91Ó°ÊÓ

If \(y=\frac{\sec x+\tan x-1}{\sec x-\tan x+1}\), find \(\frac{d y}{d x}\) at \(x=0\)

Let \(f\) be differentiable for all \(x\). If \(f(1)=-2\) and \(f^{\prime}(x) \geq 2\) for all \(x \in[1,6]\) then (a) \(f(6)<8\) (b) \(f(6)>8\) (c) \(f(6)>5\) (d) None

If \(x=\tan \left(\frac{y}{2}\right)-\left(\frac{(1+\tan (y / 2))^{2}}{\tan (y / 2)}\right)\), prove that \(2 \frac{d y}{d x}=-\sin y(1+\sin y+\cos y)\)

If \(a, b, c\) be non-zero real numbers such that $$ \begin{aligned} & \int_{0}^{1}\left(1+\cos ^{8} x\right)\left(a x^{2}+b x+c\right) d x \\ =& \int_{0}^{2}\left(1+\cos ^{8} x\right)\left(a x^{2}+b x+c\right) d x \end{aligned} $$ then the equation \(a x^{2}+b x+c=0\) will have a root between (a) \((1,3)\) (b) \((1,2)\) (c) \((2,3)\) (d) \((3,1)\)

If \(y=\cos ^{-1}\left(\sqrt{\frac{\cos 3 x}{\cos ^{3} x}}\right)\), prove that \(\frac{d y}{d x}=\sqrt{\frac{6}{\cos 2 x+\cos 4 x}}\)

If \(\frac{x^{4}+x^{2}+1}{x^{2}-x+1}\) such that \(\frac{d y}{d x}=a x+b\), find the, value of \(a+b+10\)

If \(y=\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2}\), find \(\frac{d y}{d x}\) at \(x=\frac{\pi}{6}\)

If \(\sqrt{x^{2}+y^{2}}=a e^{\tan ^{-1}} x\), where \(a>0, y(0) \neq 0\) then find the value of \(y^{\prime \prime}(0)\).

The value of \(|\cos a-\cos b|\) is (a) \(\leq|a-b|\) (b) \(\geq|a-b|\) (c) 0 (d) \(|a+b|\).

If \(y=\left(\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}}\right)\), prove that \(2 x y \frac{d y}{d x}=\frac{x}{a}-\frac{a}{x}\)