Continuity & Differentiability PYQs Set 10 (PDF)

Set 10 of 13 · Questions 271-300 of 371

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 10 (Q271-300)
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271. $A box open from top is made from a rectangular sheet of dimension axb by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :$ (1) $\frac{a + b - a ^{2} + b ^{2} - ab}{12}$ (2) $\frac{a + b - a ^{2} + b ^{2} - ab}{12}$ (3) $\frac{a + b - a ^{2} + b ^{2} - ab}{6}$ (4) $\frac{a + b + a ^{2} + b ^{2} - ab}{6}$ 272. $If lim_{x \to 0} [1 + x ln (1 + b^{2})]^{\frac{1}{x}} = 2b sin2 θ, b > 0 and θ \in (-π, π], then the value of θ is$ (1) $\pm \frac{π}{4}$ (2) $\pm \frac{π}{3}$ (3) $\pm \frac{π}{6}$ (4) $\pm \frac{π}{2}$ 273. $Let 3 be the radius and \frac{π}{3} be the semi vertical angle of the given cone. Then the height of the right circular cylinder of maximum volume that can be inscribed in the given cone is$ (1) $3$ (2) $\frac{3}{2}$ (3) $\frac{2}{3}$ (4) $\frac{1}{3}$ 274. $Let f : R \to R be a function defined by f (x) = max (x, x^{2}) . Let S denote the set of all points in R, where f is not differentiable. Then:$ (1) $ϕ (an empty set)$ (2) $(0, 1)$ (3) $(1)$ (4) $(0)$ 275. $Let f : [- 1, 2] \to R be given by f (x) = 2 x^{2} + x + [x^{2}] - [x], where [t] denotes the greatest integer less than or equal to t . The number of points, where f is not continuous, is :$ (1) $5$ (2) $6$ (3) $3$ (4) $4$ 276. $The range of a \in R for which the function f (x) = (4 a - 3) (x + log_{e} 5) + 2 (a - 7) cot (\frac{x}{2}) sin^{2} (\frac{x}{2}) x + 2 nπ, n \in N, has critical points, is$ (1) $(- 3, 1)$ (2) $(- \frac{4}{3}, 2)$ (3) $[1, \infty)$ (4) $(- \infty, - 1]$ 277. $The maximum volume (in cu. units) of the cylinder which can be inscribed in a sphere of radius 12 units is$ (1) $3843 π$ (2) $7683 π$ (3) $\frac{768 π}{3}$ (4) $\frac{1152 π}{3}$ 278. $The number of real roots of the equation e^{6 x} - e^{4 x} - 2 e^{3 x} - 12 e^{2 x} + e^{x} + 1 = 0 is :$ (1) $2$ (2) $4$ (3) $6$ (4) $1$ 279. $The value of p and q for which the function f(x) = ⎩ ⎨ ⎧ \frac{s i n ( p + 1 ) x + s i n x}{x}, q, \frac{x + x ^{2} - x}{x ^{3/2}}, x < 0 x = 0 x > 0 is continuous for all x in R, are:$ (1) $p = \frac{1}{2}, q = - \frac{3}{2}$ (2) $p = \frac{5}{2}, q = \frac{1}{2}$ (3) $p = - \frac{3}{2}, q = \frac{1}{2}$ (4) $p = \frac{1}{2}, q = \frac{3}{2}$ 280. $The radius of a circle, having minimum area, which touches the curve y = 4 - x^{2} and the lines, y = \|x\| is$ (1) $2$ (2) $2$ (3) $4 (2 - 1)$ (4) $4 (2 + 1)$ 281. $The triangle of maximum area that can be inscribed in a given circle of radius ’ r ’ is:$ (1) $A right angle triangle having two of its sides of length 2 r and r .$ (2) $An equilateral triangle of height \frac{2 r}{3} .$ (3) $An isosceles triangle with base equal to 2 r .$ (4) $An equilateral triangle having each of its side of length 3 r .$ 282. $Consider the function f : (0, \infty) \to R defined by f (x) = e^{- ∣ l o g_{6} x ∣} . If m and n be respectively the number of points at which f is not continuous and f is not differentiable, then m + n is$ (1) $0$ (2) $3$ (3) $1$ (4) $2$ 283. $A wire of length 20m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is:$ (1) $\frac{5}{2 + 3}$ (2) $\frac{10}{2 + 3 3}$ (3) $\frac{5}{3 + 3}$ (4) $\frac{10}{3 + 2 3}$ 284. $Assertion (A) : f (x) = \| x \| is differentiable at x = a \neq = 0 and continuous but not differentiable at x = 0 Reason (R): If a function is differentiable at a point then it is continuous at that point. But converse is not true.