Application of Derivatives PYQs Set 1 (PDF)

Set 1 of 10 · Questions 1-30 of 274

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Application of Derivatives · Set 1 (Q1-30)
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1. $If the tangent drawn at a point P on the curve y = 3 x^{2} - 5 x + 7 is parallel to its chord joining the points (1, y_{1}) and (2, y_{2}) on it, then the x -coordinates of the point P is$ (1) $2$ (2) $\frac{3}{2}$ (3) $\frac{5}{4}$ (4) $\frac{4}{3}$ 2. $An angle between the curves x^{2} - y^{2} = 4 and x^{2} + y^{2} = 42 is$ (1) $\frac{π}{2}$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{π}{6}$ 3. $For which value (s) of a f (x) = - x^{3} + 4 a x^{2} + 2 x - 5 is decreasing for every x ?$ (1) $(1, 2)$ (2) $(3, 4)$ (3) $R$ (4) $no value of a$ 4. $The rate of change of area of a circle with respect to its radius at r = 2 cms is$ (1) $4$ (2) $2 π$ (3) $2$ (4) $4 π$ 5. $m is the slope of a tangent to the curve e^{y} = 1 + x^{2} at x = 1 then m =$ (1) $2 log 2$ (2) $log 2$ (3) $2$ (4) $1$ 6. $The equation of the normal to the curve sin y = 3 x sin (\frac{π}{6} + y) at x = 0, is$ (1) $2 x + 3 y = 0$ (2) $2 x + y = 0$ (3) $x + 2 y = 0$ (4) $3 x + 2 y = 0$ 7. $A sphere increases its volume at the rate of π cc / s . The rate at which its surface area increases when the radius is 1 cm is$ (1) $2 π sq \cdot cm / s$ (2) $π sq . cm / s$ (3) $\frac{3 π}{2} sq \cdot cm / s$ (4) $\frac{π}{2} sq \cdot cm / s$ 8. $If f (x) = (2 k + 1) x - 3 - k e^{- x} + 2 e^{x} is monotonically increasing for all x \in R, then the least value of k is$ (1) $1$ (2) $0$ (3) $- \frac{1}{2}$ (4) $-1$ 9. $If p (x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p (0) is equal to:$ (1) $12$ (2) $6$ (3) $- 24$ (4) $- 12$ 10. $For the curve 4 x^{5} = 5 y^{4}, the ratio of the cube of the subtangent at a point on the curve to the square of the subnormal at the same point is$ (1) $y (\frac{5}{4})^{4}$ (2) $(\frac{5}{4})^{4}$ (3) $(\frac{5}{4})^{3}$ (4) $(\frac{4}{5})^{4}$ 11. $The maximum value of \frac{log _{e} x}{x}, if x > 0 is$ (1) $1$ (2) $\frac{1}{e}$ (3) $- \frac{1}{e}$ (4) $e$ 12. $The tangent of the angle between the curves xy = 1 and x^{2} + 8 y = 0, is$ (1) $\frac{1}{7}$ (2) $\frac{2}{7}$ (3) $\frac{6}{7}$ (4) $\frac{3}{7}$ 13. $If the surface area of a cube is increasing at a rate of 3.6 cm^{2} / sec retaining its shape; then the rate of change of its volume (in cm^{3} / sec)when the length of a side of the cube is 10 cm is:$ (1) $9$ (2) $20$ (3) $10$ (4) $18$ 14. $If x = t^{2} and y = 2 t are parametric equations of a curve, then equation of the normal to the curve at t = 2 is$ (1) $2 x + 3 y - 20 = 0$ (2) $x + y - 8 = 0$ (3) $2 x + y - 12 = 0$ (4) $x + 2 y - 12 = 0$ 15. $Let f (x) = x^{2025} - x^{2000}, x \in [0, 1] and the minimum value of the function f (x) in the interval [0, 1] be (80)^{80} (n)^{- 81} . Then n is equal to$ (1) $-40$ (2) $-41$ (3) $-80$ (4) $-81$ 16. $The approximate value of cos 31^{\circ} is (Take 1^{\circ} = 0.0174)$ (1) $0.7521$ (2) $0.866$ (3) $0.7146$ (4) $0.8573$ 17. $The real number k for which the equation, 2x3 + 3x + k = 0 has two distinct real roots in [0, 1]$ (1) $lies between 1 and 2$ (2) $lies between 2 and 3$ (3) $lies between -1 and 0$ (4) $does not exist.