For what value of $\mathrm{\lambda }$ the system of equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + $\mathrm{\lambda }$z = 10 is consistent ?

• 1

• 2

• - 1

• 3

Let f(x) be twice differentiable such that f''(x) = - f(x), f'(x) = g(x), where f'(x) and f''(x) represent the first and second derivatives of f(x) respectively. Also, if h(x) = [f(x)]² + [g(x)]² and h(S) = 5, then h(10) is equal to :

• 3

• 10

• 13

• 5

If a = 1 + 2 + 4 + ... to n terms, b = 1 + 3 + 9 + ... to n terms and c = 1 + 5 + 25 + ... to n terms, then

$\left|\begin{array}{ccc}\mathrm{a}& 2\mathrm{b}& 4\mathrm{c}\\ 2& 2& 2\\ 2^{\mathrm{n}}& 3^{\mathrm{n}}& 5^{\mathrm{n}}\end{array}\right|$ equals :

• (30)ⁿ

• (10)ⁿ

• 0

• 2ⁿ + 3ⁿ + 5ⁿ

The matrix $\left[\begin{array}{ccc}5& 10& 3\\ - 2& - 4& 6\\ - 1& - 2& \mathrm{b}\end{array}\right]$ is a singular matrix, if b is equal to :

• - 3

• 3

• 0

• for any value of b

For non-singular square matrices A, B and C of the same order, (AB^-1C)^-1 is equal to :

• A^-1BC^-1

• C^-1B^-1A^-1

• CBA^-1

• C^-1BA^-1

6.

Let A = $\left[\begin{array}{cc}{\cos }^2\left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right)\\ \cos \left(\mathrm{\theta }\right)\sin \left(\mathrm{\theta }\right)& {\sin }^2\left(\mathrm{\theta }\right)\end{array}\right]$ and B = $\left[\begin{array}{cc}{\cos }^2\left(\mathrm{\varphi }\right)& \sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\varphi }\right)\\ \cos \left(\mathrm{\varphi }\right)\sin \left(\mathrm{\varphi }\right)& {\sin }^2\left(\mathrm{\varphi }\right)\end{array}\right]$, then AB = 0 if :

• $\mathrm{\theta } = \mathrm{n\varphi }, \mathrm{n} = 0, 1, 2, ...$

• $\mathrm{\theta } + \mathrm{\varphi } = \mathrm{n\pi }, \mathrm{n} = 0, 1, 2, ...$

• $\mathrm{\theta } = \mathrm{\varphi } + \left(2\mathrm{n} + 1\right)\frac{\mathrm{\pi }}{2}, \mathrm{n} = 0, 1, 2, ...$

• $\mathrm{\theta } = \mathrm{\varphi } + \mathrm{n}\frac{\mathrm{\pi }}{2}, \mathrm{n} = 0, 1, 2, ...$

Let X = $\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\\ {\mathrm{x}}_3\end{array}\right], \mathrm{A} = \left[\begin{array}{ccc}1& - 1& 2\\ 2& 0& 1\\ 3& 2& 1\end{array}\right] \mathrm{and} \mathrm{B} = \left[\begin{array}{c}3\\ 1\\ 4\end{array}\right]$. If AX = B, then X is equal to

• $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$

• $\left[\begin{array}{c}- 1\\ - 2\\ 3\end{array}\right]$

• $\left[\begin{array}{c}- 1\\ - 2\\ - 3\end{array}\right]$

• $\left[\begin{array}{c}- 1\\ 2\\ 3\end{array}\right]$

If a particle is moving such that the velocity acquired is proportional to the square root of the distance covered, then its acceleration is :

• a constant

• $\propto {\mathrm{s}}^2$

• $\propto \frac{1}{{\mathrm{s}}^2}$

• $\propto \mathrm{s}$

If $\mathrm{f}\left(\mathrm{x}\right) = \frac{1 - \mathrm{x}}{1 + \mathrm{x}}\left(\mathrm{x} \ne - 1\right)$, then f^-1(x) equals to :

• f(x)

• $\frac{1}{\mathrm{f}\left(\mathrm{x}\right)}$

• - f(x)

• $- \frac{1}{\mathrm{f}\left(\mathrm{x}\right)}$

Domain of the function f(x) = sin^-1(log₂(x))in the set of real numbers is :

• $\left\{\mathrm{x} :\hspace{0.17em}1 \le \mathrm{x} \le 2\right\}$

• $\left\{\mathrm{x} :\hspace{0.17em}1 \le \mathrm{x} \le 3\right\}$

• $\left\{\mathrm{x} :\hspace{0.17em}- 1 \le \mathrm{x} \le 2\right\}$

• $\left\{\mathrm{x} :\hspace{0.17em}\frac{1}{2} \le \mathrm{x} \le 2\right\}$