CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 12 - MCQExams.com

Given $$P(x)=x^4+ax^3+bx^2+cx+d$$ such that $$x=0$$ is the only real root of $$P(x)=0$$. If $$P(-1) < P(1)$$, then in the interval $$[-1, 1]$$.
  • $$P(-1)$$ is the minimum and $$P(1)$$ is the maximum of P
  • $$P(-1)$$ is not minimum but $$P(1)$$ is the maximum of P
  • $$P(-1)$$ is the minimum and $$P(1)$$ is not the maximum of P
  • Neither $$P(-1)$$ is the minimum not $$P(1)$$ is the maximum of P
Find the slope of tangent of the curve$$x = a\,{\sin ^3}t,y = b\,\,{\cos ^3}t$$ at $$t = \frac{\pi }{2}$$
  • $$cott$$
  • $$-tant$$
  • $$-cott$$
  • $$\text{not defined  at}$$ $$\frac{\pi}{2}$$
The interval in which $$y = x^{2} e^{-x}$$ is increasing is
  • $$(-\infty, \infty)$$
  • $$(-2, 0)$$
  • $$(2, \infty)$$
  • $$(0, 2)$$
The set of all values of $$a$$ for which $$ f\left( x \right) =\left( { a }^{ 2 }-3a+2 \right) \left( \cos ^{ 2 }{ \dfrac { x }{ 4 } -\sin ^{ 2 }{ \dfrac { x }{ 4 }  }  }  \right) +\left( a-1 \right) x+\sin { 1 }$$ does not possess critical points is
  • $$(1, \infty)$$
  • $$(-2, 4)$$
  • $$(1, 3)\cup (3, 5)$$
  • $$(-\infty, 1)\cup (1, 4)$$
Let $$f(x) = \underset{x}{\overset{x + \dfrac{\pi}{3}}{\int}} |\sin \, \theta | \, d \theta \, \, (x \in [0, \pi])$$
  • $$f(x)$$ is strictly increasing in this interval
  • $$f(x)$$ is differentiable in this interval
  • Range of $$f(x)$$ is $$[2 - \sqrt{3} , 1]$$
  • $$f(x)$$ has a maxima at $$ x = \dfrac{\pi}{3}$$
The slope of the tangent to the curve $${r^2} = {a^2}\cos 2\theta$$, where $$x = r\cos \theta ,y = r\sin \theta $$, at the point $$\theta=\frac{\pi}{6}$$ is
  • $$\frac{1}{2}$$
  • $$-1$$
  • $$1$$
  • $$0$$
The line $$\dfrac{x}{a}+\dfrac{y}{b}=1$$ touches the curve $$y=be^{-x/a}$$ at the point.
  • $$(a, b/a)$$
  • $$(-a, b/a)$$
  • $$(0, b)$$
  • None of these
if $$m$$ is the slope of a tangent to the curve $$e^{y}=1+x^{2}$$, then  $$m$$ belongs to the interval
  • $$[-1, 1]$$
  • $$[-2, -1]$$
  • $$[1, 2]$$
  • $$[1, 3]$$
IF $$f(x)=\dfrac{x^2}{2-2cos x} ; \ g(x)=\dfrac{x^2}{6x-6sin x}$$ where $$0< \times  < 1$$, then 
  • both $$ 'f'$$ and$$ 'g' $$are increasing functions
  • $$'f' $$ is decreasing &$$ 'g'$$ is increasing function
  • $$'f' $$ increasing functions &$$ 'g'$$ is decreasing function
  • both $$'f'$$ & $$'g'$$ are decreasing function
A tangent drawn to the curve $$y = f\left( x \right)$$ at $$P\left( {x,y} \right)$$
cuts the x and y axes at A and B, respectively, such that $$AP:PB = 1:3$$. If $$f\left( 1 \right) = 1$$ then the curve passes through $$\left( {k,\frac{1}{8}} \right)$$ where $$k$$ is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
On the curve $${x}^{3} = 12y$$ , then the interval at which the abscissa changes at a faster rate than the ordinate ?
