WB JEE 2008

A wire of length 5 m is stretched by 1 mm by a force of 1 N. What is the value of energy stored in the wire?

Two wires each of same material and same length have 2 L 1 ratio of their radii. If the first wire has a resistance of 2 $$\Omega$$ then what will be the equivalent resistance of the two wires when they are connected in series?

A 60 W - 220 V bulb is connected in series with another bulb 40 W - 220 V. Which bulb will give more illuminance ?

Prove that if the ratio $${{z - i} \over {z - 1}}$$ is purely imaginary, the point z lies on the circle in the Argand plane whose centre is at the point $${1 \over 2}(1 + i)$$ and radius is $${1 \over {\sqrt 2 }}$$.

If an unbiased coin is tossed n times. Find the probability that head appears an odd number of times.

Find the general solution of $$\sec \theta + 1 = (2 + \sqrt 3 )\tan \theta $$.

What should be the resistance of an ideal ammeter and an ideal voltmeter?

If 2a $$-$$ 5b $$-$$ 3c = 0, show that the straight line ax + by + c = 0 always passes through a fixed point. Find the equation of one straight line passing through this fixed point and parallel to the line 9x + 12y = 20.

$$A = \left[ {\matrix{ 1 & 2 \cr 0 & 1 \cr } } \right]$$ then by the principle of mathematical induction, prove that $${A^n} = \left[ {\matrix{ 1 & {2n} \cr 0 & 1 \cr } } \right]$$

Solve : $$({x^2} + 4{y^2} + 4xy)dy = (2x + 4y + 1)dx$$

Evaluate $$\int {{{{x^2}} \over {x(1 + {x^2})}}dx} $$

The velocity of a particle (v) at an instant t is given by $$v = at + b{t^2}$$ the dimension of b is

If the tangent to the parabola y = x(2 $$-$$ x) at the point (1, 1) intersects the parabola at P. Find the co-ordinate of P.

Prove that for all values of m, except zero the st. line $$y = mx + {a \over m}$$ touches the parabola y² = 4ax

Show that sin x is a monotonic increasing function of x in 0 < x < $$\pi$$/2.

A shell of mass 10 kg is moving with a velocity of 10 ms^$$-$$1 when it blasts and forms two parts of mass 9 kg and 1 kg respectively. If the 1st mass is stationary, the velocity of the 2nd is

The distance travelled by an object along a straight line in time t is given by $$s = 3 - 4t + 5{t^2}$$, the initial velocity of the object is

Force required to move a mass of 1 kg at rest on a horizontal rough plane ($$\mu$$ = 0.1 and g = 9.8 m/s²) is

A bullet emerges from a barrel of length 1.2 m with a speed of 640 ms^$$-$$1. Assuming constant acceleration, the approximate time that it spends in the barrel after the gun is fired is

The angle subtended by the vector $$A = 4\widehat i + 3\widehat j + 12\widehat k$$ with the x-axis is

A rocket of mass 100 kg burns 0.1 kg of fuel per sec. If the velocity of exhaust gas is 1 km/sec, then it lifts with an acceleration of

A pellet of mass 1 g is moving with an angular velocity of 1 rad/s along a circle of radius 1 m. The centrifugal force is

Two point objects of masses 1.5 g and 2.5 g respectively are at a distance of 16 cm apart, the centre of gravity is at a distance x from the object of mass 1.5 gm where x is

A beaker of radius 15 cm is filled with a liquid of surface tension 0.075 N/m. Force across an imaginary diameter on the surface of the liquid is

Which of the following substances has the highest elasticity?

A body of mass 3 kg acted upon by a constant force is displaced by S meter, given by relation $$S = {1 \over 3}{t^2}$$, where t is in second. Work done by the force in 2 seconds

A wire is stretched under a force. If the wire suddenly snaps the temperature of the wire

An experiment takes 10 minutes to raise temperature of water from 0$$^\circ$$C to 100$$^\circ$$C and another 55 minutes to convert it totally into steam by a stabilized heater. The latent heat of vaporization comes out to be

When the room temperature becomes equal to the dew point, the relative humidity of the room is

A piece of wood is floating in water. When the temperature of water rises, the apparent weight of the wood will

Six molecules have speeds 2 units, 5 units, 3 units, 6 units, 3 units and 5 units respectively. The rms speed is

At what temperature will the rms speed of air molecules be double that of NTP?

Which one of the figures gives the temperatures dependance of density of water correctly?

