WB JEE 2019

Two particles A and B move from rest along a straight line with constant accelerations f and h, respectively. If A takes m seconds more than B and describes n units more than that of B acquiring the same speed, then

If the radius of a spherical balloon increases by 0.1%, then its volume increases approximately by

The area bounded by y = x + 1 and y = cos x and the X-axis, is

Let P and T be the subsets of k, y-plane defined byP = {(x, y) : x > 0, y > 0 and x2 + y2 = 1}T = {(x, y) : x > 0, y > 0 and x8 + y8 < 1}Then, P \(\cap\) T is

The number of irrational terms in the expansion of \({\left( {{3^{{1 \over 8}}} + {5^{{1 \over 4}}}} \right)^{84}}\) is

If P(0, 0), Q(1, 0) and R\(\left( {{1 \over 2},{{\sqrt 3 } \over 2}} \right)\) are three given points, then the centre of the circle for which the lines PQ, QR and RP are the tangents is

A variable circle passes through the fixed point A(p, q) and touches X-axis. The locus of the other end of the diameter through A is

If \(\theta \in R\) and \({{1 - i\cos \theta } \over {1 + 2i\cos \theta }}\) is real number, then \(\theta\) will be (when I : Set of integers)

For any non-zero complex number z, the minimum value of | z | + | z \(-\) 1 | is

The polar coordinate of a point P is \(\left( {2, - {\pi \over 4}} \right)\). The polar coordinate of the point Q which is such that line joining PQ is bisected perpendicularly by the initial line, is

Let z be a complex number such that the principal value of argument, arg z > 0. Then, arg z \(-\) arg(\(-\) z) is

The general value of the real angle \(\theta\), which satisfies the equation, \((\cos \theta + i\sin \theta )(\cos 2\theta + i\sin 2\theta )...(\cos n\theta + i\sin n\theta ) = 1\) is given by, (assuming k is an integer)

The value of the integral \(\int\limits_{ - 1}^1 {\left\{ {{{{x^{2015}}} \over {{e^{|x|}}({x^2} + \cos x)}} + {1 \over {{e^{|x|}}}}} \right\}} dx\) is equal to

The value of \(\mathop {\lim }\limits_{x \to 0} {1 \over x}\left[ {\int\limits_y^a {{e^{{{\sin }^2}t}}dt - } \int\limits_{x + y}^a {{e^{{{\sin }^2}t}}dt} } \right]\) is equal to

Let \({I_n} = \int\limits_0^1 {{x^n}} {\tan ^{ - 1}}xdx\). If \({a_n}{I_{n + 2}} + {b_n}{I_n} = {c_n}\) for all n \(\ge\) 1, then

\(\mathop {\lim }\limits_{n \to \infty } {3 \over n}\left[ {1 + \sqrt {{n \over {n + 3}}} + \sqrt {{n \over {n + 6}}} + \sqrt {{n \over {n + 9}}} + ... + \sqrt {{n \over {n + 3(n - 1)}}} } \right]\)

The value of the integration \(\int\limits_{ - {\pi \over 4}}^{\pi /4} {\left( {\lambda |\sin x| + {{\mu \sin x} \over {1 + \cos x}} + \gamma } \right)} dx\)

General solution of \({(x + y)^2}{{dy} \over {dx}} = {a^2},a \ne 0\) is (C is an arbitrary constant)

The general solution of the differential equation \(\left( {1 + {e^{{x \over y}}}} \right)dx + \left( {1 - {x \over y}} \right){e^{x/y}}dy = 0\) is (C is an arbitrary constant)

Let f(x) be a derivable function, f'(x) > f(x) and f(0) = 0. Then,

Let f and g be differentiable on the interval I and let a, b \(\in\) I, a < b. Then,

Let f(x) > 0 for all x and f'(x) exists for all x. If f is the inverse function of h and \({h'(x) = {1 \over {1 + \log x}}}\). Then, f'(x) will be

