WB JEE 2010

If A, B are two square matrices such that AB = A and BA = B, then prove that B² = B.

Prov that the equation $$\cos 2x + a\sin x = 2a - 7$$ possesses a solution if 2 $$\le$$ a $$\le$$ 6.

Find the values of x, ($$-$$ $$\pi$$, < x < $$\pi$$, x $$\ne$$ 0) satisfying the equation, $${8^{1 + |\cos x| + |{{\cos }^2}x| + ......\infty }} = {4^3}$$.

Use the formula $$\mathop {\lim }\limits_{x \to 0} {{{a^x} - 1} \over x} = {\log _e}a$$, to compute $$\mathop {\lim }\limits_{x \to 0} {{{2^x} - 1} \over {\sqrt {1 + x} - 1}}$$.

If $${{dy} \over {dx}} + \sqrt {{{1 - {y^2}} \over {1 - {x^2}}}} = 0$$, prove that $$x\sqrt {1 - {y^2}} + y\sqrt {1 - {x^2}} = A$$, where A is a constant.

Prove that the centre of the smallest circle passing through origin and whose centre lies on y = x + 1 is $$\left( { - {1 \over 2},{1 \over 2}} \right)$$.

If f(a) = 2, f'(a) = 1, g(a) = $$-$$1 and g'(a) = 2, find the value of $$\mathop {\lim }\limits_{x \to a} {{g(x)f(a) - g(a)f(x)} \over {x - a}}$$.

If N = n! (n $$\in$$ N, n > 2), then find $$\mathop {\lim }\limits_{N \to \infty } \left[ {{{({{\log }_2}N)}^{ - 1}} + {{({{\log }_3}N)}^{ - 1}} + \,\,.....\,\, + {{({{\log }_n}N)}^{ - 1}}} \right]$$.

Let R be the set of real numbers and the mapping f : R $$\to$$ R and g : R $$\to$$ R be defined by f(x) = 5 $$-$$ x² and g(x) = 3x $$-$$ 4, then the value of (fog)($$-$$1) is

If A = {1, 2, 3, 4}, B = {1, 2, 3, 4, 5, 6} are two sets, and function f : A $$\to$$ B is defined by f(x) = x + 2 $$\forall$$ x$$\in$$ A, then the function f is

The function $$f(x) = \sec \left[ {\log \left( {x + \sqrt {1 + {x^2}} } \right)} \right]$$ is

Evaluate the following integral $$\int\limits_{ - 1}^2 {|x\sin \pi x|dx} $$

Sum of n terms of the following series 1³ + 3³ + 5³ + 7³ ......... is

The value of $${{{{\log }_3}5 \times {{\log }_{25}}27 \times {{\log }_{49}}7} \over {{{\log }_{81}}3}}$$ is

If angles A, B and C are in A.P., then $${{a + c} \over b}$$ is equal to

If sum of an infinite geometric series is $${4 \over {3}}$$ and its 1^st term is $${3 \over {4}}$$, then its common ratio is

The value of n for which $${{{x^{n + 1}} + {y^{n + 1}}} \over {{x^n} + {y^n}}}$$ is the geometric mean of x and y is

If $$\alpha$$, $$\beta$$ be the roots of the quadratic equation x² + x + 1 = 0 then the equation whose roots are $$\alpha$$¹⁹, $$\beta$$⁷ is

The value of $${2 \over {3!}} + {4 \over {5!}} + {6 \over {7!}} + $$ ............ is

G.M. and H.M. of two numbers are 10 and 8 respectively. The numbers are

If $${}^{n - 1}{C_3} + {}^{n - 1}{C_4} > {}^n{C_3}$$, then n is just greater than integer

The quadratic equation $${x^2} + 15|x| + 14 = 0$$ has

The roots of the quadratic equation $${x^2} - 2\sqrt 3 x - 22 = 0$$ are

The number of permutations by taking all letters and keeping the vowels of the word COMBINE in the odd places is

$$({2^{3n}} - 1)$$ will be divisible by $$(\forall n \in N)$$

Sum of the last 30 coefficients in the expansion of (1 + x)⁵⁹, when expanded in ascending powers of x is

If in the expansion (a $$-$$ 2b)ⁿ, the sum of the 5^th and 6^th term is zero, then the value of $${a \over b}$$ is

