Practice Questions

Q83.If ∫f (x)dx = ψ (x), then ∫x5f (x3)dx, is equal to (1) 1 3 x3ψ (x3) −∫x2ψ (x3)dx + c (2) 13 [x3ψ (x3) −∫x3ψ (x3)dx] + c (3) 3 1 [x3ψ (x3) −∫x2ψ (x3)dx] + c (4) 13 x3ψ (x3) −3 ∫x3ψ (x3)dx + c π/3 dx π

Q83.The maximum area of a right angled triangle with hypotenuse h is : (1) h2 (2) h2 2√2 2 (3) h2 (4) h2 √2 4 = A(x)ecot−1 x + C , then A(x) is equal to :

Q83.For 0 ≤x ≤π2 , the value of sin2 x cos2 x sin−1(√t)dt + cos−1(√t)dt equals : ∫ 0 ∫ 0 (1) π (2) 0 4 (3) 1 (4) −π4

Q83.The cost of running a bus from A to B is Rs. (av + b/v) where vkm/h is the average speed of the bus. When the bus travels at 30 km/h, the cost comes out to be Rs. 75 while at 40 km/h, it is Rs. 65. Then the most economical speed (in km/h) of the bus is : JEE Main 2013 (23 Apr Online) JEE Main Previous Year Paper (1) 45 (2) 50 (3) 60 (4) 40

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.

Q84.If a curve passes through the point (2, 72 ) and has slope (1 − x21 ) at any point (x, y) on it, then the ordinate of the point on the curve whose abscissa is −2 is : (1) −32 (2) 23 (3) 2 5 (4) −52

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

Q84.If x = ∫y0 √1+t2dt , then dx2d2y (1) y (2) √1 + y2 (3) x (4) y2 √1+y2

Q84.Let f : [−2, 3] →[0, ∞) be a continuous function such that f(1 −x) = f(x) for all x ∈[−2, 3]. If R1 is the numerical value of the area of the region bounded by y = f(x), x = −2, x = 3 and the axis of x and R2 = ∫3−2 xf(x)dx, then : (1) 3R1 = 2R2 (2) 2R1 = 3R2 (3) R1 = R2 (4) R1 = 2R2

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

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

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

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

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.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

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.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.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. →

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