WB JEE 2009

Find the values of a for which the expression $${x^2} - (3a - 1)x + 2{a^2} + 2a - 11$$ is always positive.

The equations to the pairs of opposite sides of a parallelogram are x² $$-$$ 5x + 6 = 0 and y² $$-$$ 6y + 5 = 0. Find the equations of its diagonals.

Find the angle subtended by the double ordinate of length 2a of the parabola y² = ax at its vertex.

Show that $${{\sin \theta } \over {\cos 3\theta }} + {{\sin 3\theta } \over {\cos 9\theta }} + {{\sin 9\theta } \over {\cos 27\theta }} = {1 \over 2}(\tan 27\theta - \tan \theta )$$

If x = sin t, y = sin 2t, prove that $$(1 - {x^2}){{{d^2}y} \over {d{x^2}}} - x{{dy} \over {dx}} + 4y = 0$$

If f is differentiable at x = a, find the value of $$\mathop {\lim }\limits_{x \to a} {{{x^2}f(a) - {a^2}f(x)} \over {x - a}}$$

A train moving with constant acceleration takes t seconds to pass a certain fixed point and the front and back end of the train pass the fixed point with velocities u and v respectively. Show that the length of the train is $${1 \over 2}(u + v)t$$.

If a, b, c are G.P. (a > 1, b > 1, c > 1), then for any real number x (with x > 0, x $$\ne$$ 1) log_ax, log_bx, log_cx are in

If x = log_abc, y = log_bca, z = log_cab, then the value of $${1 \over {1 + x}} + {1 \over {1 + y}} + {1 \over {1 + z}}$$ will be

For any two sets A and B, A $$-$$ (A $$-$$ B) equals

If three positive real numbers a, b, c are in A.P. and abc = 4 then the minimum possible value of b is

The value of $$\left( {{1 \over {{{\log }_3}12}} + {1 \over {{{\log }_4}12}}} \right)$$ is

Find the sum of the first n terms of the series 0.2 + 0.22 + 0.222 + ......

If $$\alpha$$, $$\beta$$ be the roots of $${x^2} - a(x - 1) + b = 0$$, then the value of $${1 \over {{\alpha ^2} - a\alpha }} + {1 \over {{\beta ^2} - a\beta }} + {2 \over {a + b}}$$ is

The sum of the infinite series $$1 + {1 \over {2!}} + {{1\,.\,3} \over {4!}} + {{1\,.\,3\,.\,5} \over {6!}} + \,....$$ is

The coefficient of xⁿ, where n is any positive integer, in the expansion of (1 + 2x + 3x² + ....... $$\infty$$)^1/2 is

The sum of all real roots of the equation $$|x - 2{|^2} + |x - 2| - 2 = 0$$ is

For what value of m, $${{{a^{m + 1}} + {b^{m + 1}}} \over {{a^m} + {b^m}}}$$ is the arithmetic mean of a and b?

The quadratic equation whose roots are three times the roots of $$3a{x^2} + 3bx + c = 0$$ is

If C₀, C₁, C₂, ......, C_n denote the coefficients in the expansion of (1 + x)ⁿ then the value of C₁ + 2C₂ + 3C₃ + ..... + nC_n is

A polygon has 44 diagonals. The number of the sides is

If a, b, c are real, then both the roots of the equation $$(x - b)(x - c) + (x - c)(x - a) + (x - a)(x - b) = 0$$ are always

For each n $$\in$$ N, 2³ⁿ $$-$$ 1 is divisible by

here N is a set of natural numbers.

