The angle of elevation of a jet plane from a point A on the ground is 60$${}^{\circ }$$. After a flight of 20 seconds at the speed of 432 km/hour, the angle of elevation changes to 30$${}^{\circ }$$. If the jet plane is flying at a constant height, then its height is :

A man is observing, from the top of a tower, a boat speeding towards the lower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is 30$${}^{\circ }$$ (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point B, where the angle of depression is 45$${}^{\circ }$$. Then the time taken (in seconds) by the boat from B to reach the base of the tower is :

A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be {\pi \over 3}. If the radius of the circumcircle of $$\Delta$$ABC is 2, then the height of the pole is equal to :

Let in a right angled triangle, the smallest angle be $$\theta$$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then sin$$\theta$$ is equal to :

A spherical gas balloon of radius 16 meter subtends an angle 60$${}^{\circ }$$ at the eye of the observer A while the angle of elevation of its center from the eye of A is 75$${}^{\circ }$$. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is :

A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that PC = $$\sqrt{5}$$ inches and $$\angle$$PCB = tan-1(2). The acute angle through which the pencil must be rotated about C so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is :

Two poles, AB of length a metres and CD of length a + b (b $$\ne$$ a) metres are erected at the same horizontal level with bases at B and D. If BD = x and tan$$\angle$$ACB = {1 \over 2}, then :

A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, then the height of the pole (in meters) is :

The value of -15C1 + 2.15C2 – 3.15C3 + ... - 15.15C15 + 14C1 + 14C3 + 14C5 + ...+ 14C11 is :

If $$n\ge 2$$ is a positive integer, then the sum of the series $${}^{n+1}{C}_2+2\left({}^2{C}_2+{}^3{C}_2+{}^4{C}_2+...+{}^n{C}_2\right)$$ is :

The maximum value of the term independent of 't' in the expansion of {\left( {t{x^{{1 \over 5}}} + {{{{(1 - x)}^{{1 \over {10}}}}} \over t}} \right)^{10}} where x$$\in$$(0, 1) is :

If n is the number of irrational terms in the expansion of $${\left(3^{1/4}+5^{1/8}\right)}^{60}$$, then (n $$-$$ 1) is divisible by :

Let [ x ] denote greatest integer less than or equal to x. If for n$$\in$$N, $$(1-x+x^3{)}^n=\sum _{j=0}^{3n}{a}_j{x}^j$$, then \sum\limits_{j = 0}^{\left[ {{{3n} \over 2}} \right]} {{a_{2j}} + 4} \sum\limits_{j = 0}^{\left[ {{{3n - 1} \over 2}} \right]} {{a_{2j}} + 1} is equal to :

If the fourth term in the expansion of $$(x+x^{{\log }_{2}x}{)}^7$$ is 4480, then the value of x where x$$\in$$N is equal to :

The value of $$\sum _{r=0}^6\left({}^6{C}_r.{}^6{C}_{6-r}\right)$$ is equal to :

Let (1 + x + 2x2)20 = a0 + a1x + a2x2 + .... + a40x40. Then a1 + a3 + a5 + ..... + a37 is equal to

The coefficient of x256 in the expansion of (1 $$-$$ x)101 (x2 + x + 1)100 is :

For the natural numbers m, n, if $$(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+....+a_{m+n}{y}^{m+n}$$ and $$a_1=a_2=10$$, then the value of (m + n) is equal to :

If b is very small as compared to the value of a, so that the cube and other higher powers of {b \over a} can be neglected in the identity {1 \over {a - b}} + {1 \over {a - 2b}} + {1 \over {a - 3b}} + ..... + {1 \over {a - nb}} = \alpha n + \beta {n^2} + \gamma {n^3}, then the value of $$\gamma$$ is :

A possible value of 'x', for which the ninth term in the expansion of {\left\{ {{3^{{{\log }_3}\sqrt {{{25}^{x - 1}} + 7} }} + {3^{\left( { - {1 \over 8}} \right){{\log }_3}({5^{x - 1}} + 1)}}} \right\}^{10}} in the increasing powers of {3^{\left( { - {1 \over 8}} \right){{\log }_3}({5^{x - 1}} + 1)}} is equal to 180, is :