WB JEE 2020

If \({x^2} + {y^2} = {a^2}\), then \(\int\limits_0^a {\sqrt {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} dx = }\)

If c0, c1, c2, ......, c15 are the binomial coefficients in the expansion of (1 + x)15, then the value of \({{{c_1}} \over {{c_0}}} + 2{{{c_2}} \over {{c_1}}} + 3{{{c_3}} \over {{c_2}}} + ... + 15{{{c_{15}}} \over {{c_{14}}}}\) is

The equation of circle of radius \(\sqrt {17}\) unit, with centre on the positive side of X-axis and through the point (0, 1) is

The locus of the centre of the circles which touch both the circles x2 + y2 = a2 and x2 + y2 = 4ax externally is

The equation \(z\bar z + (2 - 3i)z + (2 + 3i)\bar z + 4 = 0\) represents a circle of radius

The number of complex numbers p such that \(\left| p \right| = 1\) and imaginary part of p4 is 0, is

The value of \(\sum\limits_{n = 1}^{10} {} \int\limits_{ - 2n - 1}^{ - 2n} {{{\sin }^{27}}} x\,dx + \sum\limits_{n = 1}^{10} {} \int\limits_{2n}^{2n + 1} {{{\sin }^{27}}} x\,dx\) is equal to

Let f, be a continuous function in [0, 1], then \(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 0}^n {{1 \over n}} f\left( {{j \over n}} \right)\) is

\(\int\limits_0^2 {[{x^2}]} \,dx\) is equal to

The differential equation of the family of curves y = ex (A cos x + B sin x) where, A, B are arbitrary constants is

If \(x\sin \left( {{y \over x}} \right)dy = \left[ {y\sin \left( {{y \over x}} \right) - x} \right]dx,\,x > 0\) and \(y(1) = {\pi \over 2}\), then the value of \(\cos \left( {{y \over x}} \right)\) is

Let cos\(^{ - 1}\left( {{y \over b}} \right) = \log {\left( {{x \over n}} \right)^n}\). Then

Let \(y = f(x) = 2{x^2} - 3x + 2\). The differential of y when x changes from 2 to 1.99 is

Let f be a differentiable function with \(\mathop {\lim }\limits_{x \to \infty } f(x) = 0.\) If \(y' + yf'(x) - f(x)f'(x) = 0\), \(\mathop {\lim }\limits_{x \to \infty } y(x) = 0\), then (where \(y \equiv {{dy} \over {dx}})\)

Let \(y = {1 \over {1 + x + lnx}}\), then

Let \(y = {{{x^2}} \over {{{(x + 1)}^2}(x + 2)}}\). Then \({{{d^2}y} \over {d{x^2}}}\) is

Consider a tangent to the ellipse \({{{x^2}} \over 2} + {{{y^2}} \over 1} = 1\) at any point. The locus of the mid-point of the portion intercepted between the axes is

If B and B' are the ends of minor axis and S and S' are the foci of the ellipse \({{{x^2}} \over {25}} + {{{y^2}} \over 9} = 1\), then the area of the rhombus SBS' B' will be

Consider the curve \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\). The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

Consider the curve \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\). The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

Let \(f(x) = \sqrt {{x^2} - 3x + 2}\) and \(g(x) = \sqrt x\) be two given functions. If S be the domain of fog and T be the domain of gof, then

Let \(A = \{ x \in R: - 1 \le x \le 1\}\) and \(f:A \to A\) be a mapping defined by \(f(x) = x\left| x \right|\). Then f is

The domain of \(f(x) = \sqrt {\left( {{1 \over {\sqrt x }} - \sqrt {x + 1} } \right)}\) is

Let \(f(x) = 1 - \sqrt {({x^2})}\), where the square root is to be taken positive, then

A double ordinate PQ of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) is such that \(\Delta OPQ\) is equilateral, O being the centre of the hyperbola. Then the eccentricity e satisfies the relation

\(\int {{{f(x)\phi '(x) + \phi (x)f'(x)} \over {(f(x)\phi (x) + 1)\sqrt {f(x)\phi (x) - 1} }}dx = }\)

