SECTION – A

1.      If we add 1 to the numerator of a fraction and subtract 1 from the denominator, the fraction becomes 1. If 1 is added to the denominator, the fraction reduces to ½. Find the fraction.

2.      Solve the following system of equations:

3ax – 3a – 2b – 2by = 0;          2bx + 3ay   = 2b – 3a

3.      The H.C.F. and L.C.M. of two polynomials p(x) and q(x) are respectively 10(x² + 5x + 4) and 30(x² – 16) (x² – x– 2). If p(x) = 30 (x² – 16)(x + 1), find q(x)

OR

Find the HCF. and LCM. of the polynomials p(x) = 12(x² –16) (x – 2)² and q(x) = 16x³ (64 – x³)(x – 2)⁵.

4.      Determine the A.P. the sum of whose n terms is given by S_n = 3n² + 5n

5.      A T.V. is sold for Rs.25500 cash or for Rs.4000 cash down payment together with 3 equal monthly instalments of Rs.7500. Find the rate of interest charged in the instalment plan.

6.      A card is drawn from a pack a 52 cards. What is the probability that the card drawn is neither king nor a red card?

7.      In a DABC, ÐA = ÐB, D and E are the points on the sides AB and AC such that AD = BE. Show that AB//DE.

                                                 KS          SR

SECTION-B

x – 2y – 3 = 0

Shade the triangular region bounded by these lines and y = 0. Does the origin belong to this region.

           OR

The differences of the ages of a father and his son is 30 years. Five years ago, the product of their ages ( in years ) was 400. What are their present ages?

10.  Construct a triangle ABC in which AB = 5.8cm, ÐC = 45° and median through C is 4.3cm. Write down the steps of constructions. How many such triangles are possible?

11.  If A = 1 +  x + 1 , B =  x – 1  – 1 and C =  x + x³ , find  A × B ¸ C

           x² – 1          x² – 1                      1 – x²

12.  The last term of an A.P. is –120. Its first term is 20 and common difference is –10. Find the sum of the A.P.

13.  Amit borrowed Rs.65360 from a bank. He paid back the amount in three equal quaterly instalments. If the bank charges interest at 25% p.a. compounded quaterly, find the value of each installment and the total interest paid by him.

15.  Find the distance of the point (–3,4) from the mid point of the line segment joining the points (–8,3) and (6,5).

16.  The coordinates of the centroid of the triangle whose vertices are (a, b), (–2, –5) and (10, 3) are (3a, ½b). Find the values of a and b.

17.  A right circular cylinder of radius 7cm and heigh 12cm is melted and recast into right circular cone 16cm high. What is the radius of the cone?

18.  The given pie chart represents the amount spent by a club in the year 2005. If the total amount spent by the club is Rs.216000, find the amount spent on each sport.

19.  In the figure O is the centre of the circle and BC is a chord of the circle. Line PCR is a tangent to the circle at C. If ÐCOB = 120°, find ÐBAC and ÐBCP.

21.  A cone of radius 12cm is divided into two parts by drawing a plane through the mid point of its axis parallel to the base. Compare the volumes of the two parts.

OR

A metallic dustbin in the form of a cylinder has a hemispherical lid. The internal height of the cylindrical portion is 60cm and the circumference of the bottom is 88cm. Using p = 22/7, find (a) the total area of the internal surface (b) the internal volume of the container.

22.  Prove that the angles in the same segment of a circle are equal. Use above theorem and prove the following: In a circle, two chords AQ and BC intersect each other at P such that AB = AP. Prove that CP = CQ.

OR

Prove that the ratio of the areas of two similar triangles is equal to the squares of their corresponding sides.

Using above theorem, prove that the area of an equilateral triangle described on the side of a square is half the area of the equilateral triangle describes on its diagonals.

23.  An aeroplane flying horizontally at a height of 1.5 km above the ground is observed at a certain point on earth subtends an angle of 60⁰. After 15 seconds its angle of elevation is 30⁰. Calculate the speed of the aeroplane.

24.  A solid toy in the form of a hemisphere surmounted by a right circular cone. If the height of the cone is 4cm and diameter of the base is 6cm. Calculate: (a) Volume of the toy  (b) Surface area of the toy.