WB JEE 2016

Time period T of a simple pendulum of length l is given by \(T = 2\pi \sqrt {{l \over g}}\). If the length is increased by 2%, then an approximate change in the time period is

If f(x) is a function such that f'(x) = (x \(-\) 1)2(4 \(-\) x), then

The locus of the mid-points of chords of the circle x2 + y2 = 1, which subtends a right angle at the origin, is

The equation of a line parallel to the line 3x + 4y = 0 and touching the circle x2 + y2 = 9 in the first quadrant, is

The value of \(\sum\limits_{n = 1}^{13} {({i^n} + {i^{n + 1}})}\), \(i = \sqrt { - 1}\) is

If z = sin\(\theta\) \(-\) icos\(\theta\), then for any integer n,

If \(\omega\) is an imaginary cube root of unity, then the value of (2 \(-\) \(\omega\)) (2 \(-\) \(\omega\)2) + 2(3 \(-\) \(\omega\))(3 \(-\) \(\omega\)2) + ... + (n \(-\) 1) (n \(-\) \(\omega\))(n \(-\) \(\omega\)2) is

If [x] denotes the greatest integer less than or equal to x, then the value of the integral \(\int\limits_0^2 {{x^2}[x]\,dx}\) equals

\(\int\limits_0^1 {\log \left( {{1 \over x} - 1} \right)} dx\) is equal to

If \(\phi (t) = \left\{ \matrix{ 1,\,for\,0 \le t < 1, \hfill \cr 0,\,otherwise \hfill \cr} \right.\), then \(\int\limits_{ - 300}^{3000} {\left( {\sum\limits_{r' = 2014}^{2016} {\phi (t - r')\phi (t - 2016)} } \right)} \,dt\) is

If the solution of the differential equation \(x{{dy} \over {dx}} + y = x{e^x}\,be\,xy = {e^x}\phi (x) + C\), then \(\phi\)(x) is equal to

General solution of \(y{{dy} \over {dx}} + b{y^2} = a\cos x,0 < x < 1\) is

On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y, are

The points of the ellipse 16x2 + 9y2 = 400 at which the ordinate decreases at the same rate at which the abscissa increases is/are given by

The line y = x + \(\lambda\) is tangent to the ellipse 2x2 + 3y2 = 1. Then, \(\lambda\) is

The equation of auxiliary circle of the ellipse \(16{x^2} + 25{y^2} + 32x - 100y = 284\) is

Let f : X \(\to\) X be such that f [f(x)] = x, for all x\(\in\)X and X\(\subseteq\)R, then

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

\(\int {{2^x}[f'(x) + f(x)\log 2]dx}\) is equal to

If \(f(x) = {\tan ^{ - 1}}\left[ {{{\log \left( {{e \over {{x^2}}}} \right)} \over {\log (e{x^2})}}} \right] + {\tan ^{ - 1}}\left[ {{{3 + 2\log x} \over {1 - 6\log x}}} \right]\), then the value of f''(x) is equal to

\(\mathop {\lim }\limits_{x \to 1} {\left( {{{1 + x} \over {2 + x}}} \right)^{{{(1 - \sqrt x )} \over {(1 - x)}}}}\) is equal to

The value of \(\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\sqrt {n + 1} + \sqrt {n + 2} + ... + \sqrt {2n - 1} } \over {{n^{3/2}}}}} \right\}\) is

If \(y = (1 + x)(1 + {x^2})(1 + {x^4})...(1 + {x^{2n}})\), then the value of \(\left( {{{dy} \over {dx}}} \right)\) at x = 0 is

If \({\log _{0.3}}(x - 1) < {\log _{0.09}}(x - 1)\), then x lies in the interval

For positive integer n, n3 + 2n is always divisible by

In the expansion of (x \(-\) 1) (x \(-\) 2) .... (x \(-\) 18), the coefficient of x17 is

