Chapter 13

If \(f(x)=\sqrt{x^{2}-10 x+25}\), then the derivative of \(f(x)\) on the interval \([0,7]\) is (A) 1 (B) \(-1\) (C) 0 (D) Does not exist

If the capital letters denote the cofactors of the corresponding small letters in the determinant \(\Delta=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|\) then the value of \(\Delta^{\prime}=\left|\begin{array}{ccc}A_{1} & B_{1} & C_{1} \\\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|\) is (A) 0 (B) \(2 \Delta\) (C) \(\Delta^{2}\) (D) \(\Delta\)

If \(f(x)=\cos x \cos 2 x \cos 4 x \cos 8 x\), then \(f^{\prime}\left(\frac{\pi}{4}\right)\) is (A) \(-1\) (B) 2 (C) \(\sqrt{2}\) (D) None of these

If \(y=e^{n x}\), then \(\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d^{2} x}{d y^{2}}\right)\) is equal to (A) \(n e^{n x}\) (B) \(n^{2} e^{n x}\) (C) \(-n e^{n x}\) (D) \(-n e^{-n x}\)

If the parametric equation of a curve is given by \(x=\cos \theta+\log \tan \frac{\theta}{2}\) and \(y=\sin \theta\), then the points for which \(\frac{d^{2} y}{d x^{2}}=0\) are given by (A) \(\theta=n \pi, n \in Z\) (B) \(\theta=(2 n+1) \pi / 2, n \in Z\) (C) \(\theta=(2 n+1) \pi, n \in Z\) (D) \(\theta=2 n \pi, n \in Z\).

If \(y=\tan ^{-1}\left(\frac{\log \left(e / x^{3}\right)}{\log \left(e x^{3}\right)}\right)+\tan ^{-1}\left(\frac{\log \left(e^{4} x^{3}\right)}{\log \left(e / x^{12}\right)}\right)\), then \(\frac{d^{2} y}{d x^{2}}\) is equal to (A) 1 (B) 0 (C) \(-1\) (D) None of these

Let \(f(x)=\left|\begin{array}{ccc}x^{3} & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^{2} & p^{3}\end{array}\right|\), where \(p\) is a constant. Then \(\frac{d^{3}}{d x^{3}}[f(x)]\) at \(x=0\) is (A) \(p\) (B) \(p+p^{2}\) (C) \(p+p^{3}\) (D) independent of \(p\)

The function \(y\) defined by the equation \(x y-\log y=1\) satisfies \(x\left(y y^{\prime \prime}+y^{\prime 2}\right)-y^{\prime \prime}+k y y^{\prime}=0 .\) The value \(k\) is (A) \(-3\) (B) 3 (C) 1 (D) None of these

If the function \(y(x)\) represented by \(x=\sin t\) \(y=a e^{l \sqrt{2}}+b e^{l \sqrt{2}}, t \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) satisfies the equation \(\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}=k y\), then \(k\) is equal to (A) 1 (B) \(-2\) (C) 2 (D) None of these

Let \(F(x)=f(x) g(x) h(x)\) for all real \(x\), where \(f(x), g\) \((x)\) and \(h(x)\) are differentiable functions. At some point \(x_{0}\), if \(F^{\prime}\left(x_{0}\right)=21 F\left(x_{0}\right), f^{\prime}\left(x_{0}\right)=4 f\left(x_{0}\right), g^{\prime}\left(x_{0}\right)=-7\) \(g\left(x_{0}\right)\) and \(h^{\prime}\left(x_{0}\right)=k h\left(x_{0}\right)\) then \(k\) is equal to (A) 24 (B) 12 (C) \(-12\) (D) \(-24\)

If \(f(x)\) is a polynomial of degree \(n(>2)\) and \(f(x)=\) \(f(k-x)\), (where \(k\) is a fixed real number), then degree of \(f^{\prime}(x)\) is (A) \(n\) (B) \(n-1\) (C) \(n-2\) (D) None of these

