If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line joining \left( {0,{3 \over 2}} \right) and \left( {{1 \over 2},2} \right), then :

Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is :

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness the melts at a rate of 50 cm3/min. When the thickness of ice is 5 cm, then the rate (in cm/min.) at which of the thickness of ice decreases, is :

The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy – 3y2 = 0 at the point (2,2) is

Let ƒ(x) = xcos–1(–sin|x|), x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right], then which of the following is true?

If c is a point at which Rolle's theorem holds for the function, f(x) = {\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right) in the interval [3, 4], where a $$\in$$ R, then ƒ''(c) is equal to

Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If \mathop {\lim }\limits_{x \to 0} \left( {2 + {{f\left( x \right)} \over {{x^3}}}} \right) = 4, then which one of the following is not true?

The value of c in the Lagrange's mean value theorem for the function ƒ(x) = x3 - 4x2 + 8x + 11, when x $$\in$$ [0, 1] is:

Let f : (–1, $$\infty$$) $$\to$$ R be defined by f(0) = 1 and f(x) = {1 \over x}{\log _e}\left( {1 + x} \right), x $$\ne$$ 0. Then the function f :

Let the function, ƒ:[-7, 0]$$\to$$R be continuous on [-7,0] and differentiable on (-7, 0). If ƒ(-7) = - 3 and ƒ'(x) $$\le$$ 2, for all x $$\in$$ (-7,0), then for all such functions ƒ, ƒ(-1) + ƒ(0) lies in the interval:

The area (in sq. units) of the region enclosed by the curves y = x2 – 1 and y = 1 – x2 is equal to :

The area (in sq. units) of the region A = {(x, y) : |x| + |y| $$\le$$ 1, 2y2 $$\ge$$ |x|}

The area (in sq. units) of the region A = {(x, y) : (x – 1)[x] $$\le$$ y $$\le$$ 2$$\sqrt{x}$$, 0 $$\le$$ x $$\le$$ 2}, where [t] denotes the greatest integer function, is :

The area (in sq. units) of the region { (x, y) : 0 $$\le$$ y $$\le$$ x2 + 1, 0 $$\le$$ y $$\le$$ x + 1, {1 \over 2} $$\le$$ x $$\le$$ 2 } is :

Given : f(x) = \left\{ {\matrix{ {x\,\,\,\,\,,} & {0 \le x < {1 \over 2}} \cr {{1 \over 2}\,\,\,\,,} & {x = {1 \over 2}} \cr {1 - x\,\,\,,} & {{1 \over 2} < x \le 1} \cr } } \right. and g(x) = \left( {x - {1 \over 2}} \right)^2,x \in R Then the area (in sq. units) of the region bounded by the curves, y = ƒ(x) and y = g(x) between the lines, 2x = 1 and 2x = $$\sqrt{3}$$, is :

Consider a region R = {(x, y) $$\in$$ R : x2 $$\le$$ y $$\le$$ 2x}. if a line y = $$\alpha$$ divides the area of region R into two equal parts, then which of the following is true?

The area (in sq. units) of the region {(x,y) $$\in$$ R2 : x2 $$\le$$ y $$\le$$ 3 – 2x}, is :

Area (in sq. units) of the region outside {{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1 and inside the ellipse {{{x^2}} \over 4} + {{{y^2}} \over 9} = 1 is :

The area of the region, enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = x, is:

The area (in sq. units) of the region {(x, y) $$\in$$ R2 | 4x2 $$\le$$ y $$\le$$ 8x + 12} is :