$ 18. $If the tangent at a point P on the curve y = 4 x^{4} + x is perpendicular to the tangent to the same curve at (0, 0), then the point P is$ (1) $(\frac{- 1}{2}, \frac{- 1}{4})$ (2) $(\frac{1}{2}, \frac{3}{4})$ (3) $(1, 5)$ (4) $(- 1, 3)$ 19. $The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds,, it becomes 7 units, then its radius after 9 seconds is :$ (1) $9$ (2) $10$ (3) $11$ (4) $12$ 20. $If the tangent to the curve, y = f (x) = x log_{e} x, (x > 0) at a point (c, f (c)) is parallel to the line- segment joining the points (1, 0) and (e, e), then c is equal to:$ (1) $\frac{1}{e - 1}$ (2) $\frac{e - 1}{e}$ (3) $e^{(\frac{1}{1 - e})}$ (4) $e^{(\frac{1}{e - 1})}$ 21. $The set of all a \in R for which the equation x \|x-l\| + \|x + 2\|+a = 0 has exactly one real root, is$ (1) $(-7, \infty)$ (2) $(- \infty, \infty)$ (3) $(-6,-3)$ (4) $(- \infty,-3)$ 22. $If x = f (t) and y = g (t) are differentiable functions of t, then \frac{d ^{2} y}{d x ^{2}} is$ (1) $\frac{f ^{’} ( t ) \cdot g ^{’ ’} ( t ) - g ^{’} ( t ) \cdot f ^{’ ’} ( t )}{( f ^{’} ( t ) ) ^{3}}$ (2) $\frac{f ^{’} ( t ) \cdot g ^{’ ’} ( t ) - g ^{’} ( t ) \cdot f ^{’ ’} ( t )}{( f ^{’} ( t ) ) ^{2}}$ (3) $\frac{g ^{’} ( t ) \cdot f ^{’ ’} ( t ) - f ^{’} ( t ) \cdot g ^{’ ’} ( t )}{( f ^{’} ( t ) ) ^{3}}$ (4) $\frac{g ^{’} ( t ) \cdot f ^{’ ’} ( t ) + f ^{’} ( t ) \cdot g ^{’ ’} ( t )}{( f ^{’} ( t ) ) ^{3}}$ 23. $The equation of the tangent to the curve y = x + \frac{4}{x ^{2}}, that is parallel to the x-axis,$ (1) $y = 1$ (2) $y = 2$ (3) $y = 3$ (4) $y = 0$ 24. $If x = 1 is a critical point of the function f (x) = (3 x^{2} + ax - 2 - a) e^{x}, then:$ (1) $x=1 is a local maxima and x = - \frac{2}{3} is a local minima of f$ (2) $x=1 and x = - \frac{2}{3} are local maxima of f$ (3) $x=1 and x = - \frac{2}{3} are local minima of f$ (4) $x=1 is a local minima and x = - \frac{2}{3} is a local maxima of f$ 25. $Let C be the locus of the mirror image of a point on the parabola y^{2} = 4 x with respect to the line y = x . Thenthe equation of tangent to C at P (2, 1) is :$ (1) $x - y = 1$ (2) $2 x + y = 5$ (3) $x + 3 y = 5$ (4) $x + 2 y = 4$ 26. $Let a function f : [0, 5] \to R be continuous, f (1) = 3 and F defined as: F (x) = \int_{1}^{x} t^{2} g (t) d t where g (t) = \int_{1}^{t} f (u) d u . Then for the function F, the point x = 1 is:$ (1) $a point of local minima$ (2) $a point of local maxima.$ (3) $not a critical point$ (4) $a point of inflection.$ 27. $Approximate change in the volume V of a cube of side x metres caused by increasing the side by 3 % is$ (1) $0.09 x^{3} m^{3}$ (2) $0.06 x^{3} m^{3}$ (3) $0.03 x^{3} m^{3}$ (4) $0.04 x^{3} m^{3}$ 28. $The slope of the tangent at (1, 2) to the curve x = t^{2} - 7 t + 7 and y = t^{2} - 4 t - 10, is$ (1) $\frac{8}{5}$ (2) $\frac{5}{8}$ (3) $- \frac{8}{5}$ (4) $- \frac{5}{8}$ 29. $Let f be a twice differentiable function on (1, 6) If f (2) = 8, f^{’} (2) = 5, f^{’} (x) \geq 1 and f^{’ ’} (x) \geq 4 for all x \in (1, 6) then$ (1) $f (5) + f^{’} (5) \geq 28$ (2) $f^{’} (5) + f^{’ ’} (5) \leq 20$ (3) $f(5)\leq10$ (4) $f (5) + f^{’} (5) \leq 26$ 30. $If x - 2 y + k = 0 is a tangent to the parabola y^{2} - 4 x - 4 y + 8 = 0, then the slope of the tangent drawn at (1, k) on the given parabola is$ (1) $\frac{- 5}{2}$ (2) $2$ (3) $-2$ (4) $\frac{2}{5}$