Evaluate without using trigonometric tables.

$2{\left(\frac{\tan \left(35°\right)}{\cot \left(55°\right)}\right)}^2 + \left(\frac{\cot \left(55°\right)}{\tan \left(35°\right)}\right) - 3\left(\frac{\sec \left(40°\right)}{\csc \left(50°\right)}\right)$

A Mathematics aptitude test of 50 students was recorded as follows:

Draw a histogram from the above data using a graph paper and locate the mode.

A manufacturer sells a washing machine to a wholesaler for Rs. 15000. The wholesaler sells it to a trader at a profit of Rs. 1200 and the trader in turns sells it to a consumer at a profit of Rs. 1800. If the rate of VAT is 8% find:

(i) The amount of VAT received by the state government on the sale of this machine from the manufacturer and the wholesaler.

(ii) The amount that the consumer pays for the machine.

A solid cone of radius 5 cm and height 8 cm is melted and made into small spheres of radius 0.5 cm. Find the number of spheres formed.

ABCD is a parallelogram where A(x, y), B(5, 8), C(4, 7) and D(2, - 4). Find

(i) Coordinates of A

(ii) Equation of diagonal BD

Use a graph paper to answer the following questions (Take 1 cm = 1 unit on both axes).

(i) Plot A(4,4), B(4,-6) and C(8,0), the vertices of a triangle ABC.

(ii) Reflect ABC on the y-axis and name it A’B’C’.

(iii) Write the coordinates of the images A’, B’ and C’.

(iv) Give a geometrical name for the figure AA’ C’ B’ BC.

(v) Identify the line of symmetry of AA’ C’ BC’.

Mr. Choudhury opened a Saving’s Bank Account at State Bank of India on 1st April2007. The entries of one year as shown in his pass book are given below.

If the bank pays interest at the rate of 5% per annum, find the interest paid on 1st April.2008. Give your answer correct to the nearest rupee.

Using componendo and dividendo, find the value of x.

$\frac{\sqrt{3\mathrm{x} + 4} +\sqrt{3\mathrm{x} - 5} }{\sqrt{3\mathrm{x} + 4} - \sqrt{3\mathrm{x} - 5}} = 9$

If A = $\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$, B = $\left[\begin{array}{cc}4& - 2\\ - 1& 3\end{array}\right]$ and I is the identity matrix of the same order and A^t is the transpose of matrix A, find A^t.B + BI.

20.

In the adjoining figure ABC is a right angled triangle with
$\angle$BAC = 90$°$.

(i) Prove $∆$ADB $~$ $∆$CDA.

(ii) If BD = 18 cm CD = 8 cm Find AD.

(iii) Find the ratio of the area of $∆$ADB is to area of $∆$CDA.