CBSE Questions for Class 11 Commerce Applied Mathematics Differentiation Quiz 13 - MCQExams.com

Statement I: lf $$f(\theta)=\cos\theta_{1}.\cos\theta_{2}\ldots.\cos\theta_{n}$$, then the value of $$\displaystyle\tan\theta_{1}+\tan\theta_{2}+\ldots+\tan\theta_{n}=-\frac{f'(\theta)}{f(\theta)}$$
Statement II: Differential coefficient of the function $$f(g(x))$$ w.r.t function $$g(x)$$ is $$f'(g(x))$$.

Which of the above statements is/are correct?
  • Statement $$I$$ only
  • Statement $$II$$ only
  • Both $$I$$ and $$II$$
  • Neither $$I$$ nor $$II$$
Assertion (A) If $$\sin(\mathrm{x}+\mathrm{y})=\log_{e}(\mathrm{x}+\mathrm{y})$$, then $$\displaystyle \frac{dy}{dx}=-1$$

Reason (R): The derivative of an odd function is always an even function
  • Both A and R are true R is the correct reason of A
  • Both A and R are true R is not the correct reason of A
  • A is true but R is false
  • A is false but R is true
If $$y=\cos 2x\cos 3x$$, then $$y_n$$ is equal to $$\\$$
Where, $$y_n$$ denotes the $$n^{th} $$ derivative of $$y$$.
  • $$6^{\mathrm{n}}\displaystyle \cos\left(2x+\frac{\mathrm{n}\pi}{2}\right)\cos\left(3x+\frac{\mathrm{n}\pi}{2}\right)$$
  • $$\displaystyle \frac{1}{2}\left [5^{\mathrm {n}}\cos\left ( \frac{\mathrm {n}\pi }{2}+5x \right )+\cos\left ( \frac{\mathrm{n}\pi }{2}+x \right ) \right ]$$
  • $$\displaystyle \frac{1}{2}\left[5^{\mathrm{n}}\sin\left(5x+\frac{\mathrm{n}\pi}{2}\right)+\sin\left(x+\frac{\pi}{2}\right)\right]$$
  • $$0$$
$$f(x)=|x-1|+|x-3|$$ then $$f^{'}(2)=$$
  • $$-2$$
  • $$2$$
  • $$0$$
  • $$1$$
Statement I: If $$ {x}= {e}^{  {y}+   {e}^{  {y+e}^{  {y}+\ldots\infty}}}$$, then $$ \dfrac{ {d} {y}}{ {d} {x}}=\dfrac{ {x}}{1- {x}}$$

Statement II: If $$ {y}=| \cos {x}|+|\sin  {x}|$$, then the value of $$ \dfrac{ {d} {y}}{ {d} {x}}$$ at $$  {x}=\dfrac{2\pi}{3}$$ is $$ \left( \dfrac{\sqrt{3}-1}{2}\right )$$

Which of the above statement(s) is/are correct?
  • Statement $$I$$ only
  • Statement $$II$$ only
  • Both $$I$$ and $$II$$
  • Neither $$I$$ nor $$II$$
 lf $$\displaystyle \int _{ 0 }^{ { y } } \frac { 1 }{ \sqrt { 1+9{ u }^{ 2 } }  } du=u$$, then $$\dfrac { { d }^{ 2 }{ y } }{ { d }{ u }^{ 2 } } $$ is
  • $$\sqrt{1+9{y}^{2}}$$
  • $$\displaystyle \frac{1}{\sqrt{1+9{y}^{2}}}$$
  • $$9{y}$$
  • $$9{y}^{2}$$
$$g(x+y)=g(x)+g(y)+3xy(x+y)\:\forall\:x,\:y\epsilon R$$ and $$g^\prime(0)=-4$$.For which of the following values of $$x$$ is $$\sqrt{g(x)}$$ not defined?
  • $$[-2,0]$$
  • $$[-2,\infty]$$
  • $$[-1,1]$$
  • none of these
Let $${f}({x})$$ be a polynomial of degree two which is positive for all $$ x \in R$$. lf $${g}({x})={f}({x})+{f}'({x})+{f}''(x)+xf'''({x})+{x}^{2}{f}^{{iv}}({x})$$, then for any real $$x$$
  • $$g(x)<0$$
  • $$g(x)>0$$
  • $$g(x)=0$$
  • $$g(x)\geq $$0
The $$n^{th}$$ derivative of $$h(x)=e^{3x+5}x^{2}$$ at $$x=0$$ is
  • $$e^{5}3^{n-2}n(n-1)$$
  • $$e^{5}3^{n+2}n(n-1)$$
  • $$e^{5}3^{n}n(n-1)$$
  • $$e^{5}3^{n-2}n(n+1)$$

Given that $$f (x)$$
is a differentiable function of $$ x$$ and that $$f(x)$$ . $$f (y)$$ =  $$f (x) $$+ $$f (y)$$ + $$f (xy) -2$$ and that
$$f (2) =5$$.