Let \(f(x) = {x^4} - 4{x^3} + 4{x^2} + c,\,c \in R\). Then

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

P is the extremity of the latusrectum of ellipse \(3{x^2} + 4{y^2} = 48\) in the first quadrant. The eccentric angle of P is

Let \(f:R \to R\) be defined by \(f(x) = {x^2} - {{{x^2}} \over {1 + {x^2}}}\) for all \(x \in R\). Then,

Consider the function f(x) = cos x2. Then,

Let a > b > 0 and I(n) = a1/n \(-\) b1/n, J(n) = (a \(-\) b)1/n for all n \(\ge\) 2, then

Let P(4, 3) be a point on the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). If the normal at P intersects the X-axis at (16, 0), then the eccentricity of the hyperbola is

The length of conjugate axis of a hyperbola is greater than the length of transverse axis. Then, the eccentricity e is

Equation of a tangent to the hyperbola 5x2 \(-\) y2 = 5 and which passes through an external point (2, 8) is

The equation of the directrices of the hyperbola \(3{x^2} - 3{y^2} - 18x + 12y + 2 = 0\) is

A point is in motion along a hyperbola \(y = {{10} \over x}\) so that its abscissa x increases uniformly at a rate of 1 unit per second. Then, the rate of change of its ordinate when the point passes through (5, 2)

For the hyperbola \({{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1\), which of the following remains fixed when \(\alpha\) varies?

If \(\int {\cos x\log \left( {\tan {x \over 2}} \right)} dx\) = \(\sin x\log \left( {\tan {x \over 2}} \right)\) + f(x), then f(x) is equal to (assuming c is a arbitrary real constant).

If \(\int {{2^{{2^x}}}.\,{2^x}dx} = A\,.\,{2^{{2^x}}} + C\), then A is equal to

y = \(\int {\cos \left\{ {2{{\tan }^{ - 1}}\sqrt {{{1 - x} \over {1 + x}}} } \right\}} dx\) is an equation of a family of

\(\mathop {\lim }\limits_{x \to {0^ + }} {({e^x} + x)^{1/x}}\)

Let \(a = \min \{ {x^2} + 2x + 3:x \in R\}\) and \(b = \mathop {\lim }\limits_{\theta \to 0} {{1 - \cos \theta } \over {{\theta ^2}}}\). Then \(\sum\limits_{r = 0}^n {{a^r}{b^{n - r}}}\) is

A particle starts at the origin and moves 1 unit horizontally to the right and reaches P1, then it moves \({1 \over 2}\) unit vertically up and reaches P2, then it moves \({1 \over 4}\) unit horizontally to right and reaches P3, then it mov...

The limit of the interior angle of a regular polygon of n sides as n \(\to\) \(\infty\) is

The value of \(\mathop {\lim }\limits_{x \to {0^ + }} {x \over p}\left[ {{q \over x}} \right]\) is

\(\mathop {\lim }\limits_{x \to {0^ + }} ({x^n}\ln x),\,n > 0\)

Let \(f:[1,3] \to R\) be a continuous function that is differentiable in (1, 3) an f'(x) = | f(x) |2 + 4 for all x\(\in\) (1, 3). Then,

Consider the function \(f(x) = {{{x^3}} \over 4} - \sin \pi x + 3\)

If \(\log _2^6 + {1 \over {2x}} = {\log _2}\left( {{2^{{1 \over x}}} + 8} \right)\), then the value of x are

72n + 16n \(-\)1 (n\(\in\) N) is divisible by

Let A and B be two square matrices of order 3 and AB = O3, where O3 denotes the null matrix of order 3. Then,

If M is any square matrix of order 3 over R and if M' be the transpose of M, then adj(M') \(-\) (adj M)' is equal to

If \(A = \left( {\matrix{ 5 & {5x} & x \cr 0 & x & {5x} \cr 0 & 0 & 5 \cr } } \right)\) and \(|A{|^2} = 25\), then | x | is equal to