Continuity And Differentiability Test

Continuity And Differentiability

This is Continuity and Differentiability Test-02 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
\(\mathop {Lt}\limits_{x \to 0} \;\;\frac{{\tan x}}{{\log (1 + x)}}\) is equal to
  • A
    does not exist
  • B
    0
  • C
    none of these
  • D
    1
    Correct
2
The function f (x) = 1 + | sin x l is
  • A
    continuous everywhere
    Correct
  • B
    continuous nowhere
  • C
    differentiable nowhere
  • D
    differentiable everywhere.
3
The function, f (x) = (x – a) sin \(\frac{1}{{x - a}}\)for x\( \ne \) a and f (a) = 0 is
  • A
    Not continuous at x = a
  • B
    Derivable at x = a
  • C
    Continuous but not derivable at x = a
    Correct
  • D
    None of these
4
Let f (x) = [x], then f (x) is
  • A
    differentiable for all \(x \in ({\mathbf{R}} - I).\)
    Correct
  • B
    continuous nowhere
  • C
    differentiable for all \(x \in {\mathbf{R}}\)
  • D
    continuous for all \(x \in {\mathbf{R}}\)
5
The function f (x) = [x] is
  • A
    discontinuous only for integral x
    Correct
  • B
    continuous for all x
  • C
    derivable for all x
  • D
    a constant function
6
If f (x) is a polynomial of degree m \(( \geqslant 1)\;\) , then which of the following is not true ?
  • A
    f is derivable at all \(x \in \;{\mathbf{R}}\)
  • B
    none of these
    Correct
  • C
    \(\frac{{{d^n}y}}{{d{x^n}}} = 0\)for all n > m
  • D
    f is continuous at all \(x \in \;{\mathbf{R}}\)
7
If f(x) = x \(\left| {{\text{ }}x{\text{ }}} \right|\) \(\forall x \in {\mathbf{R}},\) then
  • A
    f is derivable at x = 0 and f ‘ (0) = 1
  • B
    f is derivable at x = 0 but f’ (0) \( \ne \)
  • C
    f is discontinuous at x = 0
  • D
    none of these
    Correct
8
Let f(x) \( = \left\{ \begin{gathered} 1 + x\;if\;x > 0 \\ \;\;x\;\;\;\;if\;x \leqslant 0 \\ \end{gathered} \right.then\)\(\mathop {Lt}\limits_{x \to 0} \)f(x) is equal to
  • A
    1
  • B
    none of these
    Correct
  • C
    0
  • D
    \(\frac{1}{2}\)
9
If f (x) = (1 – x) tan \(\frac{{\pi x}}{2}\), then \(\mathop {Lt}\limits_{x \to 1} \) f(x) is equal to
  • A
    1.
  • B
    \(\frac{2}{\pi }\)
    Correct
  • C
    \(\frac{\pi }{2}\)
  • D
    0
10
If both f and g are defined in a nhd of 0 ; f(0) = 0 = g(0) and f ‘ (0) = 8 = g’ (0), then \(\mathop {Lt}\limits_{x \to 0} \;\;\;\frac{{f(x)}}{{g(x)}}\) is equal to
  • A
    0
  • B
    none of these
  • C
    1
    Correct
  • D
    16
11
The derivative of f(x) = | x | at x = 0 is
  • A
    all of these
  • B
    – 1
  • C
    1
  • D
    none of these
    Correct
12
If \(x{\text{ }} + {\text{ }}\left| {{\text{ }}y{\text{ }}} \right|{\text{ }} = {\text{ }}2y\) , then y as a function of x is
  • A
    differentiable for all x
  • B
    not defined for all real x
  • C
    not continuous at x = 0
  • D
    such that \(\frac{{dy}}{{dx}} = \frac{1}{3}for\; < 0\)
    Correct
13
Let \(f\left( x \right){\text{ }} = {\text{ }}\left| {{\text{ }}x\;--\;1} \right|,\;\;\) then
  • A
    \(f{\text{ }}\left( {{\text{ }}|{\text{ }}x{\text{ }}|} \right){\text{ }} = {\text{ }}\left| {f{\text{ }}\left( x \right){\text{ }}} \right|\)
  • B
    \(f({x^2}) = {(f(x))^2}\)
  • C
    f (x) is not derivable at x = 1.
    Correct
  • D
    f (x + y) = f (x) + f(y)
14
If f(x) = \(x\left( {\sqrt x - \sqrt {x + 1} } \right),\) then
  • A
    none of these
  • B
    f (x) is continuous but not differentiable at x = 0
  • C
    f (x) is not differentiable at x = 0
    Correct
  • D
    f (x) is differentiable at x = 0
15
In case of strict increasing functions, slope of the tangent and hence derivative is
  • A
    positive
  • B
    negative
  • C
    zero
  • D
    either positive or zero
    Correct