Then $$f (3)$$ is equal to?
  • $$6$$
  • $$24$$
  • $$15$$
  • $$19$$
lf $$\displaystyle \mathrm{f}(\mathrm{x})=log\left(\frac{1+\mathrm{x}}{1-\mathrm{x}}\right)$$ and $$\displaystyle \mathrm{g}(\mathrm{x})=\frac{3\mathrm{x}+\mathrm{x}^{3}}{1+3\mathrm{x}^{2}}$$ then $$\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{x}}(\mathrm{f}(\mathrm{g}(\mathrm{x})))$$ equals
  • $$- f^{'}(x)$$
  • $$-f(x)$$
  • $$3(f(x))^{2}$$
  • $$3f^{'}(x)$$
Let $$\mathrm{f}(\mathrm{x})$$ be a function satisfying $$f(x+y)=\mathrm{f}(\mathrm{x})\mathrm{f}(\mathrm{y})$$ for all $$x,y$$ $$\in \mathrm{R}$$ and $$\mathrm{f}(\mathrm{x})=1+\mathrm{x}\mathrm{g}(\mathrm{x})$$, where $$\displaystyle\lim_{x\rightarrow 0}\mathrm{g}(\mathrm{x})=1$$, then $$\mathrm{f}'(\mathrm{x})$$ is equal to
  • $$x\mathrm {g(x)}$$
  • $$\mathrm{g'(x)}$$
  • $$\mathrm{f(x)}$$
  • $$0$$
Suppose for a differentiable function $$f,\>f(0)=0,\>f(1)=1,\>f'(0)=4=f'(1)$$.
If $$ g(x)= f(e^{x})e^{f(x)}$$, then $$g'(0)$$ is equal to
  • $$4$$
  • $$8$$
  • $$2$$
  • $$0$$
Assertion (A): lf $$f(x)=\cos^{2}x+\cos^{2}\left(x+\dfrac{\pi}3\right)- \cos x \cos \left(x+\dfrac{\pi}3\right)$$ then $$f'(x)=0$$