  • $$x\in \left( -2,2 \right)$$
  • $$x\in \left( -2,2 \right) -\left\{ 0 \right\}$$
  • $$x\in \left( -3,3 \right) -\left\{ 0 \right\}$$
  • None of these
The point on the curve $$y = b e^{\dfrac {-x}{a}}$$ at which the tangent drawn is $$\dfrac {x}{a} + \dfrac {y}{b} = 1$$ is
  • $$(0, b)$$
  • $$\left (a, \dfrac {1}{e}\right )$$
  • $$(0, 1)$$
  • $$(1, 0)$$
Let $$f ( x ) = \frac { 1 } { 1 + x ^ { 2 } }$$. Let $$m$$ be the slope, $$'a'$$ be the $$x$$-intercept and $$'b'$$ be the $$y$$-intercept of tangent to $$y = f ( x )$$.Abscissa of point of contact of the tangent for which $$'m'$$ is greatest is:
  • $$\frac { 1 } { \sqrt { 3 } }$$
  • $$1$$
  • $$0$$
  • $$\frac { - 1 } { \sqrt { 3 } }$$
If $$V$$ is the set of points on the curve $$y^{3} - 3xy +2 = 0$$ where the tangent is vertical then $$V =$$.
  • $$\phi$$
  • $$\left \{(1 , 0)\right \}$$
  • $$\left \{(1, 1)\right \}$$
  • $$\left \{(0, 0), (1, 1)\right \}$$
Paraboals $$(y-\alpha )^{2}=4a(x-\beta )and (y-\alpha )^{2}=4a'(x-\beta ')$$ will have a common normal (other than the normal passing through vertex ofparabola)if:
  • $$\frac{2(a-a')}{\beta '-\beta }< 1$$
  • $$\frac{2(a-a')}{\beta -\beta' }< 1$$
  • $$\frac{2(a'-a)}{\beta -\beta' }< 1$$
  • $$\frac{2(a'a)}{\beta -\beta' }> 1$$
The curve $$y-{e}^{xy}+x=0$$ has a vertical tangent at the point
  • $$(1,1)$$
  • At no point
  • $$(0,1)$$
  • $$(1,0)$$
The slope of the curve $$y=\sin { x } +\cos ^{ 2 }{ x }$$ is zero at a point , whose x-coordinate can be ?
  • $$\dfrac { \pi }{ 4 } $$
  • $$\dfrac { \pi }{ 2 } $$
  • $${ \pi }$$
  • $$\dfrac { \pi }{ 3 } $$
If the curves $$\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{4} = 1$$ and $$y^{3} = 16x$$ intersect at right angles, then $$a^{2}$$ is equal to
  • $$5/3$$
  • $$4/3$$
  • $$6/11$$
  • $$3/2$$
If the slope of one of the lines given by $${a^2}{x^2} + 2hxy+by^2 = 0$$ be three times of the other , then h is equal to 
  • $$2\sqrt 3 ab$$
  • $$-2\sqrt 3 ab$$
  • $$\frac{2}{{\sqrt 3 }}ab$$
  • $$-\frac{2}{{\sqrt 3 }}ab$$
If for a curve represented parametrically by $$x={ sec }^{ 2 }t,\quad y=cot\quad t\quad $$ , the tangent  at a point $$P(t=\frac { \pi  }{ 4 } )$$ meets the curve again at the point Q, then $$\begin{vmatrix} PQ \end{vmatrix}$$is equal to 
  • $$\frac { 2\sqrt { 5 } }{ 3 } $$
  • $$\frac { 3\sqrt { 5 } }{ 2 } $$
  • $$\frac { 5\sqrt { 3 } }{ 3 } $$
  • $$\frac { 5\sqrt { 5 } }{ 4 } $$
$$\dfrac { d } { d x } \left( \sin ^ { 5 } x \cdot \sin 5 x \right) =$$
  • $$\sin ^ { 4 } x \cdot \sin 5 x$$
  • $$5\sin ^ { 4 } x \cdot \sin 6 x$$
  • $$5\sin ^ { 4 } x \cdot \sin 5 x$$
  • $$- 5 \sin ^ { 4 } x \cdot \sin 6 x$$
Let f and g be non-increasing and non-decreasing functions respectively from $$[0,\infty ]$$ vto $$[0,\infty ]$$ and $$h(x)=f(g(x)),h(0)=0$$, then in $$[0,\infty ]$$, $$h(x)-h(1)$$ is 
  • <0
  • >0
  • =0
  • none of these
Let $$f$$ and $$g$$ be differentiable function satisfying $$g'(a)=2,g(a)=b$$ and $$fog=I$$ (identity function). Then $$f'(b)$$ is equal to 
  • $$\frac{1}{2}$$
  • $$2$$
  • $$\frac{2}{3}$$
  • None of these
The function $$f\left( x \right) = \frac{{\left| {x - 1} \right|}}{{{x^2}}}$$ is
  •  One-One in $$\left( { - \infty ,1 }\right)$$
  •  One-One in $$\left( {0,\infty} \right) $$
  •  One-One in $$\left( {0,1} \right) $$
  •  One-One in $$\left( { - \infty ,0 }\right)$$
$$y = 6\tan \,x\left( {{e^x} - x - 1} \right) - 3{x^3} - {x^4} - \frac{5}{4}{x^5},\,$$ if $${n^{th}}$$ derivative at x=0 is non zero then least value of n is
  • 3
  • 4
  • 5
  • 6
If $$(x+{ y }^{ 3 })\dfrac { dy }{ dx } $$=y and y(0)=then sum of all possible value(s) of y(1) is ________________.