If \(f(x)=|x-1|\) and \(g(x)=f\\{f[f(x)]\\}\), then for \(x>2\), \(g^{\prime}(x)\) is equal to (A) \(-1\) if \(2 \leq x<3\) (B) 1 if \(2 \leq x<3\) (C) 1 for all \(x>2\) (D) None of these

Let \(f(x)=|x-a| ;(a>0)\) and \(g(x)=f\\{f[f(x)]\\}\). Then \(g^{\prime}(\alpha) ;(\alpha>3 a)\) (A) does not exist (B) equal to 3 (C) equal to 1 (D) None of these

Let \(\phi(x)\) be the inverse of the function \(f(x)\) and \(f^{\prime}(x)=\frac{1}{1+x^{5}}\) then \(\frac{d}{d x} \phi(x)\) is (A) \(\frac{1}{1+[\phi(x)]^{5}}\) (B) \(\frac{1}{1+[f(x)]^{5}}\) (C) \(1+[\phi(x)]^{5}\) (D) \(1+[f(x)]^{5}\)

If \(y=\frac{1}{x}\) then \(\frac{d y}{\sqrt{1+y^{4}}}+\frac{d x}{\sqrt{1+x^{4}}}=\) (A) 0 (B) 1 (C) \(\frac{x}{y}\) (D) \(\frac{y}{x}\)

If \(y=\frac{f(x)}{\phi(x)}\) and \(z=\frac{f^{\prime}(x)}{\phi^{\prime}(x)}\), then \(\frac{f^{\prime \prime}}{f}-\frac{\phi^{\prime \prime}}{\phi}+\frac{2(y-z)}{f \phi}\left(\phi^{\prime}\right)^{2}=\) (A) \(\frac{d^{2} y}{d x^{2}}\) (B) \(\frac{1}{y} \frac{d^{2} y}{d x^{2}}\) (C) \(y \frac{d^{2} y}{d x^{2}}\) (D) None of these

Let \(g(x)\) be the inverse of an invertible function \(f(x)\) which is differentiable for all real \(x\), then \(g^{\prime \prime}(f(x))\) equals (A) \(-\frac{f^{\prime \prime}(x)}{\left[f^{\prime}(x)\right]^{3}}\) (B) \(\frac{f^{\prime}(x) f^{\prime \prime}(x)-\left[f^{\prime}(x)\right]^{2}}{f^{\prime}(x)}\) (C) \(\frac{f^{\prime}(x) f^{\prime \prime}(x)-\left[f^{\prime}(x)\right]^{2}}{\left[f^{\prime}(x)\right]^{2}}\) (D) None of these

Let \(f(x)=x^{n}, n\) being a non-negative integer. The value of \(n\) for which equality \(f^{\prime}(a+b)=f^{\prime}(a)+f^{\prime}(b)\) is valid for all \(a, b>0\) is (A) 5 (B) 1 (C) 2 (D) 4

The solution set of \(f^{\prime}(x)>g^{\prime}(x)\) where \(f(x)=(1 / 2) 5^{2 x+1}\) and \(g(x)=5^{x}+4 x \log 5\) is (A) \((1, \infty)\) (B) \((0,1)\) (C) \((0, \infty)\) (D) \([0, \infty)\)

If \(I_{n}=\frac{d^{n}}{d x^{n}}\left(x^{n} \log x\right)\), then \(I_{n}-n I_{n-1}=\) (A) \(n\) (B) \(n-1\) (C) \(n !\) (D) \((n-1) !\)

If \(f(x)\) be a differentiable function such that \(f(x y)=f(x)\) \(+f(y)\) for all \(x\) and \(y\), then \(f(e)+f(1 / e)=\) (A) 1 (B) 0 (C) \(-1\) (D) None of these

