WB JEE 2022

Twenty metres of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle be, if the area of the flower bed be greatest?

If \(|z - 25i| \le 15\), then Maximum arg(z) \(-\) Minimum arg(z) is equal to
(arg z is the principal value of argument of z)

If z = x \(-\) iy and \({z^{{1 \over 3}}} = p + iq(x,y,p,q \in R)\), then \({{\left( {{x \over p} + {y \over q}} \right)} \over {({p^2} + {q^2})}}\) is equal to

Let z1 and z2 be two non-zero complex numbers. Then

Let f be derivable in [0, 1], then

The value of \(\int\limits_0^{{\pi \over 2}} {{{{{(\cos x)}^{\sin x}}} \over {{{(\cos x)}^{\sin x}} + {{(\sin x)}^{\cos x}}}}dx}\) is

Let \(\mathop {\lim }\limits_{ \in \to 0 + } \int\limits_ \in ^x {{{bt\cos 4t - a\sin 4t} \over {{t^2}}}dt = {{a\sin 4x} \over x} - 1,\left( {0 < x < {\pi \over 4}} \right)}\). Then a and b are given by

Let \(f(x) = \int\limits_{\sin x}^{\cos x} {{e^{ - {t^2}}}dt}\). Then \(f'\left( {{\pi \over 4}} \right)\) equals

If I is the greatest of \({I_1} = \int\limits_0^1 {{e^{ - x}}{{\cos }^2}x\,dx}\), \({I_2} = \int\limits_0^1 {{e^{ - {x^2}}}{{\cos }^2}x\,dx}\), \({I_3} = \int\limits_0^1 {{e^{ - {x^2}}}dx}\), $${I_4} = \int\limits_0^1 {{e^{ - {x^2}/2}}dx...

If \(x{{dy} \over {dx}} + y = x{{f(xy)} \over {f'(xy)}}\), then \(|f(xy)|\) is equal to

The solution of \(\cos y{{dy} \over {dx}} = {e^{x + \sin y}} + {x^2}{e^{\sin y}}\) is \(f(x) + {e^{ - \sin y}} = C\) (C is arbitrary real constant) where f(x) is equal to

If the transformation \(z = \log \tan {x \over 2}\) reduces the differential equation \({{{d^2}y} \over {d{x^2}}} + \cot x{{dy} \over {dx}} + 4y\cos e{c^2}x = 0\) into the form \({{{d^2}y} \over {d{z^2}}} + ky = 0\) then k is equal to

If \(y = {e^{{{\tan }^{ - 1}}x}}\), then

AB is a variable chord of the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\). If AB subtends a right angle at the origin O, then \({1 \over {O{A^2}}} + {1 \over {O{B^2}}}\) equals to

Chords of an ellipse are drawn through the positive end of the minor axis. Their midpoint lies on

Domain of \(y = \sqrt {{{\log }_{10}}{{3x - {x^2}} \over 2}}\) is

Let \(f(x) = {(x - 2)^{17}}{(x + 5)^{24}}\). Then

Let \(f(n) = {2^{n + 1}}\), \(g(n) = 1 + (n + 1){2^n}\) for all \(n \in N\). Then

The maximum value of \(f(x) = {e^{\sin x}} + {e^{\cos x}};x \in R\) is

\(f:X \to R,X = \{ x|0 < x < 1\}\) is defined as \(f(x) = {{2x - 1} \over {1 - |2x - 1|}}\). Then

Let \(f(x) = {x^2} + x\sin x - \cos x\). Then

Let p(x) be a polynomial with real co-efficient, p(0) = 1 and p'(x) > 0 for all x \(\in\) R. Then

Let \(P(3\sec \theta ,2\tan \theta )\) and \(Q(3\sec \phi ,2\tan \phi )\) be two points on \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1\) such that \(\theta + \phi = {\pi \over 2},0 < \theta ,\phi < {\pi \over 2}\). Then the ordinate of...

PQ is a double ordinate of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) such that \(\Delta OPQ\) is an equilateral triangle, O being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies

The line y = x + 5 touches

\(I = \int {\cos (\ln x)dx}\). Then I =

Let \(\int {{{{x^{{1 \over 2}}}} \over {\sqrt {1 - {x^3}} }}dx = {2 \over 3}g(f(x)) + c}\) ; then
(c denotes constant of integration)

The values of a, b, c for which the function $$f(x) = \left\{ \matrix{
{{\sin (a + 1)x + \sin x} \over x},x < 0 \hfill \cr
c,x = 0 \hfill \cr
{{{{(x + b{x^2})}^{{1 \over 2}}} - {x^{{1 \over 2}}}} \over {b{x^{{1 \over 2}}}}},x > 0 \h...

