Calculate the mean of the following distribution using step deviation method

In the given figure  PQ  is a tangent to the circle at  A,  AB  and  AD  are bisectors of  $\angle \mathrm{CAQ} \mathrm{and} \angle \mathrm{PAC}, \mathrm{IF} \angle \mathrm{BAQ} = 30°$, prove that: 

(i) BD is a diameter of the circle

(ii) ABC is an isosceles triangle

The printed price of an air conditioner is Rs.  45000/-. The wholesaler allows a discount of  10%  to the shopkeeper. The shopkeeper sells the article to the customer at a discount of  5%  of the marked price. Sales tax ( under VAT ) is charged at the rate of  12%  at every stage. Find:

(i) VAT paid by the shopkeeper to the government

(ii) The total amount paid by the customer inclusive of tax.

In the figure given, O is the centre of the circle. DAE = 70o, Find giving suitable reasons the measure of: 

(i) $\angle$BCD

(ii) $\angle$BOD

(iii) $\angle$OBD

A ( -1, 3 ),  B ( 4, 2 )  and  C ( 3, -2 ) are the vertices of a triangle. 

(i) Find the coordinates of the centroid G of the triangle

(ii) Find the equation of the line through  G  and parallel to  AC

Prove that 
$\frac{\sin \mathrm{\theta } - 2 {\sin }^3 \mathrm{\theta }}{2 {\cos }^3 \mathrm{\theta } - \cos \mathrm{\theta }} = \tan \mathrm{\theta }$

The sum of the ages of Vivek and his younger brother Amit is 47 years. The product of their ages in years is 550. Find their ages.

The daily wages of 80 workers in a project are given below.

Use a graph paper to draw an ogive for the above distribution. ( Use a scale of  2 cm  = Rs. 50 on  x-axis  and  2 cm = 10  workers on  y-axis ). Use your ogive to estimate:

(i) the median wage of the workers

(ii) the lower quartile wage of workers

(iii) the numbers of workers who earn more than Rs. 625 daily

The angles of depression of two ships A and B as observed from the top of a light house  60 m high are  60⁰  and  45⁰  respectively. If the two ships are on the opposite sides of the light house, find the distance between the two ships, Give your answer correct to the nearest whole number.

30.

PQR is a triangle. S is a point on the side QR of $∆$PQR such that $\angle$PSR = $\angle$QPR. Given  QP = 8 cm, PR = 6 cm and SR = 3 cm 

(i) Prove $∆$PQR $~$  $∆$SPR

(ii) Find the length of QR and PS

(iii) $\frac{\mathrm{Area} \mathrm{of} ∆\mathrm{PQR}}{\mathrm{Area} \mathrm{of} ∆\mathrm{SPR}}$