = A log|1 −x| + Bx + C , then the ordered pair (A, B) is equal to (1) (−83 , −23 ) (2) (−83 , 32 ) (3) ( 83 , 32 ) (4) ( 38 , −23 ) 2 dx k , then k is equal to

Q81.The tangent at the point (2, −2) to the curve, x2y2 −2x = 4(1 −y) does not pass through the point: (1) (−2, −7) (2) (8, 5) (3) (−4, −9) (4) (4, 13 )

Q82.Let, 𝐼𝑛= ∫tan𝑛𝑥𝑑𝑥𝑛> 1 . If 𝐼4 + 𝐼6 = 𝑎tan5𝑥+ 𝑏𝑥5 + 𝑐, then the ordered pair 𝑎, 𝑏, is equal to 1 1 (1) - 5, 1 (2) 5, 0 (3) 1 - 1 (4) -1 0 5, 5, Q83. 3𝜋4 The integral ∫ 𝑑𝑥 is equal to 𝜋 1 + cos𝑥 4 (1) -2 (2) 2 (3) 4 (4) -1

Q82.The integral ∫√1 + 2 cot x(cosec x + cot x)dx, (0 < x < π2 ) is equal to (1) 2 log sin x2 + c (2) 4 log sin x2 + c (3) 4 log cos x2 + c (4) 2 log cos x2 + c Q83. π4 The integral ∫ 8 cos 2x dx equals π (tan x+cot x)3 12 (1) 13 (2) 15 256 64 (3) 13 (4) 15 32 128

Q82.If ∫ 3 = k+5 1 (x2−2x+4) 2 (1) 4 (2) 2 (3) 3 (4) 1 lim = 601 for some positive real number a, then a is equal to 1a+2a+…+na ) (n+1)a−1[(na+1)+(na+2)+…+(na+n)]

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

Q84.Let f be a polynomial function such that f(3x) = f ′(x). f ′′(x), for all x ∈R. Then : (1) f(2) + f ′(2) = 28 (2) f ′′(2) −f ′(2) = 0 (3) f(2) −f ′(2) + f ′′(2) = 10 (4) f ′′(2) −f(2) = 4

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 ) → → →

Q85.If 2 + sin𝑥 𝑑𝑦 𝑦+ 1cos𝑥= 0 and 𝑦0 = 1, then 𝑦 𝜋 is equal to 𝑑𝑥+ 2 (1) 1 (2) -2 3 3 1 4 (3) - (4) 3 3 → →

Q85.The curve satisfying the differential equation, ydx −(x + 3y2)dy = 0 and passing through the point (1, 1) also passes through the point (1) ( 41 , −12 ) (2) (−13 , 13 ) (3) ( 41 , 12 ) (4) ( 13 , −13 )

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