Let \(f(x) = {a_0} + {a_1}|x| + {a_2}|x{|^2} + {a_3}|x{|^3}\), where \({a_0},{a_1},{a_2},{a_3}\) are real constants. Then f(x) is differentiable at x = 0

\(\mathop {\lim }\limits_{x \to 0} \left( {{1 \over x}\ln \sqrt {{{1 + x} \over {1 - x}}} } \right)\) is

Let f : [a, b] \(\to\) R be continuous in [a, b], differentiable in (a, b) and f(a) = 0 = f(b). Then

\(\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + 1} \over {x + 1}} - ax - b} \right),(a,b \in R)\) = 0. Then

If x satisfies the inequality \({\log _{25}}{x^2} + {({\log _5}x)^2} < 2\), then x belongs to

Under which of the following condition(s) does(do) the system of equations $$\left( {\matrix{
1 & 2 & 4 \cr
2 & 1 & 2 \cr
1 & 2 & {(a - 4)} \cr

} } \right)\left( {\matrix{
x \cr
y \cr
z \cr

} } \right) = \l...

If \(\Delta (x) = \left| {\matrix{ {x - 2} & {{{(x - 1)}^2}} & {{x^3}} \cr {x - 1} & {{x^2}} & {{{(x + 1)}^3}} \cr x & {{{(x + 1)}^2}} & {{{(x + 2)}^3}} \cr } } \right|\), then coefficient of x in \(\Delta\)x is

If \(p = \left[ {\matrix{ 1 & \alpha & 3 \cr 1 & 3 & 3 \cr 2 & 4 & 4 \cr } } \right]\) is the adjoint of the \(3 \times 3\) matrix A and det A = 4, then \(\alpha\) is equal to

If \(A = \left( {\matrix{ 1 & 1 \cr 0 & i \cr } } \right)\) and \({A^{2018}} = \left( {\matrix{ a & b \cr c & d \cr } } \right)\), then \((a + d)\) equals

The solution of \(\det (A - \lambda {I_2}) = 0\) be 4 and 8 and \(A = \left( {\matrix{ 2 & 2 \cr x & y \cr } } \right)\). Then
(I2 is identity matrix of order 2)

Let $$\Delta = \left| {\matrix{
{\sin \theta \cos \phi } & {\sin \theta \sin \phi } & {\cos \theta } \cr
{\cos \theta \cos \phi } & {\cos \theta \sin \phi } & { - \sin \theta } \cr
{ - \sin \theta \sin \phi } & {\sin \theta \c...

The point of contact of the tangent to the parabola y2 = 9x which passes through the point (4, 10) and makes an angle \(\theta\) with the positive side of the axis of the parabola where tan\(\theta\) > 2, is

A line passes through the point \(( - 1,1)\) and makes an angle \({\sin ^{ - 1}}\left( {{3 \over 5}} \right)\) in the positive direction of x-axis. If this line meets the curve \({x^2} = 4y - 9\) at A and B, then |AB| is equal to

Let P be a point on (2, 0) and Q be a variable point on (y \(-\) 6)2 = 2(x \(-\) 4). Then the locus of mid-point of PQ is

AB is a chord of a parabola y2 = 4ax, (a > 0) with vertex A. BC is drawn perpendicular to AB meeting the axis at C. The projection of BC on the axis of the parabola is

From the point (\(-\)1, \(-\)6), two tangents are drawn to y2 = 4x. Then the angle between the two tangents is

Let the tangent and normal at any point P(at2, 2at), (a > 0), on the parabola y2 = 4ax meet the axis of the parabola at T and G respectively. Then the radius of the circle through P, T and G is

If P1P2 and P3P4 are two focal chords of the parabola y2 = 4ax then the chords P1P3 and P2P4 intersect on the

There are n white and n black balls marked 1, 2, 3, ...... n. The number of ways in which we can arrange these balls in a row so that neighbouring balls are of different colours is

A, B, C are mutually exclusive events such that \(P(A) = {{3x + 1} \over 3}\), \(P(B) = {{1 - x} \over 4}\) and \(P(C) = {{1 - 2x} \over 2}\). Then the set of possible values of x are in

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinant chosen is non-zero is

If a, b are odd integers, then the roots of the equation \(2a{x^2} + (2a + b)x + b = 0\), \(a \ne 0\) are