Let X and Y be two events such that $$P(X) = {1 \over 3}$$, $$P(X|Y) = {1 \over 2}$$ and $$P(Y|X) = {2 \over 5}$$. Then

Let f : R $$ \to $$ (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval (0, 1) ?

Let a, b, x and y be real numbers such that a $$-$$ b = 1 and y $$ \ne $$ 0. If the complex number z = x + iy satisfies $${\mathop{\rm Im}\nolimits} \left( {{{az + b} \over {z + 1}}} \right) = y$$, then which of the following is(are) possible value(s) of x?

If $$2x - y + 1 = 0$$ is a tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {16}} = 1$$ then which of the following CANNOT be sides of a right angled triangle?

Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function $$f(x) = x\cos (\pi (x + [x]))$$ is discontinuous?

Which of the following is(are) NOT the square of a 3 $$ \times $$ 3 matrix with real entries?

If a chord, which is not a tangent, of the parabola y² = 16x has the equation 2x + y = p, and mid-point (h, k), then which of the following is(are) possible value(s) of p, h and k?

For a real number $$\alpha $$, if the system

$$\left[ {\matrix{ 1 & \alpha & {{\alpha ^2}} \cr \alpha & 1 & \alpha \cr {{\alpha ^2}} & \alpha & 1 \cr } } \right]\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr { - 1} \cr 1 \cr } } \right]$$

of linear equations, has infinitely many solutions, then 1 + $$\alpha $$ + $$\alpha $$² =

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

Let f : R $$ \to $$ R be a differentiable function such that f(0) = 0, $$f\left( {{\pi \over 2}} \right) = 3$$ and f'(0) = 1.

If $$g(x) = \int\limits_x^{\pi /2} {[f'(t)\text{cosec}\,t - \cot t\,\text{cosec}\,t\,f(t)]dt} $$

for $$x \in \left( {0,\,{\pi \over 2}} \right]$$, then $$\mathop {\lim }\limits_{x \to 0} g(x)$$ =

For how many values of p, the circle x² + y² + 2x + 4y $$-$$ p = 0 and the coordinate axes have exactly three common points?

Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let x be the number of such words where no letter is repeated; and let y be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $${y \over {9x}}$$ = ?

For $$a = \sqrt 2 $$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact ($$-$$1, 1), then which of the following options is the only CORRECT combination for obtaining its equation?

The tangent to a suitable conic (Column 1) at $$\left( {\sqrt 3 ,\,{1 \over 2}} \right)$$ is found to be $$\sqrt 3 x + 2y = 4$$, then which of the following options is the only CORRECT combination?

If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?

Which of the following options is the only INCORRECT combination?

Which of the following options is the only CORRECT combination?

Which of the following options is the only CORRECT combination?