The equation of a progressive wave can be given by $$y = 15\sin (660\pi t - 0.02\pi x)$$ cm. The frequency of the wave is

Two springs are joined and attached to a mass of 16 kg. The system is then suspended vertically from a rigid support. The spring constant of the two springs are K₁ and K₂ respectively. The period of vertical oscillations of the system will be

When sound is produced in an aeroplane with a velocity of 200 m/s horizontally its echo is heard after $$10\sqrt 5 $$ seconds. If velocity of sound in air is 300 ms^$$-$$1 the elevation of aircraft is

Two turning forks of frequencies n₁ and n₂ produces n beats per second. If n₂ and n are known, n₁ may be given by

A hollow cylinder with both sides open generates a frequency f in air. When the cylinder vertically immersed into water by half its length the frequency will be

Two stretched strings have lengths l and 2l while tensions are T and 4T respectively. If they are made of same material the ratio of their frequency is

The weight of a body on surface of earth is 12.6 N. When it is raised to a height half the radius of earth, its weight will be

4 point charges each +q is placed on the circumference of a circle of diameter 2d in such a way that they form a square. The potential at the centre is

A battery of e.m.f. E and internal resistance r is connected to an external resistance R the condition for maximum power transfer is

A car moving with a velocity of 36 km/hr crosses a siren of frequency 500 Hz. The apparent frequency of siren after passing it will be

64 identical spheres of charge q and capacitance C each are combined to form a large sphere. The charge and capacitance of the large sphere is

If the earth shrinks such that its mass does not change but radius decreases to one quarter of its original value then one complete day will take

The ionisation potential of hydrogen atom is $$-$$13.6 eV. An electron in the ground state of a hydrogen atom absorbs a photon of energy 12.75 eV. How many different spectral lines can one expect when the electron make a downward transition?

Lenz's law of electromagnetic induction corresponds to the

A thin wire of resistance 4 $$\Omega$$ is bent to form a circle. The resistance across any diameter is

The radius of curvature of a concave mirror is 24 cm and the image is magnified by 1.5 times. The object distance is

A point source of light is kept at a depth of h in water of refractive index 4/3. The radius of the circle at the surface of water through which light emits is

The refractive indices of glass and quartz w.r.t. air are 3/2 and 12/5 respectively. The refractive index of quartz w.r.t. glass is

A Si and a Ge diode has identical physical dimensions. The band gap in Si is larger than that in Ge. An identical reverse bias is applied across the diodes

The binary number 10111 is equivalent to the decimal number

Let A = {1, 2, 3} and B = {2, 3, 4}, then which of the following relations is a function from A to B?

Which of the following statements is correct?

Three sets A, B, C are such that A = B $$\cap$$ C and B = C $$\cap$$ A, then

A mapping f : N $$\to$$ N where N is the set of natural numbers is defined as

f(n) = n² for n odd

f(n) = 2n + 1 for n even

for n $$\in$$ N. Then f is

The 5^th term of the series $${{10} \over 9},{1 \over 3}\sqrt {{{20} \over 3}} ,{2 \over 3}$$ is

If $${\log _5}\,\,{\log _5}\,\,{\log _2}x = 0$$ then value of x is

A function f : A $$\to$$ B, where A = {x/$$-$$1 $$\le$$ x $$\le$$ 1} and B = {y/1 $$\le$$ y $$\le$$ 2} is defined by the rule y = f(x) = 1 + x². Which of the following statement is then true?

If the m^th term and the n^th term of an A.P. are respectively $${1 \over n}$$ and $${1 \over m}$$, then the (mn)^th term of the A.P. is

The sum of the series $$(1 + 2) + (1 + 2 + {2^2}) + (1 + 2 + {2^2} + {2^3}) + ....$$ upto n terms is

The mapping f : N $$\to$$ N given by f(n) = 1 + n², n $$\in$$ N where N is the set of natural numbers, is

If three real numbers a, b, c are in Harmonic Progression, then which of the following is true?

Out of 8 given points, 3 are collinear. How many different straight lines can be drawn by joining any two points from those 8 points?

