Continuity And Differentiability Test

Continuity And Differentiability

This is Continuity and Differentiability Test-01 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
\(\mathop {Lt}\limits_{x \to 0} \;\;\frac{{1 - \cos x}}{x}\) is equal to
  • A
    1
  • B
    none of these
  • C
    \(\frac{1}{2}\)
  • D
    0
    Correct
2
\(\mathop {\lim }\limits_{x \to 0} \frac{{2\sin x - \sin 2x}}{{{x^3}}}\)is equal to
  • A
    \(\frac{1}{2}\)
  • B
    0
  • C
    none of these
  • D
    1
    Correct
3
\(\mathop {Lt}\limits_{x \to \pi /4} \;\;\;\frac{{\cos x - \sin x}}{{x - \frac{\pi }{4}}}\) is equal to
  • A
    \(\frac{2}{{\sqrt 2 }}\)
  • B
    \( - \frac{1}{{\sqrt 2 }}\)
  • C
    – 1
  • D
    \( - \frac{2}{{\sqrt 2 }}\)
    Correct
4
If [x] stands for the integral part of x, then
  • A
    \(\mathop {Lt}\limits_{x \to {1^ + }} \;\;\;\left[ x \right] = 1\)
    Correct
  • B
    \(\mathop {Lt}\limits_{x \to 1} \;\;\;\left[ x \right] = 1\)
  • C
    \(\mathop {Lt}\limits_{x \to {1^ - }} \;\;\;\left[ x \right] = 1\)
  • D
    none of these
5
\(\mathop {Lt}\limits_{x \to \frac{\pi }{2}} \;\;[\sin x]\) is equal to
  • A
    1
  • B
    0
    Correct
  • C
    none of these
  • D
    – 1
6
\(\mathop {Lt}\limits_{x \to 0} \;\;\frac{{1 - \cos x}}{{{x^2}}}\) is equal to
  • A
    \(\frac{1}{2}\)
    Correct
  • B
    1
  • C
    – 1
  • D
    0
7
\(\mathop {Lt}\limits_{x \to - 2} \frac{{\sqrt {{x^2} + 5} - 3}}{{x + 2}}\) is equal to
  • A
    \( - \frac{2}{3}\)
    Correct
  • B
    0
  • C
    \(\frac{2}{3}\)
  • D
    none of these
8
\(\mathop {Lt}\limits_{h \to 0} \;\;\;\left( {\frac{1}{{h\sqrt[3]{{8 + h}}}} - \frac{1}{{2h}}} \right)\)is equal to
  • A
    none of these
  • B
    \(\frac{1}{{24}}\)
  • C
    \( - \frac{1}{{48}}\)
    Correct
  • D
    \(\frac{1}{{48}}\)
9
\(\mathop {Lt}\limits_{x \to \pi } \;\;\;\frac{{1 + {{\cos }^3}x}}{{{{(x - \pi )}^2}}}\)is equal to
  • A
    \(\frac{1}{3}\)
  • B
    none of these
  • C
    \(\frac{1}{2}\)
  • D
    \(\frac{3}{2}\)
    Correct
10
\(\mathop {Lt}\limits_{x \to - 3} \;\;\;\frac{{\sqrt {{x^2} + 7} - 4}}{{x + 3}}\) is equal to
  • A
    \(\frac{4}{3}\)
  • B
    \(\frac{3}{4}\)
  • C
    \( - \frac{3}{4}\)
    Correct
  • D
    none of these
11
If k be an integer, then \(\mathop {Lt}\limits_{x \to k} \;\;\) (x –[x])
  • A
    is equal to 0
  • B
    does not exist
    Correct
  • C
    is equal to 1
  • D
    is equal to – 1
12
\(\mathop {Lt}\limits_{x \to 0} \;\;\;\frac{{x\cos x - \log (1 + x)}}{{{x^2}}}\) is equal to
  • A
    \(\frac{1}{2}\)
    Correct
  • B
    0
  • C
    1
  • D
    none of these
13
\(\mathop {Lt}\limits_{x \to 0} \;\;\;\frac{{1 - \cos 4x}}{{{x^2}}}\) is equal to
  • A
    \(\frac{1}{2}\)
  • B
    0
  • C
    none of these
  • D
    \(\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos 4x}}{{{x^2}}}\) $ = \mathop {\lim }\limits_{x \to 0} \frac{{4\sin 4x}}{{2x}} $ $ = \mathop {\lim }\limits_{x \to 0} \frac{{16\cos 4x}}{2} = 8 $ ( using L’Hospital Rule)
    Correct
14
\(\mathop {Lt}\limits_{x \to 0} \;\;\;\frac{{\cos ec\;x - \cot x}}{x}\) is equal to
  • A
    0
  • B
    none of these
  • C
    \(\frac{1}{2}\)
    Correct
  • D
    1
15
\(\mathop {Lt}\limits_{x \to 0} \;\;\frac{{1 - \cos x}}{{x\sin x}}\) is equal to
  • A
    2
  • B
    \(\frac{1}{2}\)
    Correct
  • C
    1
  • D
    0