CBSE Questions for Class 12 Commerce Maths Application Of Derivatives Quiz 9 - MCQExams.com

The slope of the tangent to the curve $$x=t^2+3t-8, y=2t^2-2t-5$$ at point $$(2, -1)$$ is
  • $$\dfrac {22}{7}$$
  • $$\dfrac {6}{7}$$
  • $$-6$$
  • $$\dfrac {7}{6}$$
At what points the slope of the tangent to the curve $$x^2+y^2-2x-3=0$$ is zero?
  • $$(3, 0), (-1, 0)$$
  • $$(3, 0), (1, 2)$$
  • $$(-1, 0), (1, 2)$$
  • $$(1, 2), (1, -2)$$
The point on the curve $$y=12x-x^2$$, where the slope of the tangent is zero will be
  • $$(0, 0)$$
  • $$(2, 16)$$
  • $$(3, 9)$$
  • $$(6, 36)$$
The normal to the curve $$x^2=4y$$ passing through $$(1, 2)$$ is
  • $$x+y=3$$
  • $$x-y=3$$
  • $$x+y=1$$
  • $$x-y=1$$
The slope of the tangent to the curve $$x=3t^2+1, y=t^3-1$$ at $$x=1$$ is
  • $$\dfrac {1}{2}$$
  • $$0$$
  • $$-2$$
  • $$\infty$$
Mark the correct alternative of the following.
The point on the curve $$9y^2=x^3$$, where the normal to the curve makes equal intercepts with the axes is?
  • $$(4, \pm 8/3)$$
  • $$(-4, 8/3)$$
  • $$(-4, -8/3)$$
  • $$(8/3, 4)$$
Mark the correct alternative of the following.
The line $$y=mx+1$$ is a tangent to the curve $$y^2=4x$$, if the value of m is?
  • $$1$$
  • $$2$$
  • $$3$$
  • $$1/2$$
The slope of the tangent to the curve $$x=t^2+3t-8, y=2t^2-2t-5$$ at the point $$(2, -1)$$ is
  • $$\dfrac {22}{7}$$
  • $$\dfrac {6}{7}$$
  • $$\dfrac {7}{6}$$
  • $$-\dfrac {6}{7}$$
If the function $$f(x)=\cos\left| x \right| -2ax+b$$ increases along the entire number scale, then 
  • $$a=b$$
  • $$a=\cfrac{1}{2}b$$
  • $$a\le -\cfrac{1}{2}$$
  • $$a> -\cfrac{3}{2}$$
The function $$f(x)=\cfrac{\lambda \sin x+2\cos x}{\sin x+\cos x}$$ is increasing, if
  • $$\lambda< 1$$
  • $$\lambda> 1$$
  • $$\lambda< 2$$
  • $$\lambda> 2$$
Let $$f(x)={x}^{3}+a{x}^{2}+bx+5{\sin}^{2}x$$ be an increasing function on the set $$R$$. Then, $$a$$ and $$b$$ satisfy
  • $${a}^{2}-3b-15> 0$$
  • $${a}^{2}-3b+15> 0$$
  • $${a}^{2}-3b+15< 0$$
  • $$a> 0$$ and $$b> 0$$
If the function $$f(x)=2\tan x+(2a+1)\log _{ e }{ \left| \sec { x }  \right|  } +(a-2)x$$ is increasing on $$R$$, then
  • $$a\in \left (\dfrac {1}{2},\infty \right)$$
  • $$a\in \left (-\dfrac {1}{2},\dfrac {1}{2}\right)$$
  • $$a=\dfrac {1}{2}$$
  • $$a\in R$$
Function $$f(x)={a}^{x}$$ is increasing on $$R$$, if
  • $$a> 0$$
  • $$a< 0$$
  • $$0< a< 1$$
  • $$a> 1$$
If the function $$f(x)={x}^{3}-9k{x}^{2}+27x+30$$ is increasing on $$R$$, then 
  • $$-1< k< 1$$
  • $$k< -1$$ or $$k> 1$$
  • $$0< k< 1$$
  • $$-1< k< 0$$
If the function $$f(x)={x}^{2}-kx+5$$ is increasing on $$[2,4]$$, then 
  • $$k\in (2,\infty)$$
  • $$k\in (-\infty,2)$$
  • $$k\in (4,\infty)$$
  • $$k\in (-\infty,4)$$
Function $$f(x)=\log _{ a }{ { x }_{  } } $$ is increasing on $$R$$, if
  • $$0< a< 1$$
  • $$a> 1$$
  • $$a< 1$$
  • $$a> 0$$
The function $$f(x)={x}^{9}+3{x}^{7}+64$$ is increasing on
  • $$R$$
  • $$(-\infty,0)$$
  • $$(0,\infty)$$
  • $${R}_{0}$$
A curve $$y=me^{mx}$$ where $$m > 0$$ intersects y-axis at a point $$P$$.
