Engineering Mathematics - Calculus

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Question - 1

The value of the quantity P, where P=\(\int _{ 0 }^{ 1 }{ { xe }^{ x }dx } \quad is\)eqial to 

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

Question - 2

If S =\(\int _{ 1 }^{ \infty }{ { x }^{ -3 } } dx\quad then,\quad S\quad has\quad the\quad value\)

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

Question - 3

At t=0, the function f(t)=\(\frac { sin\quad t }{ t } has\)

  • A a minimum
  • B a discontinuity
  • C a point of inflection
  • D a maximum

Question - 4

\(The\quad \underset { x\rightarrow 0 }{ lim } \frac { sin\frac { 2 }{ 3 } x }{ x } is\)

  • A \(\frac { 2 }{ 3 } \)
  • B 1
  • C \(\frac { 1 }{ 4 } \)
  • D \(\frac { 1 }{2 } \)

Question - 5

What is the value of the definite integral \(\int _{ 0 }^{ a }{ \frac { \sqrt { x } }{ \sqrt { x } +\sqrt { a+x } } } dx?\)

  • A 0
  • B \(\frac { a }{ 2 } \)
  • C a
  • D 2a

Question - 6

Evaluate \(I=\int _{ -\infty }^{ \infty }{ \frac { dx }{ 1+{ x }^{ 2 } } } \)

  • A \(\pi \)
  • B \(\frac { \pi }{ 2 } \)
  • C \(2\pi \)
  • D \(\frac { \pi }{ 4 } \)

Question - 7

\(The\quad value\quad of\quad \underset { x\rightarrow 8 }{ lim } \frac { { x }^{ 1/3 }-2 }{ x-8 } is\)

  • A \(\frac { 1 }{ 16 } \)
  • B \(\frac { 1 }{ 12 } \)
  • C \(\frac { 1 }{ 8 } \)
  • D \(\frac { 1 }{ 4 } \)

Question - 8

\(\int _{ -a }^{ a }{ ({ sin }^{ 6 }x+{ sin }^{ 7 }x)dx\quad is\quad equal\quad to\quad } \)

  • A \(2\int _{ 0 }^{ a }{ { sin }^{ 6 } } x\quad dx\)
  • B \(2\int _{ 0 }^{ a }{ { sin }^{ 7 } } x\quad dx\)
  • C \(2\int _{ 0 }^{ a }{ { sin }^{ 6 } } x+{ sin }^{ 7 }x)\quad dx\)
  • D Zero

Question - 9

\(\underset { x\rightarrow 0 }{ lim } \frac { { sin }^{ 2 }x }{ x } is\quad equal\quad to\quad \)

  • A 0
  • B \(\infty \)
  • C 1
  • D -1

Question - 10

\(\underset { x\rightarrow 1 }{ lim } \frac { { x }^{ 2 }-1 }{ x-1 } is\)

  • A \(\infty \)
  • B 0
  • C 2
  • D 1
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