Practice Questions

Q79.Let A = {1, 2, 3, 4} and R : A →A be the relation defined by R = {(1, 1), (2, 3), (3, 4), (4, 2)} . The correct statement is : (1) R does not have an inverse. (2) R is not a one to one function. (3) R is an onto function. (4) R is not a function. x2−x

Q79.If the system of linear equations : x1 + 2x2 + 3x3 = 6 x1 + 3x2 + 5x3 = 9 2x1 + 5x2 + ax3 = b JEE Main 2013 (22 Apr Online) JEE Main Previous Year Paper is consistent and has infinite number of solutions, then : (1) a = 8, b can be any real number (2) b = 15, a can be any real number (3) a ∈R −{8} and b ∈R −{15} (4) a = 8, b = 15

Q80.The number of values of k, for which the system of equations : (k + 1)x + 8y = 4k JEE Main 2013 (07 Apr) JEE Main Previous Year Paper kx + (k + 3)y = 3k −1 has no solution, is : (1) 2 (2) 3 (3) Infinite (4) 1

Q80.Let f(x) = x ≠0, −2. Then dxd [f −1(x)] (wherever it is defined) is equal to: x2+2x (1) −1 (2) 3 (1−x)2 (1−x)2 (3) 1 (4) −3 (1−x)2 (1−x)2

Q80.Let f(1) = −2 and f ′(x) ≥4.2 for 1 ≤x ≤6 . The possible value of f(6) lies in the interval : (1) [15, 19) (2) (−∞, 12) (3) [12, 15) (4) [19, ∞)

Q80.Let f be a composite function of x defined by f(u) = 1 , u(x) = x−11 . Then the number of points x u2+u−2 where f is discontinuous is : (1) 4 (2) 3 (3) 2 (4) 1

Q80.Let f(x) = −1 + |x −2|, and g(x) = 1 −|x|; then the set of all points where fog is discontinuous is : (1) {0, 2} (2) {0, 1, 2} (3) {0} (4) an empty set π

Q81.Statement-1: The equation x log x = 2 −x is satisfied by at least one value of x lying between 1 and 2. Statement-2: The function f(x) = x log x is an increasing function in [1, 2] and g(x) = 2 −x is a decreasing function in [1, 2] and the graphs represented by these functions intersect at a point in [1, 2] (1) Statement-1 is true; Statement-2 is true; (2) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-2 is not correct explanation for Statement-1. Statement-1. (3) Statement-1 1 is false, Statement- 2 is true. (4) Statement-1 1 is true, Statement- 2 is false.

Q81.For a > 0, t ∈(0, 2 2 ), let x = √asin−1 t and y = √acos−1 t , Then, 1 + ( dxdy ) equals : (1) x2 (2) y2 y2 x2 (3) x2+y2 (4) x2+y2 y2 x2

Q81.If y = sec(tan−1 x), then dxdy at x = 1 is equal to (1) 1 (2) √2 (3) 1 (4) 1 √2 2 x

Q81.If f(x) = sin(sin x) and f ′′(x) + tan xf ′(x) + g(x) = 0, then g(x) is : (1) cos2 x cos(sin x) (2) sin2 x cos(cos x) (3) sin2 x sin(cos x) (4) cos2 x sin(sin x) y2

Q81.If an equation of a tangent to the curve, y −cos(x + f), −1 −1 ≤x ≤1 + π, is x + 2y = k then k is equal to : (1) 1 (2) 2 (3) π (4) π 4 2

Q82.The intercepts on the x-axis made by tangents to the curve, y = ∫ |t| dt, x ∈R, which are parallel to the line 0 y = 2x , are equal to (1) ±3 (2) ±4 (3) ±1 (4) ±2

Q82.Statement-1: The function x2 (ex + e−x) is increasing for all x > 0 . Statement-2: The functions x2ex and x2e−x are increasing for all x > 0 and the sum of two increasing functions in any interval (a, b) is an increasing function in (a, b). (1) Statement-1 is false; Statement-2 is true. (2) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. (3) Statement-1 is true; Statement-2 is false. (4) Statement-1is true; Statement-2 is true; Statement-2 is a correct explanation for statement-1.

Q82.If the integral cos 8x + 1 dx = A cos 8x + k ∫ cot 2x −tan 2x where k is an arbitrary constant, then A is equal to: (1) −116 (2) 161 (3) 8 1 (4) −18

Q82.If the surface area of a sphere of radius r is increasing uniformly at the rate 8 cm2/s, then the rate of change of its volume is: (1) constant (2) proportional to √r (3) proportional to r2 (4) proportional to r dx is equal to:

Q82.If the curves x2 α + 4 = 1 and y3 = 16x intersect at right angles, then a value of α is : (1) 2 (2) 4 3 (3) 1 (4) 3 2 4

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

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.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.If ∫ x+x7dx = p(x) then, ∫ x+x7x6 (1) ln |x| −p(x) + c (2) ln |x| + p(x) + c (3) x −p(x) + c (4) x + p(x) + c 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

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

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.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