Continuity & Differentiability PYQs Set 12 (PDF)

Set 12 of 13 · Questions 331-360 of 371

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 12 (Q331-360)
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331. $The set of all real values of the expression \frac{x ^{2} - x + 2}{x ^{2} + x - 2} for all x \in R - {- 2, 1} is$ (1) $(- 2, 3)$ (2) $[\frac{7}{9}, \infty)$ (3) $(- \infty, - 1] \cup [\frac{7}{9}, \infty)$ (4) $(- \infty, - 1]$ 332. $If f (x) = {\frac{3 s i n π x}{5 x}, 2 K, x \neq = 0 x = 0 is continuous at x = 0, then the value of K is$ (1) $\frac{π}{10}$ (2) $\frac{3 π}{10}$ (3) $\frac{3 π}{2}$ (4) $\frac{3 π}{5}$ 333. $Let f : S \to S where S = (0, \infty) be a twice differentiable function such that f (x + 1) = x f (x) . If g : S \to R is defined as g (x) = lo g_{e} f (x), then the value of ∣ g^{''} (5) - g^{''} (1) ∣ is equal to :$ (1) $\frac{205}{144}$ (2) $\frac{197}{144}$ (3) $\frac{187}{144}$ (4) $1$ 334. $If f (x) = {K x^{2}, 3, x \leq 2 x > 2 is continuous at x = 2, then the value of K is$ (1) $\frac{4}{3}$ (2) $\frac{3}{4}$ (3) $3$ (4) $4$ 335. $Let [\cdot] denote the greatest integer function and let f (x) = min (2 x, x^{2}) . Let S = {x \in (- 2, 2) : the function g (x) = ∣ x ∣ [x^{2}] is discontinuous at x} . Then \sum_{x \in S} f (x) equals$ (1) $6 - 22$ (2) $1 - 2$ (3) $2 - 2$ (4) $26 - 32$ 336. $If f (1) = 1, f^{'} (1) = 3, then the derivative of f (f (f (x))) + (f (x))^{2} at x = 1 is :$ (1) $33$ (2) $12$ (3) $15$ (4) $9$ 337. $If a is the point of discontinuity of the function f (x) = ⎩ ⎨ ⎧ cos 2 x, e^{3 x}, x^{2} - 4 x + 3, \frac{lo g ( 15 x - 89 )}{x - 6}, - \infty < x < 0 0 \leq x < 3 3 \leq x \leq 6 x > 6 Then lim_{x \to a} \frac{x ^{2} - 9}{x ^{3} - 5 x ^{2} + 9 x - 9} =$ (1) $1$ (2) $0$ (3) $6$ (4) $3$ 338. $The maximum value of x e^{- x} is$ (1) $- 1/ e$ (2) $e$ (3) $1/ e$ (4) $- e$ 339. $The value of 24.99 is$ (1) $4.999$ (2) $5.001$ (3) $4.899$ (4) $4.897$ 340. $Let f (x) = 5 - ∣ x - 2∣ and g (x) = ∣ x + 1∣, x \in R . If f (x) attains maximum value at α and g (x) attains minimum value at β, then lim_{x \to - α β} \frac{( x - 1 ) ( x ^{2} - 5 x + 6 )}{x ^{2} - 6 x + 8} is equal to$ (1) $\frac{1}{2}$ (2) $\frac{- 3}{2}$ (3) $\frac{- 1}{2}$ (4) $\frac{3}{2}$ 341. $The value of cot (\sum_{n = 1}^{23} cot^{- 1} (1 + \sum_{k = 1}^{n} 2 k)) is$ (1) $\frac{23}{25}$ (2) $\frac{25}{23}$ (3) $\frac{23}{24}$ (4) $\frac{24}{23}$ 342. $Let a function f (x) be continuous in an interval [a, b] . Let δ > 0 be a very small real number. Let c \in (a, b) be such that f (c - δ) < f (c) and f (c + δ) < f (c) for every δ > 0 . Let (f (α - δ) - f (α)) (f (α + δ) - f (α)) < 0 \forall α \in (a, b) and a \neq = c . Then$ (1) $f (x) has a local maximum at c and a local minimum at α .$ (2) $f (x) has a local maximum at α and a local minimum at c .$ (3) $f (x) has only one local maximum at c .$ (4) $f (x) has only one local minimum at c .$ 343. $Let f :_{(} a, b_{)} \to R be twice differentiable function such that for a differentiable function g(x). If f(x)=0 has exactly five distinct roots in (a,b), then g(x)g’(x) has at least :$ (1) $twelve roots in (a, b)$ (2) $five roots in (a, b)$ (3) $seven roots in (a, b)$ (4) $three roots in (a, b)$ 344. $Let f : R \to R be defined as f (x) = ⎩ ⎨ ⎧ \frac{λ ∣ x ^{2} - 5 x + 6∣}{μ ( 5 x - x ^{2} - 6 )}, \frac{t n ( x - 2 )}{e ^{x - [x]}}, μ, x < 2 x > 2 x = 2 where [x] is the greatest integer less than or equal to x. If f is continuous at x=2, then λ + μ is equal to$ (1) $e(-e + 1)$ (2) $e(e-2)$ (3) $e(e-1)$ (4) $(2e-1)$ 345. $If f (x) = \frac{2 x - 3}{3 x - 4}, x \neq = \frac{4}{3}, then the value of f^{- 1} (x) is$ (1) $\frac{4 x - 3}{3 x - 2}$ (2) $\frac{3 x - 2}{4 x + 3}$ (3) $\frac{3 x - 4}{4 x - 2}$ (4) $\frac{2 x + 3}{4 x - 3}$ 346. $If