CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 7 - MCQExams.com

Let $$\quad f(x)+2f(1-x)={ x }^{ 2 }+2\forall x\in R\quad $$ then the interval in which $$f(x)$$ increases is
  • $$\left( -\infty ,-2 \right) \quad $$
  • $$\left( -\infty ,\infty \right) $$
  • $$\left( 2,\infty \right) $$
  • None of these
The angle between the curves $$x^{2} + y^{2} = 25$$ and $$x^{2} + y^{2} - 2x + 3y - 43 = 0$$ at $$(-3, 4)$$ is
  • $$\tan^{-1}(1)$$
  • $$\tan^{-1}\left (\dfrac {1}{68}\right )$$
  • $$\dfrac {\pi}{2}$$
  • $$\tan^{-1}\left (\dfrac {3}{4}\right )$$
The equation of the curve satisfying the differential equation $$y_{2}(x^{2} + 1) = 2xy_{1}$$ passing through the point $$(0, 1)$$ and having slope of tangent at $$x = 0$$ as $$3$$ is
  • $$y = x^{2} + 3x + 2$$
  • $$y = x^{2} + 3x + 1$$
  • $$y = x^{3} + 3x + 1$$
  • None of these
The slope of the tangent at each point of the curve is equal to the sum of the coordinate of the point. Then, the curve that passes through the origin is
  • $$x + y = e^{x} - 1$$
  • $$e^{x} = x + y$$
  • $$y = e^{x}$$
  • $$y = e^{x} + 1$$
The set of all values of a for which the function $$f(x) = (a^{2} - 3a + 2)(\cos^{2}\dfrac{x} {4} - \sin^{2}\dfrac{x}{4}) + (a - 1)x + \sin 1$$ does not possess critical points is
  • $$[1, \infty)$$
  • $$(0, 1) \cup (1, 4)$$
  • $$(-2, 4)$$
  • $$(1, 3)\cup (3, 5)$$
If $$g\left( x \right) =2f\left( 2{ x }^{ 3 }-3{ x }^{ 2 } \right) +f\left( 6{ x }^{ 2 }-4{ x }^{ 3 }-3 \right)$$ $$\forall \ x\ \in \ R$$ and $$f^{''}\left( x \right) > 0$$, $$\forall \ x \in \ R$$, then $$g\left ( x \right)$$ is increasing in the interval
  • $$\left( \infty ,-\dfrac { 1 }{ 2 } \right)$$
  • $$\left( -\dfrac { 1 }{ 2 } ,\ 0 \right) \bigcup \left( 1,\infty \right)$$
  • $$\left( 0,\infty \right)$$
  • None of these
The slope of the tangent at the point $$(h, h)$$ of the circle $$x^{2} + y^{2} = a^{2}$$ is :
  • $$0$$
  • $$1$$
  • $$-1$$
  • Depends on $$h$$
The angle at which the curve $$y={ x }^{ 2 }$$ and the curve $$x=\cfrac { 5 }{ 3 } \cos { t } ,y=\cfrac { 5 }{ 4 } \sin { t } $$ intersect is
  • $$\tan ^{ -1 }{ \cfrac { 2 }{ 41 } } $$
  • $$\tan ^{ -1 }{ \cfrac { 41 }{ 2 } } $$
  • $$-\tan ^{ -1 }{ \cfrac { 2 }{ 41 } } $$
  • $$2\tan ^{ -1 }{ \cfrac { 41 }{ 2 } } $$
The slope of the tangent to the curve $$y=\int _{ 0 }^{ x }{ \frac { dt }{ 1+{ t }^{ 3 } }  } $$ at the point where $$x=1$$ is
  • $$\frac { 1 }{ 4 } $$
  • $$\frac { 1 }{ 3 } $$
  • $$\frac { 1 }{ 2 } $$
  • $$1$$
A point P moves such that sum of the slopes of the normal drawn from it to the hyperbola $$xy=16$$ is equal to the sum of the ordinates of the feet of the normal. Let 'P' lies on the curve C, then.
The equation of 'C' is?
  • $$x^2=4y$$
  • $$x^2=16y$$
  • $$x^2=12y$$
  • $$y^2=8x$$
A tangent PT is drawn to the circle $$x^2+y^2=4$$ at the point $$P(\sqrt{3}, 1)$$. A straight line L, perpendicular to PT is a tangent to the circle $$(x-3)^2+y^2=1$$. $$(1)$$ A possible equation of L is?
