WB JEE 2023

A missile is fired from the ground level rises x meters vertically upwards in t sec, where \(x = 100t - {{25} \over 2}{t^2}\). The maximum height reached is

The portion of the tangent to the curve \({x^{{2 \over 3}}} + {y^{{2 \over 3}}} = {a^{{2 \over 3}}},a > 0\) at any point of it, intercepted between the axes

Given \(f(x) = {e^{\sin x}} + {e^{\cos x}}\). The global maximum value of \(f(x)\)

A balloon starting from rest is ascending from ground with uniform acceleration of 4 ft/sec\(^2\). At the end of 5 sec, a stone is dropped from it. If T be the time to reach the stone to the ground and H be the height of the balloon when th...

If \(f(x) = 3\root 3 \of {{x^2}} - {x^2}\), then

If the vertices of a square are \({z_1},{z_2},{z_3}\) and \({z_4}\) taken in the anti-clockwise order, then \({z_3} =\)

Reflection of the line \(\overline a z + a\overline z = 0\) in the real axis is given by :

If z\(_1\) and z\(_2\) are two complex numbers satisfying the equation \(\left| {{{{z_1} + {z_2}} \over {{z_1} - {z_2}}}} \right| = 1\), then \({{{z_1}} \over {{z_2}}}\) may be

the expression \({{\int\limits_0^n {[x]dx} } \over {\int\limits_0^n {\{ x\} dx} }}\), where \([x]\) and \(\{ x\}\) are respectively integral and fractional part of \(x\) and \(n \in N\), is equal to

The value \(\int\limits_0^{1/2} {{{dx} \over {\sqrt {1 - {x^{2n}}} }}}\) is \((n \in N)\)

If \({I_n} = \int\limits_0^{{\pi \over 2}} {{{\cos }^n}x\cos nxdx}\), then I\(_1\), I\(_2\), I\(_3\) ... are in

\(\int\limits_0^{2\pi } {\theta {{\sin }^6}\theta \cos \theta d\theta }\) is equal to

The average ordinate of \(y = \sin x\) over \([0,\pi ]\) is :

Let f be a non-negative function defined on \(\left[ {0,{\pi \over 2}} \right]\). If \(\int\limits_0^x {(f'(t) - \sin 2t)dt = \int\limits_x^0 {f(t)\tan t\,dt} } ,f(0) = 1\) then \(\int\limits_0^{{\pi \over 2}} {f(x)dx}\) is

Which of the following statements are true?

If \(y = {x \over {{{\log }_e}|cx|}}\) is the solution of the differential equation \({{dy} \over {dx}} = {y \over x} + \phi \left( {{x \over y}} \right)\), then \(\phi \left( {{x \over y}} \right)\) is given by

Given \({{{d^2}y} \over {d{x^2}}} + \cot x{{dy} \over {dx}} + 4y\cos e{c^2}x = 0\). Changing the independent variable x to z by the substitution \(z = \log \tan {x \over 2}\), the equation is changed to

The family of curves \(y = {e^{a\sin x}}\), where 'a' is arbitrary constant, is represented by the differential equation

Suppose \(f:R \to R\) be given by \(f(x) = \left\{ \matrix{ 1,\,\,\,\,\,\,\,\,\,\,\mathrm{if}\,x = 1 \hfill \cr {e^{({x^{10}} - 1)}} + {(x - 1)^2}\sin {1 \over {x - 1}},\,\mathrm{if}\,x \ne 1 \hfill \cr} \right.\)
then

Let \({\cos ^{ - 1}}\left( {{y \over b}} \right) = {\log _e}{\left( {{x \over n}} \right)^n}\), then \(A{y_2} + B{y_1} + Cy = 0\) is possible for, where \({y_2} = {{{d^2}y} \over {d{x^2}}},{y_1} = {{dy} \over {dx}}\)

The function \(y = {e^{kx}}\) satisfies \(\left( {{{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}}} \right)\left( {{{dy} \over {dx}} - y} \right) = y{{dy} \over {dx}}\). It is valid for

If \(y = {\log ^n}x\), where \({\log ^n}\) means \({\log _e}{\log _e}{\log _e}\,...\) (repeated n times), then \(x\log x{\log ^2}x{\log ^3}x\,.....\,{\log ^{n - 1}}x{\log ^n}x{{dy} \over {dx}}\) is equal to

If \(x = \sin \theta\) and \(y = \sin k\theta\), then \((1 - {x^2}){y_2} - x{y_1} - \alpha y = 0\), for \(\alpha=\)

Let \(f(x) = {x^m}\), m being a non-negative integer. The value of m so that the equality \(f'(a + b) = f'(a) + f'(b)\) is valid for all a, b > 0 is

The tangent at point \((a\cos \theta ,b\sin \theta ),0 < \theta < {\pi \over 2}\), to the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) meets the x-axis at T and y-axis at T\(_1\). Then the value of $$\mathop {\min }\l...