If the matrices $$A = \left[ {\matrix{ 2 & 1 & 3 \cr 4 & 1 & 0 \cr } } \right]$$ and $$B = \left[ {\matrix{ 1 & { - 1} \cr 0 & 2 \cr 5 & 0 \cr } } \right]$$, then AB will be

If $$A = \left[ {\matrix{ 1 & 2 \cr { - 4} & { - 1} \cr } } \right]$$ then A^$$-$$1 is

If $${(1 - x + {x^2})^n} = {a_0} + {a_1}x + {a_2}{x^2} + \,\,....\,\,{a_{2n}}{x^{2n}}$$, then the value of $${a_0} + {a_2} + {a_4} + \,\,....\,\,{a_{2n}}$$ is

The probability that at least one of A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, then P(A') + P(B') is

If $$\omega$$ is an imaginary cube root of unity and $$\left| {\matrix{ {x + {\omega ^2}} & \omega & 1 \cr \omega & {{\omega ^2}} & {1 + x} \cr 1 & {x + \omega } & {{\omega ^2}} \cr } } \right| = 0$$, then one of the values of x is

The number of points of intersection of 2y = 1 and y = sinx, in $$-$$2$$\pi$$ $$\le$$ x $$\le$$ 2$$\pi$$ is

The value of $${{\cot x - \tan x} \over {\cot 2x}}$$ is

Two dice are tossed once. The probability of getting an even number at the first die or a a total of 8 is

The value of $${{\sin 55^\circ - \cos 55^\circ } \over {\sin 10^\circ }}$$ is

If $$z = {4 \over {1 - i}}$$, then $$\overline z $$ is (where $$\overline z $$ is complex conjugate of z)

If $$ - \pi < \arg (z) < - {\pi \over 2}$$, then $$\arg \overline z - \arg ( - \overline z )$$ is

Value of $${\tan ^{ - 1}}\left( {{{\sin 2 - 1} \over {\cos 2}}} \right)$$ is

If $${{\cos A} \over 3} = {{\cos B} \over 4} = {1 \over 5}, - {\pi \over 2} < A < 0, - {\pi \over 2} < B < 0$$, then value of $$2\sin A + 4\sin B$$ is

In a triangle PQR, $$\angle$$R = $$\pi$$/2. If $$\tan \left( {{P \over 2}} \right)$$ and $$\tan \left( {{Q \over 2}} \right)$$ are roots of ax² + bx + c = 0, where a $$\ne$$ 0, then which one is true?

If $$\sin 6\theta + \sin 4\theta + \sin 2\theta = 0$$, then the general value of $$\theta$$ is

The value of $${{\cot 54^\circ } \over {\tan 36^\circ }} + {{\tan 20^\circ } \over {\cot 70^\circ }}$$ is

In a right-angled triangle, the sides are a, b and c, with c as hypotenuse, and c $$-$$ b $$\ne$$ 1, c + b $$\ne$$ 1. Then the value of $$({\log _{c + b}}a + {\log _{c - b}}a)/(2{\log _{c + b}}a \times {\log _{c - b}}a)$$ will be

The straight line 3x + y = 9 divides the line segment joining the points (1, 3) and (2, 7) in the ratio

The equations of the lines through (1, 1) and making angles of 45$$^\circ$$ with the line x + y = 0 are

In a $$\Delta$$ABC, $$2ac\sin \left( {{{A - B + C} \over 2}} \right)$$ is equal to

If the sum of distances from a point P on two mutually perpendicular straight lines is 1 unit, then the locus of P is a/an

The equations $$y = \pm \sqrt {3x} $$, y = 1 are the sides of

If the three points (3q, 0) (0, 3p) and (1, 1) are collinear then which one is true?

If t₁ and t₂ be the parameters of the end points of a focal chord for the parabola y² = 4ax, then which one is true?

