WB JEE 2021

Let f : R \(\to\) R be such that f(0) = 0 and \(\left| {f'(x)} \right| \le 5\) for all x. Then f(1) is in

Two particles A and B move from rest along a straight line with constant accelerations f and f' respectively. If A takes m sec. more than that of B and describes n units more than that of B in acquiring the same velocity, then

If the tangent at the point P with co-ordinates (h, k) on the curve y2 = 2x3 is perpendicular to the straight line 4x = 3y, then

The greatest and least value of \(f(x) = {\tan ^{ - 1}} - {1 \over 2}\,ln \,x\,on\,\left[ {{1 \over {\sqrt 3 }},\sqrt 3 } \right]\) are

The straight the through the origin which divides the area formed by the curves y = 2x \(-\) x2, y = 0 and x = 1 into two equal halves is

The area bounded by the parabolas \(y = 4{x^2},y = {{{x^2}} \over 9}\) and the straight line y = 2 is

For x\(\in\)R, x \(\ne\) \(-\)1, if \({(1 + x)^{2016}} + x{(1 + x)^{2015}} + {x^2}{(1 + x)^{2014}} + ..... + {x^{2016}} = \sum\limits_{i = 0}^{2016} {{a_i}\,.\,{x^i}}\), then a17 is equal to

The coefficient of a3b4c5 in the expansion of (bc + ca + ab)6 is

The remainder when \({7^{{7^{{7^{{{..}^7}}}}}}}\) (22 time 7) is divided by 48 is

Two tangents to the circle x2 + y2 = 4 at the points A and B meet at M(\(-\)4, 0). The area of the quadrilateral MAOB, where O is the origin is

Let P be a variable point on a circle C and Q be a fixed point outside C. If R is the midpoint of the line segment PQ, then locus of R is

If |z| = 1 and z \(\ne\) \(\pm\) 1, then all the points representing \({z \over {1 - {z^2}}}\) lie on

Let C denote the set of all complex numbers. Define A = {(z, w) | z, w\(\in\)C and |z| = |w|}, B = {z, w} | z, w\(\in\)C and z2 = w2}. Then

If \(\left| {z + i} \right| - \left| {z - 1} \right| = \left| z \right| - 2 = 0\) for a complex number z, then z is equal to

\(\int\limits_1^3 {{{\left| {x - 1} \right|} \over {\left| {x - 2} \right| + \left| {x - 3} \right|}}dx}\) is equal to

The value of the integral \(\int\limits_{ - {1 \over 2}}^{{1 \over 2}} {{{\left\{ {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{x - 1} \over {x + 1}}} \right)}^2} - 2} \right\}}^{1/2}}} dx\) is equal to

If \(\int\limits_{{{\log }_e}2}^x {{{({e^x} - 1)}^{ - 1}}dx = {{\log }_e}{3 \over 2}}\), then the value of x is

The value of \(\int\limits_0^5 {\max \{ {x^2},6x - 8\} \,dx}\) is

Let f(x) be continuous periodic function with period T. Let \(I = \int\limits_a^{a + T} {f(x)\,dx}\). Then

If \(b = \int\limits_0^1 {{{{e^t}} \over {t + 1}}dt}\), then \(\int\limits_{a - 1}^a {{{{e^{ - t}}} \over {t - a - 1}}}\) is

Let \(I = \int_{\pi /4}^{\pi /3} {{{\sin x} \over x}dx}\). Then

Whichever of the following is/are correct?

Let \(f(x) = \left\{ {\matrix{ {0,} & {if} & { - 1 \le x \le 0} \cr {1,} & {if} & {x = 0} \cr {2,} & {if} & {0 < x \le 1} \cr } } \right.\) and let \(F(x) = \int\limits_{ - 1}^x {f(t)dt}\), \(-\)1 \(\le\) x \(\le\) 1, then

The differential equation of all the ellipses centred at the origin and have axes as the co-ordinate axes is where \(y^{\prime}\equiv{{{dx}\over {dy}}},y^{\prime\prime}\equiv{{{d^2}y\over {dx^2}}}\)

If \(x{{dy} \over {dx}} + y = {{xf(xy)} \over {f'(xy)'}}\), then | f(xy) | is equal to (where k is an arbitrary positive constant).

The differential of \(f(x) = {\log _e}(1 + {e^{10x}}) - {\tan ^{ - 1}}({e^{5x}})\) at x = 0 and for dx = 0.2 is

Let \(g(x) = \int\limits_x^{2x} {{{f(t)} \over t}dt}\) where x > 0 and f be continuous function and f(2x) = f(x), then

A bulb is placed at the centre of a circular track of radius 10 m. A vertical wall is erected touching the track at a point P. A man is running along the track with a speed of 10 m/sec. Starting from P the speed with which his shadow is run...

