WB JEE 2018

A ladder 20 ft long leans against a vertical wall. The top end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is

A particle is in motion along a curve 12y = x3. The rate of change of its ordinate exceeds that of abscissa in

The normal to the curve \(y = {x^2} - x + 1\), drawn at the points with the abscissa \({x_1} = 0\), \({x_2} = - 1\) and \({x_3} = {5 \over 2}\)

The law of motion of a body moving along a straight line is x = \({1 \over 2}\) vt. x being its distance from a fixed point on the line at time t and v is its velocity there. Then

Let \(f(x) = \cos \left( {{\pi \over x}} \right),x \ne 0\), then assuming k as an integer,

If n is even positive integer, then the condition that the greatest term in the expansion of (1 + x)n may also have the greatest coefficient, is

The number (101)100 \(-\) 1 is divisible by

If one of the diameter of the circle, given by the equation x2 + y2 + 4x + 6y \(-\) 12 = 0, is a chord of a circle S, whose centre is (2, \(-\) 3), the radius of S is

A chord AB is drawn from the point A(0, 3) on the circle x2 + 4x + (y \(-\) 3)2 = 0, and is extended to M such that AM = 2AB. The locus of M is

The angle between a pair of tangents drawn from a point P to the circle x2 + y2 + 4x \(-\) 6y + 9sin2\(\alpha\) + 13cos2\(\alpha\) = 0 is 2\(\alpha\). The equation of the locus of the point P is

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