$ (1) $A is correct, R is correct, R is correct explanation of A .$ (2) $A is correct, R is correct, but R is not correct explanation of A .$ (3) $A is correct, R is false.$ (4) $A is false, R is correct.$ 285. $The function f: R \to R defined by f (x) = lim_{n \to \infty} \frac{c o s ( 2 π x ) - x ^{2 n} s ( x - 1 )}{1 + x ^{2 n + 1} - x ^{2 n}} is continuous for all x in$ (1) $R-{-1}$ (2) $R-{-l, 1}$ (3) $R-{1)$ (4) $R-{0)$ 286. $Let f : R \to R be the function defined by f (x) = ⎩ ⎨ ⎧ 5, a + b x, b + 5 x, 30, if x \leq 1 if 1 < x < 3 if 3 \leq x < 5 if x \geq 5 then f is$ (1) $continuous if a = 5 and b=5$ (2) $continuous if a = 0 and b=5$ (3) $continuous if a = - 5 and b = 10$ (4) $not continuous for any values of a and b$ 287. $A circular sector of perimeter 60 meter with maximum area is to be constructed. The radius of the circular arc in meter must be$ (1) $10 m$ (2) $15 m$ (3) $5 m$ (4) $20 m$ 288. $A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then$ (1) $(4 - π) x = πr$ (2) $x = 2r$ (3) $2x = r$ (4) $2x = (π + 4) r$ 289. $Let f (x) = min {1, 1 + x sin x}, 0 \leq x \leq 2 π . If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to$ (1) $(2, 0)$ (2) $(1, 0)$ (3) $(1, 1)$ (4) $(2, 1)$ 290. $A function f is defined on [- 3, 3] as f (x) = (min (\| x \|, 2 - x^{2}), - 2 \leq x \leq 2 [\| x \|], 2 < \| x \| \leq 3) where [x] denotes the greatest integer \leq x . The number of points, where f is not differentiable in (- 3, 3) is$ 291. $If [t] denotes the greatest integer \leq t, then number of points, at which the function f(x) = 4\|2x + 3\| + 9 -12 [x + 20] is not differentiable in the open interval (-20, 20), is___.$ 292. $Let a, b \inR, b \neq 0, Define a function f (x) = {a sin \frac{π}{2} (x - 1), \frac{t a n 2 x - s i n 2 x}{b x ^{3}}, for x \leq 0 for x > 0 If f is continuous at x=0, then 10-ab is equal to .$ 293. $For the curve C : (x^{2} + y^{2} - 3) + (x^{2} - y^{2} - 1)^{5} = 0, the value of 3y' - y^{3} y", at the point (α, α), α > 0, on C, is equal to________.$ 294. $Let m and n be the number of points at which the function f (x) = max (x, x^{3}, x^{5}, \dots x^{21}), x \in R, is not differentiable and not continuous, respectively. Then m + n is equal to _____.$ 295. $If y = y(x) is an implicit function of x such that l o g_{e} (x + y) = 4xy, then \frac{d ^{2} y}{d x ^{2}} at x=0 is equal to.$ 296. $Let p(x) be a polynomial of degree 4 having extremum at x = 1, 2 and lim_{x \to 0} = 2. Then the value of p(2) is$ 297. $Let a \in Z and [t] be the greatest integer \leq t, then the number of points, where the function f(x)=[a + 13 sinx], x \in_{(} 0, π_{)} is not differentiable, is$ 298. $Let the function f (x) = {- 3 a x^{2} - 2, a^{2} + b x, x < 1 x \geq 1 be differentiable for all x \in R, where a > 1, b \in R . If the area of the region enclosed by y = f (x) and the line y = - 20 is α + β 3, α, β \in Z, then the value of α + β is$ 299. $The number of distinct real roots of x4 - 4x3 + 12x2 + x - 1 = 0 is$ 300. $The largest value of the non-negative integer a for which lim_{x \to 1} {\frac{- a x + s i n ( x - 1 ) + a}{x + s i n ( x - 1 ) - 1}}^{\frac{1 - x}{1 - x}} = \frac{1}{4} is$