Find the principal value of \(sin^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \).

An airplane propeller rotates 1000 times per minute. Find the number of degree that a point on the edge of the propeller will rotate in 1 second

Find the principal solution and general solutions of the following : sin\(\theta\) = \(-\frac { 1 }{ \sqrt { 2 } } \)

Prove that \(\sin { 4\alpha } =4\tan { \alpha } \frac { 1-\tan ^{ 2 }{ \alpha } }{ { \left( 1+\tan ^{ 2 }{ \alpha } \right) }^{ 2 } } \)

Show that \(\cot { \left( 7\frac { 1° }{ 2 } \right) } =\sqrt { 2 } +\sqrt { 3 } +\sqrt { 4 } +\sqrt { 6 } \)

Solve the following equations sin 5x - sin x = cos3x

Find the principal solution of cos \(\theta ={1\over 2}\)

A circular wire of radius 3 cm is cut and bent so as to lie along the circumference of a sector whose radius is 48 cm, Find in degrees the angle which is subtended at the centre of the sector.

Prove that \(\frac { sin(A-B) }{ sin(A+B) } =\frac { { a }^{ 2 }-{ b }^{ 2 } }{ { c }^{ 2 } } \)

Prove that \({ tan }^{ -1 }\left( \frac { m }{ n } \right) -{ tan }^{ -1 }\left( \frac { m-n }{ m+n } \right) =\frac { \pi }{ 4 } \)

A spring was hung from a hook in the ceiling. A number of different weights were attached to the spring to make it stretch, and the total length of the spring was measured each time shown in the following table.

Weight, (kg)	2	4	5	8
Length, (cm)	3	4	4.5	6

(i) Draw a graph showing the results.
(ii) Find the equation relating the length of the spring to the weight on it.
(iii) What is the actual length of the spring.
(iv) If the spring has to stretch to 9 cm long, how much weight should be added?
(v) How long will the spring be when 6 kilograms of weight on it?

A 150 m long train is moving with constant velocity of 12.5 m/s. Find
(i) the equation of the motion of the train
(ii) time taken to cross a pole
(iii) The time taken to cross the bridge of length 850m is?

A straight line cuts intercepts from the axes of co-ordinates the sum of whose reciprocals is a constant. Show that it always passes through a fixed point.

Find the equation of a straight line cutting an intercept of 5 from the negative direction of the y-axis and is inclined at an angle 150⁰ to the x-axis.

Find the equations of the straight lines, making the y-intercept of 7 and angle between the line and the y-axis is 30°.

A student when walks from his house, at an average speed of 6 kmph, reaches his school ten minutes before the school starts. When his average speed is 4 kmph, he reaches his school five minutes late. If he starts to school every day at 8.00 A.M, then find
(i) the distance between his house and the school
(ii) the minimum average speed to reach the school on time and time taken to reach the school
(iii) the time the school gate closes
(iv) the pair of straight lines of his path of walk.

Find the equation of the straight line passing through intersection of the straight lines 5x - 6y = 1 and 3x + 2y + 5 = 0 and perpendicular to the straight line 3x - 5y + 11=0.

Show that 3x²+10xy+8y²+14x+22y+15=0 represents a pair of straight lines and the angle between them is tan^-1\(\left( \frac { 2 }{ 11 } \right) \).

A problem in Mathematics is given to three students whose chances of solving  \(\frac { 1 }{ 3 } ,\frac { 1 }{ 4 } \) and \(\frac { 1 }{ 5 } \) (i) What is the probability that the problem is solved? (ii) What is the probability that exactly one of them will solve it?

A town has 2 fire engines operating independently. The probability that a fire engine is available when needed is 0.96.
(i) What is the probability that a fire engine is available when needed?
(ii) What is the probability that neither is available when needed?