If for all \(x, y\) the function \(f\) is defined by \(f(x)+f(y)+\) \(f(x) \cdot f(y)=1\) and \(f(x)>0\) then (A) \(f^{\prime}(x)\) does not exist (B) \(f^{\prime}(x)=0\) for all \(x\) (C) \(f^{\prime}(0)

Let \(3 f(x)-2 f(1 / x)=x\), then \(f^{\prime}(2)\) is equal to (A) \(\frac{2}{7}\) (B) \(\frac{1}{2}\) (C) 2 (D) \(\frac{7}{2}\)

\(\frac{d^{2} x}{d y^{2}}\) equals (A) \(-\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3}\) (B) \(\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}\) (C) \(-\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}\left(\frac{d y}{d x}\right)^{-3}\) (D) \(\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-2}\)

Let \(y\) be an implicit function of \(x\) defined by, \(x^{2 x}-2 x^{x} \cot y-1=0\) Then \(y^{\prime}(1)\) equals (A) \(-1\) (B) 1 (C) \(\log 2\) (D) \(-\log 2\)

If \(S_{n}\) denotes the sum of \(n\) terms of a G.P. whose common ratio is \(r\), then \((r-1) \frac{d S_{n}}{d r}\) is equal to (A) \((n-1) S_{n}+n S_{n-1}\) (B) \((n-1) S_{n}-n S_{n-1}\) (C) \((n-1) S_{n}\) (D) None of these

Let \(f\left(\frac{x_{1}+x_{2}+\ldots+x_{n}}{n}\right)=\) \(\frac{f\left(x_{1}\right)+f\left(x_{2}\right)+\ldots+f\left(x_{n}\right)}{n}\) where all \(x_{i} \in R\) are independent to each other and \(n \in N\). If \(f(x)\) is differentiable and \(f^{\prime}(0)=a, f(0)=b\) then \(f^{\prime}(x)\) is equal to (A) \(a\) (B) 0 (C) \(b\) (D) None of these

If \(y^{2}=P(x)\), a polynomial of degree \(n \geq 3\), then \(2 \frac{d}{d x}\left(y^{3} \frac{d^{2} y}{d x^{2}}\right)=\) (A) \(-P(x) \times P^{\prime \prime \prime}(x)\) (B) \(P(x) \times P^{\prime \prime \prime}(x)\) (C) \(P(x) \times P^{\prime \prime}(x)\) (D) None of these

A function \(f(x)\) is so defined that for all \(x,[f(x)]^{n}=\) \(f(n x)\). If \(f^{\prime}(x)\) denotes derivative of \(f(x)\) with respect to \(x\), then \(f^{\prime}(x) \times f(n x)=\) (A) \(f(x)\) (B) 0 (C) \(f(x) \times f^{\prime}(n x)\) (D) None of these

If \(f, g, h\) are differentiable functions of \(x\) and \(\Delta=\left|\begin{array}{ccc}f & g & h \\ (x f)^{\prime} & (x g)^{\prime} & (x h)^{\prime} \\ \left(x^{2} f\right)^{\prime \prime} & \left(x^{2} g\right)^{\prime \prime} & \left(x^{2} h\right)^{\prime \prime}\end{array}\right|\) then \(\Delta^{\prime}\) (the derivative of \(\Delta\) with respect to \(x\) ) is given by (A) \(\left|\begin{array}{ccc}f^{\prime} & g^{\prime} & h^{\prime} \\ f & g & h \\ \left(x^{3} f^{\prime \prime}\right)^{\prime} & \left(x^{3} g^{\prime \prime}\right)^{\prime} & \left(x^{3} h^{\prime \prime}\right)^{\prime}\end{array}\right|\) (B) \(\left|\begin{array}{ccc}f & g & h \\ f^{\prime} & g^{\prime} & h^{\prime} \\ \left(x^{2} f^{\prime \prime}\right)^{\prime} & \left(x^{2} g^{\prime \prime}\right)^{\prime} & \left(x^{2} h^{\prime \prime}\right)^{\prime}\end{array}\right|\) (C) \(\left|\begin{array}{ccc}f & g & h \\ f^{\prime} & g^{\prime} & h^{\prime} \\ \left(x^{3} f^{\prime \prime}\right)^{\prime} & \left(x^{3} g^{\prime \prime}\right)^{\prime} & \left(x^{3} h^{\prime \prime}\right)^{\prime}\end{array}\right|\) (D) None of these

