JEE Maths - Definite Integration

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Question - 1

Let p (x) be a polynomial of least degree whose graph has three points of inflection (-1, -l), (1, l) and a point with abscissa 0 at which the curve is inclined to the axis of abscissa at an angle of 60o. Then \(\int_{0}^{1} p(x) d x\) is equal to

  • A \(\frac{3 \sqrt{3}+4}{14}\)
  • B \(\frac{3 \sqrt{3}}{7}\)
  • C \(\frac{\sqrt{3}+\sqrt{7}}{14}\)
  • D \(\frac{\sqrt{3}+2}{7}\)

Question - 2

\(\int_{-\pi}^{\pi}(\cos p x-\sin q x)^{2} d x\) where p,q are integers is equal to 

  • A -\(\pi\)
  • B 0
  • C \(\pi\)
  • D 2\(\pi\)

Question - 3

The value of the definite integral \(\int_{0}^{2 \pi} e^{\cos \theta} \cos \theta(\sin \theta) d \theta\) is

  • A 0
  • B \(\pi\)
  • C 2\(\pi\)
  • D e\(\pi\)

Question - 4

If p, q, r, s are in arithmetic progression and \(f(x)=\left|\begin{array}{ccc} p+\sin x & q+\sin x & p-r+\sin x \\ q+\sin x & r+\sin x & -1+\sin x \\ r+\sin x & s+\sin x & s-q+\sin x \end{array}\right|\) such that \(\int_{0}^{2} f(x) d x=-4\) then the common difference of the progession is 

  • A \(\pm 1\)
  • B \(\frac{1}{2}\)
  • C \(\pm 2\)
  • D None of these

Question - 5

If \(\int_{0}^{\pi / 2} \frac{d x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}=\frac{\pi}{16}\) Then minimum value of a cosx + b sinx is

  • A -4
  • B -8
  • C -2
  • D \(-2 \sqrt{2}\)

Question - 6

\(\text { Let } I=\int_{\pi / 4}^{\pi / 3} \frac{\sin x}{x} d x\) then I belongs to

  • A \(\left(\frac{\sqrt{3}}{8}, \frac{\sqrt{2}}{6}\right)\)
  • B \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}\right)\)
  • C \(\left(\frac{1}{2}, \frac{\sqrt{2}}{2}\right)\)
  • D None of these

Question - 7

Let f(x) be positive, continuous and differentiable on the interval (a, b) and \(\lim _{x \rightarrow a^{+}} f(x)=1, \lim _{x \rightarrow b^{-}} f(x)=3^{1 / 4}\). If \(f^{\prime}(x) \geq f^{3}(x)+\frac{1}{f(x)}\) then the greatest value of b-a is

  • A 1
  • B 31/4
  • C \(\left(3^{1 / 4}-1\right) \frac{\pi}{24}\)
  • D \(\frac{\pi}{24}\)

Question - 8

suppose the limit L \(=\lim _{n \rightarrow \infty} \sqrt{n} \int_{0}^{1} \frac{1}{\left(1+x^{2}\right)^{n}} d x\) exists and is larger than \(\frac{1}{2}\) Then

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

Question - 9

Suppose a continuous function \(f:[0, \infty) \rightarrow \mathbf{R}\) satisfies \(f(x)=2 \int_{0}^{x} t f(t) d t+1 \text { for all } x \geq 0\)  Then f(l) equals

  • A e
  • B e2
  • C e4
  • D e6

Question - 10

Let \(J=\int_{0}^{1} \frac{x}{1+x^{8}} d x\) .
Consider the following assertions 
\(I. J>\frac{1}{4} II. J<\frac{\pi}{8}\)
Then

  • A only I is true
  • B only II is true
  • C both I and II are true
  • D neither I nor II is true
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