$$\begin{gathered} \mathrm{What} \mathrm{is} \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{straight} \mathrm{line} \mathrm{passing} \mathrm{through} \mathrm{the} \mathrm{point} \mathrm{of} \mathrm{intersection} \mathrm{of} \mathrm{the} \mathrm{lines} \frac{\mathrm{x}}{2} +\hspace{0.17em}\frac{\mathrm{y}}{3} = 1 \mathrm{and} \frac{\mathrm{x}}{3} + \frac{\mathrm{y}}{2} = 1, \\ \mathrm{and} \mathrm{parallel} \mathrm{to} \mathrm{the} \mathrm{line} 4\mathrm{x} + 5\mathrm{y} - 6 = 0 ? \end{gathered}$$

• 20x + 25 - 54 = 0

• 25x + 20y - 54 = 0

• 4x + 5y - 54 = 0

• 4x + 5y - 45 = 0

Coordinates of the points O, P, Q and R are respectively (0, 0, 0), (4, 6, 2m), (2, 0, 2n) and (2, 4, 6). Let L, M, N and K be points on the sides OR, OP, PQ and QR respectively such that LMNK is a parallelogram whose two adjacent sides LK and LM are each of length $\sqrt{2}$.What are the values of m anon respectively ?

• 6, 2

• 1, 3

• 3, 1

• None of the above

Consider the following statements :

Statement I : If the line segment joining the points P(m, n) and Q(r, s) subtends an angle a at the origin then

$$\begin{gathered} \cos \left(\mathrm{\alpha }\right) = \frac{\mathrm{ms} - \mathrm{nr}}{\sqrt{\left({\mathrm{m}}^2 + {\mathrm{n}}^2\right)\left({\mathrm{r}}^2 +\hspace{0.17em}{\mathrm{s}}^2\right)}} \\ \mathrm{Statements} \mathrm{II} : \mathrm{In} \mathrm{any} \mathrm{triangle} \mathrm{ABC}, \mathrm{it} \mathrm{is} \mathrm{true} \mathrm{that} {\mathrm{a}}^2 = {\mathrm{b}}^2 + {\mathrm{c}}^2 - 2\mathrm{bccos}\left(\mathrm{A}\right) \end{gathered}$$

Which one of the following is correct in respect of the above two statements?

• Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I

• Both Statement I and Statement II are true, but Statement II is not the correct explanation of Statement I

• Statement I is true, but Statement II is false

• Statement I is false, but Statement II is true

24.

$$\begin{gathered} \mathrm{Let} \mathrm{the} \mathrm{slope} \mathrm{of} \mathrm{the} \mathrm{cureve} \mathrm{y} = {\cos }^{-1}\left(\sin \left(\mathrm{x}\right)\right) \mathrm{be} \tan \left(\mathrm{\theta }\right). \\ \mathrm{Then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{\theta } \mathrm{in} \mathrm{the} \mathrm{interval} \left(0, \mathrm{\pi }\right) \mathrm{is} \end{gathered}$$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{3\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

From the top of a lighthouse, 100 m high, the angle of depression of a boat is ${\tan }^{-1}\left(\frac{5}{12}\right)$. What is the distance between the boat and the lighthouse ?

• 120m

• 180m

• 240m

• 360m

If a vertex of a triangle is (1, 1) and the midpoints of two sides of the triangle through this vertex are ( - 1, 2) and (3, 2), then the centroid of the triangle is

• $\left( - \frac{1}{3}, \frac{7}{3}\right)$

• $\left( - 1, \frac{7}{3}\right)$

• $\left(\frac{1}{3}, \frac{7}{3}\right)$

• $\left(1, \frac{7}{3}\right)$

What is the ratio in which the point $\mathrm{C}\left( - \frac{2}{7}, - \frac{20}{7}\right)$ divides the line joining the  points A( - 2, - 2) and B(2, - 4) ?

• 1 : 3

• 3 : 4

• 1 : 2

• 2 : 3

What is the equation of the straight line parallel to 2x + 3y + 1 = 0 and passes through the point( - 1, 2) ?

• 2x + 3y - 4 = 0

• 2x + 3y - 5 = 0

• x + y - 1 = 0 

• 3x - 2y + 7 = 0 

What is the acute angle between the pair of straight lines $\sqrt{2}\mathrm{x} + \sqrt{3}\mathrm{y} = 1 \mathrm{and} \sqrt{3}\mathrm{x} + \sqrt{2}\mathrm{y} = 2 ?$

• ${\tan }^{-1}\left(\frac{1}{2\sqrt{6}}\right)$

• ${\tan }^{-1}\left(\frac{1}{\sqrt{2}}\right)$

• ${\tan }^{-1}\left(3\right)$

• ${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

If the centroid of a triangle formed by (7, x), (y, - 6) and (9, 10) is (6, 3), then the values of x and y are respectively

• 5, 2

• 2, 5

• 1, 0

• 0, 0