WB JEE 2022

A particle moving in a straight line starts from rest and the acceleration at any time t is \(a - k{t^2}\) where a and k are positive constants. The maximum velocity attained by the particle is

From a balloon rising vertically with uniform velocity v ft/sec a piece of stone is let go. The height of the balloon above the ground when the stone reaches the ground after 4 sec is [g = 32 ft/sec2]

Area of the figure bounded by the parabola \({y^2} + 8x = 16\) and \({y^2} - 24x = 48\) is

Let f be a non-negative function defined in \([0,\pi /2]\), f' exists and be continuous for all x and \(\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int\limits_0^x {f(t)dt} }\) and f (0) = 0. Then

The number of zeros at the end of \(\left| \!{\underline {\, {100} \,}} \right.\) is

A curve passes through the point (3, 2) for which the segment of the tangent line contained between the co-ordinate axes is bisected at the point of contact. The equation of the curve is

If the equation of one tangent to the circle with centre at (2, \(-\)1) from the origin is 3x + y = 0, then the equation of the other tangent through the origin is

The side AB of \(\Delta\)ABC is fixed and is of length 2a unit. The vertex moves in the plane such that the vertical angle is always constant and is \(\alpha\). Let x-axis be along AB and the origin be at A. Then the locus of the vertex is

Two circles \({S_1} = p{x^2} + p{y^2} + 2g'x + 2f'y + d = 0\) and \({S_2} = {x^2} + {y^2} + 2gx + 2fy + d' = 0\) have a common chord PQ. The equation of PQ is

A straight line meets the co-ordinate axes at A and B. A circle is circumscribed about the triangle OAB, O being the origin. If m and n are the distances of the tangent to the circle at the origin from the points A and B respectively, the d...

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