What is the slope of the curve at the point of intersection $$P$$ ? 
  • $$m$$
  • $$m^2$$
  • $$2m$$
  • $$2m^2$$
Consider the equation $$x^y=e^{x-y}$$
What is $$\dfrac{d^2y}{dx^2}$$ at $$x=1$$ equal to ?
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
Consider the equation $$x^y=e^{x-y}$$
What is $$\dfrac{dy}{dx}$$ at $$x=1$$ equal to ?
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
A curve $$y=me^{mx}$$ where $$m > 0$$ intersects y-axis at a point $$P$$.
How much angle does the tangent at $$P$$ make with y-axis ? 
  • $$\tan^{-1}m^2$$
  • $$\cot^{-1}(a+m^2)$$
  • $$\sin^{-1}(\dfrac{1}{\sqrt{1+m^4}})$$
  • $$\sec^{-1}\sqrt{1+m^4}$$
The function $$f(x)=4-3x+3x^2-x^3$$ is
  • decreasing on $$R$$
  • increasing on $$R$$
  • strictly decreasing on $$R$$
  • strictly increasing on $$R$$
The real value of $$k$$ for which $$f(x)=x^2+kx+1$$ is increasing on $$(1, 2)$$, is 
  • $$-2$$
  • $$-1$$
  • $$1$$
  • $$2$$
Consider the equation $$az^2 + z + 1 = 0$$ having purely imaginary root where $$a$$ = cos$$\theta + i$$ sin $$\theta$$, $$i = \sqrt{-1}$$ and function $$f(x) = x^3 - 3x^2 + 3(1$$ + cos $$\theta)x + 5$$, then answer the following questions. 
Which of the following is true about $$f(x)$$?
  • $$f(x)$$ decreases for $$x$$ $$\epsilon$$ $$[2 n \pi, (2n + 1)\pi]$$, $$n$$ $$\epsilon$$ $$Z$$
  • $$f(x)$$ decreases for $$x$$ $$\epsilon$$ $$\left [ (2n - 1)\frac{\pi}{2}, (2n + 1)\frac{\pi}{2}\right ]$$ $$n$$ $$\epsilon$$ $$Z$$
  • $$f(x)$$ is non-monotonic function
  • $$f(x)$$ increases for $$x$$ $$\epsilon$$ $$R$$.
