WB JEE 2017

The chord of the curve \(y = {x^2} + 2ax + b\), joining the points where x = \(\alpha\) and x = \(\beta\), is parallel to the tangent to the curve at abscissa x is equal to

The value of K in order that f(x) = sin x \(-\) cos x \(-\) kx + 5 decreases for all positive real values of x is given by

Let, \(F(x) = {e^x},G(x) = {e^{ - x}}\) and \(H(x) = G(F(x))\), where x is a real variable. Then, \({{dH} \over {dx}}\) at x = 0 is

Two particles move in the same straight line starting at the same moment from the same point in the same direction. The first moves with constant velocity u and the second starts from rest with constant acceleration f. Then,

If the line ax + by + c = 0, ab \(\ne\) 0, is a tangent to the curve xy = 1 \(-\) 2x, then

The area of the figure bounded by the parabolas x = \(-\) 2y2 and x = 1 \(-\) 3y2 is

Let \({(1 + x + {x^2})^9} = {a_0} + {a_1}x + {a_2}{x^2} + ... + {a_{18}}{x^{18}}\). Then,

The locus of the mid-points of the chords of the circle \({x^2} + {y^2} + 2x - 2y - 2 = 0\), which make an angle of 90\(^\circ\) at the centre is

If one of the diameters of the curve x2 + y2 \(-\) 4x \(-\) 6y + 9 = 0 is a chord of a circle with centre (1, 1), the radius of this circle is

The common chord of the circles \({x^2} + {y^2} - 4x - 4y = 0\) and \(2{x^2} + 2{y^2} = 32\) subtends at the origin an angle equal to

The complex number z satisfying the equation | z \(-\) 1 | = | z + 1 | = 1 is

Let z = x + iy, where x and y are real. The points (x, y) in the X-Y plane for which \({{{z + i} \over {z - i}}}\) is purely imaginary, lie on

The expression \({{{{(1 + i)}^n}} \over {{{(1 - i)}^{n - 2}}}}\) equals

If \(f(x) = \int_{ - 1}^x {|t|} \,dt\), then for any \(x \ge 0,\,f(x)\) is equal to

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

The value of the integral \(\int_0^1 {{e^{{x^2}}}} dx\)

Let f be a non-constant continuous function for all x \(\ge\) 0. Let f satisfy the relation f(x) f(a \(-\) x) = 1 for some a \(\in\) R+. Then, \(I = \int_0^a {{{dx} \over {1 + f(x)}}}\) is equal to

Let \({I_1} = \int_0^n {[x]} \,dx\) and \({I_2} = \int_0^n {\{ x\} } \,dx\), where [x] and {x} are integral and fractional parts of x and n \(\in\) N \(-\) {1}. Then I1 / I2 is equal to

Let \(I = \int_0^{100\pi } {\sqrt {(1 - \cos 2x)} } \,dx\), then

\(\int_0^{100} {{e^{x - [x]}}} dx\) is equal to

The integrating factor of the first order differential equation \({x^2}({x^2} - 1){{dy} \over {dx}} + x({x^2} + 1)y = {x^2} - 1\) is

If \(y = {e^{m{{\sin }^{ - 1}}x}}\) then \((1 - {x^2}){{{d^2}y} \over {d{x^2}}} - x{{dy} \over {dx}} -\)ky = 0, where k is equal to

Solution of \({(x + y)^2}{{dy} \over {dx}} = {a^2}\) ('a' belong a constant) is

If \(f(x) = {\log _5}{\log _3}x\), then f'(e) is equal to

If f(x) = xn, being a non-negative integer, then the values of n for which f'(\(\alpha\) + \(\beta\)) = f'(\(\alpha\)) + f'(\(\beta\)) for all \(\alpha\), \(\beta\) > 0 is

Tangents are drawn to the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 5} = 1\) at the ends of both latusrectum. The area of the quadrilateral, so formed is

B is an extremity of the minor axis of an ellipse whose foci are S and S'. If \(\angle SBS'\) is a right angle, then the eccentricity of the ellipse is

Let \(f:R \to R\) be such that f is injective and \(f(x)f(y) = f(x + y)\) for \(\forall x,y \in R\). If f(x), f(y), f(z) are in G.P., then x, y, z are in

Let \(f(x) = {x^{13}} + {x^{11}} + {x^9} + {x^7} + {x^5} + {x^3} + x + 19\). Then, f(x) = 0 has

The line segment joining the foci of the hyperbola \({x^2} - {y^2} + 1 = 0\) is one of the diameters of a circle. The equation of the circle is

Let P be the foot of the perpendicular from focus S of hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) on the line bx \(-\) ay = 0 and let C be the centre of the hyperbola. Then, the area of the rectangle whose sides are...

