The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y is :

The angle of elevation of the summit of a mountain from a point on the ground is 45°. After climbing up one km towards the summit at an inclination of 30° from the ground, the angle of elevation of the summit is found to be 60°. Then the height (in km) of the summit from the ground is :

The angle of elevation of a cloud C from a point P, 200 m above a still lake is 30°. If the angle of depression of the image of C in the lake from the point P is 60°,then PC (in m) is equal to :

If the constant term in the binomial expansion of {\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}} is 405, then |k| equals :

If {p} denotes the fractional part of the number p, then \left\{ {{{{3^{200}}} \over 8}} \right\}, is equal to :

If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1 + x)n + 5 are in the ratio 5 : 10 : 14, then the largest coefficient in this expansion is :

The value of $$\sum _{r=0}^{20}{}^{50-r}{C}_6$$ is equal to:

If the term independent of x in the expansion of {\left( {{3 \over 2}{x^2} - {1 \over {3x}}} \right)^9} is k, then 18 k is equal to :

If the number of integral terms in the expansion of (31/2 + 51/8)n is exactly 33, then the least value of n is :

The greatest positive integer k, for which 49k + 1 is a factor of the sum 49125 + 49124 + ..... + 492 + 49 + 1, is:

The coefficient of x7 in the expression (1 + x)10 + x(1 + x)9 + x2(1 + x)8 + ......+ x10 is:

In the expansion of {\left( {{x \over {\cos \theta }} + {1 \over {x\sin \theta }}} \right)^{16}}, if $${\ell }_1$$ is the least value of the term independent of x when {\pi \over 8} \le \theta \le {\pi \over 4} and $${\ell }_2$$ is the least value of the term independent of x when {\pi \over {16}} \le \theta \le {\pi \over 8}, then the ratio $${\ell }_2$$ : $${\ell }_1$$ is equal to :

Let $$\alpha$$ > 0, $$\beta$$ > 0 be such that $$\alpha$$3 + $$\beta$$2 = 4. If the maximum value of the term independent of x in the binomial expansion of {\left( {\alpha {x^{{1 \over 9}}} + \beta {x^{ - {1 \over 6}}}} \right)^{10}} is 10k, then k is equal to :

If $$\alpha$$ and $$\beta$$ be the coefficients of x4 and x2 respectively in the expansion of $${\left(x+\sqrt{x^2-1}\right)}^6+{\left(x-\sqrt{x^2-1}\right)}^6$$, then

If \int {{{\cos \theta } \over {5 + 7\sin \theta - 2{{\cos }^2}\theta }}} d\theta = A$${\log }_e\left|B\left(\theta \right)\right|+C$$, where C is a constant of integration, then {{{B\left( \theta \right)} \over A}} can be :

If $$\int \left(e^{2x}+2e^x-e^{-x}-1\right)e^{\left(e^x+e^{-x}\right)}dx$$ = $$g\left(x\right)e^{\left(e^x+e^{-x}\right)}+c$$ where c is a constant of integration, then g(0) is equal to :

The integral \int {{{\left( {{x \over {x\sin x + \cos x}}} \right)}^2}dx} is equal to (where C is a constant of integration):

Let f\left( x \right) = \int {{{\sqrt x } \over {{{\left( {1 + x} \right)}^2}}}dx\left( {x \ge 0} \right)}. Then f(3) – f(1) is eqaul to :

If \int {{{d\theta } \over {{{\cos }^2}\theta \left( {\tan 2\theta + \sec 2\theta } \right)}}} = \lambda \tan \theta + 2{\log _e}\left| {f\left( \theta \right)} \right| + C where C is a constant of integration, then the ordered pair ($$\lambda$$, ƒ($$\theta$$)) is equal to :

If \int {{{\sin }^{ - 1}}\left( {\sqrt {{x \over {1 + x}}} } \right)} dx = A(x)$${\tan }^{-1}\left(\sqrt{x}\right)$$ + B(x) + C, where C is a constant of integration, then the ordered pair (A(x), B(x)) can be :