JEE Main & Advanced Mathematics Applications of Derivatives Question Bank

done Mock Test - Application of Derivatives Total Questions - 25

• question_answer1)
  The slope of the tangent to the curve \[y=\sqrt{4-{{x}^{2}}}\] at the point where the ordinate and abscissa equal is
  A)
  -1 done clear
  B)
  1 done clear
  C)
  0                      done clear
  D)
  none of these done clear
  View Solution play_arrow

• question_answer2)
  The equation of the tangent to the curve \[y=b{{e}^{-x/a}}\] at the point where it crosses the y-axis is
  A)
  \[\frac{x}{a}-\frac{y}{b}=1\] done clear
  B)
  \[ax+by=1\] done clear
  C)
  \[ax-by=1\]        done clear
  D)
  \[\frac{x}{a}+\frac{y}{b}=1\] done clear
  View Solution play_arrow

• question_answer3)
  If \[x+4y=14\] is a normal to the curve \[{{y}^{2}}=a{{x}^{3}}-\beta \] at  (2, 3), then the value of \[\alpha +\beta \]is
  A)
  9                      done clear
  B)
  \[-\,5\] done clear
  C)
  7                      done clear
  D)
  \[-\,7\] done clear
  View Solution play_arrow

• question_answer4)
  The normal to the curve \[2{{x}^{2}}+{{y}^{2}}=12\] at the point (2, 2) cuts the curve again at
  A)
  \[\left( -\frac{22}{9},-\frac{2}{9} \right)\]  done clear
  B)
  \[\left( \frac{22}{9},\frac{2}{9} \right)\] done clear
  C)
  \[\left( -\,2,\,\,-2 \right)\]   done clear
  D)
  none of these done clear
  View Solution play_arrow

• question_answer5)
  If a variable tangent to the curve \[{{x}^{2}}y={{c}^{3}}\]makes intercepts a and b on x-and y-axis, respectively, then the value of \[{{a}^{2}}b\]is
  A)
  27\[{{c}^{3}}\]            done clear
  B)
  \[\frac{4}{27}{{c}^{3}}\] done clear
  C)
  \[\frac{27}{4}{{c}^{3}}\]         done clear
  D)
  \[\frac{4}{9}{{c}^{3}}\] done clear
  View Solution play_arrow

• question_answer6)
  If the length of sub-normal is equal to the length of sub-tangent at my point (3, 4) on the curve y=f(x) and the tangent at (3, 4) to y=f(x) meets the coordinate axes at A and B, then the maximum area of the triangle OAB, where O is origin, is
  A)
  45/2                  done clear
  B)
  49/2 done clear
  C)
  25/2                  done clear
  D)
  81/2             done clear
  View Solution play_arrow

• question_answer7)
  Let \[f\,\left( 1 \right)=-\,2\] and \[f'(x)\ge 4.2\] for \[1\le x\le 6\]. The smallest possible value of f(6) is
  A)
  9                      done clear
  B)
  12 done clear
  C)
  15                     done clear
  D)
  19 done clear
  View Solution play_arrow

• question_answer8)
  A curve is represented by the equations \[x={{\sec }^{2}}\]t and\[y=\cot \,t\], where t is a parameter. If the tangent at the point P on the curve where \[t=\pi /4\] meets the curve again at the point Q, Then \[\left| PQ \right|\] is equal to
  A)
  \[\frac{5\sqrt{3}}{2}\]    done clear
  B)
  \[\frac{5\sqrt{5}}{2}\] done clear
  C)
  \[\frac{2\sqrt{5}}{3}\]    done clear
  D)
  \[\frac{3\sqrt{5}}{2}\] done clear
  View Solution play_arrow

• question_answer9)
  Let f(x) and g(x) be differentiable for \[0\le x\le 1\]such that f(0)=0, g(0)=0, f(1)=6. Let there exists a real number c in (0, 1) such that f?= 2g?. then the value of g(1) must be
  A)
  1 done clear
  B)
  3 done clear
  C)
  \[-\,2\]               done clear
  D)
  \[-\,1\] done clear
  View Solution play_arrow

