Let a vector $\alpha \hat i + \beta \hat j $ be obtained by rotating the vector $\sqrt 3 \hat i + \hat j $ by an angle $45^\circ $ about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices $(\alpha , \beta )$, $(0, \beta )$ and (0, 0) is equal to:
(A) $2\sqrt 2 $
(B) $\frac {1}{2}$
(C) 1
(D) $\frac {1}{\sqrt 2 }$ Continue reading Let a vector $\alpha \hat i + \beta \hat j $ be obtained by ….
Consider an arithmetic series and a geometric series having four initial terms from the set {11,8,21,16,26,32,4}. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to _ _ _ _ . Continue reading Consider an arithmetic series and a geometric series ….
If y = y(x) is the solution to the differential equation, $\frac {dy}{dx} + 2y tan x = sin x $, $y (\frac {\pi}{3})=0$, then the maximum value of the function y(x) over R is equal to:
(A) $\frac {1}{2} $
(B) $\frac {1}{8} $
(C) $-\frac {15}{4}$
(D) 8 Continue reading $\frac {dy}{dx} + 2y tan x = sin x $
$y (\frac {\pi}{3})=0$
${y_{\max }} = ?$
If for a > 0, the feet of perpendiculars from the points A (a, -2a, 3) and B (0, 4, 5) on the plane lx + my + nz = 0 are points C (0, -a, -1) and D respectively, then the length of line segment CD is equal to:
(A) $\sqrt {41} $
(B) $\sqrt {31} $
(C) $\sqrt {66} $
(D) $\sqrt {55} $ Continue reading If for a > 0, the feet of perpendiculars from the points ….
Let P be a plane lx + my + nz = 0 containing the line, $\frac{{1 – x}}{1} = \frac{{y + 4}}{2} = \frac{{z + 2}}{3}$. If plane P divides the line segment AB joining points A (-3, -6, 1) and B (2, 4, -3) in ratio k : 1 then the value of k is equal to:
(A) 4
(B) 2
(C) 1.5
(D) 3 Continue reading Let P be a plane lx + my + nz = 0 containing the line ….
Let z and $\omega $ be two complex numbers such that $\omega = z\bar z – 2z + 2$, $\left| {\frac{{z + i}}{{z – 3i}}} \right| = 1$ and $Re(\omega )$ has minimum value. Then, the minimum value of $n \in N $ for which $\omega ^n $ is real, is equal to _ _ _ _ . Continue reading $\omega = z\bar z – 2z + 2$
$\left| {\frac{{z + i}}{{z – 3i}}} \right| = 1$
$Re(\omega ) $=min.
$n_{min.} \in N =?$
$\omega ^n \in R$
Let $f: (0, 2) \to R $ be defined as $f(x) = {\log _2}\left( {1 + \tan \left( {\frac{{\pi x}}{4}} \right)} \right)$. Then $\mathop {\lim }\limits_{n \to \infty } \frac{2}{n}\left( {f\left( {\frac{1}{n}} \right) + f\left( {\frac{2}{n}} \right) + ………… + f(1)} \right)$ is equal to _ _ _ _ . Continue reading $f(x) = {\log _2}\left( {1 + \tan \left( {\frac{{\pi x}}{4}} \right)} \right)$
$\mathop {\lim }\limits_{n \to \infty } \frac{2}{n}\left( {f\left( {\frac{1}{n}} \right) + f\left( {\frac{2}{n}} \right) + ………… + f(1)} \right)$=?
