If the tangents on the ellipse 4x2 + y2 = 8 at the points (1, 2) and (a, b) are perpendicular to each other, then a2 is equal to :

Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $$\Delta$$S'BS is a right angled triangle with right angle at B and area ($$\Delta$$S'BS) = 8 sq. units, then the length of a latus rectum of the ellipse is :

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

Let S = \left\{ {\left( {x,y} \right) \in {R^2}:{{{y^2}} \over {1 + r}} - {{{x^2}} \over {1 - r}}} \right\};r \ne \pm 1. Then S represents :

If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve :

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at (0,5$$\sqrt{3}$$), then the length of its latus rectum is :

Let A, B and C be sets such that $$\varphi$$ $$\ne$$ A $$\cap$$ B $$\subseteq$$ C. Then which of the following statements is not true ?

Two newspapers A and B are published in a city. It is known that 25% of the city populations reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisement is :-

Let Z be the set of integers. If A = {x $$\in$$ Z : 2(x + 2) (x2 $$-$$ 5x + 6) = 1} and B = {x $$\in$$ Z : $$-$$ 3 < 2x $$-$$ 1 < 9}, then the number of subsets of the set A $$\times$$ B, is

Let S = {1, 2, 3, … , 100}. The number of non-empty subsets A of S such that the product of elements in A is even is :

In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is

\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x} \over {\sqrt 2 - \sqrt {1 + \cos x} }} equals:

Let ƒ : R $$\to$$ R be a differentiable function satisfying ƒ'(3) + ƒ'(2) = 0. Then \mathop {\lim }\limits_{x \to 0} {\left( {{{1 + f(3 + x) - f(3)} \over {1 + f(2 - x) - f(2)}}} \right)^{{1 \over x}}} is equal to

Let ƒ : [–1,3] $$\to$$ R be defined as f(x) = \left\{ {\matrix{ {\left| x \right| + \left[ x \right]} & , & { - 1 \le x < 1} \cr {x + \left| x \right|} & , & {1 \le x < 2} \cr {x + \left[ x \right]} & , & {2 \le x \le 3} \cr } } \right. where [t] denotes the greatest integer less than or equal to t. Then, ƒ is discontinuous at:

Let ƒ(x) = 15 – |x – 10|; x $$\in$$ R. Then the set of all values of x, at which the function, g(x) = ƒ(ƒ(x)) is not differentiable, is :

If the function ƒ defined on , \left( {{\pi \over 6},{\pi \over 3}} \right) by $f(x) = \left\{ {\matrix{ {{{\sqrt 2 {\mathop{\rm cosx}\nolimits} - 1} \over {\cot x - 1}},} & {x \ne {\pi \over 4}} \cr {k,} & {x = {\pi \over 4}} \cr } } \right.$ is continuous, then k is equal to

If f(x) = [x] - \left[ {{x \over 4}} \right] ,x $$\in$$ 4 , where [x] denotes the greatest integer function, then

If the function $$f(x)=\{\text{matrix}a|\pi -x|+1,x\le 5\text{cr}b|x-\pi |+3,x>5\text{cr}$$ is continuous at x = 5, then the value of a – b is :-

If \mathop {\lim }\limits_{x \to 1} {{{x^4} - 1} \over {x - 1}} = \mathop {\lim }\limits_{x \to k} {{{x^3} - {k^3}} \over {{x^2} - {k^2}}}, then k is :

Let f : R $$\to$$ R be differentiable at c $$\in$$ R and f(c) = 0. If g(x) = |f(x)| , then at x = c, g is :