Let A(\(-\) 1, 0) and B(2, 0) be two points. A point M moves in the plane in such a way that \(\angle MBA\) = 2\(\angle MAB\). Then, the point M moves along

\(\int {\cos (\log x)dx}\) = F(x) + C, where C is an arbitrary constant. Here, F(x) is equal to

Let I = \(\left| {\int {_{10}^{19}{{\sin x} \over {1 + {x^8}}}dx} } \right|\). Then,

\(\int {{{{x^2} - 1} \over {{x^4} + 3{x^2} + 1}}dx}\) (x > 0) is

The possible values of x, which satisfy the trigonometric equation \({\tan ^{ - 1}}\left( {{{x - 1} \over {x - 2}}} \right) + {\tan ^{ - 1}}\left( {{{x + 1} \over {x + 2}}} \right) = {\pi \over 4}\) are

Let \(f(x) = \left\{ {\matrix{ {{{{x^p}} \over {{{(\sin x)}^q}}},} & {if\,0 < x \le {\pi \over 2}} \cr {0,} & {if\,x = 0} \cr } } \right.\), \((p,q \in R)\). Then, Lagrange's mean value theorem is applicable to f(x) in closed i...

\(\mathop {\lim }\limits_{x \to 0} {(\sin x)^{2\tan x}}\) is equal to

Let f : R \(\to\) R be twice continuously differentiable. Let f(0) = f(1) = f'(0) = 0. Then,

If f'' (0) = k, k \(\ne\) 0, then the value of \(\mathop {\lim }\limits_{x \to 0} {{2f(x) - 3f(2x) + f(4x)} \over {{x^2}}}\) is

Consider the non-constant differentiable function f one one variable which obeys the relation \({{f(x)} \over {f(y)}} = f(x - y)\). If f' (0) = p and f' (5) = q, then f' (\(-\)5) is

Let for all x > 0, \(f(x) = \mathop {\lim }\limits_{n \to \infty } n({x^{1/n}} - 1)\), then

If \(({\log _5}x)({\log _x}3x)({\log _{3x}}y) = {\log _x}{x^3}\), then y equals

Let a, b, c be such that b(a + c) \(\ne\) 0. If $$\left| {\matrix{
a & {a + 1} & {a - 1} \cr
{ - b} & {b + 1} & {b - 1} \cr
c & {c - 1} & {c + 1} \cr

} } \right| + \left| {\matrix{
{a + 1} & {b + 1} & {c - 1} \cr
...

Let \(A = \left( {\matrix{ {x + 2} & {3x} \cr 3 & {x + 2} \cr } } \right),\,B = \left( {\matrix{ x & 0 \cr 5 & {x + 2} \cr } } \right)\). Then all solutions of the equation det (AB) = 0 is

Let P be the set of all non-singular matrices of order 3 over R and Q be the set of all orthogonal matrices of order 3 over R. Then,

Let \(A = \left( {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right)\). Then, for positive integer n, An is

The value of det A, where \(A\, = \left( {\matrix{ 1 & {\cos \theta } & 0 \cr { - \cos \theta } & 1 & {\cos \theta } \cr { - 1} & { - \cos \theta } & 1 \cr } } \right)\), lies

The linear system of equations\(\left. \matrix{ 8x - 3y - 5z = 0 \hfill \cr 5x - 8y + 3z = 0 \hfill \cr 3x + 5y - 8z = 0 \hfill \cr} \right\}\) has

If the tangent to \({y^2} = 4ax\) at the point \((a{t^2},2at)\) where | t | > 1 is a normal to \({x^2} - {y^2} = {a^2}\) at the point \((a\sec \theta ,a\tan \theta )\), then