  • $$x-\sqrt{3}y=1$$
  • $$x+\sqrt{3}y=1$$
  • $$x-\sqrt{3}y=-1$$
  • $$x+\sqrt{3}y=5$$
$$f(x) = |x\log_{e}x|$$ monotonically decreases in
  • $$(0, 1/e)$$
  • $$(1/e, 1)$$
  • $$(1, \infty)$$
  • $$(1/e, \infty)$$
The points of the curve $$y={ x }^{ 3 }+x-2$$ at which its tangent are parallel to the straight line $$y=4x-1$$ are
  • $$\left( 2,7 \right) ,\left( -2,-11 \right) $$
  • $$\left( 0,-2 \right) ,\left( { 2 }^{ 1/3 },{ 2 }^{ 1/3 } \right) $$
  • $$\left( { -2 }^{ 1/3 },{ -2 }^{ 1/3 } \right) \left( 0,-4 \right) $$
  • $$\left( 1,0 \right) ,\left( -1,-4 \right) $$
If the tangent at $$(1, 7)$$ to the curve $$x^{2} = y - 6$$ touches the circle $$x^{2} + y^{2} + 16x + 12y + c = 0$$ then the value of $$c$$ is
  • $$85$$
  • $$95$$
  • $$195$$
  • $$185$$
The percentage error in the surface area of a cube with edge x cm, when the edge is increased by $$11\%$$ is _________.
  • $$11$$
  • $$22$$
  • $$10$$
  • $$44$$
Consider the following statements:
Statement I:
$$x > \sin x$$ for all $$x > 0$$
Statement II:
$$f(x) = x - \sin x$$ is an increasing function for all $$x > 0$$
Which one of the following is correct in respect of the above statements?
  • Both Statements I and II are true and Statement II is the correct explanation of Statement I
  • Both Statements I and II are true and Statement II is the not correct explanation of Statement I
  • Statement I is true but Statement II is false
  • Statement I is true but Statement II is true
The critical points of the function $$f(x)={ x }^{ 3/5 }\left( 4-x \right) ,x\in { R }^{ + }\cup \left\{ 0 \right\} $$ is _______
  • $$0,-\cfrac { 3 }{ 2 } $$
  • $$0,\cfrac { 3 }{ 2 } $$
  • $$0,\cfrac { 2 }{ 3 } $$
  • $$0,-\cfrac { 2 }{ 3 } $$
The tangent to $$\left( a{ t }^{ 2 },2at \right) $$ is perpendicular to X-axis at _____ point $$t\in R$$.
  • $$(4a,4a)$$
  • $$(a,2a)$$
  • $$(0,0)$$
  • $$(a,-2a)$$
Let, $$f:A\rightarrow B$$ be an invertible function. If $$f(x)=2x^3+3x^2+x-1$$, then $$f'^{-1}(5)$$=
  • $$\dfrac{1}{13}$$
  • $$1$$
  • $$6$$
  • can not be determined
Number of critical points of the function $$\displaystyle f\left( x \right) =\dfrac { 2 }{ 3 } \sqrt { { x }^{ 3 } } -\dfrac { x }{ 2 } +\int _{ 1 }^{ x }{ \left( \dfrac { 1 }{ 2 } +\dfrac { 1 }{ 2 } \cos { 2t } -\sqrt { t }  \right)  } dt$$ which lie in the interval $$\left[ -2\pi ,2\pi  \right] $$ is:
  • $$2$$
  • $$6$$
  • $$4$$
  • $$8$$
The point on the curve $$y^{2}=x$$ where tangent makes $$45^{o}$$ angle with $$x-$$axis ?
  • $$(\dfrac{1}{2},\dfrac{1}{4})$$
  • $$(\dfrac{1}{4},\dfrac{1}{2})$$
  • $$(4,2)$$
  • $$(1,1)$$
Line $$y=x$$ and curve $$y=x^2+bx+c$$ touches at $$(1, 1)$$ then __________.
  • $$b=-1, c=1$$
  • $$b=1, c=2$$
  • $$b=1, c=1$$
  • $$b=0, c=1$$
If $$f(x)=min(|x|^{2}-5|x|,1)$$ then $$f(x)$$ is non differentiable at $$\lambda$$ points, then $$\lambda+13$$ equals
  • $$16$$
  • $$14$$
  • $$13$$
  • $$15$$
Find the angle between tangent of the curve $$y = (x + 1) (x - 3)$$ at the point where it cuts the axis of $$x$$.
