Mathematics - Application of Derivatives

Question - 1

The time T of oscillation of a simple pendulum of length l is given by \(T = 2\pi\sqrt{1\over g}\).  The percentage error in T corresponding to an error of 2% in the value of l is

  • A 2%
  • B 1%
  • C 3%
  • D 1.2%

Question - 2

The line y = mx + 1 is a tangent to the curve y2= 4x, if the value of m is

  • A 1
  • B 2
  • C 3
  • D 1/2

Question - 3

The normal to the curve x2 = 4y passing through (1,2) is

  • A x + y = 3
  • B x  - y = 3
  • C x + y = 1
  • D x - y = 1

Question - 4

The points on the the curve \(y=x^3-x^2-x+3\), where the tangents are parallel to the X-axis, are

  • A \((+{1\over 3},{-88\over 27}) and (1,2)\)
  • B \((-{1\over 3},{86\over 27}) and (1,2)\)
  • C \(({-1\over 3},{86\over 27}) and (-1,-2)\)
  • D \(({-1\over 3},{-88\over 27}) and (-1,2)\)

Question - 5

If the line ax + by + c = 0 is normal to curve xy + 5 = 0, then

  • A a + b = 0
  • B a > 0
  • C a < 0, b < 0
  • D a = - 2b

Question - 6

The line \({x\over a}+{y \over b}=2\) touches then curve \(({X\over a})^n=({y\over b})^n=2\) at the point (a, b) for

  • A n = 2 only
  • B n = -3 only
  • C n is any real number
  • D None of the above

Question - 7

The length of subnormal to the curve \(y = {x\over 1-x^2}\) at the point having abscissa \(\sqrt 2\) is

  • A \(5\sqrt2\)
  • B \(3\sqrt3\)
  • C \(\sqrt3\)
  • D \(3\sqrt2\)

Question - 8

The two curves \(x^3-3xy^2+2=0\) and  \(3x^2y-y^3=2\)

  • A touch each other
  • B cut at right angle
  • C cut at an angle \(\pi\over 3\)
  • D cut at an angle \(\pi\over 4\)

Question - 9

The interval on which \(f(x)=5x^3-15x^2-120x+3\) is strictly increasing or decreasing are respectively

  • A \((-\infty, + 2) \cup (4, \infty) and (-2,3)\)
  • B \((-\infty, -1) (4, \infty) and (-2,5)\)
  • C \((-\infty, - 2) \cup (4, \infty) and (-2,4)\)
  • D \((-\infty, -3) \cup (5, \infty) and (-2,4)\)

Question - 10

The coordinates of the point on the curve \((x^2+1)(y-3)=x\) where a tangent to the curve has the greatest slope are 

  • A \((\sqrt3, - \sqrt3 + 2)\)
  • B \((3,-2)\)
  • C \((0,3)\)
  • D \(-2\sqrt3,\sqrt3\)
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