If the line ax + by + c = 0 is a tangent to the curve xy = 4, then

S and T are the foci of an ellipse and B is an end point of the minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is

The straight line x + y $$-$$ 1 = 0 meets the circle x² + y² $$-$$ 6x $$-$$ 8y = 0 at A and B. Then the equation of the circle of which AB is a diameter is

The domain of the function $$f(x) = \sqrt {{{\cos }^{ - 1}}\left( {{{1 - |x|} \over 2}} \right)} $$ is

For different values of $$\alpha$$, the locus of the point of intersection of the two straight lines $$\sqrt 3 x - y - 4\sqrt 3 \alpha = 0$$ and $$\sqrt 3 \alpha x + \alpha y - 4\sqrt 3 = 0$$ is

The minimum value of $$f(x) = {e^{({x^4} - {x^3} + {x^2})}}$$ is

If $$y = (1 + x)(1 + {x^2})(1 + {x^4})\,.....\,(1 + {x^{2n}})$$, then the value of $${\left( {{{dy} \over {dx}}} \right)_{x = 0}}$$ is

The value of $$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x + \cos x - 1} \over {{x^2}}}$$ is

The value of $$\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + 5{x^2}} \over {1 + 3{x^2}}}} \right)^{{1 \over {{x^2}}}}}$$ is

For what values of x, the function $$f(x) = {x^4} - 4{x^3} + 4{x^2} + 40$$ is monotone decreasing?

If $$f(5) = 7$$ and $$f'(5) = 7$$, then $$\mathop {\lim }\limits_{x \to 5} {{x\,f(5) - 5f(x)} \over {x - 5}}$$ is given by

The point in the interval [0, 2$$\pi$$], where $$f(x) = {e^x}\sin x$$ has maximum slope, is

The value of f(0) so that the function $$f(x) = {{1 - \cos (1 - \cos x)} \over {{x^4}}}$$ is continuous everywhere is

If the normal to the curve y = f(x) at the point (3, 4) makes an angle 3$$\pi$$/4 with the positive x-axis, then f'(3) is

In which of the following functions, Rolle's theorem is applicable?

The equation of normal of $${x^2} + {y^2} - 2x + 4y - 5 = 0$$ at (2, 1) is

The co-ordinates of the point on the curve $$y = {x^2} - 3x + 2$$ where the tangent is perpendicular to the straight line y = x are

$$\int {\sqrt {1 + \cos x} dx} $$ is equal to

$$\int {{{\log \sqrt x } \over {3x}}dx} $$ is equal to

The value of the integral $$\int\limits_0^{\pi /2} {{{\sin }^5}xdx} $$ is

The value of the integral $$\int {{{dx} \over {{{({e^x} + {e^{ - x}})}^2}}}} $$ is

$$\int {{e^x}\left( {{2 \over x} - {2 \over {{x^2}}}} \right)dx} $$ is equal to

The area of the region bounded by y² = x and y = | x | is

If $$I = \int\limits_0^1 {{{dx} \over {1 + {x^{\pi /2}}}}} $$, then

The area enclosed by y = 3x $$-$$ 5, y = 0, x = 3 and x = 5 is

The value of $$I = \int\limits_{ - \pi /2}^{\pi /2} {|\sin x|dx} $$ is

If $${d \over {dx}}\{ f(x)\} = g(x)$$, then $$\int\limits_a^b {f(x)g(x)dx} $$ is equal to

If $${I_1} = \int\limits_0^{3\pi } {f({{\cos }^2}x)dx} $$ and $${I_2} = \int\limits_0^\pi {f({{\cos }^2}x)dx} $$, then

The equation of one of the curves whose slope at any point is equal to y + 2x is

If $$y'' - 3y' + 2y = 0$$ where y(0) = 1, y'(0) = 0, then the value of y at $$x = {\log _e}2$$ is

The general solution of the differential equation $$100{{{d^2}y} \over {d{x^2}}} - 20{{dy} \over {dx}} + y = 0$$ is

Solution of the differential equation xdy $$-$$ ydx = 0 represents a

The degree of the differential equation $$x = 1 + \left( {{{dy} \over {dx}}} \right) + {1 \over {2!}}{\left( {{{dy} \over {dx}}} \right)^2} + {1 \over {3!}}{\left( {{{dy} \over {dx}}} \right)^3} + .....$$

The area bounded by the parabolas y = 4x², $$y = {{{x^2}} \over 9}$$ and the line y = 2 is

Prove by induction that for all n $$\in$$ N, n² + n is an even integer (n $$\ge$$ 1).

The displacement of a particle at time t is x, where x = t⁴ $$-$$ kt³. If the velocity of the particle at time t = 2 is minimum, then

If the displacement, velocity and acceleration of a particle at time t be x, v and f respectively, then which one is true?

The displacement x of a particle at time t is given by x = At² + Bt + C, where A, B, C are constants and v is velocity of a particle, then the value of 4Ax $$-$$ v² is