### 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 $-2\sqrt3,\sqrt3$