If a, b, c are in A.P. and a2, b2, c2 are in G.P. such that a < b < c and a + b + c = {3 \over 4}, then the value of a is :

If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :

If x1, x2, . . ., xn and {1 \over {{h_1}}}, {1 \over {{h_2}}}, . . . , {1 \over {{h_n}}} are two A.P..s such that x3 = h2 = 8 and x8 = h7 = 20, then x5.h10 equals :

Let {1 \over {{x_1}}},{1 \over {{x_2}}},...,{1 \over {{x_n}}}\,\, (xi $$\ne$$ 0 for i = 1, 2, ..., n) be in A.P. such that x1=4 and x21 = 20. If n is the least positive integer for which $$x_n>50,$$ then \sum\limits_{i = 1}^n {\left( {{1 \over {{x_i}}}} \right)} is equal to :

Let $$a_1$$, $$a_2$$, $$a_3$$, ......... ,$$a_{49}$$ be in A.P. such that $$\sum _{k=0}^{12}{a}_{4k+1}=416$$ and $$a_9+a_{43}=66$$. $$a_1^2+a_2^2+.......+a_{17}^2=140m$$, then m is equal to

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 12 + 2.22 + 32 + 2.42 + 52 + 2.62 ........... If B - 2A = 100$$\lambda$$, then $$\lambda$$ is equal to

Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f(x) = 2x3 $$-$$ 9x2 + 12x + 5 in the interval [0, 3]. Then M $$-$$m is equal to :

If the curves y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is :

Let f\left( x \right) = {x^2} + {1 \over {{x^2}}} and g\left( x \right) = x - {1 \over x}, $$x\in R-\left\{-1,0,1\right\}$$. If h\left( x \right) = {{f\left( x \right)} \over {g\left( x \right)}}, then the local minimum value of h(x) is

If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm2) of this cone is :

If $$\beta$$ is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points (3 cos $$\theta$$, $$\sqrt{3}\sin \theta$$) and ($$-$$ 3 sin $$\theta$$, $$\sqrt{3}\cos \theta$$); \theta \in \left( {0,{\pi \over 2}} \right); then {{2\,\cot \beta } \over {\sin 2\theta }} is equal to :

The area (in sq. units) of the region {x $$\in$$ R : x $$\ge$$ 0, y $$\ge$$ 0, y $$\ge$$ x $$-$$ 2 and y $$\le$$ $$\sqrt{x}$$}, is :

If the area of the region bounded by the curves, y = {x^2},y = {1 \over x} and the lines y = 0 and x= t (t >1) is 1 sq. unit, then t is equal to :

Let g(x) = cosx2, f(x) = $$\sqrt{x}$$ and $$\alpha ,\beta \left(\alpha <\beta \right)$$ be the roots of the quadratic equation 18x2 - 9$$\pi$$x + $${\pi }^2$$ = 0. Then the area (in sq. units) bounded by the curve y = (gof)(x) and the lines $$x=\alpha$$, $$x=\beta$$ and y = 0 is :

The Boolean expression $$\sim \left(p\vee q\right)\vee \left(\sim p\wedge q\right)$$ is equvalent to :

Consider the following two statements : Statement p : The value of sin 120o can be derived by taking $$\theta =240^o$$ in the equation 2sin{\theta \over 2} = \sqrt {1 + \sin \theta } - \sqrt {1 - \sin \theta } Statement q : The angles A, B, C and D of any quadrilateral ABCD satisfy the equation cos\left( {{1 \over 2}\left( {A + C} \right)} \right) + \cos \left( {{1 \over 2}\left( {B + D} \right)} \right) = 0 Then the truth values of p and q are respectively :

If (p $$\wedge$$ $$\sim$$ q) $$\wedge$$ (p $$\wedge$$ r) $$\to$$ $$\sim$$ p $$\vee$$ q is false, then the truth values of $$p,q$$ and $$r$$ are, respectively :

If p $$\to$$ ($$\sim$$ p$$\vee$$ $$\sim$$ q) is false, then the truth values of p and q are respectively :

The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants G, h and c. Which of the following correctly gives the Planck length ?

In a screw gauge, $$5$$ complete rotations of the screw cause it to move a linear distance of $$0.25$$ $$cm.$$ There are $$100$$ circular scale divisions. The thickness of a wire measured by this screw gauge gives a reading of $$4$$ main scale divisions and $$30$$ circular scale divisions. Assuming negligible zero error, the thickness of the wire is :