Practice Questions

Q73.The proposition (~p) ∨(p ∧~q) is equivalent to (1) p →∼q (2) p∧∼q (3) q →p (4) none

Q74.Let a vertical tower 𝐴𝐵 have its end 𝐴 on the level ground. Let 𝐶 be the mid-point of 𝐴𝐵 and 𝑃 be a point on the ground such that 𝐴𝑃= 2𝐴𝐵. If ∠𝐵𝑃𝐶= 𝛽, then tan𝛽 is equal to: (1) 6 (2) 1 7 4 2 4 (3) (4) 9 9

Q74.The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is (1) 35 (2) 40 (3) 25 (4) 30

Q74.For two 3 × 3 matrices A and B , let A + B = 2B′ and 3A + 2B = I3, where B′ is the transpose of B and I3 is 3 × 3 identity matrix. Then : (1) 10A + 5B = 3I3 (2) 3A + 6B = 2I3 (3) 5A + 10B=2I3 (4) B + 2A = I3

Q75.If x = a, y = b, z = c is a solution of the system of linear equations x + 8y + 7z = 0 9x + 2y + 3z = 0 x + y + z = 0 Such that the point (a, b, c) lies on the plane x + 2y + z = 6 , then 2a + b + c equals: (1) 2 (2) −1 (3) 1 (4) 0

Q75.If 𝐴= 2 -3 , then Adj3𝐴2 + 12𝐴 is equal to: -4 1 (1) 72 -84 (2) 51 63 -63 51 84 72 (3) 51 84 (4) 72 -63 63 72 -84 51

Q75.Let A be any 3 × 3 invertible matrix. Then which one of the following is not always true? (1) adj (adj (A)) = |A|2. (adj (A))−1 (2) adj (adj (A)) = |A|. (adj (A))−1 (3) adj (adj (A)) = |A| . A (4) adj (A) = |A|. A−1

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

Q76.The number of real values of λ for which the system of linear equations, 2x + 4y −λz = 0 , 4x + λy + 2z = 0 and λx + 2y + 2z = 0 , has infinitely many solutions, is: (1) 3 (2) 1 (3) 2 (4) 0 Q77. ⎧ 0 cos x −sin x ⎫ π If S = x ∈[0, 2π] : sin x 0 cos x = 0 , then ∑x ∈S tan( 3 + x) is equal to: ⎨ ⎬ ⎩ cos x sin x 0 ⎭ (1) 4 + 2√3 (2) −4 -2 √3 (3) −2 + √3 (4) -2 −√3 |x| < 12 , x ≠0, is equal to:

Q76.If 𝑆 is the set of distinct values of 𝑏 for which the following system of linear equations 𝑥+ 𝑦+ 𝑧= 1 𝑥+ 𝑎𝑦+ 𝑧= 1 𝑎𝑥+ 𝑏𝑦+ 𝑧= 0 has no solution, then 𝑆 is: (1) An empty set (2) An infinite set (3) A finite set containing two or more elements (4) A singleton

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.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

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

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.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 )

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

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.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