The equation of the circle whose diameter is the common chord of the circles x² + y² + 2x + 2y + 1 = 0 and x² + y² + 4x + 6 y + 4 = 0 is

• $10{\mathrm{x}}^2 + 10{\mathrm{y}}^2 + 14\mathrm{x} + 8\mathrm{y} +\hspace{0.17em}1 = 0$

• $3{\mathrm{x}}^2 + 3{\mathrm{y}}^2 - 3\mathrm{x} + 6\mathrm{y} - 8 = 0$

• $2{\mathrm{x}}^2 +\hspace{0.17em}2{\mathrm{y}}^2 - 2\mathrm{x} + 4\mathrm{y} +\hspace{0.17em}1 = 0$

• $2{\mathrm{x}}^2 + 2{\mathrm{y}}^2 - 2\mathrm{x} + 4\mathrm{y} + 1 = 0$

If a cylindrical vessel of given volume V with no lid on the top is to be made from a sheet of metal, then the radius (r) and height(h) of the vessel so that the metal sheet used is minimum is

• $\mathrm{r} = \sqrt[3]{\frac{\mathrm{\pi }}{\mathrm{V}}}, \mathrm{h} = \sqrt[3]{\frac{\mathrm{\pi }}{\mathrm{V}}}$

• $\mathrm{r} = \sqrt{\mathrm{\pi V}}, \mathrm{h} = \sqrt{\mathrm{\pi V}}$

• $\mathrm{r} = \sqrt[3]{\frac{\mathrm{V}}{\mathrm{\pi }}}, \mathrm{h} = \sqrt[3]{\frac{\mathrm{V}}{\mathrm{\pi }}}$

• $\mathrm{r} = \sqrt{\frac{\mathrm{V}}{\mathrm{\pi }}}, \mathrm{h} = \sqrt{\frac{\mathrm{V}}{\mathrm{\pi }}}$

$\mathrm{In} ∆\mathrm{ABC} \mathrm{if} \mathrm{\theta } \mathrm{is} \mathrm{any} \mathrm{angle}, \mathrm{then} \mathrm{bcos}\left(\mathrm{C} + \mathrm{\theta }\right) + \mathrm{ccos}\left(\mathrm{B} - \mathrm{\theta }\right) = ?$

• $\mathrm{acot}\left(\mathrm{\theta }\right)$

• $\mathrm{acos}\left(\mathrm{\theta }\right)$

• $\mathrm{atan}\left(\mathrm{\theta }\right)$

• $\mathrm{asin}\left(\mathrm{\theta }\right)$

$$\begin{gathered} \mathrm{The} \mathrm{number} \mathrm{of} \mathrm{solutions} \mathrm{of} \mathrm{the} \mathrm{trigonometric} \mathrm{equation} \\ 1 + \cos \left(\mathrm{x}\right) . \cos \left(5\mathrm{x}\right) = {\sin }^2\left(\mathrm{x}\right) \mathrm{in} [0, 2\mathrm{\pi }] \mathrm{is} \end{gathered}$$

• 8

• 12

• 10

• 6

$\mathrm{If} \cot \left({\cos }^{-1}\left(\mathrm{x}\right)\right) = \sec \left\{{\tan }^{-1}\left(\frac{\mathrm{a}}{\sqrt{{\mathrm{b}}^2 - {\mathrm{a}}^2}}\right)\right\} : \mathrm{b} > \mathrm{a}, \mathrm{then} \mathrm{x} =$

• $\frac{\mathrm{b}}{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}$

• $\frac{\sqrt{{\mathrm{b}}^2 - {\mathrm{a}}^2}}{\mathrm{ab}}$

• $\frac{\mathrm{a}}{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}$

• $\frac{\sqrt{{\mathrm{b}}^2 - {\mathrm{a}}^2}}{\mathrm{a}}$

If $∆ = \left|\begin{array}{ccc}1& \cos \left(\mathrm{\theta }\right)& 1\\ - \cos \left(\mathrm{\theta }\right)& 1& \cos \left(\mathrm{\theta }\right)\\ - 1& - \cos \left(\mathrm{\theta }\right)& 1\end{array}\right|$, then $∆$ lies in the interval

• [2, 4]

• (2, 4)

• [1, 4]

• [- 1, 1]

If a, b, c are non-zero real numbers and if the equations (a - 1) x = y + z, (b - 1)y = z + x, (c - 1)z = x + y have a non-trivial solution, then ab + be + ca = ?

• a²b²c²

• 0

• abc

• a + b + c

If a system of three linear equations in three unknowns, which is in the matrix equation form of AX = D, is in consistent, then rank of A/rank of AD is

• Less than one

• Greater than or equal to one

• One

• Greater than one

The area (in sq. units) enclosed between the curves y = x² and y = $\left|\mathrm{x}\right|$ is

• $\frac{2}{3}$

• $\frac{1}{6}$

• $\frac{1}{3}$

• 1

60.

 $∆$ABC is formed by A(1, 8, 4), B (0, - 11, 4) and C(2, - 3,1). If D is the foot of the perpendicular from A to BC. Then the coordinates of D are

• ( - 4, 5, 2)

• (4, 5, - 2)

• (4,  - 5, 2)

• (4,  - 5, - 2)