If \(\alpha\) is a repeated root of a quadratic equation \(f(x)=0\) and \(A(x), B(x), C(x)\) be polynomials of degree \(>2\), then the determinant \(\left|\begin{array}{ccc}A(x) & B(x) & C(x) \\ A(\alpha) & B(\alpha) & C(\alpha) \\ A^{\prime}(\alpha) & B^{\prime}(\alpha) & C^{\prime}(\alpha)\end{array}\right|\) is divisible by (A) \(A(x)\) (B) \(B(x)\) (C) \(C(x)\) (D) \(f(x)\)

If the capital letters denote the cofactors of the corresponding small letters in the determinant \(\Delta=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|\), then the value of \(\Delta^{\prime}=\left|\begin{array}{ccc}A_{1} & B_{1} & C_{1} \\\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|\) is (A) 0 (B) \(2 \Delta\) (C) \(\Delta^{2}\) (D) \(\Delta\)

If \(f(x)\) is a polynomial of degree \(n(>2)\) and \(f(x)=f(k-x)\), (where \(k\) is a fixed real number), then degree of \(f^{\prime}(x)\) is (A) \(n\) (B) \(n-1\) (C) \(n-2\) (D) None of these

If \(y=\frac{f(x)}{\phi(x)}\) and \(z=\frac{f^{\prime}(x)}{\phi^{\prime}(x)}\), then \(\frac{f^{\prime \prime}}{f}-\frac{\phi^{\prime \prime}}{\phi}+\frac{2(y-z)}{f \phi}\left(\phi^{\prime}\right)^{2}=\) (A) \(\frac{d^{2} y}{d x^{2}}\) (B) \(\frac{1}{y} \frac{d^{2} y}{d x^{2}}\) (C) \(y \frac{d^{2} y}{d x^{2}}\) (D) None of these

The solution set of \(f^{\prime}(x)>g^{\prime}(x)\) where \(f(x)=(1 / 2) 5^{2 x+1}\) and \(g(x)=5^{x}+4 x \log 5\) is (A) \((1, \infty)\) (B) \((0,1)\) (C) \((0, \infty)\) (D) \([0, \infty)\)

If for all \(x, y\) the function \(f\) is defined by \(f(x)+f(y)+\) \(f(x) \cdot f(y)=1\) and \(f(x)>0\), then (A) \(f^{\prime}(x)\) does not exist (B) \(f^{\prime}(x)=0\) for all \(x\) (C) \(f^{\prime}(0)

If \(\sqrt{x+y}+\sqrt{y-x}=c\) then \(\frac{d^{2} y}{d x^{2}}\) equals (A) \(\frac{2}{c^{2}}\) (B) \(\frac{-2}{c^{2}}\) (C) \(\frac{2}{c}\) (D) \(\frac{-2}{c}\)

Let \(f(x)=\prod_{k=1}^{n}(\cos (2 k-1) x+i \sin (2 k-1) x)\), then \((\operatorname{Re} f(x))^{\prime \prime}+i(\operatorname{Im} f(x))^{\prime \prime}\) is equal to (A) \(n^{2} f(x)\) (B) \(-n^{4} f(x)\) (C) \(-n^{2} f(x)\) (D) \(n^{4} f(x)\)