If the lines joining the focii of the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) where \(a > b\), and an extremity of its minor axis is inclined at an angle 60\(^\circ\), then the eccentricity of the ellipse is

Let f be a strictly decreasing function defined on R such that \(f(x) > 0,\forall x \in R\). Let \({{{x^2}} \over {f({a^2} + 5a + 3)}} + {{{y^2}} \over {f(a + 15)}} = 1\) be an ellipse with major axis along the y-axis. The value of 'a' can ...

In the interval \(( - 2\pi ,0)\), the function \(f(x) = \sin \left( {{1 \over {{x^3}}}} \right)\).

If a hyperbola passes through the point P(\(\sqrt2\), \(\sqrt3\)) and has foci at (\(\pm\) 2, 0), then the tangent to this hyperbola at P is

The average length of all vertical chords of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1,a \le x \le 2a\), is :

Let \(A(2\sec \theta ,3\tan \theta )\) and \(B(2\sec \phi ,3\tan \phi )\) where \(\theta + \phi = {\pi \over 2}\) be two points on the hyperbola \({{{x^2}} \over 4} - {{{y^2}} \over 9} = 1\). If (\(\alpha,\beta\)) is the point of interse...

If \(I = \int {{{{x^2}dx} \over {{{(x\sin x + \cos x)}^2}}} = f(x) + \tan x + c}\), then \(f(x)\) is

If \(\int {{{dx} \over {(x + 1)(x - 2)(x - 3)}} = {1 \over k}{{\log }_e}\left\{ {{{|x - 3{|^3}|x + 1|} \over {{{(x - 2)}^4}}}} \right\} + c}\), then the value of k is

\(\mathop {\lim }\limits_{x \to \infty } \left\{ {x - \root n \of {(x - {a_1})(x - {a_2})\,...\,(x - {a_n})} } \right\}\) where \({a_1},{a_2},\,...,\,{a_n}\) are positive rational numbers. The limit

Let \(f:[1,3] \to R\) be continuous and be derivable in (1, 3) and \(f'(x) = {[f(x)]^2} + 4\forall x \in (1,3)\). Then

f(x) is a differentiable function and given \(f'(2) = 6\) and \(f'(1) = 4\), then \(L = \mathop {\lim }\limits_{h \to 0} {{f(2 + 2h + {h^2}) - f(2)} \over {f(1 + h - {h^2}) - f(1)}}\)

Let \(f(x) = \left\{ {\matrix{ {x + 1,} & { - 1 \le x \le 0} \cr { - x,} & {0 < x \le 1} \cr } } \right.\)

Let \(f(x) = [{x^2}]\sin \pi x,x > 0\). Then

The value of $$\mathop {\lim }\limits_{n \to \infty } \left[ {\left( {{1 \over {2\,.\,3}} + {1 \over {{2^2}\,.\,3}}} \right) + \left( {{1 \over {{2^2}\,.\,{3^2}}} + {1 \over {{2^3}\,.\,{3^2}}}} \right)\, + \,...\, + \,\left( {{2 \over {{2^n...

Let \(P(n) = {3^{2n + 1}} + {2^{n + 2}}\) where \(n \in N\). Then

Let A and B are orthogonal matrices and det A + det B = 0. Then

Let \(A = \left( {\matrix{ 2 & 0 & 3 \cr 4 & 7 & {11} \cr 5 & 4 & 8 \cr } } \right)\). Then

If the matrix Mr is given by \({M_r} = \left( {\matrix{ r & {r - 1} \cr {r - 1} & r \cr } } \right)\) for r = 1, 2, 3, ... then det (M1) + det (M2) + ... + det (M2008) =

Let \(\alpha,\beta\) be the roots of the equation \(a{x^2} + bx + c = 0,a,b,c\) real and \({s_n} = {\alpha ^n} + {\beta ^n}\) and $$\left| {\matrix{
3 & {1 + {s_1}} & {1 + {s_2}} \cr
{1 + {s_1}} & {1 + {s_2}} & {1 + {s_3}} \cr
...

Let \(A = \left( {\matrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right),B = \left( {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right)\) and $$P\left( {\matrix{
0 & 1 & 0 \cr
x ...

Let A be the point (0, 4) in the xy-plane and let B be the point (2t, 0). Let L be the midpoint of AB and let the perpendicular bisector of AB meet the y-axis M. Let N be the midpoint of LM. Then locus of N is

Let O be the vertex, Q be any point on the parabola x\(^2\) = 8y. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is :

From the focus of the parabola \({y^2} = 12x\), a ray of light is directed in a direction making an angle \({\tan ^{ - 1}}{3 \over 4}\) with x-axis. Then the equation of the line along which the reflected ray leaves the parabola is

The number of ways in which the letters of the word 'VERTICAL' can be arranged without changing the order of the vowels is

n objects are distributed at random among n persons. The number of ways in which this can be done so that at least one of them will not get any object is