The values of x, y and z for the system of equations x + 2y + 3z = 6, 3x - 2y + z = 2 and 4x + 2y + z = 7 are respectively

• 1, 1, 1

• 1, 2, 3

• 1, 3, 2

• 2, 3, 1

If the determinant $∆ = \left|\begin{array}{ccc}3& - 2& \sin \left(3\mathrm{\theta }\right)\\ - 7& 8& \cos \left(2\mathrm{\theta }\right)\\ - 11& 14& 2\end{array}\right|$ = 0, then the value of $\sin \left(\mathrm{\theta }\right)$ is

• $\frac{1}{3}\mathrm{or} 1$

• $\frac{1}{\sqrt{2}} \mathrm{or} \frac{\sqrt{3}}{2}$

• $0 \mathrm{or} \frac{1}{2}$

• None of these

The relation R in R defined by R = {(a, b): a $\le$ b³), is

• reflexive

• symmetric

• transitive

• None of these

The value of $2{\tan }^{-1}\left(\csc \left({\tan }^{-1}\left(\mathrm{x}\right)\right) - \tan \left({\cot }^{-1}\left(\mathrm{x}\right)\right)\right)$ is

• ${\tan }^{-1}\left(\mathrm{x}\right)$

• tan(x)

• cot(x)

• ${\csc }^{-1}\left(\mathrm{x}\right)$

Let f (x + y) = f(x) + f(y) for all x and y. If the function f(x) is continuous at x = 0, then f(x) is continuous

• only at x = 0

• $\mathrm{at} \mathrm{x} \in \mathrm{R} - \left\{0\right\}$

• for all x

• None of these

6.

Let $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{x}}^2\sin \left(\frac{1}{\mathrm{x}}\right), \mathrm{x} \ne 0 \\ 0, \mathrm{x} = 0\right\} \end{gathered}$$. Then, f(x) is continuous but not differentiable at x = 0, if

• $\mathrm{n} \in \left(0, 1\right)$

• $\mathrm{n} \in [1, \infty )$

• $\mathrm{n} \in \left(- \infty , 0\right)$

• n = 0

The altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is

• $\frac{\mathrm{r}}{2}$

• $\frac{\mathrm{r}}{3}$

• $\frac{3\mathrm{r}}{4}$

• $\frac{4\mathrm{r}}{3}$

If in a $∆\mathrm{ABC}$, ${\sin }^3\left(\mathrm{A}\right) + {\sin }^3\left(\mathrm{B}\right) + {\sin }^3\left(\mathrm{C}\right) = 3\sin \left(\mathrm{A}\right)\sin \left(\mathrm{B}\right)\sin \left(\mathrm{C}\right)$, then the value of determinant $\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right|$ is equal to

• 0

• 1

• 2

• 3

Let f(x) = x(x - 1)², the point at which f(x) assumes maximum and minimum are respectively

• $\frac{1}{3}, 1$

• $1, \frac{1}{3}$

• 3, 1

• None of these

Rectangles are inscribed ina circle of radius r. The dimensions of the rectangle which has the maximum area, are

• r, r

• 2r, 2r

• $\sqrt{2}\mathrm{r}, \sqrt{2}\mathrm{r}$

• None of the above