An empty tank can be filled by one pipe in 'x' minutes and emptied by another pipe in (x + 5) minutes. Both the pipes, when opened together, can fill the empty tank in 16.8 minutes. Find the value of x.

Use ruler and compass only to construct a triangle ABP such that AB = 8 cm, BP = 5 cm and angle ABP = 30°.

(i) Complete the rhombus ABCD such that P is equidistant from AB and BC.

(ii) Locate a point Q on line BP such that Q is equidistant from A and B.

Each of the letters of the word 'AUTHORIZES' is written on identical circular discs and put in a bag. If a disc is drawn at random from the bag, what is the probability that the letter is :

(i) a vowel ?

(ii) one of the first 9 letters of the English alphabet which appears in the given word ?

(iii) one of the last 9 letters of the English alphabet which appears in the given word ?

Use graph paper for this question :

Use 1 cm = 1 unit on both the axes.

(i) Plot the points A(4, 5) and B(-1, -3)

(ii) A' is the image of A when reflected in the y-axis. Write down the co-ordinates of A' and plot it on the graph paper.

(iii) B' is the image of B when reflected in the x-axis followed by reflection in the origin. Write down the co-ordinates of B' and plot it on the graph paper.

(iv) Write down the geometrical name of the figure AA'BB'.

(v) Name two invariant points under reflection in y-axis.