Let f : R \(\to\) R be twice continuously differentiable (or f" exists and is continuous) such that f(0) = f(1) = f'(0) = 0. Then

Let \(\phi (x) = f(x) + f(1 - x)\) and \(f(x) < 0\) in [0, 1], then

Let \(f(x) = {1 \over 3}x\sin x - (1 - \cos \,x)\). The smallest positive integer k such that \(\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^k}}} \ne 0\) is

Let \(0 < \alpha < \beta < 1\). Then, \(\mathop {\lim }\limits_{n \to \infty } \int\limits_{1/(k + \beta )}^{1/(k + \alpha )} {{{dx} \over {1 + x}}}\) is

\(\mathop {\lim }\limits_{x \to 1} \left( {{1 \over {1nx}} - {1 \over {(x - 1)}}} \right)\)

If \(\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + cx} \over {1 - cx}}} \right)^{{1 \over x}}} = 4\), then \(\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + 2cx} \over {1 - 2cx}}} \right)^{{1 \over x}}}\) is

The equation \({x^{{{(\log 3x)}^2}}} - {9 \over 2}\log 3\,x + 5 = 3\sqrt 3\) has

If 2 log(x + 1) \(-\) log(x2 \(-\) 1) = log 2, then x =

Let \(A = \left( {\matrix{ a & b \cr c & d \cr } } \right)\) be a 2 \(\times\) 2 real matrix with det A = 1. If the equation det (A \(-\) \(\lambda\)I2) = 0 has imaginary roots (I2 be the identity matrix of order 2), then

Let \(A = \left[ {\matrix{ {12} & {24} & 5 \cr x & 6 & 2 \cr { - 1} & { - 2} & 3 \cr } } \right]\). The value of x for which the matrix A is not invertible is

If \(\left| {\matrix{ {{a^2}} & {bc} & {{c^2} + ac} \cr {{a^2} + ab} & {{b^2}} & {ca} \cr {ab} & {{b^2} + bc} & {{c^2}} \cr } } \right| = k{a^2}{b^2}{c^2}\),then K =

If f : S \(\to\) R, where S is the set of all non-singular matrices of order 2 over R and \(f\left[ {\left( {\matrix{ a & b \cr c & d \cr } } \right)} \right] = ad - bc\), then

Let A = $$\left( {\matrix{
{3 - t} \cr
{ - 1} \cr
0 \cr

} \matrix{
{} \cr
{} \cr
{} \cr

} \,\matrix{
1 \cr
{3 - t} \cr
{ - 1} \cr

} \matrix{
{} \cr
{} \cr
{} \cr

} \matrix{...

If the vectors \(\alpha = \widehat i + a\widehat j + {a^2}\widehat k,\,\beta = \widehat i + b\widehat j + {b^2}\widehat k\) and \(\,\gamma = \widehat i + c\widehat j + {c^2}\widehat k\) are three non-coplanarvectors and $$\left| {\matrix...

The length of the chord of the parabola y2 = 4ax(a > 0) which passes through the vertex and makes an acute angle \(\alpha\) with the axis of the parabola is

If the line y = x is a tangent to the parabola y = ax2 + bx + c at the point (1, 1) and the curve passes through (\(-\)1, 0), then

The equation of the latusrectum of a parabola is x + y = 8 and the equation of the tangent at the vertex is x + y = 12. Then, the length of the latusrectum is

In a 12 storied building, 3 persons enter a lift cabin. It is known that they will leave the lift at different floors. In how many ways can they do so if the lift does not stop at the second floor?

If the total number of m-element subsets of the set A = {a1, a2, ..., an} is k times the number of m element subsets containing a4, then n is

A and B are independent events. The probability that both A and B occur is \({1 \over {20}}\) and the probability that neither of them occurs is \({3 \over {5}}\). The probability of occurrence of A is

A rifleman is firing at a distant target and has only 10% chance of hitting it. The least number of rounds he must fire to have more than 50% chance of hitting it at least once, is

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

The expression ax2 + bx + c (a, b and c are real) has the same sign as that of a for all x if

Let z1 and z2 be two imaginary roots of z2 + pz + q = 0, where p and q are real. The points z1, z2 and origin form an equilateral triangle if