Let \(f(x)=\sqrt{x-1}+\sqrt{x+24-10 \sqrt{x-1}} ; 1

If \(f(x)=|x-2|\) and \(g(x)=f\\{f(x)\\}\), then \(g^{\prime}(x)\) for \(x>2\) is (A) \(-1\) (B) 1 (C) 0 (D) does not exist

The derivative of the function represented parametrically as \(x=2 t-|t|, y=t^{3}+t^{2}|t|\) at \(t=0\) is (A) 0 (B) 1 (C) \(-1\) (D) does not exist

A polynomial \(f(x)\) leaves remainder 15 when divided by \((x-3)\) and \((2 x+1)\) when divided by \((x-1)^{2} .\) When \(f\) is divided by \((x-3)(x-1)^{2}\), the remainder is (A) \(2 x^{2}+2 x+3\) (B) \(2 x^{2}-2 x-3\) (C) \(2 x^{2}-2 x+3\) (D) None of these

If for a non-zero \(x\), the function \(f(x)\) satisfies the equation \(a f(x)+b f\left(\frac{1}{x}\right)=\frac{1}{x}-5(a \neq b)\) then \(f^{\prime}(x)\) is equal to (A) \(\frac{1}{b^{2}-a^{2}}\left(\frac{a}{x^{2}}+b\right)\) (B) \(\frac{1}{a^{2}-b^{2}}\left(\frac{a}{x^{2}}+b\right)\) (C) \(\frac{1}{a^{2}-b^{2}}\left(\frac{a}{x^{2}}-b\right)\) (D) None of these

If \(f(x)=\cos ^{-1}\left(\frac{x^{-1}-x}{x^{-1}+x}\right)\), then \(f^{\prime}(x)\) is (A) odd (B) even (C) periodic (D) None of these

If \(f(x)=(1-x)^{n}\), then the value of \(f(0)+f^{\prime}(0)+\frac{f^{\prime \prime}(0)}{2 !}+\ldots+\frac{f^{n}(0)}{n !}\) is (A) \(n\) (B) 0 (C) \(2^{n}\) (D) \(2^{n}-1\)

If \(y=\cot ^{-1}\left(\frac{x^{x}-x^{-x}}{2}\right)\), then \(\frac{d y}{d x}\) at \(x=1\), equals (A) 0 (B) 1 (C) \(-1\) (D) None of these

If \(y=\sqrt{\frac{1+\cos 2 \theta}{1-\cos 2 \theta}}\), then (A) \(y^{\prime}\left(\frac{\pi}{4}\right)=y^{\prime}\left(\frac{3 \pi}{4}\right)\) (B) \(y^{\prime}\left(\frac{\pi}{4}\right) \cdot y^{\prime}\left(\frac{3 \pi}{4}\right)=-4\) (C) \(y^{\prime}\left(\frac{\pi}{4}\right)\) and \(y^{\prime}\left(\frac{3 \pi}{4}\right)\) do not exist (D) None of these

If \(5 f(x)+3 f\left(\frac{1}{x}\right)=x+2\) and \(y=x f(x)\) then \(\frac{d y}{d x}\) at \(x=1\) is equal to (A) 1 (B) \(-1\) (C) \(\frac{7}{8}\) (D) \(-\frac{7}{8}\)

Let \(f(x)\) be a polynomial function of degree 2 and \(f(x)\) \(>0\) for all \(x \in R\). If \(g(x)=f(x)+f^{\prime}(x)+f^{\prime \prime}(x)\), then for any \(x\) (A) \(g(x)>0\) (B) \(g(x)<0\) (C) \(g(x)=0\) (D) \(g(x) \leq 0\)

Let \(f(x+y)=f(x)+f(y)+2 x y-1 \forall x, y \in R\). If \(f(x)\) is differentiable and \(f^{\prime}(0)=\sin \theta\), then (A) \(f(x)>0 \forall x \in R\) (B) \(f(x)<0 \forall x \in R\) (C) \(f(x)=\sin \theta \forall x \in R\) (D) None of these