WB JEE 2019

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

Let \(A = \left[ {\matrix{ 3 & 0 & 3 \cr 0 & 3 & 0 \cr 3 & 0 & 3 \cr } } \right]\). Then, the roots of the equation det \((A - \lambda {I_3})\) = 0 (where I3 is the identity matrix of order 3) are

Let A be a square matrix of order 3 whose all entries are 1 and let I3 be the identity matrix of order 3. Then, the matrix \(A - 3{I_3}\) is

The system of equations\(\eqalign{ & \lambda x + y + 3z = 0 \cr & 2x + \mu y - z = 0 \cr & 5x + 7y + z = 0 \cr}\)has infinitely many solutions in R. Then,

A candidate is required to answer 6 out of 12 questions which are divided into two parts A and B, each containing 6 questions and he/she is not permitted to attempt more than 4 questions from any part. In how many different ways can he/she ...

There are 7 greeting cards, each of a different colour and 7 envelopes of same 7 colours as that of the cards. The number of ways in which the cards can be put in envelopes, so that exactly 4 of the cards go into envelopes of respective col...

A problem in mathematics is given to 4 students whose chances of solving individually are \({{1 \over 2}}\), \({{1 \over 3}}\), \({{1 \over 4}}\) and \({{1 \over 5}}\). The probability that the problem will be solved at least by one student...

If X is a random variable such that \(\sigma\)(X) = 2.6, then \(\sigma\)(1 \(-\) 4X) is equal to

The three sides of a right angled triangle are in GP (geometric progression). If the two acute angles be \(\alpha\) and \(\beta\), then tan\(\alpha\) and tan\(\beta\) are

The angles of a triangle are in the ratio 2 : 3 : 7 and the radius of the circumscribed circle is 10 cm. The length of the smallest side is

Let a, b, c be real numbers such that a + b + c < 0 and the quadratic equation ax2 + bx + c = 0 has imaginary roots. Then,