f (x) = x^{3} and g (x) = x^{3} - 4 x in - 2 \leq x \leq 2, then consider the statements: (i) f (x) and g (x) satisfy mean value theorem. (ii) f (x) and g (x) both satisfy Rolle's theorem. (iii) Only g (x) satisfies Rolle's theorem. Of these statements$ (1) $(i) and (ii) are correct$ (2) $(i) alone is correct$ (3) $None is correct$ (4) $(i) and (iii) are correct$ 347. $If a cylindrical vessel of given volume V with no lid on the top is to be made from a sheet of metal, then the radius (r) and height (h) of the vessel so that the metal sheet used is minimum, is$ (1) $r = 3 \frac{π}{V}, h = 3 \frac{π}{V}$ (2) $r = πV, h = πV$ (3) $r = 3 \frac{V}{π}, h = 3 \frac{V}{π}$ (4) $r = \frac{V}{π}, h = \frac{V}{π}$ 348. $If x^{2} + y^{2} = 25, then log_{5} [max (3 x + 4 y)] is$ (1) $2$ (2) $3$ (3) $4$ (4) $5$ 349. $The set of all values of x and the set of all values of a for which the real valued function f (x) = log_{a} (x - [x]) is defined are respectively$ (1) $R - Z and (0, 1)$ (2) $Z and R - {0, 1}$ (3) $Z^{-} and (1, \infty)$ (4) $R and R$ 350. $A closed cylinder of given volume will have least surface area when the ratio of its height and base radius is$ (1) $2 : 1$ (2) $1 : 2$ (3) $2 : 3$ (4) $3 : 2$ 351. $The values of p and q so that the function f (x) = ⎩ ⎨ ⎧ (1 + ∣ sin x ∣)^{\frac{p}{s i n x}}, q, e^{\frac{s i n 2 x}{s i n 3 x}}, - \frac{π}{6} < x < 0 x = 0 0 < x < \frac{π}{6} is continuous at x = 0, are$ (1) $p = \frac{1}{3}, q = e^{2/3}$ (2) $p = 0, q = e^{2/3}$ (3) $p = \frac{2}{3}, q = e^{- 2/3}$ (4) $p = - \frac{2}{3}, q = e^{2/3}$ 352. $The function f (x) = \frac{log ( 1 + ax ) - log ( 1 - bx )}{x} is not defined at x = 0 . The value which should be assigned to ' f' at x = 0 so that it is continuous at x = 0 is$ (1) $log a + log b$ (2) $0$ (3) $a - b$ (4) $a + b$ 353. $Let S be the set of points where the function f (x) = ∣ 2 - ∣ x - 3∣ ∣, x \in R is not differentiable. Then \sum_{x \in S} f (f (x)) is equal to ....$ 354. $Let f : (0, π) \to R be a function given by f (x) = ((\frac{8}{7})^{\frac{tan 8 x}{tan 7 x}}, 0 < x < \frac{π}{2} a - 8, x = \frac{π}{2} (1 + \| cot x \|)^{\frac{b}{\| c \|} \| tan x \|}, \frac{π}{2} < x < π) where a, b \in Z . If f is continuous at x = \frac{π}{2}, then a^{2} + b^{2} is equal to$ 355. $Let f (x) = ⎩ ⎨ ⎧ 3 x, min {1 + x + [x], x + 2 [x]}, 5, x < 0 0 \leq x \leq 2 x > 2 where [\cdot] denotes greatest integer function. If α and β are the number of points, where f is not continuous and is not differentiable, respectively, then α + β equals .$ 356. $Let f : [- 1, 1] \to R be defined as f (x) = a x^{2} + bx + c for all x \in [- 1, 1], where a, b, c \in R such that f (- 1) = 2, f^{’} (- 1) = 1 and for x \in (- 1, 1) the maximumvalue of f^{’ ’} (x) is \frac{1}{2} . If f (x) \leq α x \in [- 1, 1], then the least value of α is equal to_____$ 357. $A cylindrical container is to be made from certain solid material with the following constraints: It has fixed inner volume of V mm3, has a 2 mm thick solid wall and is open at the top. The bottom of the container is solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of \frac{V}{250 π} is$ 358. $Let {4 x^{2} - 8 x + 5, [4 x^{2} - 8 x + 5], i f 8 x^{2} - 6 x + 1 \geq 0 i f 8 x^{2} - 6 x + 1 < 0, where [α] denotes the greatest integer less than or equal to α . Then the number of points in R where f is not differentiate is_______.$ 359. $Let f : ΙR \to ΙR be defined as f(x) = \|x\| + \|x2 - 1\|. The total number of points at which f attains either a local maximum or a local minimum is$ 360. $A man starts walking from the point P(-3,4) touches the x-axis at R, and then turns to reach at the point Q(0,2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then 5 0_{((} P R_{)}^{2} +_{(} R Q_{)}^{2}_{)} is equal to.$