The co-ordinate of a point on the auxiliary circle of the ellipse x2 + 2y2 = 4 corresponding to the point on the ellipse whose eccentric angle is 60\(^\circ\) will be

The points of intersection of two ellipses \({x^2} + 2{y^2} - 6x - 12y + 20 = 0\) and \(2{x^2} + {y^2} - 10x - 6y + 15 = 0\) lie on a circle. The centre of the circle is

Let f : R \(\to\) R be given by f(x) = | x2 \(-\) 1 |, x\(\in\)R. Then,

f(x) is real valued function such that 2f(x) + 3f(\(-\)x) = 15 \(-\) 4x for all x\(\in\)R. Then f(2) =

Consider the functions f1(x) = x, f2(x) = 2 + loge x, x > 0. The graphs of the functions intersect

Given that f : S \(\to\) R is said to have a fixed point at c of S if f(c) = c. Let f : [1, \(\infty\)) \(\to\) R be defined by f(x) = 1 + \(\sqrt x\). Then

Let f and g be periodic functions with the periods T1 and T2 respectively. Then f + g is

The normal to a curve at P(x, y) meets the X-axis at G. If the distance of G from the origin is twice the abscissa of P then the curve is

The locus of the centre of a variable circle which always touches two given circles externally is

If \(\int {{{\sin 2x} \over {{{(a + b\cos x)}^2}}}dx} = \alpha \left[ {{{\log }_e}\left| {a + b\cos x} \right| + {a \over {a + b\cos x}}} \right] + c\), then \(\alpha\) is equal to

For \(y = {\sin ^{ - 1}}\left\{ {{{5x + 12\sqrt {1 - {x^2}} } \over {13}}} \right\};\left| x \right| \le 1\), if \(a(1 - {x^2}){y_2} + bx{y_1} = 0\) then (a, b) =

If \(I = \mathop {\lim }\limits_{x \to 0} sin\left( {{{{e^x} - x - 1 - {{{x^2}} \over 2}} \over {{x^2}}}} \right)\), then limit

Let \({S_n} = {\cot ^{ - 1}}2 + {\cot ^{ - 1}}8 + {\cot ^{ - 1}}18 + {\cot ^{ - 1}}32 + ....\) to nth term. Then \(\mathop {\lim }\limits_{n \to \infty } {S_n}\) is

Let f : D \(\to\) R where D = [\(-\)0, 1] \(\cup\) [2, 4] be defined by \(f(x) = \left\{ {\matrix{ {x,} & {if} & {x \in [0,1]} \cr {4 - x,} & {if} & {x \in [2,4]} \cr } } \right.\) Then,

The \(\mathop {\lim }\limits_{x \to \infty } {\left( {{{3x - 1} \over {3x + 1}}} \right)^{4x}}\) equals

$$\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\sqrt n } \over {\sqrt {{n^3}} }} + {{\sqrt n } \over {\sqrt {{{(n + 4)}^3}} }} + {{\sqrt n } \over {\sqrt {{{(n + 8)}^3}} }} + .... + {{\sqrt n } \over {\sqrt {{{[n + 4(n - 1)]}^3}} }}} \...

If M is a 3 \(\times\) 3 matrix such that (0, 1, 2) M = (1 0 0), (3, 4 5) M = (0, 1, 0), then (6 7 8) M is equal to

Let \(A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & {\cos t} & {\sin t} \cr 0 & { - \sin t} & {\cos t} \cr } } \right)\)Let \(\lambda\)1, \(\lambda\)2, \(\lambda\)3 be the roots of \(\det (A - \lambda {I_3}) = 0\), where I3 denote...

Let A and B two non singular skew symmetric matrices such that AB = BA, then A2B2(ATB)\(-\)1(AB\(-\)1)T is equal to

If an (> 0) be the nth term of a G.P. then$$\left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr
{\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr
{\log {a_{n + 6}}} & {\log {a_{n + 7}}} & ...

Let T and U be the set of all orthogonal matrices of order 3 over R and the set of all non-singular matrices of order 3 over R respectively. Let A = {\(-\)1, 0, 1}, then

The determinant \(\left| {\matrix{ {{a^2} + 10} & {ab} & {ac} \cr {ab} & {{b^2} + 10} & {bc} \cr {ac} & {bc} & {{c^2} + 10} \cr } } \right|\) is