  • $$\tan^{-1} \left(\dfrac{8}{15}\right)$$
  • $$\tan^{-1} \left(\dfrac{15}{8}\right)$$
  • $$\tan^{-1} 4$$
  • $$None\ of\ these$$
If the curves $$y^2 = 4ax$$ and $$xy = c^2$$ cut orthogonally then $$\dfrac{c^4}{a^4} =$$
  • $$4$$
  • $$8$$
  • $$16$$
  • $$32$$
An equation of the tangent to the curve $$y=x^{4}$$ from the point $$(2,0)$$ not on the curve is:
  • $$y=0$$
  • $$x=0$$
  • $$x+y=0$$
  • $$none\ of\ these$$
The equation of the tangent to the curve $$y = b{e^{ -\dfrac{x}{a}}}$$ at a point , where $$x=0$$ is 
  • $$\dfrac{x}{a} - \dfrac{y}{b} = 1$$
  • $$\dfrac{y}{b} - \dfrac{x}{a} = 1$$
  • $$\dfrac{x}{a} + \dfrac{y}{b} = 1$$
  • $$\dfrac{x}{b} + \dfrac{y}{a} = 1$$
The function $$y = \dfrac{2x^2 - 1}{x^4}$$ is
  • Always increasing
  • Always decreasing
  • Neither increasing nor decreasing
  • None of these
The equation of the curve passing through $$(1,3)$$ whose slope at any point $$(x,y)$$ on it is  $$\dfrac { y }{ { x }^{ 2 } }$$ is given by
  • $$y={ 3e }^{ -1/x }$$
  • $$y={ 3e }^{ 1-1/x }$$
  • $$y={ ce }^{ 1/x }$$
  • $$y={ 3e }^{ 1/x }$$
Let $$f:R\rightarrow  R$$  be a function defined by $$f\left(x\right)=Min \left\{x+1, \left|x\right|+1\right\}$$. Then which of the following is true?
  • $$f(x)\ge 1$$ for all $$x\in R$$
  • $$f(x)\ge 1$$ is not differentiable at $$x=1$$
  • $$f(x)\ge 1$$ is differentiable at $$x=1$$
  • $$f(x)\ge 1$$ is not differentiable at $$x=0$$
Area of the triangle formed by the tangent at $$x=2$$ on the curve $$y= \dfrac{8}{4+x^2}$$ with the coordinate axes is (in sq. units)
  • $$4$$
  • $$3$$
  • $$2$$
  • $$1$$
Find the points on the ellipse $$\dfrac{{{x^2}}}{4} + \dfrac{{{y^2}}}{9}=1$$ , on which the normals are parallel to the line $$3x-y=1$$.
  • $$(\pm\dfrac{6}{\sqrt {10}},\pm\dfrac{3}{\sqrt {10}})$$
  • $$(\pm\dfrac{3}{\sqrt {10}},\pm\dfrac{1}{\sqrt {10}})$$
  • $$(\pm\dfrac{1}{\sqrt {10}},\pm\dfrac{2}{\sqrt {10}})$$
  • None of these
The entire graph of the equation $$y=x^{2}+kx-x+9$$ is strictly above the $$x-$$axis if and only if : 
  • $$k<7$$
  • $$-5< k <7$$
  • $$k>-5$$
  • $$none\ of\ these$$
If $$y=e^{4x}+2e^{-x}$$ satisfied the equation $$\dfrac{d^{3}y}{dx^{3}}+A\dfrac{dy}{dx}+By=0$$ then value of $$|A+B|$$ is 
  • $$36$$
  • $$25$$
  • $$24$$
  • $$15$$
Let $$f(x)$$ be a differentiate function and $$f(\alpha)=f(\beta)=0(\alpha < \beta)$$, then in the interval $$(\alpha, \beta)$$.
  • $$f(x)+f'(x)=0$$ has at least one root
  • $$f(x)-f'(x)=0$$ has at least one real root
  • $$f(x)\cdot f'(x)=0$$ has at least one real root
  • None of these
If the tangent to the curve $$y=x\log { x } $$ at $$\left( c,f\left( x \right)  \right) $$ is parallel to the line-segment joining $$A\left(1,0\right)$$ and $$B\left(e,e\right)$$, then c=...... .
  • $$\dfrac {e-1}{e}$$
  • $$\log { \dfrac { e-1 }{ e } } $$
  • $${ e }^{ \dfrac { 1 }{ 1-e } }$$
  • $${ e }^{ \dfrac { 1 }{ e-1 } }$$
If $$f(x) = g(x) (x - \lambda)^2 \,$$ and $$\, g (\lambda)$$, where  $$0 < x \le 1$$, then in this interval 
  • Both $$f(x)$$ and $$g(x)$$ are increasing functions
  • Both $$f(x)$$ and $$g(x)$$ are decreasing function
  • $$f(x)$$ is an increasing function
  • $$g(x)$$ is an increasing function
Let $$h(x) = f(x) - (f (x) )^2 + (f(x))^3$$ for every real number $$x$$, then 
  • $$h$$ is increasing whenever $$f$$ is increasing and decreasing whenever $$f$$ is decreasing
  • $$h$$ is increasing whenever $$f$$ is decreasing
  • $$h$$ is decreasing whenever $$ f$$ is increasing
  • Nothing can be said in general
Find the slope of the normal to the curve $$2x^{2} - xy + 3y^{2} = 18$$ at $$(3,1)$$.
