Class 12 continuity and differentiability multiple choice

Multiple choice (Continuity and differentiability)

Choose and write the correct option in the following question:(continuity and differentiability multiple choice)

continuity and differentiability multiple choice

1.) The function f:R → R given by f(x) = -|x – 1| is

(a) Continuous as well as differentiable at x = 1

(b) Not continuous but differentiable at x = 1

(c) Continuous but not differentiable at x = 1

(d) Neither continuous nor differentiable at x = 1

Answer (c)

2.) The function f(x) = e^{|x|} is

(a) Continuous everywhere but not differentiable at x = 1

(b) Continuous and differentiable everywhere

(c) Not continuous at x= 0

(d) None of these

Answer (a)

3.) The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at

(a) 4                         (b) 2

(c) 1                           (d) None of these

Answer (d)

4.) The number of points at which the function f(x) = \frac{1}{x-[x]} is not continuous  is

(a) 1                           (b) 2

(c) 3                           (d) None of these

Answer (d)

5.) The function f(x) = \left\{\begin{array}{l} -\frac{\sin x}{x}+\cos x, \text { if } x \neq 0 \\ k, \text { if } x = 0 \end{array}\right\} is continuous at x = 0 then the value of k is

(a) 3                             (b) 2

(c) 1                              (d) 1.5

Answer (b)

6.) If the function f defined by f(x)  = \left\{\begin{array}{l} -\frac{k\cos x}{\pi - 2x}, \text { if } x \neq \frac{\pi}{2} \\ 3, \text { if } x = \frac{\pi}{2} \end{array}\right\}, is continuous at x =π/2, then the value of

(a) 2                              (b) 3

(c) 6                              (d) -6

Answer (c)

7.) The function f(x) = cot x is discontinuous on the set

(a) {x = nπ: n∈ Z}           (b) {x = 2nπ: n∈ Z}

(c) {x = (2nπ+1)π/2: n∈ Z}   (d) {x = nπ/2: n∈ Z}

Answer (a)

8.)Let f(x) = |sin x|. then

(a) f is everywhere differentiable.

(b) f is everywhere continuous but not differentiable at x = nπ: n∈Z

(c) f is everywhere continuous but not differentiable at x =(2n + 1)π/2, n∈ Z.

(d) None of these

Answer (b)

9.) The function f(x) = \frac{x-1}{x(x^2-1)} is discontinuous at

(a) Exactly one point              (b) Exactly two points

(c) Exactly three points         (d)  No point

Answer (c)

10.) If f(x)  = x^2\sin\frac{1}{x}, where x ≠ 0, thenthe value of the function f at x = 0, so that the function is

(a) 0                     (b)  -1

(c) 1                       (d) None of these

Answer (a)

11.) The function f(x) =\left\{\begin{array}{l} \frac{e^{3x}-e^{-5x}}{x}, \text { if } x \neq 0 \\ 3, \text { if } x = 0 \end{array}\right\} is continuous at x = 0 for the value of k, as

(a) 3                   (b) 5

(c) 2                    (d) 8

Answer (d)

12.) If y = \sin(m\sin^{-1}x), then which of the following question is true ?

(a) (1-x^2)\frac{d^y}{dx^2}+x\frac{dy}{dx}+m^2y = 0

(b) (1-x^2)\frac{d^y}{dx^2}-x\frac{dy}{dx}+m^2y = 0

(c) (1+x^2)\frac{d^y}{dx^2}-x\frac{dy}{dx}-m^2y = 0

(d) (1+x^2)\frac{d^y}{dx^2}+x\frac{dy}{dx}-m^2y = 0

Answer (b)

13.) Differential of \log[\log(\log(x^5))] w.r.t. , is

(a) \frac{5}{x\log(x^5)\log(\log x^5)}       (b) \frac{5}{x\log(x^5)}

(c) \frac{5x^4}{\log(x^5)\log(\log x^5)}  (d) \frac{5x}{\log(x^5)\log(\log x^5)}

Answer (a)

14.) If \sin y = x\cos(a+y), then \frac{dx}{dy} is

(a) \frac{\cos a}{\cos^2 (a+y)}           (b) \frac{-\cos a}{\cos^2 (a+y)}

(c) \frac{\cos a}{\sin^2 y}                    (d) \frac{-\cos a}{\sin^2 y}

Answer (a)

15.) If (x^2+y^2)^2 = xy, then \frac{dy}{dx} is

(a) \frac{y + 4x(x^2+y^2)}{4y(x^2+y^2)-x}

(b) \frac{y - 4x(x^2+y^2)}{x+y(x^2+y^2)}

(c) \frac{y - 4x(x^2+y^2)}{4y(x^2+y^2)-x}

(d) \frac{ 4y(x^2+y^2)-x}{y-4x(x^2+y^2)}

Answer (c)

16.) The set of points where the functions f given by f(x) = |x – 3| cos x is differentiable is

(a) R                          (b) R – {3}

(c) (0, ∞)                  (d) None of these

Answer (b)

17.) Differential coefficient of \sec(\tan^{-1} x) w.r.t. x is

(a) \frac{x}{\sqrt{1+x^2}}            (b) \frac{x}{1+x^2}

(c) x\sqrt{1+x^2}                             (d) \frac{1}{\sqrt{1+x^2}}

Answer (a)

18.) If u = \sin^{-1}(\frac{2x}{1+x^2}) and v = \tan^{-1}(\frac{2x}{1-x^2}), then \frac{du}{dx} is

(a) 1/2                                           (b) x

(c) \frac{1-x^2}{1+ x^2}{4, -4}, \phi        (d) 1

Answer (d)

19.) If y = \log{\sqrt{\tan x}}, then the value of \frac{dy}{dx} at x = π/4 is

(a) 0                      (b) 1

(c) 1/2                   (d) ∞

Answer (b)

20.) If y = \sqrt{\sin x + y}, then \frac{dy}{dx} is equal to

(a) \frac{\cos x}{2y - 1}          (b) \frac{\cos x}{1-2y}

(c) \frac{\sin x}{1 - 2y}            (d) \frac{\sin x}{2y - 1}

Answer (a)

 

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