WB JEE 2023

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

Let A be a set containing n elements. A subset P of A is chosen, and the set A is reconstructed by replacing the elements of P. A subset Q of A is chosen again. The number of ways of choosing P and Q such that Q contains just one element mo...

A letter lock consists of three rings with 15 different letters. If N denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock, then

Let A and B are two independent events. The probability that both A and B happen is \({1 \over {12}}\) and probability that neither A and B happen is \({1 \over 2}\). Then

Let S be the sample space of the random experiment of throwing simultaneously two unbiased dice and \(\mathrm{E_k=\{(a,b)\in S:ab=k\}}\). If \(\mathrm{p_k=P(E_k)}\), then the correct among the following is :

ABC is an isosceles triangle with an inscribed circle with centre O. Let P be the midpoint of BC. If AB = AC = 15 and BC = 10, then OP equals

If one root of \({x^2} + px - {q^2} = 0,p\) and \(q\) are real, be less than 2 and other be greater than 2, then

If the n terms \({a_1},{a_2},\,......,\,{a_n}\) are in A.P. with increment r, then the difference between the mean of their squares & the square of their mean is

If \(1,{\log _9}({3^{1 - x}} + 2),{\log _3}({4.3^x} - 1)\) are in A.P., then x equals

Consider a quadratic equation \(a{x^2} + 2bx + c = 0\) where a, b, c are positive real numbers. If the equation has no real root, then which of the following is true?

Let \({a_1},{a_2},{a_3},\,...,\,{a_n}\) be positive real numbers. Then the minimum value of \({{{a_1}} \over {{a_2}}} + {{{a_2}} \over {{a_3}}}\, + \,...\, + \,{{{a_n}} \over {{a_1}}}\) is