If sin4$$\alpha$$ + 4 cos4$$\beta$$ + 2 = 4$$\sqrt{2}$$ sin $$\alpha$$ cos $$\beta$$; $$\alpha$$, $$\beta$$ $$\in$$ [0, $$\pi$$], then cos($$\alpha$$ + $$\beta$$) $$-$$ cos($$\alpha$$ $$-$$ $$\beta$$) is equal to :

If nC4, nC5 and nC6 are in A.P., then n can be :

The sum of the following series 1 + 6 + {{9\left( {{1^2} + {2^2} + {3^2}} \right)} \over 7} + {{12\left( {{1^2} + {2^2} + {3^2} + {4^2}} \right)} \over 9} + {{15\left( {{1^2} + {2^2} + ... + {5^2}} \right)} \over {11}} + ..... up to 15 terms, is :

Let a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A.P. If these are also three consecutive terms of a G.P., then {a \over c} equal to :

Let $$a_1,a_2,.......,a_{30}$$ be an A.P., $$S=\sum _{i=1}^{30}{a}_i$$ and $$T=\sum _{i=1}^{15}{a}_{\left(2i-1\right)}$$. If $$a_5$$ = 27 and S - 2T = 75, then $$a_{10}$$ is equal to :

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is -

Let a1, a2, a3, ..... a10 be in G.P. with ai > 0 for i = 1, 2, ….., 10 and S be the set of pairs (r, k), r, k $$\in$$ N (the set of natural numbers) for which \left| {\matrix{ {{{\log }_e}\,{a_1}^r{a_2}^k} & {{{\log }_e}\,{a_2}^r{a_3}^k} & {{{\log }_e}\,{a_3}^r{a_4}^k} \cr {{{\log }_e}\,{a_4}^r{a_5}^k} & {{{\log }_e}\,{a_5}^r{a_6}^k} & {{{\log }_e}\,{a_6}^r{a_7}^k} \cr {{{\log }_e}\,{a_7}^r{a_8}^k} & {{{\log }_e}\,{a_8}^r{a_9}^k} & {{{\log }_e}\,{a_9}^r{a_{10}}^k} \cr } } \right| $$=$$ 0. Then the number of elements in S, is -

If a, b, c be three distinct real numbers in G.P. and a + b + c = xb , then x cannot be

Let a1, a2, . . . . . ., a10 be a G.P. If {{{a_3}} \over {{a_1}}} = 25, then {{{a_9}} \over {{a_5}}} equals

If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is :

Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression {{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}} is :

Let Sk = {{1 + 2 + 3 + .... + k} \over k}. If S_1^2 + S_2^2 + .....\, + S_{10}^2 = {5 \over {12}}A, then A is equal to :

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :

If the sum of the first 15 terms of the series {\left( {{3 \over 4}} \right)^3} + {\left( {1{1 \over 2}} \right)^3} + {\left( {2{1 \over 4}} \right)^3} + {3^3} + {\left( {3{3 \over 4}} \right)^3} + .... is equal to 225 k, then k is equal to :

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is {{27} \over {19}}.Then the common ratio of this series is :

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is :

If m is the minimum value of k for which the function f(x) = x$$\sqrt{kx-x^2}$$ is increasing in the interval [0,3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to :

If the tangent to the curve y = {x \over {{x^2} - 3}} , $$x\in \rho ,\left(x\ne \pm \sqrt{3}\right)$$, at a point ($$\alpha$$, $$\beta$$) $$\ne$$ (0, 0) on it is parallel to the line 2x + 6y – 11 = 0, then :

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

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is {\tan ^{ - 1}}\left( {{1 \over 2}} \right). Water is poured into it at a constant rate of 5 cubic meter per minute. The the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is :-