If a, b, c be in Arithmetic progression, then the value of $$(a + 2b - c)(2b + c - a)(a + 2b + c)$$ is

If one root of the equation $${x^2} + (1 - 3i)x - 2(1 + i) = 0$$ is $$-$$1 + i, then the other root is

The sum of the infinite series $${\left( {{1 \over 3}} \right)^2} + {1 \over 3}{\left( {{1 \over 3}} \right)^4} + {1 \over 5}{\left( {{1 \over 3}} \right)^6} + ...$$ is

The equation $${x^2} - 3|x| + 2 = 0$$ has

How many odd numbers of six significant digits can be formed with the digits 0, 1, 2, 5, 6, 7 when no digit is repeated?

If the magnitude of the coefficient of x⁷ in the expansion of $${\left( {a{x^2} + {1 \over {bx}}} \right)^8}$$, where a, b are positive numbers, is equal to the magnitude of the coefficient of x⁷ in the expansion of $${\left( {ax + {1 \over {b{x^2}}}} \right)^8}$$, then a and b are connected by the relation

The number of ways four boys can be seated around a round table in four chairs of different colours is

The values of x for which the given matrix $$\left[ {\matrix{ { - x} & x & 2 \cr 2 & x & { - x} \cr x & { - 2} & { - x} \cr } } \right]$$ will be non-singular are

If $${}^{16}{C_r} = {}^{16}{C_{r + 1}}$$, then the value of $${}^r{P_{r - 3}}$$ is

The coefficient of x^$$-$$10 in $${\left( {{x^2} - {1 \over {{x^3}}}} \right)^{10}}$$ is

The value of $${(1 - \omega + {\omega ^2})^5} + {(1 + \omega - {\omega ^2})^5}$$, where $$\omega$$ and $$\omega$$² are the complex cube roots of unity is

If the matrix $$\left[ {\matrix{ a & b \cr c & d \cr } } \right]$$ is commutative with the matrix $$\left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]$$ then

For two complex numbers z₁, z₂ the relation $$\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|$$ holds if

A person draws out two balls successively from a bag containing 6 red and 4 white balls. The probability that at least one of them will be red is

The principal amplitude of $${(\sin 40^\circ + i\cos 40^\circ )^5}$$ is

Let $$\alpha$$, $$\beta$$ be the roots of $${x^2} - 2x\cos \phi + 1 = 0$$, then the equation whose roots are $${\alpha ^n},{\beta ^n}$$ is

The value of $$\cos {\pi \over {15}}\cos {{2\pi } \over {15}}\cos {{4\pi } \over {15}}\cos {{8\pi } \over {15}}$$ is

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

If 1, $$\omega$$, $$\omega$$² are cube roots of unity, then $$\left| {\matrix{ 1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr {{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr {{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr } } \right|$$ has value

A and B are two points on the Argand plane such that the segment AB is bisected at the point (0, 0). If the point A, which is in the third quadrant has principal amplitude $$\theta$$, then the principal amplitude of the point B is

The principal value of $${\sin ^{ - 1}}\tan \left( { - {{5\pi } \over 4}} \right)$$ is

If $$\tan \left( {{{\alpha \pi } \over 4}} \right) = \cot \left( {{{\beta \pi } \over 4}} \right)$$ then

The distance between the lines $$5x - 12y + 65 = 0$$ and $$5x - 12y - 39 = 0$$ is

The equation $$(x - {x_1})(x - {x_2}) + (y - {y_1})(y - {y_2}) = 0$$ represents a circle whose centre is

The equation of the circle which passes through the points of intersection of the circles $${x^2} + {y^2} - 6x = 0$$ and $${x^2} + {y^2} - 6y = 0$$, and has its centre at $$\left( {{3 \over 2},{3 \over 2}} \right)$$ is

One possible condition for the three points (a, b), (b, a) and (a², $$-$$ b²) to be collinear is

The co-ordinates of the foot of perpendicular from (a, 0) on the line $$y = mx + {a \over m}$$ are

The value of $$\tan \alpha + 2\tan (2\alpha ) + 4\tan (4\alpha ) + ... + {2^{n - 1}}\tan ({2^{n - 1}}\alpha ) + {2^n}\cot ({2^n}\alpha )$$ is

The two parabolas x² = 4y and y² = 4x meet in two distinct points. One of these is the origin and the other is

If P(at², 2at) be one end of a focal chord of the parabola y² = 4ax, then the length of the chord is

The vertex of the parabola x² + 2y = 8x $$-$$ 7 is

The length of the common chord of the parabolas y² = x and x² = y is

The circles $${x^2} + {y^2} + 6x + 6y = 0$$ and $${x^2} + {y^2} - 12x - 12y = 0$$

