JEE Maths - Application Of Derivatives

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Question - 1

If y = a sin 3x + b cos 3x satisfies \(\frac{d^{2} y}{d x^{2}}+4 \frac{d y}{d x}+3 y=10 \cos 3 x\) then find the value of 3(a + b).

  • A 0
  • B 1
  • C 3
  • D 4

Question - 2

If y = tan-1 (sec x - tanx), then frnd dy/dx.

  • A -0.5
  • B 0.5
  • C 1
  • D 1.5

Question - 3

Let f = (-1, 1) \(\rightarrow\) R be a differentiable function with f (0) = -1 and f '(0) = l. If \(g(x)=[f\{2 f(x)+2\}]^{2}, \text { then find } g^{\prime}(0)\)

  • A 1
  • B 4
  • C -4
  • D -1

Question - 4

\(\text { If } y=\tan ^{-1}\left(\frac{1}{x^{2}+x+1}\right)+\tan ^{-1}\left(\frac{1}{x^{2}+3 x+3}\right) +\tan ^{-1}\left(\frac{1}{x^{2}+5 x+7}\right)+\tan ^{-1}\left(\frac{1}{x^{2}+7 x+13}\right) x>0 \text { and }\left(\frac{d y}{d x}\right)_{x=0}=\frac{-k}{1+k}, \text { find the value of } k . \)

  • A 12
  • B 14
  • C 16
  • D 18

Question - 5

Let f (x) = x(x + 3)(x - 2), x \(\in\) [-l , 4] If f '(c) = 10, then find the value of c in (-1,4).

  • A 1
  • B 2
  • C 3
  • D 4

Question - 6

If \(\frac{d}{d x}\left(\frac{1+x^{4}+x^{8}}{1+x^{2}+x^{4}}\right)=a x^{3}+b x\) then find the value of a+b.

  • A 2
  • B 4
  • C 6
  • D 8

Question - 7

Let g (x) = log f (x), where f (x) is twice differentiable positive function on (0, \(\infty\)) such that f(x+ 1) = x f(x).Find \(g^{\prime \prime}\left(\frac{3}{2}\right)-g^{\prime \prime}\left(\frac{1}{2}\right)\)

  • A 1
  • B 2
  • C 3
  • D -4

Question - 8

lt f (x) = x3 + ex/2 and g(x) is the inverse of f (x), then find g'(l).

  • A 1
  • B 2
  • C 3
  • D 4

Question - 9

If \(2 x=y^{\frac{1}{3}}+y^{\frac{-1}{3}} \text { and }\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+k y=0\) then find the value of k.

  • A 7
  • B 8
  • C -9
  • D 9

Question - 10

Let f = R \(\rightarrow\) R be differentiable at x = 0. If f(0)  = 0 and f''(0) = 2, then find \(\lim _{x \rightarrow 0} \frac{1}{x}[f(x)+f(2 x)+f(3 x)+\ldots .+f(15 x)]\)

  • A 120
  • B 240
  • C 360
  • D 480
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