Let A be a 3 \(\times\) 3 matrix and B be its adjoint matrix. If | B | = 64, then | A | is equal to

If x, y and z are greater than 1, then the value of \(\left| {\matrix{ 1 & {{{\log }_x}y} & {{{\log }_x}z} \cr {{{\log }_y}x} & 1 & {{{\log }_y}z} \cr {{{\log }_z}x} & {{{\log }_z}y} & 1 \cr } } \right|\) is

The locus of the mid-points of all chords of the parabola y2 = 4ax through its vertex is another parabola with directrix

A line passing through the point of intersection of x + y = 4 and x \(-\) y = 2 makes an angle \({\tan ^{ - 1}}\left( {{3 \over 4}} \right)\) with the X-axis. It intersects the parabola \({y^2} = 4(x - 3)\) at points \(({x_1},{y_1})\) and $...

If the vertex of the conic \({y^2} - 4y = 4x - 4a\) always lies between the straight lines \(x + y = 3\) and \(2x + 2y - 1 = 0\), then

The letters of the word COCHIN are permuted and all permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

If \({1 \over {{}^5{C_r}}} + {1 \over {{}^6{C_r}}} = {1 \over {{}^4{C_r}}}\), then the value of r is

If \({}^n{C_{r - 1}} = 36,{}^n{C_r} = 84\) and \({}^n{C_{r + 1}} = 126\), then the value of \({}^n{C_8}\) is

In a group of 14 males and 6 females. 8 and 3 of the males and females, respectively are aged above 40 yr. The probability that a person selected at random from the group is aged above 40 yr given that the selected person is a female, is

Let A and B be two events such that P(A \(\cap\) B) = \({1 \over 6}\), P(A \(\cup\) B) = \({31 \over 45}\) and P(\(\overline B\)) = \({7 \over 10}\), then

If A, B are two events such that P(A \(\cup\) B) \(\ge\) \({3 \over 4}\) and \({1 \over 8}\) \(\le\) P (A \(\cap\) B) \(\le\) \({3 \over 8}\), then

If in a \(\Delta\)ABC, AD, BE and CF are the altitudes and R is the circumradius, then the radius of the circumcircle of \(\Delta\)DEF is

If the equation x2 + y2 \(-\) 10x + 21 = 0 has real roots x = \(\alpha\) and y = \(\beta\), then

If p, q are the roots of the equation x2 + px + q = 0, then

If \(\alpha\) and \(\beta\) are roots of ax2 + bx + c = 0, then the equation whose roots are \(\alpha\)2 and \(\beta\)2, is

The number of values of k, for which the equation x2 \(-\) 3x + k = 0 has two distinct roots lying in the interval (0, 1), are

If a, x are real numbers and | a | < 1, | x | < 1, then 1 + (1 + a) x + (1 + a + a2) x2 + ..... \(\infty\) is equal to

The sum of n terms of the following series \({1^3} + {3^3} + {5^3} + {7^3} + ...\) is

Let R be a relation defined on the set Z of all integers and xRy, when x + 2y is divisible by 3, then

If A = {5n \(-\) 4n \(-\) 1 : n\(\in\)N} and B = {16(n \(-\) 1) : n\(\in\)N}, then

Standard deviation of n observations a1, a2, a3, ....., an is \(\sigma\). Then, the standard deviation of the observations \(\lambda\)a1, \(\lambda\)a2, ....., \(\lambda\)an is

x + 8y \(-\) 22 = 0, 5x + 2y \(-\) 34 = 0, 2x \(-\) 3y + 13 = 0 are the three sides of a triangle. The area of the triangle is

The coordinates of a point on the line x + y + 1 = 0, which is at a distance \({1 \over 5}\) unit from the line 3x + 4y + 2 = 0, are

The line AB cuts off equal intercepts 2a from the axes. From any point P on the line AB perpendiculars PR and PS are drawn on the axes. Locus of mid-point of RS is