The number of tangents to the cure $$x^{3/2}+y^{3/2}=2a^{3/2}, a> 0$$, which are equally inclined to the axes, is 
  • 2
  • 1
  • 0
  • 4
If m is the slope of a tangent to the curve $$e^{y}=1+x^{2},$$ then 
  • $$\left | m \right |> 1$$
  • $$m> 1$$
  • $$m> -1$$
  • $$\left | m \right |\leq 1$$
Let $$f(x)=\displaystyle \int e^x  (x-1)(x-2)dx$$. Then $$f$$ decreases in the interval
  • $$(-\infty , -2)$$
  • $$(-2, -1)$$
  • $$(1, 2)$$
  • $$(2, \infty)$$
The function $$f(x)=3x+\cos 3x$$ is
  • increasing on $$R$$
  • decreasing on $$R$$
  • strictly increasing on $$R$$
  • strictly decreasing on $$R$$
$$f(x)=\sin x-kx $$ is decreasing for all $$x \in R$$, when
  • $$k < 1$$
  • $$k \le 1$$
  • $$k > 1$$
  • $$k \ge 1$$
For $$x > 1, y=\log_e x$$ satisfies the inequality 
  • $$x-1 > y$$
  • $$x^2 -1 >y$$
  • $$y > x-1$$
  • $$\dfrac {x-1}{x} < y$$
The slope of the tangent to the curve $$y = \sqrt{4-x^{2}}$$ at the point, where the ordinate and the abscissa are equal , is
  • -1
  • 1
  • 0
  • None of these
At the point $$P(a, a^{n})$$ on the graph of $$y = x^{n}(n \epsilon  n)$$ in the first quadrant, a normal is drawn. the normal intersects the y-axis at the point (0, b) . if $$\underset{a\rightarrow b}{lim}b=\dfrac{1}{2}$$, then n equals
  • 1
  • 3
  • 2
  • 4
The curve given by $$x + y = e^{xy}$$ has a tangent parallel to the y-axis at the point
  • $$(0,1)$$
  • $$( 1, 0 )$$
  • $$(1, 1)$$
  • None of these
The abscissa of points P and Q in the curve $$y = e^{x}+e^{-x}$$ such that tangents at P and Q make $$60^{o}$$ with the x-axis
  • ln $$\left ( \dfrac{\sqrt{3}+\sqrt{7}}{7} \right )$$ and ln $$\left ( \dfrac{\sqrt{3}+\sqrt{5}}{2} \right )$$
  • ln $$\left ( \dfrac{\sqrt{3}+\sqrt{7}}{2} \right )$$
  • ln $$\left ( \dfrac{\sqrt{7}+\sqrt{3}}{2} \right )$$
  • $$\pm$$ ln $$\left ( \dfrac{\sqrt{3}+\sqrt{7}}{2} \right )$$
If x=4 y = 14 is a normal to the curve $$y^{2}=ax^{3}-\beta $$ at (2,3) then the value of $$\alpha +\beta $$ is 
  • 9
  • -5
  • 7
  • -7
At what points of curve $$y = \dfrac{2}{3}x^{3}+\dfrac{1}{2}x^{2}$$, the tangent makes the equal with the axis?
  • $$(\dfrac{1}{2},\dfrac{5}{24})$$ and $$\left ( -1,\dfrac{-1}{6} \right )$$
  • $$(\dfrac{1}{2},\dfrac{4}{9})$$ and $$ ( -1,0)$$
  • $$\left ( \dfrac{1}{3},\dfrac{1}{7} \right )$$ and$$ \left ( -3, \dfrac{1}{2} \right )$$
  • $$\left ( \dfrac{1}{3},\dfrac{4}{47} \right )$$ and $$\left ( -1, \dfrac{1}{2} \right )$$
The curve represented parametrically by the equations x = 2 in $$\cot t+1$$ and $$y=\tan t+\cot t$$ 
  • tanfent and normal intersect at the point (2, 1)
  • normal at $$t = \pi /4$$ is parallel to the y-axis
  • tangent at $$t = \pi /4$$ is parallel to the line y = x
  • tangent at $$t = \pi /4$$ is parallel to the x-axis
If a variable tangent to the curve $$x^{2}y=c^{3}$$ makes intercepts a, b on x-and y-axes, respectively, then the value of $$a^{2}b$$ is
  • $$27c^{3}$$
  • $$\dfrac{4}{27}c^{3}$$
  • $$\dfrac{27}{4}c^{3}$$
  • $$\dfrac{4}{9}c^{3}$$
The x-intercept of the tangent at any arbitrary point of the curve $$\dfrac{a}{x^{2}}+\dfrac{b}{y^{2}}=1$$ is proportion to
  • square of the abscissa of the point of tangency
  • square root of the abscissa of the point of tangency
