Continuity & Differentiability PYQs Set 11 (PDF)

Set 11 of 13 · Questions 301-330 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 11 (Q301-330)
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.
301. $If the function g(x) = {k x + 1, m x + 2, 0 \leq x \leq 3 3 < x \leq 5 is differentiable, then the value of k+ m is;$ (1) $2$ (2) $\frac{16}{5}$ (3) $\frac{10}{3}$ (4) $4$ 302. $Let f : R \to R be defined as f (x) = ⎩ ⎨ ⎧ [e^{x}], a e^{x} + [x - 1], b + [sin (π x)], [e^{- x}] - c, x < 0 0 \leq x < 1 1 \leq x < 2 x \geq 2 where a,b,c \in M and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true ?$ (1) $There exists a,b,c \in M such that f is continuous of R.$ (2) $If f is discontinuous at exactly one point, then a + b + c= 1.$ (3) $If f is discontinuous at exactly one point, then a+b+c \neq = 1.$ (4) $f is discontinuous at atleast two points, for any values of a, b and c.$ 303. $The domain of the real valued function f (x) = \frac{2 x ^{2} - 7 x + 5}{3 x ^{2} - 5 x - 2} is$ (1) $(- \infty, - \frac{1}{3}) \cup [1, 2) \cup (\frac{5}{2}, \infty)$ (2) $(- \infty, 1) \cup (2, \infty)$ (3) $(- \frac{1}{3}, \frac{5}{2})$ (4) $(- \infty, \frac{- 1}{3}) \cup (\frac{5}{2}, \infty)$ 304. $If for p \neq = q \neq = 0, then function f (x) = \frac{7 p ( 729 + x ) - 3}{3 729 + q x - 9} is continuous at x = 0, then:$ (1) $7pq f(0) -1 = 0$ (2) $63q f(0) - p^{2} = 0$ (3) $21q f(0) - p^{2} = 0$ (4) $7pq f(0) -9 = 0$ 305. $If x = a^{sin^{- 1} t} and y = a^{cos^{- 1} t}, then \frac{dy}{dx} =$ (1) $- \frac{y}{x}$ (2) $\frac{y}{x}$ (3) $\frac{- x}{y}$ (4) $\frac{x}{y}$ 306. $If y = t a n^{- 1} (sec x^{3} - t an x^{3}). \frac{π}{2} < x^{3} < \frac{3 π}{2}, then$ (1) $xy" + 2y' = 0$ (2) $x^{2} y^{''} - 6 y + \frac{3 π}{2} = 0$ (3) $x^{2} y" - 6y + 3 π = 0$ (4) $xy" - 4y' = 0$ 307. $If f : R \to R is a twice differentiable function such that f ''(x) > 0 for all x \in R, and f (\frac{1}{2}) = \frac{1}{2}, f(1) = 1, then$ (1) $f '(1) \leq 0$ (2) $f '(1) > 1$ (3) $0 < f '(1) \leq \frac{1}{2}$ (4) $\frac{1}{2} < f '(1) \leq 1$ 308. $If f (x) = (x, 0 \leq x \leq 12 - x, 1 \leq x \leq 2), then Rolle's theorem is not applicable to f (x) because$ (1) $f (x) is not defined everywhere on [0, 2]$ (2) $f (x) is not continuous on [0, 2]$ (3) $f(x) is not differentiable on (1, 2)$ (4) $f(x) is not differentiable on (0, 2)$ 309. $A ladder of length 13 mts has one end, resting against a vertical wall and the other on the ground. If the lower end moves away from the wall at a speed of 2 mts / minute, then the speed (in mts / min) at which upper end falls when the bottom is 5 mts away from the wall is$ (1) $\frac{6}{5}$ (2) $\frac{12}{5}$ (3) $\frac{5}{6}$ (4) $\frac{5}{12}$ 310. $Let A(4, -4) and B(9, 6) be points on the parabola, y^{2} = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of ΔACB is maximum. Then, the area (in sq. units) of ΔACB, is :$ (1) $30 \frac{1}{2}$ (2) $31 \frac{3}{4}$ (3) $31 \frac{1}{4}$ (4) $32$ 311. $Let g (x) be a linear function and f (x) = {g (x), (\frac{1 + x}{2 + x})^{1/ x}, x \leq 0 x > 0, is continuous at x = 0 . If f^{'} (1) = f (- 1), then the value of g (3) is$ (1) $\frac{1}{3} log_{e} (\frac{4}{9 e ^{1/3}})$ (2) $\frac{1}{3} log_{e} (\frac{4}{9}) + 1$ (3) $lo g_{e} (\frac{4}{9}) - 1$ (4) $lo g_{e} (\frac{4}{9 e ^{1/3}})$ 312. $Let f : R \to R be a function defined as f (x) = ⎩ ⎨ ⎧ \frac{sin ( a + 1 ) x + sin 2 x}{2 x}, b, \frac{x + b x ^{3} - x}{b x ^{5/2}}, x < 0 x = 0 x > 0 If f is continuous at x = 0, then the value of a+b is equal to:$ (1) $- \frac{5}{2}$ (2) $- 2$ (3) $- 3$ (4) $- \frac{3}{2}$ 313. $The range of the real valued function f (x) = \frac{x ^{2} + 2 x + 8}{x ^{2} + 2 x + 4} is$ (1) $[\frac{7}{3}, \infty)$ (2) $(0, \infty)$ (3) $(1, \infty)$ (4) $(1, \frac{7}{3}]$ 314. $If [x] represents the greatest integer \leq x, then the range of the real valued function f (x) = \frac{1}{[ x ] ^{2} + [ x ] - 2} is$ (1) $(- \infty, 0] \cup (\frac{1}{2}, \infty)$ (2) $(0, \frac{1}{2}]$ (3) $(- \infty, 0) \cup [2, \infty)$ (4) $(0, 2]$ 315. $If f (x) = b e^{a x} + a e^{b x}, then f^{''} (0) =$ (1) $ab (a + b)$ (2) $ab$ (3) $0$ (4) $2 ab$ 316. $For what value of k, the function defined by f (x) = ⎩ ⎨ ⎧ \frac{lo g ( 1 + 2 x ) sin x ^{\circ}}{x ^{2}}, k, x \neq = 0 x = 0 is continuous at x = 0 ?