21.

As observed from the top of a 80 m tall lighthouse, the angles of depression of two ships on the same side of the light house of horizontal line with its base are 30 and 40 respectively. Find the distance between the two ships. Give your answer correct to the nearest meter.

A man invests Rs. 9600 on Rs. 100 shares at Rs. 80. If the company pays him 18% dividend find:

(i) The number of shares he buys.

(ii) His total dividend.

(iii) His percentage return on the shares.

In the given figure ΔABC and ΔAMP are right angled at B and M respectively. Given AC = 10 cm, AP = 15 cm and PM = 12 cm.

(i) Prove ΔABC $~$ ΔAMP

(ii) Find AB and BC.

If x = $\frac{\sqrt{\mathrm{a} + 1} + \sqrt{\mathrm{a} - 1}}{\sqrt{\mathrm{a} +1} - \sqrt{\mathrm{a} - 1}}$, Using properties of proportion show that x² - 2ax + 1 = 0

The line through A(–2, 3) and B(4, b) is perpendicular to the line 2x – 4y = 5. Find the value of b.

Prove that $\frac{{\tan }^2\left(\mathrm{\theta }\right)}{\left(\sec \right(\mathrm{\theta }) - 1{)}^2} = \frac{1 + \cos \left(\mathrm{\theta }\right)}{1 - \cos \left(\mathrm{\theta }\right)}$

A car covers a distance of 400 km at a certain speed. Had the speed been 12 km/h more, the time taken for the journey would have been 1 hour 40 minutes less. Find the orginal speed of the car.

Construct a triangle ABC in which base BC = 6 cm, AB = 5.5 cm and ABC = 120$°$.

(i) Construct a circle circumscribing the triangle ABC.

(ii) Draw a cyclic quadrilateral ABCD so that D is equidistant from B and C

In triangle PQR, PQ = 24 cm, QR = –7 cm and PQR = 90$°$. Find the radius of the inscribed circle.

Find the mode and median of the following frequency distribution:

x	10	11	12	13	14	15
f	1	4	7	5	9	3