If the coefficients of x² and x³ in the expansion of (3 + ax)⁹ be same, then the value of a is

using binomial theorem, the value of (0.999)³ correct to 3 decimal places is

Three numbers are chosen at random from 1 to 20. The probability that they are consecutive is

If A² $$-$$ A + I = 0, then the inverse of the matrix A is

If A is a square matrix. Then

A and B are two independent events such that P(A $$\cup$$ B') = 0.8 and P(A) = 0.3. Then P(B) is

If A and B are square matrices of the same order and AB = 3I, then A^$$-$$1 is equal to

If $$i = \sqrt { - 1} $$ and n is positive integer, then $${i^n} + {i^{n + 1}} + {i^{n + 2}} + {i^{n + 3}}$$ is equal to

The equation $$\sqrt 3 \sin x + \cos x = 4$$ has

General solution of $$\sin x + \cos x = \mathop {\min }\limits_{a \in IR} \{ 1,{a^2} - 4a + 6\} $$ is

For any complex number z, the minimum value of $$|z| + |z - 1|$$ is

The modulus of $${{1 - i} \over {3 + i}} + {{4i} \over 5}$$ is

The value of $$\cos 15^\circ \cos 7{1 \over 2}^\circ \sin 7{1 \over 2}^\circ $$ is

Simplest form of $${2 \over {\sqrt {2 + \sqrt {2 + \sqrt {2 + 2\cos 4x} } } }}$$ is

A positive acute angle is divided into two parts whose tangents are 1/2 and 1/3. Then the angle is

$$P = {1 \over 2}{\sin ^2}\theta + {1 \over 3}{\cos ^2}\theta $$, then

In triangle ABC, a = 2, b = 3 and sin A = 2/3, thne B is equal to

The smallest value of $$5\cos \theta + 12$$ is

The value of $$\left( {1 + \cos {\pi \over 6}} \right)\left( {1 + \cos {\pi \over 3}} \right)\left( {1 + \cos {{2\pi } \over 3}} \right)\left( {1 + \cos {{7\pi } \over 6}} \right)$$ is

The coordinates of the foot of the perpendicular from (0, 0) upon the line x + y = 2 are

If C is a point on the line segment joining A($$-$$3, 4) and B(2, 1) such that AC = 2BC, then the coordinate of C is

If a = 2$$\sqrt2$$, b = 6, A = 45$$^\circ$$, then

If $$5\cos 2\theta + 2{\cos ^2}\theta /2 + 1 = 0$$, when $$(0 < \theta < \pi )$$, then the values of $$\theta$$ are

$$\tan \left[ {{\pi \over 4} + {1 \over 2}{{\cos }^{ - 1}}\left( {{a \over b}} \right)} \right] + \tan \left[ {{\pi \over 4} - {1 \over 2}{{\cos }^{ - 1}}\left( {{a \over b}} \right)} \right]$$ is equal to

In triangle ABC, if $$\sin A\sin B = {{ab} \over {{c^2}}}$$, then the triangle is

The circles x² + y² $$-$$ 10x + 16 = 0 and x² + y² = a² intersect at two distinct points if

The equation of the chord of he circle $${x^2} + {y^2} - 4x = 0$$ whose midpoint is (1, 0) is

The point ($$-$$4, 5) is the vertex of a square and one of its diagonals is 7x $$-$$ y + 8 = 0. The equation of the other diagonal is

A line through the point A(2, 0) which makes an angle of 30$$^\circ$$ with the positive direction of x-axis is rotated about A in clockwise direction through an angle 15$$^\circ$$. Then the equation of the straight line in the new position is

The number of points on the line x + y = 4 which are unit distance apart from the line 2x + 2y = 5 is

If C is the reflection of A(2, 4) in x-axis and B is the reflection of C in y-axis, then $$|AB|$$ is

For the two circles x² + y² = 16 and x² + y² $$-$$ 2y = 0 there is/are

The total number of tangents through the point (3, 5) that can be drawn to the ellipse 3x² + 5y² = 32 and 25x² + 9y² = 450 is

The coordinates of the focus of the parabola described parametrically by x = 5t² + 2, y = 10t + 4 are

The angle between the line joining the foci of an ellipse to one particular extremity of the minor axis is 90$$^\circ$$. The eccentricity of the ellipse is

A mapping from IN to IN is defined as follows:

$$f:IN \to IN$$

$$f(n) = {(n + 5)^2},\,n \in IN$$

(IN is the set of natural numbers). Then

The line y = 2t² intersects the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ in real point if

The domain of definition of the function $$f(x) = \sqrt {1 + {{\log }_e}(1 - x)} $$ is

Let $$f(x) = {{\sqrt {x + 3} } \over {x + 1}}$$, then the value of $$\mathop {\lim }\limits_{x \to - 3 - 0} f(x)$$ is

The value of $$\mathop {\lim }\limits_{x \to 1} {{\sin ({e^{x - 1}} - 1)} \over {\log x}}$$ is

The second order derivative of a sin³t with respect to a cos³t at $$t = {\pi \over 4}$$ is

If $$y = {\tan ^{ - 1}}\sqrt {{{1 - \sin x} \over {1 + \sin x}}} $$, then the value of $${{dy} \over {dx}}$$ at $$x = {\pi \over 6}$$ is

$$f(x) = x + |x|$$ is continuous for

The equation of the tangent to the conic $${x^2} - {y^2} - 8x + 2y + 11 = 0$$ at (2, 1) is

Angle between y² = x and x² = y at the origin is

The distance covered by a particle in t seconds is given by x = 3 + 8t $$-$$ 4t². After 1 second its velocity will be

The Rolle's theorem is applicable in the interval $$-$$1 $$\le$$ x $$\le$$ 1 for the function

A particle is moving in a straight line. At time t, the distance between the particle from its starting point is given by x = t $$-$$ 6t² + t³. Its acceleration will be zero at

If the rate of increase of the radius of a circle is 5 cm/sec., then the rate of increase of its area, when the radius is 20 cm, will be

$$\int {{{dx} \over {x(x + 1)}}} $$ equals

where c is arbitrary constant.

The value of $$\int\limits_{ - 1}^1 {{{|x + 2|} \over {x + 2}}dx} $$ is

The value of $$\int\limits_0^\infty {{{dx} \over {({x^2} + 4)({x^2} + 9)}}} $$ is

$$\int {{{{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx} $$ equal to

where c is an arbitrary constant

The value of $$\mathop {\lim }\limits_{n \to \infty } \left[ {{n \over {{n^2} + {1^2}}} + {n \over {{n^2} + {2^2}}} + ..... + {n \over {{n^2} + {n^2}}}} \right]$$ is

$$\int {{{dx} \over {\sin x + \sqrt 3 \cos x}}} $$ equals

If $$f(x) = f(a - x)$$, then $$\int\limits_0^a {xf(x)dx} $$ is equal to

The slope at any point of a curve y = f(x) is given by $${{dy} \over {dx}} = 3{x^2}$$ and it passes through ($$-$$1, 1). The equation of the curve is

$$\int\limits_{ - 1}^4 {f(x)dx = 4} $$ and $$\int\limits_2^4 {\{ 3 - f(x)\} dx = 7} $$, then the value of $$\int\limits_{ - 1}^2 {f(x)dx} $$ is

The integrating factor of the differential equation $$x\log x{{dy} \over {dx}} + y = 2\log x$$ is given by

$$\int\limits_0^{1000} {{e^{x - [x]}}dx} $$ is equal to

If $${I_1} = \int\limits_0^{\pi /4} {{{\sin }^2}xdx} $$ and $${I_2} = \int\limits_0^{\pi /4} {{{\cos }^2}xdx} $$, then

The general solution of the differential equation $${{dy} \over {dx}} = {e^{y + x}} + {e^{y - x}}$$ is

where c is an arbitrary constant

If x² + y² = 1, then

Show that, for a positive integer n, the coefficient of x^k (0 $$\le$$ k $$\le$$ n) in the expansion of 1 + (1 + x) + (1 + x)² + ....... + (1 + x)ⁿ is $${}^{n + 1}{C_{n - k}}$$.

If m, n be integers, then find the value of $$\int\limits_{ - \pi }^\pi {{{(\cos mx - \sin nx)}^2}dx} $$

The order of the differential equation $${{{d^2}y} \over {d{x^2}}} = \sqrt {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} $$ is

Product of any r consecutive natural numbers is always divisible by