Practice Questions

Q62.If the system of linear equations x + y + 3z = 0 x + 3y + k2z = 0 3x + y + 3z = 0 has a non-zero solution (x, y, z) for some k ∈R, then x + ( yz ) is equal to : (1) −3 (2) 9 (3) 3 (4) −9

Q62.Let S , be the set of all functions f : [0, 1] →R, which are continuous on [0, 1], and differentiable on (0, 1). Then for every f in S , there exists c ∈(0, 1), depending on f , such that. f '(c) (1) |f(c) −f(1)| < (1 −c) f '(c) (2) f(1)−f(c)1−c = (3) |f(c) + f(1)| < (1 + c) f '(c) (4) |f(c) −f(1)| < f '(c)

Q62.The domain of the function f(x) = sin−1( |x|+5x2+1 ) is (−∞, −a] ∪[a, ∞), then a is equal to (1) √17 (2) √17−1 2 2 (3) 1+√17 (4) √17 2 2 + 1 Q63. ⎧ aex + be−x, −1 ≤x < 1 If a function f(x) defined by f(x) = 1 ≤x ≤3 be continuous for some a, b, c ∈R and ⎨ cx2, ⎩ ax2 + 2cx, 3 < x ≤4 f ′(0) + f ′(2) = e, then the value of a is (1) 1 (2) e e2−3e+13 e2−3e−13 (3) e (4) e e2+3e+13 e2−3e+13

Q62.Let y = y(x) be a function of x satisfying y√1 −x2 = k −x√1 −y2 where k is a constant and y( 21 ) = −14 .Then dx dy at x = 12 , is equal to (1) −√54 (2) −√52 (3) 2 (4) √5 √5 2

Q63.Suppose f(x) is a polynomial of degree four having critical points at −1, 0, 1. If T = {x ∈R |f(x) = f(0)}, then the sum of squares of all the elements of T is : (1) 4 (2) 6 (3) 2 (4) 8

Q63.Let xk + yk = ak, (a, k > 0) and dx 1 dy + ( xy ) 3 = 0, then k is (1) 3 (2) 4 2 3 (3) 32 (4) 13

Q63.The set of all real values λ for which the function f(x) = (1 −cos2 x). (λ + sin x), xε (−π2 2 ), has exactly one maxima and exactly one minima, is : (1) (−12 , 12 ) −{0} (2) (−32 , 32 ) (3) (−12 , 12 ) (4) (−32 , 32 ) −{0}

Q63.The value of c, in the Lagrange’s mean value theorem for the function f(x) = x3 −4x2 + 8x + 11, when x ∈[0,1], is (1) 4−√5 (2) 4−√7 3 3 (3) 2 (4) √7−2 3 3

Q63.Let f(x) = (sin(tan−1 x) + sin(cot−1 x))2 −1 , |x| > 1. If dxdy = 12 dxd (sin−1(f(x))) and y(√3) y(−√3) is equal to: (1) 2π 3 (2) −π6 (3) 5π (4) π 6 3 [3, 4], where

Q63.Let f be any function continuous on [a, b] and twice differentiable on (a, b) . If all x ∈(a, b), f '(x) > 0 and f ''(x) < 0 , then for any c ∈(a, b), f(c)−f(a)f(b)−f(c) (1) b+a (2) 1 b−a (3) b−c (4) c−a c−a b−c

Q63.Let λ ∈R. The system of linear equations 2x1 −4x2 + λx3 = 1 x1 −6x2 + x3 = 2 λx1 −10x2 + 4x3 = 3 is inconsistent for : (1) exactly one positive value of λ (2) exactly one negative value of λ (3) every value of λ (4) exactly two values of λ

Q63.If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then x a + y x + a y b + y y + b is equal to : z c + y z + c (1) y(b – a) (2) y (a – b) (3) 0 (4) y(a – c) at x = 21 is :

Q63.The length of the perpendicular from the origin, on normal to the curve, x2 + 2xy −3y2 = 0, at the point (2, 2), is. (1) √2 (2) 4√2 (3) 2 (4) 2√2 ∫x0 tsin(10t)dt , is equal to

Q63.If y2 + loge(cos2 x) = y, x ∈(−π2 , π2 ) then : (1) y′′(0) = 0 (2) |y′(0)| + |y′′(0)| = 1 (3) |y′′(0)| = 2 (4) |y′(0)| + |y′′(0)| = 3

Q63.The minimum value of 2sin x + 2cos x is : −1+ (1) √2 1 (2) 2−1+√2 2 (3) 21−√2 (4) 2 1−1√2 4 + tan−1 x, |x| ≤1 is :

Q63.If f(x + y) = f(x) f(y) and x=1f(x) of f(4) is f(2) (1) 2 (2) 1 3 9 (3) 1 (4) 4 3 9

Q63.If A = [ cosisinθθ cosisinθθ ], true? (1) 0 ≤a2 + b2 ≤1 (2) a2 −d2 = 0 (3) a2 −c2 = 1 (4) a2 −b2 = 12 cos = a2 −b2 , where a > b > 0, then dxdy at ( π4 , π4 ) is:

Q63.If x = 2 sin θ −sin 2θ and y = 2 cos θ −cos 2θ , θ ∈[0, 2π], then d2y at θ = π is: dx2 (1) 4 3 (2) −38 (3) 2 3 (4) −34

Q64.If c is a point at which Rolle’s theorem holds for the function, f(x) = loge( x2+α7x ) in the interval α ∈R, then f ''(c) is equal to (1) −112 (2) 121 (3) −124 (4) √37

Q64. lim x x→0 (1) 0 (2) 101 (3) −15 (4) −110 2 dx

Q64.The derivative of tan−1( √1+x2−1x ) with respect to tan−1( 2x√1−x21−2x2 ) (1) 2√3 (2) √3 5 12 (3) 2√3 (4) √3 3 10

Q64.Let f : R →R be a function which satisfies f(x + y) = f(x) + f(y), ∀x, y ∈R . If f(1) = 2 and g(n) = ∑(n−1)k=1 f(k), n ∈N then the value of n, for which g(n) = 20, is (1) 5 (2) 20 (3) 4 (4) 9

Q64.Let f(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If x→0(2lim + f(x)x3 ) = 4, then which one of the following is not true? (1) f is an odd function (2) f(1) −4f(−1) = 4 . x = 1 is a point of maximum and x = −1 (3) x = 1is a point of local minimum and x = −1 is (4) x = 1 is a point of local maxima of f a point of local maximum JEE Main 2020 (07 Jan Shift 2) JEE Main Previous Year Paper

Q64.Let f and g be differentiable functions on R such that fog is the identity function. If for some a, b ∈R, g'(a) = 5 and g(a) = b, then f '(b) is equal to: (1) 1 (2) 1 5 (3) 5 (4) 52

Q64.If the surface area of a cube is increasing at a rate of 3. 6cm2/sec, retaining its shape; then the rate of change of its volume (in cm3/sec), when the length of a side of the cube is 10cm, is: (1) 20 (2) 10 (3) 18 (4) 9 = A(x) tan−1(√x) + B(x) + C , where C is a constant of integration, then the