WB JEE 2018

Let A be the centre of the circle \({x^2} + {y^2} - 2x - 4y - 20 = 0\). Let B(1, 7) and D(4, \(-\)2) be two points on the circle such that tangents at B and D meet at C. The area of the quadrilateral ABCD is

Without changing the direction of the axes, the origin is transferred to the point (2, 3). Then the equation x2 + y2 \(-\) 4x \(-\) 6y + 9 = 0 changes to

If \({a_r} = {(\cos 2r\pi + i\sin 2r\pi )^{1/9}}\), then the value of \(\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{a_4}} & {{a_5}} & {{a_6}} \cr {{a_7}} & {{a_8}} & {{a_9}} \cr } } \right|\) is equal to

Let z1 and z2 be complex numbers such that z1 \(\ne\) z2 and |z1| = |z2|. If Re(z1) > 0 and Im(z2) < 0, then \({{{z_1} + {z_2}} \over {{z_1} - {z_2}}}\) is

If z1 and z2 be two non-zero complex numbers such that \({{{z_1}} \over {{z_2}}} + {{{z_2}} \over {{z_1}}} = 1\), then the origin and the points represented by z1 and z2

If \({Z_r} = \sin {{2\pi r} \over {11}} - i\cos {{2\pi r} \over {11}}\), then \(\sum\limits_{r = 0}^{10} {{Z_r}}\) is equal to

The value of \(I = \int_{\pi /2}^{5\pi /2} {{{{e^{{{\tan }^{ - 1}}(\sin x)}}} \over {{e^{{{\tan }^{ - 1}}(\sin x)}} + {e^{{{\tan }^{ - 1}}(\cos x)}}}}} dx\), is

Let \(I = \int\limits_0^I {{{{x^3}\cos 3x} \over {2 + {x^2}}}dx}\), then

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

The value of \(\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left\{ {{{\sec }^2}{\pi \over {4n}} + {{\sec }^2}{{2\pi } \over {4n}} + ... + {{\sec }^2}{{n\pi } \over {4n}}} \right\}\) is

If \(M = \int\limits_0^{\pi /2} {{{\cos x} \over {x + 2}}dx}\), \(N = \int\limits_0^{\pi /4} {{{\sin x\cos x} \over {{{(x + 1)}^2}}}dx}\), then the value of M \(-\) N is

The value of the integral \(I = \int_{1/2014}^{2014} {{{{{\tan }^{ - 1}}x} \over x}} dx\) is

The differential equation representing the family of curves \({y^2} = 2d(x + \sqrt d )\), where d is a parameter, is of

Let y(x) be a solution of \((1 + {x^2}){{dy} \over {dx}} + 2xy - 4{x^2} = 0\). Then y(1) is equal to

The equation x log x = 3 \(-\) x

Let \({f_1}(x) = {e^x}\), \({f_2}(x) = {e^{{f_1}(x)}}\), ......, \({f_{n + 1}}(x) = {e^{{f_n}(x)}}\) for all n \(\ge\) 1. Then for any fixed n, \({d \over {dx}}{f_n}(x)\) is

Let P be a point on the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\) and the line through P parallel to the Y-axis meets the circle x2 + y2 = 9 at Q, where P, Q are on the same side of the X-axis. If R is a point on PQ such that $$...

Consider the function \(y = {\log _a}(x + \sqrt {{x^2} + 1} ),a > 0,a \ne 1\). The inverse of the function

If f : R \(\to\) R be defined by f (x) = ex and g : R \(\to\) R be defined by g(x) = x2. The mapping gof : R \(\to\) R be defined by (gof) (x) = g[f(x)] \(\forall\)x\(\in\)R. Then,

The domain of definition of \(f(x) = \sqrt {{{1 - |x|} \over {2 - |x|}}}\) is

For 0 \(\le\) p \(\le\) 1 and for any positive a, b; let I(p) = (a + b)p, J(p) = ap + bp, then

A hyperbola, having the transverse axis of length 2sin\(\theta\) is confocal wit6h the ellipse 3x2 + 4y2 = 12. Its equation is

Let the eccentricity of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) be reciprocal to that of the ellipse x2 + 9y2 = 9, then the ratio of a2 : b2 equals to

If \(\int {f(x)} \sin x\cos xdx = {1 \over {2({b^2} - {a^2})}}\log (f(x)) + c\), where c is the constant of integration, then f(x) is equal to

If \(\int {{e^{\sin x}}} .\left[ {{{x{{\cos }^3}x - \sin x} \over {{{\cos }^2}x}}} \right]dx = {e^{\sin x}}f(x) + c\), where c is constant of integration, then f(x) is equal to

If \(0 \le A \le {\pi \over 4}\), then \({\tan ^{ - 1}}\left( {{1 \over 2}\tan 2A} \right) + {\tan ^{ - 1}}(\cot A) + {\tan ^{ - 1}}({\cot ^3}A)\)