  • -4
  • 4
  • 0
  • 2
The function $$f(x)=x-ln|2x+1|,x\epsilon \left(-100,\dfrac{-1}{2}\right)\cup \left(\dfrac{-1}{2},\dfrac{1}{2}\right)$$ is decreasing in interval
  • $$\left(\dfrac{-1}{2},\dfrac{1}{2}\right)$$
  • $$\left(-100,\dfrac{-1}{2}\right)$$
  • $$\left(\dfrac{-1}{2},0\right)$$ only
  • $$\left(0,\dfrac{1}{2}\right)$$ only
For what values of a , $$f(x) = -x^3 +4ax^2 +2x-5$$ is decreasing $$\forall$$ x
  • $$(1, 2)$$
  • $$(3, 4)$$
  • R
  • no value of a
Equation of the tangent line at $$y=\dfrac{a}{4}$$ to the curve $$y\left( { x }^{ 2 }+{ a }^{ 2 } \right) ={ ax }^{ 2 }$$.
  • $$8y=3\sqrt { 3 }x -a$$
  • $$8y=3\sqrt { 3 }x -5a$$
  • $$8y=3\sqrt { 3 }x +a$$
  • None
Which of the following is not always correct for the function $$f(x)$$ and $$g(x)$$ these are inverse to each other.
  • If $$f(x)$$ is an increasing function, then $$g(x)$$ will also be increasing.
  • If $$f(x)$$ is an odd function, then $$g(x)$$ will also be an odd function.
  • Tangent at $$(\alpha,\ \beta )$$ to $$f(x)$$ is parallel to tangent at $$(\beta,\ \alpha)$$ to $$g(x)$$
  • Tangent at $$(\alpha,\ \beta )$$ to $$f(x)$$ and tangent at is parallel to $$(\beta,\ \alpha)$$ to $$g(x)$$ forms complementary angles with x-axis
The increasing function in $$(0,\ \pi /4)$$ is
  • $$\cos x+\sin x$$
  • $$\cos x-\sin x$$
  • $$\dfrac {\sin x}{x}$$
  • $$\dfrac {x}{\sin x}$$
Let $$f :\left[ 2,4 \right] \rightarrow \left[ 3,5 \right] $$ be a bijective decreasing function, then find $$\int _{ 2 }^{ 4 }{ f(t)dt- } \int _{ 3 }^{ 5 }{ { f }^{ -1 }(t)dt. } $$
  • 2
  • 14
  • 0
  • 10/3
The increasing function in $$\left(0,\pi/4\right)$$ is 
  • $$\cos x+\sin x$$
  • $$\cos x-\sin x$$
  • $$\dfrac{\sin x}{x}$$
  • $$\dfrac{x}{\sin x}$$
If $$f(x)=\cos x+a^{2}x+b$$ is an increasing function for all values of $$x$$, then the value which $$'a'$$ can take. 
  • $$a\in [-1,1]$$
  • $$a\in(-\infty,-1]\cup [1,\infty)$$
  • $$a\in [-1,\infty)$$
  • $$a\in(-\infty,1]$$
Which of the following statements is/are correct ?
