11.

Prove that  $\sqrt{ {\sec }^2 \mathrm{\theta } + {\mathrm{cosec}}^2 \mathrm{\theta }} = \tan \mathrm{\theta } + \cot \mathrm{\theta }$

Using a graph paper draw a histogram for the given distribution showing the number of runs scored by 50 batsmen. Estimate the mode of the data:

If the straight lines 3x – 5y = 7 and 4x + ay + 9 = 0 are perpendicular to one another, find the value of a.

Solve  x² + 7x = 7  and give your answer correct to two decimal places.

Use graph paper for this question (Take 2 cm = 1 unit along both x and y axis). ABCD
is a quadrilateral whose vertices are A(2, 2), B(2, –2), C(0, –1) and D(0, 1). 

(i) Reflect quadrilateral ABCD on the y-axis and name it as A'B'CD.

(ii) Write down the coordinates of A' and B'.

(iii) Name two points which are invariant under the above reflection.

(iv) Name the polygon A'B'CD.

Using properties of proportion, solve for x. Given that x is positive:

$\frac{2 \mathrm{x} + \sqrt{ 4 {\mathrm{x}}^2 - 1}}{2 \mathrm{x} - \sqrt{ 4 {\mathrm{x}}^2 - 1}} = 4$

If  $\mathrm{A} = \left[ \begin{array}{cc}2& 3\\ 5& 7\end{array} \right], \mathrm{B} = \left[ \begin{array}{cc}0& 4\\ - 1 & 7\end{array} \right] \mathrm{and} \mathrm{C} = \left[\begin{array}{cc}1& 0\\ - 1 & 4\end{array} \right],$  find  AC + B² - 10 C.

Prove that  $( 1 + \cot \mathrm{\theta } - \mathrm{cosec} \mathrm{\theta } ) ( 1 + \tan \mathrm{\theta } + \sec \mathrm{\theta } ) = 2$

Find the value of k for which the following equation has equal roots.  x² +  4 k x + ( k² – k + 2) = 0

On a map drawn to a scale of 1 : 50,000, a rectangular plot of land ABCD has the following dimensions. AB = 6 cm; BC = 8 cm and all angles are right angles. Find: 

(i) the actual length of the diagonal distance AC of the plot in km.

(ii) the actual area of the plot in sq. km.