Practice Questions

Q83.The integral ∫(1 + x −1x )ex+ 1 JEE Main 2014 (06 Apr) JEE Main Previous Year Paper (1) (x + 1)ex+ x1 + c (2) −xex+ x1 + c (3) (x −1)ex+ x1 + c (4) xex+ x1 + c π x 2 −4 sin x2 dx equals

Q83.If m is a non-zero number and ∫x5m−1+2x4m−1 dx = f(x) + c, then f(x) is equal to (x2m+xm+1)3 (1) (x5m−x4m) (2) 1 x4m 2m(x2m+xm+1)2 2m (x2m+xm+1)2 (3) x5m (4) 2m(x5m+x4m) 2m(x2m+xm+1)2 (x2m+xm+1)2

Q83.The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius = √3 is: (1) 3 4 √3π (2) 83 √3π (3) 4π (4) 2π > 0) is equal to:

Q83. sin2 x cos2 x The integral dx is equal to: ∫ 2 (sin3 x + cos3 x) 1 1 (1) + c (2) − + x (1+cot3 x 3(1+tan3 c) (3) sin3 x + c (4) − cos3 x + c (1+cos3 x 3(1+sin3 x

Q84.The integral ∫x cos−1 ( 1+x21−x2 )dx(x (1) −x + (1 + x2) tan−1 x + c (2) x −(1 + x2) cot−1 x + c (3) −x + (1 + x2) cot−1 x + c (4) x −(1 + x2) tan−1 x + c

Q84.Let, the function F be defined as F(x) = ∫x1 ett dt, x > 0, then the value of the integral ∫x1 t+aet dt, where a > 0, is (1) ea[F(x) −F(1 + a)] (2) e−a[F(x + a) −F(a)] (3) ea[F(x + a) −F(1 + a)] (4) e−a[F(x + a) −F(1 + a)]

Q84.If [ ] denotes the greatest integer function, then the integral ∫π0 [cos xdx is equal to: (1) π (2) 0 2 (3) −1 (4) −π2

Q84.The integral ∫ √1 + 4 sin2 0 (1) 4√3 −4 (2) 4√3 −4 −π3 (3) π −4 (4) 2π3 −4 −4√3

Q85.Let A = {(x, y) : y2 ≤4x, y −2x ≥−4}. The area of the region A in square units is (1) 10 (2) 8 (3) 9 (4) 11

Q85.The area (in sq. unit) of the region described by A = {(x, y) : x2 + y2 ≤1 and y2 ≤1 −x} is (1) π 2 −23 (2) π2 + 32 (3) π 2 + 34 (4) π2 −43

Q85.The area of the region (in square units ) above the x-axis bounded by the curve y = tan x, 0 ≤x ≤π2 and the tangent to the curve at x = π4 is (1) 2 1 (log 2 −12 ) (2) 12 (1 + log 2) (3) 1 2 (1 −log 2) (4) 12 (log 2 + 12 )

Q85.If for n ≥1, Pn = ∫e1 (log xn)dx, then P10 −90P8 is equal to: (1) −9 (2) 10e (3) −9e (4) 10 Φ, is given by

Q85.If for a continuous function f(x), ∫t−π(f(x) + xdx) = π2 −t2 , for all t ≥−π, then f (−π3 ) is equal to: (1) π (2) π2 (3) π (4) π 3 6

Q86.If the general solution of the differential equation y′ = xy + Φ ( xy ), for some function y ln |cx| = x, where c is an arbitrary constant, then Φ(2) is equal to: (1) 4 (2) 1 4 (3) −4 (4) −14 →

Q86.If the differential equation representing the family of all circles touching x-axis at the origin is (x2 −y2) dxdy = g(x)y, then g(x) equals (1) 1 x2 (2) 2x 2 (3) 1 x (4) 2x2 2 → → →

Q86.Let the population of rabbits surviving at a time t be governed by the differential equation dp(t) . If p(0) = 100, then p(t) equals dt = 12 {p(t) −400} (1) 600 − 500 e 2t (2) 400 −300 e −t2 (3) 400 − 300 et/2 (4) 300 −200 e −t2 2 → → → → a b then λ is equal to

Q86.If dxdy + ytan x=sin 2x and y(0) = 1, then y(π) is equal to (1) −1 (2) 5 (3) 1 (4) −5 → → →

Q86.The general solution of the differential equation, sin 2x −y = 0, is : dx ( dy −√tan x) (1) y√tan x = x + c (2) y√cot x = tan x + c (3) y√tan x = cot x + c (4) y√cot x = x + c

Q87.If ^x, ^y and ^z are three unit vectors in threedimensional space, then the minimum value of |^x + ^y|2 + |^y + ^z|2 + |^z + ^x|2 (1) 3 (2) 3 2 (3) 3√3 (4) 6

Q87.If |→c|2 = 60 and →c × (^i + 2^j + 5^k) = 0, then a value of →c ⋅(−7^i + 2^j + 3^k) is: (1) 4√2 (2) 12 (3) 24 (4) 12√2 y−2

Q87.If ×→b →b ×→c c = λ [→a ×→a] [ c] (1) 0 (2) 1 (3) 2 (4) 3 y−3

Q87.If →a = 2, b = 3 and 2→a− b = 5, then 2→a+ b equals : (1) 5 (2) 7 (3) 17 (4) 1 y−2

Q87.If x = 3ˆi −6ˆj −ˆk , y = ˆi + 4ˆj −3ˆk and →z= 3ˆi −4ˆj −12ˆk, then the magnitude of the projection of x ×→y on →zis (1) 14 (2) 12 (3) 15 (4) 10

Q88.If the angle between the line 2(x + 1) = y = z + 4 and the plane 2x −y + √λz + 4 = 0 is π6 , then the value of λ is (1) 45 (2) 135 7 11 (3) 135 (4) 45 7 11 y

Q88.The image of the line x−1 3 = 1 = z−4−5 in the plane 2x −y + z +3=0 is the line (1) x−3 3 = y+51 = z−2−5 (2) x−3−3 = y+5−1 = z−25 (3) x+3 3 = y−51 = z−2−5 (4) x+3−3 = y−5−1 = z+25