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Application of Derivatives

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  1. 51.If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, …
  2. 52.The minimum distance of a point on the curve y = x2-4 from the origin is :
  3. 53.A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceler…
  4. 54.Let f:R R be a twice differentiable function such that the quadratic equation f(x)m^2-2f^'…
  5. 55.Let A(4,-2),B(1,1) and C(9,-3) be the vertices of a triangle ABC .
  6. 56.If the set of all values of a, for which the equation 5 x^3-15 x-a=0 has three distinct re…
  7. 57.Let the set of all values of p, for which f(x)=(p^2-6p+8)( ^22x- ^22x)+2(2-p)x+7 does not …
  8. 58.Let the set of all positive values of , for which the point of local minimum of the functi…
  9. 59.Let A be the region enclosed by the parabola y^2=2x and the line x=24.
  10. 60.Let the maximum and minimum values of (√8x-x^2-12-4)^2+(x-7)^2,x R be M and m, respectivel…
  11. 61.Let f(x)=2^x-x^2,x R.
  12. 62.Let for a differentiable function f:(0,) R,f(x)-f(y) _e((x)/(y))+x-y,,y (0,).
  13. 63.Consider the triangles with vertices A(2,1), B(0,0) and C(t, 4), t [0,4].
  14. 64.Let the quadratic curve passing through the point (-1,0) and touching the line y=x at (1,1…
  15. 65.If a_ is the greatest term in the sequence _n=n^3n^4+147,n=1,2,3,, then is equal to ______…
  16. 66.Let a curve y=f(x),x (0,) pass through the points P(1,(3)/(2)) and Q(a,(1)/(2)).
  17. 67.The number of points, where the curve y=x^5-20x^3+50x+2 crosses the x-axis, is ___________…
  18. 68.Let the lines y+2x=√11+7√7 and 2y+x=2√11+6√7 be normal to a circle C:(x-h)^2+(y-k)^2=r^2.
  19. 69.Consider a curve y=y(x) in the first quadrant as shown in the figure.
  20. 70.Let the function f(x)=2x^2-_e x, x>0, be decreasing in (0,a) and increasing in (a,4).
  21. 71.If the curves x^2a+y^2b=1 and x^2c+y^2d=1 intersect each other at an angle of 90^, then wh…
  22. 72.If the normal to the ellipse 3x^2+4y^2=12 at a point P on it is parallel to the line 2x+y=…
  23. 73.If 2a+3b+6c=0, [( a,b,c R )] then the quadratic equation ax^2 + bx + c = 0 has
  24. 74.The real number x when added to its inverse gives the minimum sum at x equal :
  25. 75.If the function [f( x ) = 2x^3 - 9ax^2 + 12a^2x + 1,] where a>0, attains its maximum and m…
  26. 76.A point on the parabola y^2 = 18x at which the ordinate increases at twice the rate of the…
  27. 77.A function y=f(x) has a second order derivative [f''( x ) = 6( x - 1 ).] If its graph pass…
  28. 78.If 2a+3b+6c=0, then at least one root of the equation ax^2 + bx + c = 0 lies in the interv…
  29. 79.The normal to the curve x = a(1 + cos [ ]), [y = a ] at [' '] always passes through the fi…
  30. 80.A function is matched below against an interval where it is supposed to be increasing.
  31. 81.If the equation [a_nx^n + a_n - 1x^n - 1 + ...........
  32. 82.The normal to the curve [x = a( + ),y = a( - )] at any point [, '] is such that
  33. 83.A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness t…
  34. 84.Area of the greatest rectangle that can be inscribed in the ellipse [x^2 a^2 + y^2 b^2 = 1…
  35. 85.A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceler…
  36. 86.The function [f( x ) = x 2 + 2 x] has a local minimum at
  37. 87.Angle between the tangents to the curve y = x^2 - 5x + 6 at the points (2,0) and (3,0) is
  38. 88.A value of c for which conclusion of Mean Value Theorem holds for the function [f( x ) = _…
  39. 89.The function [f( x ) = ^ - 1( x + x )] is an incresing function in
  40. 90.Suppose the cubic x^3 - px + q has three distinct real roots where p>0 and q>0.
  41. 91.How many real solutions does the equation [x^7 + 14x^5 + 16x^3 + 30x - 560 = 0] have?
  42. 92.Given [P( x ) = x^4 + ax^3 + bx^2 + cx + d] such that x=0 is the only real root of [P',( x…
  43. 93.The equation of the tangent to the curve [y = x + 4 x^2], that is parallel to the x-axis, …
  44. 94.Let [f:R R] be defined by [ f( x ) = pmatrix k - 2x,,,if & x - 1 2x + 3,,,if & x > - 1 pma…
  45. 95.Let [f:R R] be a continuous function defined by [ f( x ) = 1 e^x + 2e^ - x]
  46. 96.For [x ( 0,5 2 ),] define [f( x ) = _0^x t t,dt. ] Then f has
  47. 97.Let [a,b R] be such that the function f given by [f( x ) = In| x | + bx^2 + ax,,x 0] has e…
  48. 98.A spherical balloon is filled with [4500 ] cubic meters of helium gas.
  49. 99.A line is drawn through the point (1, 2) to meet the coordinate axes at P and Q such that …
  50. 100.The real number k for which the equation, 2x^3 + 3x + k = 0 has two distinct real roots in…