Continuity & Differentiability PYQs Set 9 (PDF)

Set 9 of 13 · Questions 241-270 of 371

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 9 (Q241-270)
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241. $The constant ' c' of Lagrange's mean value theorem for the function f (x) = \frac{2 x + 3}{4 x - 1} defined on [1, 2] is$ (1) $\frac{1 + 15}{3}$ (2) $\frac{1 + 21}{4}$ (3) $\frac{5}{3}$ (4) $\frac{3}{2}$ 242. $An enemy fighter jet is flying along the curve given by y = x^{2} + 2 . A soldier is placed at (3, 2) wants to shoot down the jet when it is nearest to him. Then the nearest distance is$ (1) $2 units$ (2) $3 units$ (3) $5 units$ (4) $6 units$ 243. $If f (x) = (\frac{1 - cos Kx}{x sin x}, if x \neq = 0 \frac{1}{2}, if x = 0) is continuous at x = 0, then the value of K is$ (1) $0$ (2) $\pm 2$ (3) $\pm 1$ (4) $\pm 1 2$ 244. $f (x) = {\frac{7 2 ^{x} - 9 ^{x} - 8 ^{x} + 1}{2 - 1 + c o s x}, K lo g 2 lo g 3, x \neq = 0 x = 0 Find the value of k for which the function f is continuous.$ (1) $2$ (2) $24$ (3) $183$ (4) $242$ 245. $Let f be any function defined on R and let it satisfy the condition: \| f (x) - f (y) \| \leq ((x - y)^{2}), \forall (x, y) \in R If f (0) = 1, then:$ (1) $f (x) < 0, \forall x \in R$ (2) $f (x) can take any value in R$ (3) $f (x) = 0, \forall x \in R$ (4) $f (x) > 0, \forall x \in R$ 246. $If y = (tan - 1 x)^{2}, then (x^{2} + 1)^{2} \frac{d ^{2} y}{d x ^{2}} + 2 x (x^{2} + 1) \frac{dy}{dx} =$ (1) $4$ (2) $2$ (3) $1$ (4) $0$ 247. $Let f (x) = a x^{3} + b x^{2} + c x + 41 be such that f (1) = 40, f^{'} (1) = 2 and f^{''} (1) = 4 . Then a^{2} + b^{2} + c^{2} is equal to:$ (1) $73$ (2) $62$ (3) $51$ (4) $54$ 248. $Let y = f (x) = sin^{3} (\frac{π}{3} (cos (\frac{π}{3 2} (- 4 x^{3} + 5 x^{2} + 1)^{\frac{3}{2}}))), then at x = 1,$ (1) $2 y^{’} + 3 π^{2} y = 0$ (2) $2 y^{’} + 3 π^{2} y = 0$ (3) $2 y^{’} - 3 π^{2} y = 0$ (4) $y^{’} + 3 π^{2} y = 0$ 249. $The range of the real valued function f (x) = 9 - x^{2} is$ (1) $[- 3, 3]$ (2) $[- 3, 0]$ (3) $[0, 3]$ (4) $[- 2, 2]$ 250. $The approximate value of log_{10} 998 is (given that log_{10} e = 0.4343)$ (1) $3.0008686$ (2) $1.9991314$ (3) $2.0008686$ (4) $2.9991314$ 251. $The number of points in (- \infty, \infty), for which x2 - x sinx - cosx = 0, is$ (1) $6$ (2) $4$ (3) $2$ (4) $0$ 252. $The function f (x) = {\frac{π}{4} + tan^{- 1} x, \frac{1}{2} (∣ x ∣ - 1), ∣ x ∣ \leq 1 ∣ x ∣ > 1 is:$ (1) $both continuous and differentiable on R - (- 1)$ (2) $both continuous and differentiable on R - (1)$ (3) $continuous on R - (- 1) and differentiable on R - (- 1, 1)$ (4) $continuous on R - (1) and differentiable on R - (- 1, 1)$ 253. $Let f (x) = ⎩ ⎨ ⎧ \frac{s i n ( x - [ x ])}{x - [ x ]}, max {2 x, 3 [∣ x ∣]}, 1, x \in (- 2, - 1) ∣ x ∣ < 1 otherwise where [t] denotes greatest integer \leq t. If m is the number of points where f is not continuous and n is the number of points where f is not differentiate, then the ordered pair (m, n) is :$ (1) $(3, 3)$ (2) $(2, 4)$ (3) $(2, 3)$ (4) $(3, 4)$ 254. $Let f, g : [-1, 2] \to R be continuous function which are twice differentiable on the interval (-1, 2). Let the values of f and g at the points -1, 0 and 2 be as given in the following table : In each of the intervals (-1, 0) and (0, 2) the function (f - 3g)" never vanishes. Then the correct statement(s) is (are)$ (1) $f'(x) - 3g'(x) = 0 has exactly three solutions