$$\begin{gathered} \mathrm{In} \mathrm{a} ∆\mathrm{ABC} \\ \frac{\left( \mathrm{a} + \mathrm{b} + \mathrm{c}\right)\left(\mathrm{b} + \mathrm{c} - \mathrm{a}\right)\left(\mathrm{c} + \mathrm{a} - \mathrm{b}\right)\left( \mathrm{a} + \mathrm{b} - \mathrm{c}\right)}{4{\mathrm{b}}^2{\mathrm{c}}^2} = ? \end{gathered}$$

• cos²A

• cos²B

• sin²A

• sin²B

P is a point on the segment joining the feet of two vertical poles of heights a and b. The angles of elevation of the tops of the poles from P are 45° each. Then, the square of the distance between the tops of the poles is

• $\frac{{\mathrm{a}}^2 + {\mathrm{b}}^2}{2}$

• ${\mathrm{a}}^2 + {\mathrm{b}}^2$

• $2\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)$

• $4\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)$

The probability of choosing randomly a numberc from the set {1, 2, 3, . . . , 9} such that the quadratic equation  x² + 4x + c = 0 has real roots is

• $\frac{1}{9}$

• $\frac{2}{9}$

• $\frac{3}{9}$

• $\frac{4}{9}$

If m and σ² are the mean and variance of the random variable X, whose distribution is given by

Then

• m = σ² = 2

• m = 1, σ² = 2

• m = σ² = 1

• m = 2, σ² = 2

If X is a bmomial variate with the range {0, 1, 2, 3, 4, 5, 6} and P(X = 2) = 4P(X = 4), then the parameter p of X is

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{2}{3}$

• $\frac{3}{4}$

The transformed equation of x² + y² = r² when the axes are rotated through an angle 36° is

• $\sqrt{5}{\mathrm{X}}^2 - 4\mathrm{XY} + {\mathrm{Y}}^2 = {\mathrm{r}}^2$

• ${\mathrm{X}}^2 + 2\mathrm{XY} - \sqrt{5}{\mathrm{Y}}^2 = {\mathrm{r}}^2$

• ${\mathrm{X}}^2 - {\mathrm{Y}}^2 = {\mathrm{r}}^2$

• ${\mathrm{X}}^2 + {\mathrm{Y}}^2 = {\mathrm{r}}^2$

27.

The area (in square unit) of the circle which touches the lines 4x + 3y = 15 and 4x + 3y = 5 is

• 4$\mathrm{\pi }$

• $\mathrm{3\pi }$

• $\mathrm{2\pi }$

• $\mathrm{\pi }$

The point on the line 3x + 4y = 5 which is equidistant from (1, 2) and (3, 4) is

• (7, - 4)

• (15, - 10)

• $\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$7$}\right., \raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$7$}\right.\right)$

• $\left(0, \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)$

The equanon of the straight line perpendicular to the straight line 3x + 2y = 0 and passing through the point of intersection of the lines x + 3y - 1 = 0 and x - 2y + 4 = 0 is

• 2x - 3y + 1 = 0

• 2x - 3y + 3 = 0

• 2x - 3y + 5 = 0

• 2x - 3y + 7 = 0

The value of $\mathrm{\lambda }$ with $\left|\mathrm{A}\right|$ < 16 such that 2x² - 10xy + 12y² + 5x + $\mathrm{\lambda }$y - 3 = 0 represents a pair of straight lines, is

• - 10

• - 9

• 10

• 9