  • x + sinx is increasing function
  • tanx is an increasing function
  • x + sinx is decreasing function
  • sec x is an increasing function
If $$y = \log _ { \sin x } ( \tan x ) ,$$ then $$\frac { d y } { d x }$$ at $$x = \frac { \pi } { 4 }$$ is:
  • $$\frac { 4 } { \log 2 }$$
  • $$-\frac { 4 } { \log 2 }$$
  • $$\frac { 1 } { \log 2 }$$
  • none of these
$$f(x)$$ is differentiable function satisfying the relation $$f(x)=x^{2}+\displaystyle \int^{x}_{0}e^{-t}f(x-t)dt$$, then $$\displaystyle \sum^{9}_{k=1}f(k)$$ equals
  • $$1060$$
  • $$1260$$
  • $$960$$
  • $$1224$$
$$f(x)=\frac{x}{log x}-\frac{log}{x}$$ is increasing in 
  • $$(e,\infty )$$
  • $$(0,1)\epsilon (1,e)$$
  • (0,1)
  • (1,e)
Solve it:-
$$y = x + \dfrac{1}{x},$$
  • $$x=1$$ is a point of local maximum
  • $$x=-1$$ is a point of local minimum
  • Local maximum value > Local minimum value
  • Local maximum value < Local minimum value
In the interval $$\left( {7,\infty } \right),f(x) = \left| {x - 5} \right| + 2\left| {x - 7} \right|$$ is 
  • Increasing
  • Decreasing
  • Constant
  • Cannot be estimated
If $$f(x)=\dfrac {x}{\sin x}$$ and $$g(x)=\dfrac {x}{\tan x}$$ where $$0 < x < 1$$, then in this interval 
  • $$g(x)$$ are decreasing functions
  • both $$f(x)$$ and $$g(x)$$ are decreasing functions
  • $$f(x)$$ is an increasing function
  • $$g(x)$$ is an increasing function
If $$\theta$$ is angle of intersection between $$y=10-x^{2}$$ and $$y=4+x^{2}$$ then $$|\tan \theta|$$ is-
  • $$\dfrac {5\sqrt {3}}{11}$$
  • $$\dfrac {7\sqrt {3}}{15}$$
  • $$\dfrac {4\sqrt {3}}{11}$$
  • $$none$$
The function $$f(x)=\sqrt{25-4x^{2}}$$ is increasing in
  • (-3,0)
  • (4,0)
  • (-5/20)
  • R
The function log (log x) increases in 
  • $$(1,\infty )$$
  • $$(0,\infty )$$
  • $$\infty $$
  • R
The function f defined by $$f(x)=(x+2)e^{-x}$$ si
  • decreasing for all x
  • decreasing in $$(-\infty ,-1)$$ and increasing in $$(1,\infty )$$
  • increasing for all x
  • decreasing in $$(-1,-\infty )$$ and increasing in $$(-\infty ,-1)$$
The function $$\frac{ln(1+x)}{x}$$ in $$(0,\infty )$$ is
  • increasing
  • decreasing
  • not decreasing
  • not increasing
Let $$f(x)=\left\{\begin{matrix} max \{|x|, x^2\}, & |x|\leq 2\\ 8-2|x|, & 2 < |x|\leq 4\end{matrix}\right.$$
Let S be the set of points in the interval $$(-4, 4)$$ at which f is not differentiable. Then S?
  • Is an empty set
  • Equals $$\{-2, -1, 1, 2\}$$
  • Equals $$\{-2, -1, 0, 1, 2\}$$
  • Equals $$\{-2, 2\}$$
If the subnormal to the curve $${ x }^{ 2 }.{ y }^{ n }={ a }^{ 2 }$$ is a constant then n=
  • $$-4$$
  • $$-3$$
  • $$-2$$
  • $$-1$$
Let $$f(x)$$ be a function satisfying $$f'(x)=f(x)$$ with $$f(0)=1$$ and g be the function satisfying $$f(x)+g(x)=x^2$$ the value of the integral $$\displaystyle\int^1_0f(x)g(x)dx$$ is?
  • $$\dfrac{1}{4}(e-7)$$
  • $$\dfrac{1}{4}(e-2)$$
  • $$\dfrac{1}{2}(e-3)$$
  • None of these
Define $$f(x) = \dfrac{1}{2} [ |\sin x| + \sin x], 0 < x \le 2\pi$$. The $$f$$ is
  • increasing in $$\left(\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right)$$
  • decreasing in $$\left(0, \dfrac{\pi}{2}\right)$$ and increasing in $$\left(\dfrac{\pi}{2}, \pi\right)$$
  • increasing in $$\left(0, \dfrac{\pi}{2}\right)$$ and decreasing in $$\left(\dfrac{\pi}{2}, \pi\right)$$
  • increasing in $$\left(0, \dfrac{\pi}{4}\right)$$ and decreasing in $$\left(\dfrac{\pi}{4}, \pi\right)$$
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