• question_answer10)
  The lines tangent to the curves \[{{y}^{3}}-{{x}^{2}}y+5y-2x=0\]and \[{{x}^{4}}-{{x}^{3}}{{y}^{2}}+5x+2y=0\]at the origin intersect at an angles \[\theta \]equal to
  A)
  \[\frac{\pi }{6}\]             done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{\pi }{3}\]             done clear
  D)
  \[\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer11)
  If the function \[f(x)=a{{x}^{3}}+b{{x}^{2}}+11x-6\] satisfies condition of Rolle's theorem in [1, 3] for \[x=2+\frac{1}{\sqrt{3}}\], then values of a and b , respectively, are
  A)
  \[-\,3,\text{ }2\]  done clear
  B)
  \[2,\,\,-4\] done clear
  C)
  1, 6                   done clear
  D)
  none of these done clear
  View Solution play_arrow

• question_answer12)
  Let f(x) be a twice differentiable function for all real values of x and satisfies f(1)=1, f(2)=4, f(3)=9. Then which of the following is definitely true?
  A)
  \[f''(x)=2\forall x\in (1,3)\] done clear
  B)
  \[f''(x)=f'(x)=5\] for some \[x\in (2,3)\] done clear
  C)
  \[f''(x)=3\,\forall \,x\in (2,\,\,3)\] done clear
  D)
  \[f''(x)=2\] for some \[x\in (1,3)\] done clear
  View Solution play_arrow

• question_answer13)
  A lamp of negligible height is placed on the ground \[{{\ell }_{1}}\] away from a wall. A man \[{{\ell }_{2}}\]m tall is walking at a speed of \[\frac{{{\ell }_{1}}}{10}\]m/s form the lamp to the nearest point on the wall. When he is midway between the lamp and the wall, the rate of change in the length of this shadow on the all is
  A)
  \[-\frac{5{{\ell }_{2}}}{2}m/s\]  done clear
  B)
  \[-\frac{2{{\ell }_{2}}}{5}m/s\] done clear
  C)
  \[-\frac{{{\ell }_{2}}}{2}m/s\]    done clear
  D)
  \[-\frac{{{\ell }_{2}}}{5}m/s\] done clear
  View Solution play_arrow

• question_answer14)
  In which of the following function is rolle's theorem applicable?
  A)
  \[f(x)=\left\{ \begin{matrix}    x,\,  \\    0,\,  \\ \end{matrix} \right.\,\begin{matrix}    \,\,\,\,\,0\le x<1  \\    x=1  \\ \end{matrix}\] on [0, 1] done clear
  B)
  \[f(x)=\left\{ \begin{matrix}    \frac{\sin x}{x},\,  \\    1,\,  \\ \end{matrix} \right.\,\,\,\,\begin{matrix}    -\pi \le x<0  \\    x=0  \\ \end{matrix}\] on [-\[\pi \],0] done clear
  C)
  \[f(x)=\frac{{{x}^{2}}-x-6}{x-1}\] on [-2, 3] done clear
  D)
  \[f(x)=\left\{ \begin{matrix}    \frac{{{x}^{3}}-2{{x}^{2}}-5x+6}{x-1},\,\,\,if\,\,x\ne 1,  \\    -\,6,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,x=1  \\ \end{matrix} \right.\]on [-2, 3] done clear
  View Solution play_arrow

• question_answer15)
  If f(x) and g(x) are differentiable functions for \[0\le x\le 1\] such that f(0)=10, g(0)=2,  f(1)=2, g(1)=4, then in the interval (0, 1),
  A)
  f'(x) =0 for all x done clear
  B)
  f'(x)+4g'(x)=0 for at least one x done clear
  C)
  f'(x)=2g'(x) for at most done x done clear
  D)
  none of these done clear
  View Solution play_arrow