in (-1, 0) \cup (0, 2)$ (2) $f'(x) - 3g'(x) = 0 has exactly one solution in (-1, 0)$ (3) $f'(x) - 3g'(x) = 0 has exactly one solution in (0, 2)$ (4) $f'(x) - 3g'(x) = 0 has exactly two solutions in (-1, 0) and exactly two solutions in (0, 2)$ 255. $Let the function f (x) = (x^{2} + 1) (x^{2} - ax + 2) + cos \| x \| be not differentiable at the two points x = α = 2 and x = β . Then the distance of the point (α, β) from the line 12 x + 5 y + 10 = 0 is equal to:$ (1) $5$ (2) $4$ (3) $3$ (4) $2$ 256. $If f (x) = Max 3-x, 3+x, 6 is not differentiable at x = a, and x = b, then \| a \| + \| b \| =$ (1) $4$ (2) $5$ (3) $6$ (4) $8$ 257. $If x = -1 and x = 2 are extreme points of f(x) = αlog\|x\| + βx2 + x then :$ (1) $α = 2, β = - \frac{1}{2}$ (2) $α = 2, β = \frac{1}{2}$ (3) $α = -6, β = \frac{1}{2}$ (4) $α = -6, β = - \frac{1}{2}$ 258. $A wire of length 20 cm is bent in the form of a sector of a circle. The maximum area that can be enclosed by the wire is$ (1) $25sq. cm$ (2) $20 sq. cm$ (3) $30 sq \cdot cm$ (4) $10 sq. cm$ 259. $Consider the function f : (0, 2) \to R defined by f (x) = \frac{x}{2} + \frac{2}{x} and the function g (x) defined by g (x) = (min f (t), 0 < t \leq x and 0 < x \leq 1 \frac{3}{2} + x, 1 < x < 2) . Then$ (1) $g is continuous but not differentiable at x = 1$ (2) $g is not continuous for all x \in (0, 2)$ (3) $g is neither continuous nor differentiable at x = 1$ (4) $g is continuous and differentiable for all x \in (0, 2)$ 260. $If 2 x^{y} + 3 y^{x} = 20, then \frac{d y}{d x} at (2, 2) is equal to:$ (1) $- (\frac{2 + l o g _{e} 8}{3 + l o g _{e} 4})$ (2) $- (\frac{3 + l o g _{e} 16}{4 + l o g _{e} 8})$ (3) $- (\frac{3 + l o g _{e} 8}{2 + l o g _{e} 4})$ (4) $- (\frac{3 + l o g _{e} 4}{2 + l o g _{e} 8})$ 261. $The range of the real valued function f (x) = \frac{1}{x - \| x \|} is$ (1) $(0, \infty)$ (2) $(- \infty, 0)$ (3) $(- \infty, 0) \cup (0, \infty)$ (4) $(- \infty, \infty)$ 262. $Let f (x) = a a x a x^{2} - 1 a a x 0 - 1 a, a \in R . Then the sum of which the squares of all the values of a for 2f'(10)-f'(5)+100=0 is:$ (1) $117$ (2) $106$ (3) $125$ (4) $136$ 263. $Let f (x) = (2 x^{2} + 5) ∣ x ∣ - 3, x \in R . If m and n denote the number of points where f is not continuous and not differentiable respectively, then m + n is equal to :$ (1) $5$ (2) $2$ (3) $0$ (4) $3$ 264. $Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f (x) = [x] + \| x - 2 \|, - 2 < x < 3, is not continuous and not differentiable. Then m + n is equal to:$ (1) $6$ (2) $8$ (3) $9$ (4) $7$ 265. $Let f : R \to R satisfy f(x + y) = 2^{x} f(y) + 4^{y} f(x), \forall x, y \in R . If f(2) = 3, then 14. \frac{f ^{'} ( 4 )}{f ^{'} ( 2 )} is equal to_____.$ 266. $The minimum value of α for which the equation \frac{4}{sin x} + \frac{1}{1 - sin x} = α has at least one solution in (0, \frac{π}{2}) is$ 267. $Let m and n be two positive integers greater than 1. If lim_{α \to 0} (\frac{e ^{c o s (α^{n})} - e}{α ^{m}}) = - (\frac{e}{2}), then \frac{m}{n} the value of is$ 268. $The maximum value of the function f(x) = 2x3 - 15x2 + 36x - 48 on the set A = {x \|x2 + 20 \leq 9x} is$ 269. $Let [x] be the greatest integer \leq x. Then the number of points in the interval (-2, 1) where the function f (x) = ∣ [x] ∣ + x - [x] is discontinuous, is_____.$ 270. $If y (x) = (x^{x^{x}}), x > 0 then \frac{d ^{2} x}{d y ^{2}} + 20 at x = 1 is equal to:$