WB JEE 2021

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

\(\left| {\matrix{ x & {3x + 2} & {2x - 1} \cr {2x - 1} & {4x} & {3x + 1} \cr {7x - 2} & {17x + 6} & {12x - 1} \cr } } \right| = 0\) is true for

The locus of the vertices of the family of parabolas \(6y = 2{a^3}{x^2} + 3{a^2}x - 12a\) is

From a point (d, 0) three normal are drawn to the parabola y2 = x, then

Five letter words, having distinct letters, are to be constructed using the letters of the word 'EQUATION' so that each word contains exactly three vowels and two consonants. How many of them have all the vowels together?

What is the number of ways in which an examiner can assign 10 marks to 4 questions, giving not less than 2 marks to any question?

Four persons A, B, C and D throw and unbiased die, turn by turn, in succession till one gets an even number and win the game. What is the probability that A wins the game if A begins?

Let \(\alpha\), \(\beta\) be the roots of the equation x2 \(-\) 6x \(-\) 2 = 0 with \(\alpha\) > \(\beta\). If an = \(\alpha\)n \(-\) \(\beta\)n for n \(\ge\) 1, then the value of \({{{a_{10}} - 2{a_8}} \over {2{a_9}}}\) is

Let a, b, c be real numbers, each greater than 1, such that \({2 \over 3}{\log _b}a + {3 \over 5}{\log _c}b + {5 \over 2}{\log _a}c = 3\). If the value of b is 9, then the value of 'a' must be

Consider the real valued function h : {0, 1, 2, ...... 100} \(\to\) R such that h(0) = 5, h(100) = 20 and satisfying h(p) = \({1 \over 2}\) {h(p + 1) + h(p \(-\) 1)} for every p = 1, 2 ..... 99. Then the value of h(1) is

The digit in the unit's place of the number 1! + 2! + 3! + .... + 99! is