Continuity & Differentiability PYQs Set 2 (PDF)

Set 2 of 13 · Questions 31-60 of 371

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 2 (Q31-60)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
31. $Rolle's theorem is not applicable in which one of the following cases?$ (1) $f (x) = x^{2} - 4 x + 5 in [1, 3]$ (2) $f (x) = \| x \| in [- 2, 2]$ (3) $f (x) = x^{2} - x in [0, 1]$ (4) $f (x) = [x] in [2.5, 2.7]$ 32. $If y = 5 x^{2} + 6 x + 6, x = 2, Δ x = 0.001, then value of dy is$ (1) $0.026$ (2) $0.0026$ (3) $0.062$ (4) $0.0062$ 33. $For m > 1, n > 1, the value of c for which the Rolle's theorem is applicable for the function f (x) = x^{2 m - 1} (a - x)^{2 n} in (0, a) is$ (1) $\frac{2 a ( m - 1 )}{m + 2 n - 1}$ (2) $\frac{a ( m - n + 1 )}{2 m + 2 n}$ (3) $\frac{a ( 2 m - 1 )}{2 m + 2 n - 1}$ (4) $\frac{a ( 2 m + 1 )}{m + n - 1}$ 34. $The curve 3 y^{2} = 2 a x^{2} + 6 b passes through the point P (3, - 1) and the gradient of the curve at P is " -1 ", then the values of a and b are$ (1) $a = \frac{1}{2}, b = - 1$ (2) $a = \frac{- 1}{2}, b = 1$ (3) $a = \frac{1}{2}, b = 1$ (4) $a = \frac{- 1}{2}, b = - 1$ 35. $α is the maximum value of 1 - 2 x - 5 x^{2} and β is the minimum value of x^{2} - 2 x + r . If 5 α x^{2} + βx + 6 > 0 for all real values x, then the interval in which r lies is$ (1) $(0, 5)$ (2) $(- 5, \infty)$ (3) $(- \infty, 7)$ (4) $(- 11, 13)$ 36. $Determine the value of k for which the function f (x) is continuous at x = 4 . f (x) = (\frac{x ^{2} - 16}{x - 4}, x \neq = 4 k, x = 4)$ (1) $2$ (2) $4$ (3) $6$ (4) $8$ 37. $Let f (x) = ⎩ ⎨ ⎧ 3 - x 6 3 + x if x < - 3 if - 3 \leq x \leq 3 if x > 3 . Let α be the number of points of discontinuity of f and β be the number of points where f is not differentiable. Then α + β =$ (1) $6$ (2) $3$ (3) $2$ (4) $0$ 38. $The function f (x) = [x] \cdot cos (\frac{2 x - 1}{2}) π, where 1.] denotes the greatest integer function, is discontinuous at$ (1) $all irrational numbers x .$ (2) $no x .$ (3) $all integer points.$ (4) $every rational x which is not an integer.$ 39. $If log_{e} y = 3 sin^{- 1} x, then (1 - x^{2}) y^{’ ’} - x y^{’} at x = \frac{1}{2} is equal to$ (1) $3 e^{π /6}$ (2) $9 e^{π /2}$ (3) $3 e^{π /2}$ (4) $9 e^{π /6}$ 40. $Let f (x) = x^{5} + 2 e^{x /4} for all x \in R . Consider a function g (x) such that (g \circ f) (x) = x for all x \in R . Then the value of 8 g^{’} (2) is :$ (1) $2$ (2) $8$ (3) $4$ (4) $16$ 41. $Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function f(x) = x cos(π(x + [x])) is discontinuous ?$ (1) $x = - 1$ (2) $x = 1$ (3) $x = 0$ (4) $x = 2$ 42. $Suppose f (x) = \frac{( 2 ^{x} + 2 ^{- x} ) tan x tan ^{- 1} ( x ^{2} - x + 1 )}{( 7 x ^{2} + 3 x + 1 ) ^{3}} Then the value of f^{’} (0) is equal to$ (1) $π$ (2) $0$ (3) $π$ (4) $\frac{π}{2}$ 43. $The derivative of f (tan x) w.r.t. g (sec x) at x = \frac{π}{4}, where f^{’} (1) = 2 and g^{’} (2) = 4, is$ (1) $\frac{1}{2}$ (2) $2$ (3) $1$ (4) $0$ 44. $If 2 y = (cot^{- 1} (\frac{3 cos x + sin x}{cos x - 3 sin x}))^{2}, x \in (0, \frac{π}{2}) then \frac{dy}{dx} is equal to :$ (1) $\frac{dy}{dx} = (x - \frac{7 π}{6} if x \in (\frac{π}{6}, \frac{π}{2}) x - \frac{π}{6} if x \in (0, \frac{π}{6}))$ (2) $x - \frac{π}{6}$ (3) $\frac{π}{3} - x$ (4) $2 x - \frac{π}{3}$ 45. $Let f (x) = (x^{2} sin (\frac{1}{x}), x \neq = 00, x = 0), Then at x = 0$ (1) $f is continuous but not differentiable$ (2) $f is continuous but f ’ is not continuous$ (3) $f and f ’ both are continuous$ (4) $f ’ is continuous but not differentiable$ 46. $Let f : R \to R be a function given by f (x) = ⎩ ⎨ ⎧ \frac{1 - c o s 2 x}{x ^{2}}, α, \frac{β 1 - c o s x}{x}, x < 0 x = 0 x > 0 where α, β \in R . If f is continuous at x = 0, then α^{2} + β^{2} is equal to :$ (1) $3$ (2) $12$ (3) $48$ (4) $6$ 47. $Let f : R \to R be a function such that f (x) = x^{3} + x^{2} f^{’} (1) + x f^{’ ’} (2) + f^{’ ’ ’} (3), x \in R . Then f (2) equals:$ (1) $- 4$ (2) $30$ (3) $- 2$ (4) $8$ 48. $If P = (x 11 x) and Q = (x 111 x 111 x), then \frac{dQ}{dx}$ (1) $1 - 3 p$ (2) $3 P$ (3) $3 p + 1$ (4) $- 3 p$ 49. $A particle starts from rest and its angular displacement (in radians) is given by θ = \frac{t ^{2}}{20} + \frac{t}{5} . If the angular velocity at the end of t = 4 is k, then the value of 5 k is$ (1) $0.6$ (2) $5$ (3) $50$ (4) $3$ 50. $If the curves 2 x = y^{2} and 2 xy = K intersect perpendicularly, then the value of K^{2} is$ (1) $22$ (2) $2$ (3) $8$ (4) $4$ 51. $If the extreme value of 3 x - 2 x^{2} + 1 is k then the set of all real values of x for which k x^{2} + 2 x + 1 > 0 is$ (1) $(\frac{1}{2}, 1)$ (2) $(- \infty, \frac{1}{2}) \cup (1, \infty)$ (3) $(- \infty, \infty)$ (4) $(- \infty, \frac{17}{8})$ 52. $If f (n) = n! (31 - n)!, where n \in 0, 1, 2, \dots, 31, then the minimum value of f (n) is$ (1) $(15!) (15!)$ (2) $(15!) (14!)$ (3) $(14!) (16!)$ (4) $(15!) (16!)$ 53. $The minimum value of (1 + \frac{1}{sin ^{n} α}) (1 + \frac{1}{cos ^{n} α}) is$ (1) $1$ (2) $2$ (3) $(1 + 2^{n})^{2}$ (4) $(1 + 2^{n /2})^{2}$ 54. $Let f (x) = (1 + 6 x - 3 x^{2}, x \leq 1 x + log_{2} (b^{2} + 7), x > 1) . Then the set of all possible values of b such that f (1) is the maximum value of f (x) is$ (1) $[- 1, 1]$ (2) $[0, 1]$ (3) $[0, 2]$ (4) $[- 1, 0]$ 55. $For real value of x, the range of \frac{x ^{2} + 2 x + 1}{x ^{2} + 2 x - 1} is$ (1) $(- \infty, 0] \cup (1, \infty)$ (2) $(\frac{1}{2}, 2)$ (3) $(- \infty, \frac{- 2}{9}) \cup (1, \infty)$ (4) $(- \infty, - 6] \cup (- 2, \infty)$ 56. $Let α, β \in R be such that the function f (x) = (2 α (x^{2} - 2) + 2 βx, x < 1 (α + 3) x + (α - β), x \geq 1) be differentiable at all x \in R . Then 34 (α + β) is equal to$ (1) $36$ (2) $24$ (3) $84$ (4) $48$ 57. $The local minimum value of the function f^{’} given by f (x) = 3 + \| x \|, x \in R is$ (1) $-1$ (2) $3$ (3) $none of the above$ (4) $0$ 58. $If the function f (x) = ⎩ ⎨ ⎧ \frac{2}{x} (sin (k_{1} + 1) x + sin (k_{2} - 1) x), 4, \frac{2}{x} lo g_{e} (\frac{2 + k _{1} x}{2 + k _{2} x}), x < 0 x = 0 x > 0 is continuous at x = 0, then k_{1}^{2} + k_{2}^{2} is equal to$ (1) $20$ (2) $5$ (3) $8$ (4) $10$ 59. $Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x = 3. If p(1) = 6 p(3) = 2 , then p'(0) is$ 60. $If the curves x = y^{4} and xy = k cut at right angles, then (4 k)^{6} is equal to$