The locus of the centres of the circles which touch both the axes is given by

The latus rectum of an ellipse is equal to one-half of its minor axis. The eccentricity of the ellipse is

The range of the function $$f(x) = {\log _e}\sqrt {4 - {x^2}} $$ is given by

The equation of the ellipse having vertices at ($$\pm$$ 5, 0) and foci ($$\pm$$ 4, 0) is

If 2y = x and 3y + 4x = 0 are the equations of a pair of conjugate diameters of an ellipse, then the eccentricity of the ellipse is

If t is a parameter, then $$x = a\left( {t + {1 \over t}} \right)$$, $$y = b\left( {t - {1 \over t}} \right)$$ represents

The equation e^x + x $$-$$ 1 = 0 has, apart from x = 0

The function $$f(x) = \log \left( {{{1 + x} \over {1 - x}}} \right)$$ satisfies the equation

The $$\mathop {\lim }\limits_{x \to 2} {5 \over {\sqrt 2 - \sqrt x }}$$ is

$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{{a^{\cot x}} - {a^{\cos x}}} \over {\cot x - \cos x}},a > 0$$

The value of the limit $$\mathop {\lim }\limits_{x \to 2} {{{e^{3x - 6}} - 1} \over {\sin (2 - x)}}$$ is

Rolle's theorem is not applicable to the function $$f(x) = |x|$$ for $$ - 2 \le x \le 2$$ because

A function f(x) is defined as follows for real x

$$f(x) = \left\{ {\matrix{ {1 - {x^2}} & , & {for\,x < 1} \cr 0 & , & {for\,x = 1} \cr {1 + {x^2}} & , & {for\,x > 1} \cr } } \right.$$

Then

A particle is projected vertically upwards and is at a height h after t₁ seconds and again after t₂ seconds then

The value of $${{dy} \over {dx}}$$ at $$x = {\pi \over 2}$$, where y is given by $$y = {x^{\sin x}} + \sqrt x $$ is

Select the correct statement from (a), (b), (c), (d). The function $$f(x) = x{e^{1 - x}}$$

If $$x = {e^t}\sin t$$, $$y = {e^t}\cos t$$ then $${{{d^2}y} \over {d{x^2}}}$$ at x = $$\pi$$ is

If $$h(x) = \int\limits_0^x {{{\sin }^4}t\,dt} $$, then $$h(x + \pi )$$ equals

The function $$f(x) = {e^{ax}} + {e^{ - ax}},a > 0$$ is monotonically increasing for

$$\int\limits_{ - \pi /2}^{\pi /2} {{{\sin }^9}x{{\cos }^5}x\,dx} $$ equals

If $$I = \int\limits_{ - \pi }^\pi {{{{e^{\sin x}}} \over {{e^{\sin x}} + {e^{ - \sin x}}}}dx} $$, then I equals

The value of the integral $$\int\limits_{ - a}^a {{{x{e^{{x^2}}}} \over {1 + {x^2}}}dx} $$ is

The value of $$\int\limits_{ - 3}^3 {(a{x^5} + b{x^3} + cx + k)dx} $$, where a, b, c, k are constants, depends only on

The value of $$\int\limits_0^\pi {|\cos x|dx} $$ is

The value of the integral $$\int\limits_0^2 {|{x^2} - 1|dx} $$ is

The degree of the differential equation $${\left[ {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} \right]^{5/3}} = {{{d^2}y} \over {d{x^2}}}$$ is

The area enclosed between the curve y = 1 + x², the y-axis, and the straight line y = 5 is given by

The function f(x) which satisfies $$f(x) = f( - x) = {{f'(x)} \over x}$$ is given by

The area included between the parabolas y² = 4x and x² = 4y is

The line which is parallel to x-axis and crosses the curve y = $$\sqrt x$$ at an angle 45$$^\circ$$ is

The value of the $$\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + ... + {1 \over {6n}}} \right)$$ is

The differential equation of the family of curves $$y = {e^{2x}}(a\cos x + b\sin x)$$, where a and b are arbitrary constants, is given by

The differential equation of all parabolas whose axes are parallel to y-axis is

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

The order and degree of the following differential equation $${\left[ {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} \right]^{5/2}} = {{{d^3}y} \over {d{x^3}}}$$ are respectively

The differential equation of the family of circles passing through the fixed points (a, 0) and ($$-$$a, 0) is

Write down the relation between tesla and gauss.