• question_answer16)
  The radius of a right curular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. the rate of change of the volume of the cylinder, in \[c{{m}^{2}}\]/min when the radius is 2 cm and the height is 3 cm is
  A)
  \[-2\pi \]             done clear
  B)
  \[-\frac{8\pi }{5}\] done clear
  C)
  \[-\frac{3\pi }{5}\]          done clear
  D)
  \[\frac{2\pi }{5}\] done clear
  View Solution play_arrow

• question_answer17)
    A function\[y=f\left( x \right)\]has a second order derivative\[f''\left( x \right)=6\left( x-1 \right)\]. If its graph passes through the point (2, 1) and at that point the tangent to the graph is \[y=3x-5,\]then the function is
  A)
  \[{{(x-1)}^{2}}\]          done clear
  B)
  \[{{(x-1)}^{3}}\] done clear
  C)
  \[{{(x+1)}^{3}}\]        done clear
  D)
  \[{{(x+1)}^{2}}\] done clear
  View Solution play_arrow

• question_answer18)
  If \[2a+3b+6c=0,\] then at least one root of the equation \[a{{x}^{2}}+bx+c=0\]lies in the interval
  A)
  (0, 1)     done clear
  B)
  (1, 3) done clear
  C)
  (2, 3)     done clear
  D)
  (1, 3) done clear
  View Solution play_arrow

• question_answer19)
  A point on the parabola \[{{y}^{2}}=18x\] at which the ordinate increases at twice the rate of the abscissa is
  A)
  (2, 4)     done clear
  B)
  (2, -4) done clear
  C)
  \[\left( \frac{-9}{8},\frac{9}{2} \right)\]    done clear
  D)
  \[\left( \frac{9}{8},\frac{9}{2} \right)\] done clear
  View Solution play_arrow

• question_answer20)
  The maximum distance from the origin of a point on the curve \[x=a\text{ }sin\text{ }t-b\]\[\sin \left( \frac{at}{b} \right),\] \[y=a\,\cos \,\,t-b\cos \left( \frac{at}{b} \right),\] both a, b>0, is
  A)
  a - b                 done clear
  B)
  a+b done clear
  C)
  \[\sqrt{{{a}^{2}}+{{b}^{2}}}\]         done clear
  D)
  \[\sqrt{{{a}^{2}}-{{b}^{2}}}\] done clear
  View Solution play_arrow

• question_answer21)
  The radius of the base of a cone is increasing at the rate of 3 cm/min and altitude is decreasing at the rate of 4 cm/min the rate of change (\[c{{m}^{2}}/\sec \]) of lateral surface when the radius is 7 cm and altitude is 24 cm is _______.
  View Solution play_arrow

• question_answer22)
  The altitude of a cone is 20 cm and its semi-vertical angle is \[30{}^\circ \]if the semi-vertical angle is increasing at the rate of \[2{}^\circ \]per second, then the radius of the base is increasing at the rate (cm/sec) of _____.
  View Solution play_arrow

• question_answer23)
  A function f is differentiable in the interval \[0\le x\le 5\] such that f (0) = 4 and\[f\left( 5 \right)=-1\]. If \[g(x)=\frac{f(x)}{x+1}\], then there exists some \[c\in (0,5)\]such that g'=____.
  View Solution play_arrow

• question_answer24)
  The surface area of a cube is increasing at the rate of \[2c{{m}^{2}}\text{/}\sec \]. When its edge is 90 cm, the volume (\[c{{m}^{3}}/\sec \]) is increasing at the rate of ____.
  View Solution play_arrow

• question_answer25)
  Suppose that water is emptied form a spherical tank of radius 10 cm. if the depth of the water in the tank is 4 cm and is decreasing at the rate of 3 cm/sec, then the radius of the top surface of water id decreasing at the rate of ____.
  View Solution play_arrow