The number of solutions of the system of equations 2x + y - z = 7, x - 3y + 2z = 1 and x + 4y - 3z = 5 is

• 0

• 1

• 2

• 3

If $\left|\begin{array}{ccc}\mathrm{x}& 2& 8\\ 2& 8& \mathrm{x}\\ 8& \mathrm{x}& 2\end{array}\right| = \left|\begin{array}{ccc}3& \mathrm{x}& 7\\ \mathrm{x}& 7& 3\\ 7& 3& \mathrm{x}\end{array}\right| = \left|\begin{array}{ccc}5& 5& \mathrm{x}\\ 5& \mathrm{x}& 5\\ \mathrm{x}& 5& 5\end{array}\right|$ = 0, then x is equal to

• 0

• - 10

• 3

• None of these

Let R = {(1, 1), (1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is

• a function

• transitive

• not symmetric

• reflexive

If $\mathrm{\alpha } = {\sin }^{-1}\left(\cos \left({\sin }^{-1}\left(\mathrm{x}\right)\right)\right)$ and $\mathrm{\beta } = {\cos }^{-1}\left(\sin \left({\cos }^{-1}\left(\mathrm{x}\right)\right)\right)$, then $\tan \left(\mathrm{\alpha }\right) . \tan \left(\mathrm{\beta }\right)$ is equal to

• 1

• - 1

• 2

• $\frac{1}{2}$

At which point the function f(x) = $\frac{{\mathrm{x}}^2}{\left[\mathrm{x}\right]}$, where [.] is greatest integer function, is discontinuous ?

• Only positve integers

• All postive and negative integers and (0,1)

• all rational numbers

• None of these

The function y = $\left|2\sin \left(\mathrm{x}\right)\right|$ is continuous for any x but it is not differentiable at

• only x = 0

• only x = $\mathrm{\pi }$

• only x = $\frac{\mathrm{\pi }}{2}$

• x = $\mathrm{k\pi } \left(\mathrm{k} \mathrm{is} \mathrm{integer}\right)$

If $\mathrm{y} = {\mathrm{e}}^{\log \left(1 + \mathrm{x} +\hspace{0.17em}{\mathrm{x}}^2 + {\mathrm{x}}^3 + \right)}, \mathrm{where} \left|\mathrm{x}\right| < 1$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{- 1}{{\left(1 - \mathrm{x}\right)}^2}$

• $\frac{1}{{\left(1 - \mathrm{x}\right)}^2}$

• $\frac{1}{{\left(1 + \mathrm{x}\right)}^2}$

• None of these

8.

If two sides of a triangle are given, then the area of the triangle will be maximum, if. the angle between the given sides is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

If f(x) = $\frac{80}{3{\mathrm{x}}^4 + 8{\mathrm{x}}^3 - 18{\mathrm{x}}^2 + 60}$, then the points of local maxima for the function f(x) are

• 1, 3

• - 3, 1

• - 1, 3

• - 1, - 3

The adjacent sides of a rectangle with given parameter as 200 cm and enclosing minimum area are

• 20 cm and 80 cm

•  50 cm and 50 cm40 cm and 60 cm

• 50 cm and 50 cm

• 30 cm and 70 cm