Reason(R): Derivative of constant function is zero
  • Both A & R are true, R is correct explanation for A
  • Both A & R are true,R is not correct explanation for A
  • A is true but R is false
  • A is false but R is true
lf $$y^2=p(x)$$ , a polynomial of degree $$3$$, then $$\displaystyle 2\frac{d}{dx}\left(y^3\dfrac{d^2y}{dx^2}\right)$$, is equal to
  • $$p'''(x)+p'(x)$$
  • $$p''(x)+p'''(x)$$
  • $$p(x)p'''(x)$$
  • constant
lf $$\mathrm{f}(\mathrm{x})$$ is a quadratic expression which is positive for all real vaues of $$\mathrm{x}$$ and $$\mathrm{g}(\mathrm{x})=\mathrm{f}(\mathrm{x})+\mathrm{f}'(\mathrm{x})+\mathrm{f}''(\mathrm{x})$$ then for any real value of $$\mathrm{x}$$
  • $$\mathrm{g}(\mathrm{x})<0$$
  • $$\mathrm{g}(\mathrm{x})>0$$
  • $$\mathrm{g}(\mathrm{x})=0$$
  • $$\mathrm{g}(\mathrm{x})\underline{>}0$$
Let $$f$$ be a twice differentiable function such that $$f''\left( x \right) =-f\left( x \right) $$ and $$f'(x)=g(x).$$.
If $$h'\left( x \right) ={ \left[ f\left( x \right) \right]  }^{ 2 }+{ \left[ g\left( x \right) \right]  }^{ 2 },h\left( 1 \right) =6$$ and $$h(0)=4$$ then $$h(4)$$ is equal to?
  • $$16$$
  • $$12$$
  • $$13$$
  • None of these
If $$y=x^{\displaystyle x^{\displaystyle x^{\displaystyle \dots^{\displaystyle\infty}}}}$$, find $$\displaystyle\frac{dy}{dx}$$.
  • $$\displaystyle\frac{y^2}{x(1-y\log{x})}$$
  • $$\displaystyle\frac{y}{x(1-\log{x})}$$
  • $$\displaystyle\frac{y^2}{x(y-\log{x})}$$
  • None of these
If $${ e }^{ y }+xy=e$$ then at $$x=0$$,$$\displaystyle \frac{d^2y}{dx^2}=e^{-\lambda,}$$ then numerical quantity $$-\lambda$$ should be equal to
  • $$2$$
  • $$3$$
  • $$4$$
  • $$5$$
Derivative of $$({\log{x}})^{\displaystyle\cos{x}}$$ with respect to $$x$$ is
  • $$\displaystyle({\log{x}})^{\displaystyle\cos{x}}\left[\frac{\cos{x}}{x\log{x}}-\sin{x}\log{(\log{x})}\right]$$
  • $$\displaystyle({\log{x}})^{\displaystyle\cos{x}}\left[\frac{\cos{x}}{x\log{x}}-\cos{x}\log{(\log{x})}\right]$$
  • $$\displaystyle({\log{x}})^{\displaystyle\sin{x}}\left[\frac{\sin{x}}{x\log{x}}-\cos{x}\log{(\log{x})}\right]$$
  • None of these
Let $$\mathrm{f}$$: $$(-1,1 )\rightarrow \mathrm{R}$$ be a differentiable function with $$\mathrm{f}(\mathrm{0})=-1$$ and $$\mathrm{f}'(\mathrm{0})=1$$. Let $$\mathrm{g}(\mathrm{x})=[\mathrm{f}(2\mathrm{f}(\mathrm{x})+2)]^{2}$$. Then $$\mathrm{g}'(\mathrm{0})=$$
  • $$-4$$
  • $$0$$
  • $$-2$$
  • $$4$$
If $$x^m y^n=(x+y)^{m+n}$$, then $$\displaystyle\frac{dy}{dx}$$ is
  • $$\displaystyle -\frac{y}{x}$$
  • $$\displaystyle\frac{y}{x}$$
  • $$\displaystyle\frac{x}{y}$$
  • None of these
If y is a function of x and $$ln (x + y) - 2xy = 0,$$ then the value of y'(0) is equal to
  • 1
  • 2
  • 3
  • 4
If $$f(x)={|x|}^{|\sin{x}|}$$, then $$\displaystyle f^\prime\left(-\frac{\pi}{4}\right)=$$
  • $$\displaystyle {\left(\frac{\pi}{5}\right)}^{\tfrac{1}{\sqrt{2}}}\left(\frac{\sqrt{2}}{2}\log{\frac{5}{\pi}}-\frac{2\sqrt{2}}{\pi}\right)$$
  • $$\displaystyle {\left(\frac{\pi}{4}\right)}^{\tfrac{1}{\sqrt{2}}}\left(\frac{\sqrt{2}}{2}\log{\frac{4}{\pi}}-\frac{2\sqrt{2}}{\pi}\right)$$
  • $$\displaystyle {\left(\frac{\pi}{3}\right)}^{\tfrac{1}{\sqrt{2}}}\left(\frac{\sqrt{2}}{2}\log{\frac{3}{\pi}}-\frac{3\sqrt{3}}{\pi}\right)$$
  • None of these
Let $$y=\sqrt{x+\sqrt{x+\sqrt{x+\cdots\infty}}}$$, then $$\displaystyle\frac{dy}{dx}$$ is equal to
  • $$\displaystyle\frac{1}{2y-1}$$
  • $$\displaystyle\frac{x}{x+2y}$$
  • $$\displaystyle\frac{1}{\sqrt{1+4x}}$$
  • $$\displaystyle\frac{y}{2x+y}$$
 Let $$\mathrm{f}(\mathrm{x})=\mathrm{x}+\tan^{-1}\mathrm{x}, \displaystyle \mathrm{g}(\mathrm{x})=\frac{\mathrm{x}}{1+\mathrm{x}^{2}}(\mathrm{x}>0)$$ Then
  • $$\mathrm{f}(\mathrm{x})<\mathrm{g}(\mathrm{x}), \mathrm{x}>0$$
  • $$\mathrm{f}(\mathrm{x})>\mathrm{g}(\mathrm{x}), \mathrm{x}>0$$
  • $$\mathrm{f}(\mathrm{x})<\mathrm{g}(\mathrm{x})$$ in $$[1, \infty)$$
  • None of these
If $$x<1$$,  then $$\displaystyle\frac{1}{1+x}+\frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\cdots\infty=$$
  • $$\displaystyle -\frac{1}{1+x}$$
  • $$\displaystyle\frac{1}{1-x}$$
  • $$\displaystyle -\frac{1}{1-x}$$
  • $$\displaystyle\frac{1}{1+x}$$
$$g(x+y)=g(x)+g(y)+3xy(x+y)\:\forall\:x,\:y\epsilon R$$ and $$g^\prime(0)=-4$$.The value of $$g^\prime(1)$$ is
  • $$0$$
  • $$1$$
  • $$-1$$
  • none of these
If $$y=e^{\displaystyle\sqrt{x}}+e^{\displaystyle -\sqrt{x}}$$, then $$\displaystyle\frac{dy}{dx}$$ is equal to
  • $$\displaystyle\frac{e^{\displaystyle\sqrt{x}}-e^{\displaystyle -\sqrt{x}}}{2\sqrt{x}}$$
  • $$\displaystyle\frac{e^{\displaystyle\sqrt{x}}-e^{\displaystyle -\sqrt{x}}}{2x}$$
  • $$\displaystyle\frac{1}{2\sqrt{x}}\sqrt{y^2-4}$$
  • $$\displaystyle\frac{1}{2\sqrt{x}}\sqrt{y^2+4}$$
$$g(x+y)=g(x)+g(y)+3xy(x+y)\:\forall\:x,\:y\epsilon R$$ and $$g^\prime(0)=-4$$.The number of real roots of the equation $$g(x)=0$$ is
  • $$2$$
  • $$0$$
  • $$1$$
  • $$3$$
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