Let \(f(x) = 3{x^{10}} - 7{x^8} + 5{x^6} - 21{x^3} + 3{x^2} - 7\). Then \(\mathop {\lim }\limits_{h \to 0} {{f(1 - h) - f(1)} \over {{h^3} + 3h}}\)

Let f : [a, b] \(\to\) R be such that f is differentiable in (a, b), f is continuous at x = a and x = b and moreover f(a) = 0 = f(b). Then

Let f : [a, b] \(\to\) R be differentiable on [a, b] and k \(\in\) R. Let f(a) = 0 = f(b). Also let J(x) = f'(x) + kf(x). Then

Let f : R \(\to\) R be a twice continuously differentiable function such that f(0) = f(1) = f'(0) = 0. Then

Let \(f(x) = \left\{ {\matrix{ { - 2\sin x,} & {if\,x \le - {\pi \over 2}} \cr {A\sin x + B,} & {if\, - {\pi \over 2} < x < {\pi \over 2}} \cr {\cos x} & {if\,x \ge {\pi \over 2}} \cr } } \right.\). Then,

If \(x + {\log _{10}}(1 + {2^x}) = x{\log _{10}}5 + {\log _{10}}6\), then the value of x is

If \({S_r} = \left| {\matrix{ {2r} & x & {n(n + 1)} \cr {6{r^2} - 1} & y & {{n^2}(2n + 3)} \cr {4{r^3} - 2nr} & z & {{n^3}(n + 1)} \cr } } \right|\), then the value of \(\sum\limits_{r = 1}^n {{S_r}}\) is independent of

If the following three linear equations have a non-trivial solution, thenx + 4ay + az = 0x + 3by + bz = 0x + 2cy + cz = 0

In a third order matrix A, aij denotes the element in the ith row and jth column. If aij = 0 for i = j= 1 for i > j= \(-\) 1 for i < jThen the matrix is

The least positive integer n such that \({\left( {\matrix{ {\cos \pi /4} & {\sin \pi /4} \cr { - \sin {\pi \over 4}} & {\cos {\pi \over 4}} \cr } } \right)^n}\) is an identity matrix of order 2 is

If \(\left| {\matrix{ { - 1} & 7 & 0 \cr 2 & 1 & { - 3} \cr 3 & 4 & 1 \cr } } \right| = A\), then $$\left| {\matrix{
{13} & { - 11} & 5 \cr
{ - 7} & { - 1} & {25} \cr
{ - 21} & { - 3} & { - 15} \cr

} } \rig...

If the polynomial \(f(x) = \left| {\matrix{ {{{(1 + x)}^a}} & {{{(2 + x)}^b}} & 1 \cr 1 & {{{(1 + x)}^a}} & {{{(2 + x)}^b}} \cr {{{(2 + x)}^b}} & 1 & {{{(1 + x)}^a}} \cr } } \right|\), then the constant term of f(x) is

The area of the region lying above X-axis, and included between the circle x2 + y2 = 2ax and the parabola y2 = ax, a > 0 is

Number of common tangents of y = x2 and y = \(-\)x2 + 4x \(-\) 4 is

Let P(at2, 2at), Q, R(ar2, 2ar) be three points on a parabola y2 = 4ax. If PQ is the focal chord and PK, QR are parallel where the co-ordinates of K is (2a, 0), then the value of r is

Let A, B the two distinct points on the parabola y2 = 4x. If the axis of the parabola touches a circle of radius r having AB as diameter, the slope of the line AB is

Consider the parabola y2 = 4x. Let P and Q be points on the parabola where P(4, \(-\) 4) and Q(9, 6). Let R be a point on the arc of the parabola between P and Q. Then, the area of \(\Delta\)PQR is largest when

The number of selection of n objects from 2n objects of which n are identical and the rest are different, is

If (2 \(\le\) r \(\le\) n), then \({}^n{C_r}\) + 2 . \({}^n{C_{r + 1}}\) + \({}^n{C_{r + 2}}\) is equal to

From a collection of 20 consecutive natural numbers, four are selected such that they are not consecutive. The number of such selections is

On the occasion of Dipawali festival each student of a class sends greeting cards to others. If there are 20 students in the class, the number of cards sends by students is

In order to get a head at least once with probability \(\ge\) 0.9, the minimum number of times a unbiased coin needs to be tossed is

A student appears for tests I, II and III. The student is successful if he passes in tests I, II or I, III. The probabilities of the student passing in tests I, II and III are respectively p, q and 1/2. If the probability of the student to ...

If the equation \({x^2} - cx + d = 0\) has roots equal to the fourth powers of the roots of \({x^2} + ax + b = 0\), where \({a^2} > 4b\), then the roots of \({x^2} - 4bx + 2{b^2} - c = 0\) will be