  • $$\dfrac {-11}{3}$$
  • $$\dfrac {3}{11}$$
  • $$\dfrac {11}{3}$$
  • $$\dfrac {-3}{11}$$
The radius of the sphere is measured as $$ \left( {10 \pm 0.02} \right)cm$$. The error in the measurement of its volume is 
  • $$25.1 cc$$
  • $$25.21 cc$$
  • $$2.51 cc$$
  • $$251.2 cc$$
The curve $$y = ax^3 + bx^2 + cx + 8$$ touches $$x-$$ axis at $$P(-2, 0)$$ and cuts $$y-$$ axis at a point $$Q$$ where its gradient is $$3$$. The values of $$a, b, c$$ are respectively ?
  • $$-\dfrac{5}{4}, -3, 3$$
  • $$0, \dfrac{1}{4},3$$
  • $$\dfrac{1}{4}, 0, 3$$
  • $$\dfrac{1}{4}, - \dfrac{1}{4}, 3$$
The curve satisfying D.E $$y dx - (x+3{y}^{2})dy=0$$ and passing through the point $$(1,1)$$ also passes through the point:
  • $$\left( \cfrac { 1 }{ 4 } ,-\cfrac { 1 }{ 2 } \right) $$
  • $$\left( -\cfrac { 1 }{ 3 } ,\cfrac { 1 }{ 3 } \right) $$
  • $$\left( \cfrac { 1 }{ 3 } ,-\cfrac { 1 }{ 3 } \right) $$
  • $$\left(- \cfrac { 1 }{ 4 } ,-\cfrac { 1 }{ 2 } \right) $$
Slope of the line $$ \sqrt { { x }^{ 2 }+{ 4y }^{ 2 }-4xy+4 } +x-2y=1$$ equals to
  • $$ \dfrac { 1 }{ 2 }$$
  • $$2$$
  • $$ -\dfrac { 1 }{ 2 }$$
  • $$None\ of\ these$$
If $$f(x)=x^{3/2}(3x-10), x\ge-0)$$, then $$f(x)$$ is decreasing in
  • $$(-\infty,0)\cup(0,\infty)$$
  • $$(2,\infty)$$
  • $$(-\infty,-1]\cup[1,\infty)$$
  • $$(-\infty,0]\cup[2,\infty)$$
The numbers of tangent to the curve $$y - 2 = {x^5}$$  which are drawn
from point $$\left( {2,2} \right)$$ is/are 
  • $$30$$
  • $$80$$
  • $$20$$
  • $$50$$
If the normal to the curve $$y=f\left( x \right)$$ at $${(3,4)}$$ makes angle $$\dfrac {3\pi}{4}$$ with $$\bar {OX}$$ then $$f^{ 1 }\left( 3 \right)=$$
  • $$-1$$
  • $$1$$
  • $${(-3/4)}$$
  • $${(4/3)}$$
If the curves $$\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{4}=1$$ and $$y^{2}=16x$$ intersect at right angles then value of $$a^{2}$$ is
  • $$2$$
  • $$4$$
  • $$\dfrac{8}{3}$$
  • $$\dfrac{16}{3}$$
Tangents are drawn from a point on the circle $$x^2+y^2=25$$ to the ellipse $$9x^2+16y^2-144=0$$ then the angle between the tangents is 
  • $$\frac{\pi}{4}$$
  • $$\frac{3\pi}{4}$$
  • $$\frac{\pi}{2}$$
  • $$\frac{2\pi}{3}$$
Area of the triangle formed by the tangent, normal to the curve $$x^{2}/a^{2}+y^{2}/b^{2}=1$$ at the point $$(a/\sqrt{2} , b/\sqrt{2})$$ and the $$x-$$axis is
  • $$\dfrac{ab}{4}\sqrt{a^{2}+b^{2}}$$
  • $$4ab$$
  • $$\dfrac{b}{4a}({a^{2}+b^{2}})$$
  • $$none$$
The slope of normal to the curve y= log (logx) at x = e is 
  • e
  • -e
  • $$\frac{1}{e}$$
  • -$$\frac{1}{e}$$
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