If cos($$\alpha$$ + $$\beta$$) = 3/5 ,sin ( $$\alpha$$ - $$\beta$$) = 5/13 and 0 < $$\alpha ,\beta$$ < \pi \over 4, then tan(2$$\alpha$$) is equal to :

The maximum value of 3cos$$\theta$$ + 5sin \left( {\theta - {\pi \over 6}} \right) for any real value of $$\theta$$ is :

The value of \cos {\pi \over {{2^2}}}.\cos {\pi \over {{2^3}}}\,.....\cos {\pi \over {{2^{10}}}}.\sin {\pi \over {{2^{10}}}} is -

For any \theta \in \left( {{\pi \over 4},{\pi \over 2}} \right), the expression $$3(\cos \theta -\sin \theta {)}^4$$$$+6(\sin \theta +\cos \theta {)}^2+4{\sin }^6\theta$$ equals :

Let A = \left( {\matrix{ {\cos \alpha } & { - \sin \alpha } \cr {\sin \alpha } & {\cos \alpha } \cr } } \right), ($$\alpha$$ $$\in$$ R) such that {A^{32}} = \left( {\matrix{ 0 & { - 1} \cr 1 & 0 \cr } } \right) then a value of $$\alpha$$ is

The greatest value of c $$\in$$ R for which the system of linear equations x – cy – cz = 0 cx – y + cz = 0 cx + cy – z = 0 has a non-trivial solution, is :

Let the number 2,b,c be in an A.P. and A = \left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4 & {{b^2}} & {{c^2}} \cr } } \right]. If det(A) $$\in$$ [2, 16], then c lies in the interval :

Let $$\alpha$$ and $$\beta$$ be the roots of the equation x2 + x + 1 = 0. Then for y $$\ne$$ 0 in R, $\left| {\matrix{ {y + 1} & \alpha & \beta \cr \alpha & {y + \beta } & 1 \cr \beta & 1 & {y + \alpha } \cr } } \right|$ is equal to

If \left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]\left[ {\matrix{ 1 & 2 \cr 0 & 1 \cr } } \right]\left[ {\matrix{ 1 & 3 \cr 0 & 1 \cr } } \right]....\left[ {\matrix{ 1 & {n - 1} \cr 0 & 1 \cr } } \right] = \left[ {\matrix{ 1 & {78} \cr 0 & 1 \cr } } \right], then the inverse of \left[ {\matrix{ 1 & n \cr 0 & 1 \cr } } \right] is

If the system of equations 2x + 3y – z = 0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trival solution (x, y, z), then {x \over y} + {y \over z} + {z \over x} + k is equal to :-

The total number of matrices A = \left( {\matrix{ 0 & {2y} & 1 \cr {2x} & y & { - 1} \cr {2x} & { - y} & 1 \cr } } \right) (x, y $$\in$$ R,x $$\ne$$ y) for which ATA = 3I3 is :-

If {\Delta _1} = \left| {\matrix{ x & {\sin \theta } & {\cos \theta } \cr { - \sin \theta } & { - x} & 1 \cr {\cos \theta } & 1 & x \cr } } \right| and {\Delta _2} = \left| {\matrix{ x & {\sin 2\theta } & {\cos 2\theta } \cr { - \sin 2\theta } & { - x} & 1 \cr {\cos 2\theta } & 1 & x \cr } } \right|, $$x\ne 0$$ ; then for all \theta \in \left( {0,{\pi \over 2}} \right) :

If the system of linear equations x + y + z = 5 x + 2y + 2z = 6 x + 3y + $$\lambda$$z = $$\mu$$, ($$\lambda$$, $$\mu$$ $$\in$$ R), has infinitely many solutions, then the value of $$\lambda$$ + $$\mu$$ is :

Let $$\lambda$$ be a real number for which the system of linear equations x + y + z = 6, 4x + $$\lambda$$y – $$\lambda$$z = $$\lambda$$ – 2, 3x + 2y – 4z = – 5 has infinitely many solutions. Then $$\lambda$$ is a root of the quadratic equation:

The sum of the real roots of the equation \left| {\matrix{ x & { - 6} & { - 1} \cr 2 & { - 3x} & {x - 3} \cr { - 3} & {2x} & {x + 2} \cr } } \right| = 0, is equal to :

If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = \left[ {\matrix{ 2 & 3 \cr 5 & { - 1} \cr } } \right], then AB is equal to :

If B = \left[ {\matrix{ 5 & {2\alpha } & 1 \cr 0 & 2 & 1 \cr \alpha & 3 & { - 1} \cr } } \right] is the inverse of a 3 × 3 matrix A, then the sum of all values of $$\alpha$$ for which det(A) + 1 = 0, is :

A value of \theta \in \left( {0,{\pi \over 3}} \right), for which \left| {\matrix{ {1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\cos 6\theta } \cr {{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\cos 6\theta } \cr {{{\cos }^2}\theta } & {{{\sin }^2}\theta } & {1 + 4\cos 6\theta } \cr } } \right| = 0, is :

If \left| {\matrix{ {a - b - c} & {2a} & {2a} \cr {2b} & {b - c - a} & {2b} \cr {2c} & {2c} & {c - a - b} \cr } } \right| = (a + b + c) (x + a + b + c)2, x $$\ne$$ 0, then x is equal to :

If A = \left[ {\matrix{ 1 & {\sin \theta } & 1 \cr { - \sin \theta } & 1 & {\sin \theta } \cr { - 1} & { - \sin \theta } & 1 \cr } } \right]; then for all $$\theta$$ $$\in$$ \left( {{{3\pi } \over 4},{{5\pi } \over 4}} \right), det (A) lies in the interval :