12th Standard cbse -- Maths - Continuity and Differentiability

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Question - 1

Given functions f(x) = \(\frac { { x }^{ 2 }-4 }{ x-2 } \) and g(x) = x + 2, x <= R. Then which of the following is

  • A f is continuous at x = 2, g is continuous at x = 2
  • B f is continuous at x = 2, g is not continuous at x = 2
  • C f is not continuous at x = 2, g is continuous at x = 2
  • D f is not continuous at x = 2, g is not continuous at x = 2

Question - 2

\(\lim _{ x\rightarrow 0 }{ \frac { \sqrt { \frac { 1 }{ 2 } (1-cosx) } }{ x } } \) is equal to

  • A 1
  • B -1
  • C 0
  • D none of this

Question - 3

If \(f(x)=\frac { sin({ e }^{ x-2 }-1) }{ log(x-1) } \), x ≠ 2 and f(x) = k for x = 2, then value of k for which f is continuous is

  • A -2
  • B -1
  • C 0
  • D 1

Question - 4

A function /is said to be continuous for x ∈ R, if

  • A it is continuous at x = 0
  • B differentiable at x = 0
  • C continuous at two points
  • D differentiable for x ∈ R

Question - 5

A function \(f(x)=\begin{cases} \frac { sinx }{ x } +cosx,x\neq 0 \\ 2k\quad \quad \quad \quad ,x=0 \end{cases}\) is continuous at x = 0 for

  • A  k = 1
  • B k = 2
  • C K = \(\frac12\)
  • D k = \(\frac32\)

Question - 6

Write the number of points where f(x) = |x + 2| + |x – 3| is not differentiable

  • A 2
  • B 3
  • C 0
  • D 1

Question - 7

Derivative of cot x° with respect to x is

  • A cosec x°
  • B cosec x° cot x°
  • C -1° cosec2 x°
  • D -1° cosec x° cot x°

Question - 8

If y = sin-1 \(\left( \frac { 3x }{ 2 } -\frac { { x }^{ 3 } }{ 2 } \right) \), then \(\frac { dy }{ dx } \) is

  • A \(\frac { 3 }{ \sqrt { 4-{ x }^{ 2 } } } \)
  • B \(\frac { -3 }{ \sqrt { 4-{ x }^{ 2 } } } \)
  • C \(\frac { 1 }{ \sqrt { 4-{ x }^{ 2 } } } \)
  • D \(\frac {- 1 }{ \sqrt { 4-{ x }^{ 2 } } } \)

Question - 9

If f(x) = logx2 (log x), then f(e) is

  • A 0
  • B 1
  • C \(\frac1e\)
  • D \(\frac{1}{2e}\)

Question - 10

If f(x) = ex and g(x) = loge x, then (gof)’ (x) is

  • A 0
  • B 1
  • C e
  • D 1 + e
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