Continuity And Differentiability Test

Continuity And Differentiability

This is Continuity and Differentiability Test-03 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
Let f(x) = x – [x], then f ‘ (x) = 1 for
  • A
    all \(x \in {\mathbf{R}}\)
  • B
    all \(x \in {\mathbf{I}}\)
  • C
    all\(x \in {\mathbf{R}}\) – {0]
  • D
    all \(x \in ({\mathbf{R - I)}}\)
    Correct
2
Let f (x + y) = f(x) + f(y) \(\forall \) x, y \( \in {\mathbf{R}}\). Suppose that f (6) = 5 and f ‘ (0) = 1, then f ‘ (6) is equal to
  • A
    25
  • B
    none of these
  • C
    30
  • D
    1
    Correct
3
If x sin (a + y) = sin y, then \(\frac{{dy}}{{dx}}\) is equal to
  • A
    \(\frac{{{{\sin }^2}(a + y)}}{{\sin a}}\)
    Correct
  • B
    \(\frac{{\sin a}}{{{{\sin }^2}(a + y)}}\)
  • C
    \(\frac{{\sin (a + y)}}{{\sin a}}\)
  • D
    \(\frac{{\sin a}}{{\sin (a + y)}}\)
4
If f (x) \( = \frac{1}{{3x + 1}},\) then f ‘ (0)
  • A
    is positive
  • B
    is negative
    Correct
  • C
    vanishes
  • D
    does not exist.
5
Let f be a function satisfying f(x + y) = f(x) + f(y) for all x, y \( \in {\mathbf{R}},\)then f ‘ (x) =
  • A
    f ‘ (0) for all \(x \in {\mathbf{R}}\)
    Correct
  • B
    f (0) for all \(x \in {\mathbf{R}}\)
  • C
    none of these
  • D
    0 for all \(x \in {\mathbf{R}}\)
6
Let f and g be differentiable functions such that fog = I, the identity function. If g’ (a) = 2 and g (a) = b, then f ‘ (b) =
  • A
    2
  • B
    – 2
  • C
    \(\frac{1}{2}\)
    Correct
  • D
    none of these
7
Differential coefficient of a function f (g (x)) w.r.t. the function g (x) is
  • A
    f ‘ (g (x))
    Correct
  • B
    none of these
  • C
    f ‘ (g (x)) g’ (x)
  • D
    \(\frac{{f'(g(x))}}{{g'(x)}}\)
8
If a function f is derivable at x = a, then \(\mathop {Let}\limits_{h \to 0} \;\;\;\frac{{f(a - h) - f(a)}}{h}\) is equal to
  • A
    none of these.
  • B
    f ‘ (a)
  • C
    does not exist
  • D
    – f ‘ (a)
    Correct
9
If \(y = {\tan ^{ - 1}}\)x and \(z = {\cot ^{ - 1}}x\) then \(\frac{{dy}}{{dz}}\)is equal to
  • A
    none of these
  • B
    1
  • C
    \(\frac{\pi }{2}\)
  • D
    – 1
    Correct
10
\(\frac{d}{{dx}}({\cos ^{ - 1}}x) = - \frac{1}{{\sqrt {1 - {x^2}} }}\)where
  • A
    \( - 1 < x \leqslant 1\)
  • B
    \( - 1 < x < 1\)
    Correct
  • C
    \( - 1 \leqslant x < 1\)
  • D
    \( - 1 \leqslant x \leqslant 1\)
11
If f (x) = \(xta{n^{ - 1}}\) x then f ‘ (1 ) is equal to
  • A
    \(\frac{1}{2} - \frac{\pi }{4}\)
  • B
    \(\frac{\pi }{4} - \frac{1}{2}\)
  • C
    none of these
  • D
    \(\frac{\pi }{4} + \frac{1}{2}\)
    Correct
12
\(\frac{d}{{dx}}({\tan ^{ - 1}}(\cot x))\)is equal to
  • A
    \( - cosse{c^2}\)x
  • B
    – 1
    Correct
  • C
    \(si{n^2}\)x
  • D
    none of these
13
\(\frac{d}{{dx}}({\tan ^{ - 1}}(\sec x + \tan x)\) is equal to
  • A
    \(\frac{1}{2}\)
    Correct
  • B
    \( - \frac{1}{2}\)
  • C
    none of these
  • D
    \(\frac{1}{{2\sec x(\sec x + \tan x)}}\)
14
\(\mathop {Lt}\limits_{x \to 0} \;\;\;{(1 + 2x)^{\frac{{x + 3}}{x}}}\) is equal to
  • A
    \({e^3}\)
  • B
    none of these
  • C
    \({e^6}\)
    Correct
  • D
    \({e^{3/2}}\)
15
Let f(x) = \(\left\{ \begin{gathered} {e^{1/x}},x < 0 \\ \;\;\;\;\;x,x \geqslant 0 \\ \end{gathered} \right.,then\mathop {Lt}\limits_{x \to 0} \;\;f(x)\)
  • A
    is equal to 0
    Correct
  • B
    none of these
  • C
    does not exist
  • D
    is equal to non – zero real number