  • cube of the abscissa pf the point of tangency
  • cube root of the abscissa of the point of tangency
The angle between the tangent to the curves $$y = x^{2}$$ and $$x = y^{2}$$ at (1, 1) is 
  • $$\cos ^{-1}\dfrac{4}{5}$$
  • $$\sin ^{-1}\dfrac{3}{5}$$
  • $$\tan ^{-1}\dfrac{3}{4}$$
  • $$\tan ^{-1}\dfrac{1}{3}$$
Point on the curve $$f(x)=\dfrac{x}{1-x^{2}}$$ where the tangent is inclined at an angle of $$\dfrac{\pi }{4}$$ ot the x-axis are 
  • (0, 0)
  • $$\left ( \sqrt{3},\dfrac{-\sqrt{3}}{2} \right )$$
  • $$\left ( -2 ,\dfrac{2}{3}\right )$$
  • $$\left (- \sqrt{3},\dfrac{\sqrt{3}}{2} \right )$$
If the tangent at any point $$P(4m^{2}, 8m^{3})$$ of $$x^{3}-y^{3}=0$$ is also a normal to the curve  $$x^{3}-y^{3}=0$$ , then value of m is
  • $$m = \dfrac{\sqrt{2}}{3}$$
  • $$m = -\dfrac{\sqrt{2}}{3}$$
  • $$m = \dfrac{3}{\sqrt{2}}$$
  • $$m = -\dfrac{3}{\sqrt{2}}$$
A curve passes through $$(2,1)$$ and is such that the square of the ordinate is twice the contained by the abscissa and the intercept of the normal. Then the equation of curve is
  • $$x^2 +y^2=9x$$
  • $$4x^2 +y^2=9x$$
  • $$4x^2 +2y^2=9x$$
  • None of these
The tangent to the curve $$y = e^{x}$$ drawn at the point $$(c, e^{c})$$ intersects the line joining the points $$(c-1, e^{c-1})$$ and $$(c+1, e^{c+1})$$
  • on the left of x =c
  • on the right of x = c
  • at no point
  • at all point
Let $$f:[1, \infty) \rightarrow R$$ and $$f(x)=x \int_{1}^{x} \dfrac{e^{t}}{t} d t-e^{x},$$ then
  • $$f(x)$$ is an increasing function
  • $$\lim _{x \rightarrow \infty} f(x) \rightarrow \infty$$
  • $$f^{\prime}(x)$$ has a maxima at $$x=e$$
  • $$f(x)$$ is a decreasing function
If the line ax +by + c = 0 is a normal to the curve xy = 1, then 
  • $$a > 0, b> 0$$
  • $$a > 0, b < 0$$
  • $$a < 0, b > 0$$
  • $$a < 0, b < 0$$
Consider the following statement is $$S$$ and $$R$$ 
$$S$$. Both $$\sin x$$ and $$\cos x$$ are decreasing function in the interval $$\left(\dfrac {\pi}{2}, \pi \right)$$
$$R:$$ If a differentiable function decreases in an interval $$(a, b)$$ then its derivative also decreases in $$(a, b)$$, which of the following is true?
  • Both $$S$$ and $$R$$ are wrong
  • Both $$S$$ and $$R$$ are correct but $$R$$ is not the correct explanation of $$S$$
  • $$S$$ is the correct and $$R$$ is the correct explanation of $$S$$
  • $$S$$ is the correct and $$R$$ is the wrong
The slope of the tangent to the curve $$y = f(x)$$ at $$\left [ x, f(x) \right ]$$ is 2x +If the curve passes through the point (1, 2)then the area bounded by the curve, the x-axis and the line x = 1 is
  • $$\dfrac{5}{6}$$
  • $$\dfrac{6}{5}$$
  • $$\dfrac{1}{6}$$
  • 6
The point(s) on the curve $$y^{3} + 3x^{2} = 12y,$$ where the tangent is vertical, is (are)
  • $$\left ( \pm \dfrac{4}{\sqrt{3}}, -2 \right )$$
  • $$\left ( \pm \sqrt{\dfrac{11}{3}}, 1 \right )$$
  • (0, 0)
  • $$\left ( \pm \dfrac{4}{\sqrt{3}}, 2 \right )$$
The normal to the curve $$x = a (\cos 0 + 0\sin 0), y= a (\sin 0- 0\cos 0)$$ at any point 0 is such that
  • it makes a constant angle with x-axis
  • it passes through the origin
  • it is at a constant distance from the origin
  • none of these
0:0:1


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