Practice Questions

Q84.If ∫x2−x+1x2+1 ecot−1 xdx (1) −x (2) x (3) √1 −x (4) √1 + x xdx is equal to :

Q84.Statement - I : The value of the integral ∫ is equal to 6 . 1+√tan x π/6 b b Statement - II : ∫ f(x)dx = ∫ f(a + b −x)dx. a a (1) Statement - I is true; Statement - II is false. (2) Statement - I is false; Statement - II is true. (3) Statement - I true; Statement - II is true; (4) Statement - I is true; Statement - II is true; Statement - II is a correct explanation for Statement - II is not a correct explanation for Statement - I. Statement - I.

Q85.The integral ∫ xdx equals : 2−x2+√2−x2 (1) log 1 + √2 + x2 + c (2) −log 1 + √2 −x2 + c (3) −x log 1 −√2 −x2 + c (4) x log 1 −√2 + x2 + c dx is :

Q85.The area (in square units) bounded by the curves y=√x, 2y −x + 3 = 0 , X -axis and lying in the first quadrant is (1) 18 sq. units (2) 274 sq. units (3) 9 sq. units (4) 36 sq. units

Q85.The area bounded by the curve y = ln(x) and the lines y = 0, y = ln(3) and x = 0 is equal to: (1) 3 (2) 3 ln(3) −2 (3) 3 ln(3) + 2 (4) 2

Q85.The integral ∫7π/37π/4 √tan2 (1) log 2√2 (2) log 2 (3) 2 log 2 (4) log √2

Q85.The equation of the curve passing through the origin and satisfying the differential equation (1 + x2) dxdy + 2xy = 4x2 is (1) (1 + x2)y = x3 (2) 3 (1 + x2)y = 2x3 (3) (1 + x2)y = 3x3 (4) 3 (1 + x2)y = 4x3

Q86.The value of ∫π/2−π/2 sin21+2xx (1) π (2) π 2 (3) 4π (4) π4

Q86.The area of the region (in sq. units), in the first quadrant bounded by the parabola y = 9x2 and the lines x = 0, y = 1 and y = 4 , is : (1) 7/9 (2) 14/3 (3) 7/3 (4) 14/9

Q86.Let →a = 2^i −^j + ^k,→b = ^i + 2^j −^k and →c = ^i + ^j −2^k be three vectors. A vector of the type →b + λ→c for some scalar λ, whose projection on →a is of magnitude is : √23 (1) 2^i + ^j + 5^k (2) 2^i + 3^j −3^k (3) 2^i −^j + 5^k (4) 2^i + 3^j + 3^k

Q86.At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers x is given by dP dx = 100 −12√x. If the firm employs 25 more workers, then the new level of production of items is (1) 3500 (2) 4500 (3) 2500 (4) 3000 −−

Q86.Let →a = 2^i + ^j −2^k,→b = ^i + ^j. If →c is a vector such that →a ∙→c = |→c|, |→c −→a| = 2√2 and the angle between →a × →b and →c is 30∘ , then |(→a × →b) × →c| equals: (1) 1 (2) 3√3 2 2 (3) 3 (4) 23

Q87.Let A(−3, 2) and B(−2, 1) be the vertices of a triangle ABC. If the centroid of this triangle lies on the line 3x + 4y + 2 = 0 , then the vertex C lies on the line : JEE Main 2013 (25 Apr Online) JEE Main Previous Year Paper (1) 4x + 3y + 5 = 0 (2) 3x + 4y + 3 = 0 (3) 4x + 3y + 3 = 0 (4) 3x + 4y + 5 = 0

Q87.The area under the curve y = | cos x −sin x|, 0 ≤x ≤π2 , and above x-axis is : (1) 2√2 (2) 2√2 −2 (3) 2√2 + 2 (4) 0

Q87.If the vectors AB→ = 3ˆi + 4ˆk and AC→ = 5ˆi −2ˆj + 4ˆk are the sides of a triangle ABC, then the length of the median through A is: (1) √33 (2) √45 (3) √18 (4) √72

Q87.Consider the differential equation : dy y3 = dx 2 (xy2 −x2) JEE Main 2013 (22 Apr Online) JEE Main Previous Year Paper Statement-1: The substitution z = y2 transforms the above equation into a first order homogenous differential equation. Statement-2: The solution of this differential equation is y2e−y2/x = C . (1) Both statements are false. (2) Statement-1 is true and statement- 2 is false. (3) Statement-1 is false and statement-2 is true. (4) Both statements are true. →

Q87.The vector (^i × →a ⋅→b)^i + (^j × →a→b)^j + (^k × →a ⋅→b)^k is equal to: (1) →b × →a (2) →a (3) →a × →b (4) →b JEE Main 2013 (09 Apr Online) JEE Main Previous Year Paper

Q88.If ^a,^b and ^c are unit vectors satisfying ^a −√3^b + ^c = 0, then the angle between the vectors ^a and ^c is : (1) π (2) π 4 3 (3) π (4) π 6 2

Q88.A vector →n is inclined to x-axis at 45∘ , to y-axis at 60∘ and at an acute angle to z-axis. If →n is a normal to a plane passing through the point (√2, −1, 1) then the equation of the plane is : (1) 4√2x + 7y + z −2 (2) 2x + y + 2z = 2√2 + 1 (3) 3√2x −4y −3z = 7 (4) √2x −y −z = 2

Q88.If the lines x−2 1 = y−31 = z−4−k and x−1k = y−42 = z−51 are coplanar, then k can have JEE Main 2013 (07 Apr) JEE Main Previous Year Paper (1) exactly two values. (2) exactly three values. (3) any value. (4) exactly one value.

Q88.Let ABC be a triangle with vertices at points A (2, 3, 5), B (−1, 3, 2) and C(λ, 5, μ) in three dimensional space. If the median through A is equally inclined with the axes, then (λ, μ) is equal to: (1) (10, 7) (2) (7, 5) (3) (7, 10) (4) (5, 7)

Q88.If →a and →b are non-collinear vectors, then the value of α for which the vectors →u = (α −2)→a + →b and →v = (2 + 3α)→a −3→b are collinear is : (1) 3 (2) 2 2 3 (3) −32 (4) −23

Q89.Let Q be the foot of perpendicular from the origin to the plane 4x −3y + z + 13 = 0 and R be a point (−1, −6) on the plane. Then length QR is : (1) √14 (2) √192 (3) 3√72 (4) √23

Q89.Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is (1) 7 (2) 9 2 2 (3) 3 (4) 5 2 2

Q89.If the lines x+1 2 = y−11 = z+13 and x+22 = y−k3 = 4z are coplanar, then the value of k is : (1) 11 2 (2) −112 (3) 2 9 (4) −92