$ (1) $2$ (2) $\frac{1}{2}$ (3) $\frac{π}{90}$ (4) $\frac{90}{π}$ 317. $If x = e^{s i n^{- 1} t} and y = e^{c o s^{- 1} t}, then \frac{d ^{2} y}{d x ^{2}} is$ (1) $\frac{- y}{x ^{2}}$ (2) $\frac{y ^{2}}{2 x ^{2}}$ (3) $\frac{2 y}{x ^{2}}$ (4) $\frac{- 2 y}{x ^{2}}$ 318. $If f (x) = {lo g (sec^{2} x)^{c o t^{2} x}, K + 1, x \neq = 0 x = 0 is continuous at x = 0, then value of K is$ (1) $0$ (2) $e$ (3) $e^{- 1}$ (4) $1$ 319. $Let f (x) be a real differentiable function such that f (0) = 1 and f (x + y) = f (x) f^{'} (y) + f^{'} (x) f (y) for all x, y \in R . Then \sum_{n = 1}^{100} lo g_{e} f (n) is equal to:$ (1) $2525$ (2) $5220$ (3) $2384$ (4) $2406$ 320. $Let f and g be twice differentiable functions on R such that f^{''} (x) = g^{''} (x) + 6 x, f^{'} (1) = 4 g^{'} (1) - 3 = 9, f (2) = 3 g (2) = 12 . Then which of the following is NOT true?$ (1) $g (- 2) - f (- 2) = 20$ (2) $If - 1 < x < 2, then ∣ f (x) - g (x) ∣ < 8$ (3) $∣ f^{'} (x) - g^{'} (x) ∣ < 6 \Rightarrow - 1 < x < 1$ (4) $There exists x_{0} ϵ (1, \frac{3}{2}) such that f (x_{0}) = g (x_{0})$ 321. $If f (x) = x^{3} - x^{2} f^{'} (1) + x f^{''} (2) - f^{'''} (3), x \in R, then$ (1) $3 f (1) + f (2) = f (3)$ (2) $f (3) - f (2) = f (1)$ (3) $2 f (0) - f (1) + f (3) = f (2)$ (4) $f (1) + f (2) + f (3) = f (0)$ 322. $Let f : R \to R be a differentiable function such that ∣ f (x) - f (y) ∣ \leq 2∣ x - y ∣^{3/2} \forall x, y \in R . If f (0) = 1, then \int_{0}^{1} f^{2} (x) d x =$ (1) $-2$ (2) $\frac{1}{2}$ (3) $0$ (4) $1$ 323. $Which of the following statements is correct for the function g (α) for α \in R such that g (α) = \int_{π /6}^{π /3} \frac{s i n ^{α} x}{c o s ^{α} x + s i n ^{α} x} d x$ (1) $g (α) is a strictly increasing function$ (2) $g (α) has an inflection point at α = - \frac{1}{2}$ (3) $g (α) is a strictly decreasing function$ (4) $g (α) is an even function$ 324. $If f and g are differentiable functions in [0, 1] satisfying f (0) = 2 = g (1), g (0) = 0 and f (1) = 6, then for some c \in] 0, 1 [:$ (1) $f'(c) = g'(c)$ (2) $f'(c) = 2g'(c)$ (3) $2f'(c) = g'(c)$ (4) $2f'(c) = 3g'(c)$ 325. $If a function f : R - {l} \to R - {m} defined by f (x) = \frac{x + 3}{x - 2} is a bijection, then 3 l + 2 m =$ (1) $10$ (2) $12$ (3) $8$ (4) $14$ 326. $If f : R \to R is defined by f (x) = ⎩ ⎨ ⎧ x - 1, 2 - x^{2}, x - 10, 2 x, x \leq 1 1 < x \leq 3 3 < x < 5 x \geq 5 then the set of points of discontinuity of f is$ (1) $R - {1, 5}$ (2) ${1, 3, 5}$ (3) ${1, 5}$ (4) $R - {1, 3, 5}$ 327. $If f is defined by f (x) = {x, 2 - x, 0 \leq x < 1 x \geq 1, then at x = 1, f (x) is$ (1) $continuous and differentiable$ (2) $continuous but not differentiable$ (3) $discontinuous but differentiable$ (4) $neither continuous nor differentiable$ 328. $A wire of length 36m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then (\frac{4}{π} + 1) k is equal to .$ 329. $Let f:(-2, 2) \to R be defined by f (x) = {x [x] (x - 1) [x], - 2 < x < 0, 0 \leq x < 2 where [x] denotes the greatest integer function. If m and n respectively are the number of points in (-2, 2) at which y =\|f[x]\| is not continuous and not differentiable, then m + n is equal to_____$ 330. $Let f : R \to R be a differentiable function that satisfies the relation f (x + y) = f (x) + f (y) - 1, x, y \in R . If f^{'} (0) = 2, then ∣ f (- 2) ∣ is equal to$