If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ..... is (102)m, then m is equal to :

Let ƒ : R $$\to$$ R be such that for all x $$\in$$ R (21+x + 21–x), ƒ(x) and (3x + 3–x) are in A.P., then the minimum value of ƒ(x) is

If the 10th term of an A.P. is {1 \over {20}} and its 20th term is {1 \over {10}}, then the sum of its first 200 terms is

The product {2^{{1 \over 4}}}{.4^{{1 \over {16}}}}{.8^{{1 \over {48}}}}{.16^{{1 \over {128}}}} ... to $$\infty$$ is equal to :

If the sum of first 11 terms of an A.P., a1, a2, a3, .... is 0 (a $$\ne$$ 0), then the sum of the A.P., a1 , a3 , a5 ,....., a23 is ka1 , where k is equal to :

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :

If |x| < 1, |y| < 1 and x $$\ne$$ y, then the sum to infinity of the following series (x + y) + (x2+xy+y2) + (x3+x2y + xy2+y3) + ....

Let an be the nth term of a G.P. of positive terms. $$\sum _{n=1}^{100}{a}_{2n+1}=200$$ and $$\sum _{n=1}^{100}{a}_{2n}=100$$, then $$\sum _{n=1}^{200}{a}_n$$ is equal to :

The function, f(x) = (3x – 7)x2/3, x $$\in$$ R, is increasing for all x lying in :

The equation of the normal to the curve y = (1+x)2y + cos 2(sin–1x) at x = 0 is :

If the surface area of a cube is increasing at a rate of 3.6 cm2/sec, retaining its shape; then the rate of change of its volume (in cm3/sec), when the length of a side of the cube is 10 cm, is :

Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $$\ge$$ 1 and f''(x) $$\ge$$ 4, for all x $$\in$$ (1, 6), then :

The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x2–1 below the x-axis, is :

If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0, -4), then PQ2 is equal to:

Which of the following points lies on the tangent to the curve x4ey + 2$$\sqrt{y+1}$$ = 3 at the point (1, 0)?

If x = 1 is a critical point of the function f(x) = (3x2 + ax – 2 – a)ex , then :

The position of a moving car at time t is given by f(t) = at2 + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average speed of the car over the time interval [t1 , t2 ] is attained at the point :

The set of all real values of $$\lambda$$ for which the function f(x) = \left( {1 - {{\cos }^2}x} \right)\left( {\lambda + \sin x} \right),x \in \left( { - {\pi \over 2},{\pi \over 2}} \right) has exactly one maxima and exactly one minima, is :

If the tangent to the curve, y = f (x) = xloge x, (x > 0) at a point (c, f(c)) is parallel to the line-segment joining the points (1, 0) and (e, e), then c is equal to :

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :