SECTION A
Question numbers 1 to 10 carry 3 marks each.
Q. 1. Prove that 
Q. 2. Find the GCD and LCM of the following polynomials 
Ans. GCD 
LCM 
Or
If the GCD of 
is 
find the values of 
Ans. (a = 5; b = -24)
Q. 3. Find the value of k for which the following system of linear equations has infinite number of solutions 
Ans. (K = 2)
Q. 4. Express the following as a rational expression in lowest form:
Ans. 
Q. 5. Eleven balls are serially numbered and placed in an urn. Find the probability that a ball drawn will be: (a) odd numbered ball (b) even numbered ball (c) prime numbered ball.
Ans. 
Q. 6. If the mean of the following frequency distribution is 8, find the value of p.
x |
3 |
5 |
7 |
9 |
11 |
13 |
y |
6 |
8 |
15 |
p |
8 |
4 |
Q. 7. Find the co-ordinates of the points which divide internally the line joining (1, -3) and (-3, 9) in the ratio 1 : 3. Ans. (0, 0)
Q. 8. The 4th term of an A.P. is equal to 3 times the first term and the 7th term exceeds twice the 3rd term by 1. Find the1st term and common difference.
Ans. (3 : 2)
Q. 9. D and E are points on the sides AB and AC respectively of 
such that DE is parallel to BC, and AD : DB = 4 : 5. CD and BE intersect each other at F. Find the ratio of the areas of 
Ans. (16/81)
Q. 10. From Figure 1, find 
Ans. 
SECTION B
Question numbers 11 to 20 carry 4 marks each.
Q. 11. Solve the following system of linear equations graphically:
Does the point (4, 9) lie on any of the lines? Write its equation.
Ans. 
Q. 12. Solve the following system of linear equations:
Ans. 
Or
The sum of the numerator and denominator of a fraction is 8. If 3 is added to both the numerator and the denominator, the fraction becomes. Find the fraction.
Ans. (3/5)
Q. 13. Determine value(s) of p for which the quadratic equation 
= 0 has real roots.
Ans. 
Or
*Determine the 2nd term and the rth term of an A.P., whose 6 th term is 12 and 8 th term is 22.
Ans. (5r -18)
Q. 14. The total annual income of Rajesh (excluding HRA) is Rs.1,83,000. He pays Rs.3,500 semi annually towards life insurance premium. How much should he contribute per month towards Provident Fund to get maximum rebate. Also find his tax liability for the financial year.
Assume the following for calculating income tax:
(a) Standard Deduction : |
(i) 40% of the total income subject to a maximum of Rs.30,000 in case the total annual income is up to Rs.1,00,000 |
(b) Rates of income tax : |
|
Slab |
Income tax |
(i) Upto Rs.50,000 |
No tax |
(ii) From Rs.50,001 to Rs. 60,000 |
10% of the amount exceeding Rs. 50,000 |
(iii)From Rs.60,001 to Rs. 1,50,000 |
Rs.1,000+20% of the amount exceeding Rs. 60,000 |
(iv) Above Rs.1,50,000 |
Rs.19,000+30% of the amount exceeding Rs.1,50,000 |
(c) Rebate in tax
|
(i) 20% of the amount of saving Subject to maximum Rs. 14,000, if gross income is upto Rs. 1,50,000 |
Q. 15. A housing society charges for a flat Rs. 16,00,000 cash or Rs. 5,85,500 cash down payment and three equally half yearly instalments. If the society charges 16% per annum interest compounded half-yearly, calculate the value of each instalment. Find also the total interest charged.
Ans. (Rs. 3,93,660; Rs.1,66,480)
Q 16. In what ratio does the point 
divide the line segment joining the points 
?
Ans. (1 : 3)
Q. 17. Without using trigonometric tables, evaluate the following:
Ans. 
Q. 18. In 
is acute. BD and CE are perpendiculars on AC and AB respectively. Prove that 
Q. 19. Construct a 
in which BC = 4 cm, LA = 500 and altitude through A is 3 cm. Also write the steps of construction.
Q. 20. In the month of January 2003, a householder spent his monthly salary amounting to Rs. 14,400 on different items as given below:
Items |
Clothing |
Food |
House Rent |
Education |
Misc. |
Amount spent (in Rs.) |
1,200 |
8,000 |
2,400 |
800 |
2,0000 |
SECTION C
Question numbers 21 to 25 carry 6 marks each.
Q. 21. The length of the hypotenuse of a rt. angled triangle exceeds the base by 1 cm and also exceeds twice the length of the altitude by 3 cm. Find the length of each side of & 
.
Ans. (12 cm; 13 cm; 5 cm)
Or
"If twice the area of smaller square is subtracted from the area of a large square, the result is 14 cm 2 However, if twice the area of the larger square is added to three times the area of the smaller square, the result is 203 cm 2 Determine the sides of squares.
Ans. (smaller side = 5 cm; larger side = 8cm)
Q. 22. In a triangle, if the square on one side is equal to the sum of the squares on the other two sides, prove that the angle opposite the first side is a right angle.
Use the above theorem and prove the following:
In triangle 
and BD = 3CD. Prove that 2AB2 = 2AC2 + BC2 .
Q. 23. If 
is a secant to a circle intersecting the circle at A and B and PT is a tangent segment, prove that PA x PB = PT22
Using the above theorem, prove the following:
Two circles intersect each other at P and Q. From a point R on PQ produced, two tangents RB and RC are drawn to the two circles touching them at B and C. Prove that RB = RC.
Q. 24. The internal and external radii of a hollow sphere are 3 cm and 5 cm respectively. The sphere is melted to form a solid cylinder of height 
cm. Find the diameter and curved surface area of the cylinder.
Ans. (14 cm; 117.3 3 cm3)
Q. 25. The angles of elevation and depression of the top and bottom of a light-house from the top of a building, 60 m high, are 300 and 600 respectively. Find
(i) the difference between the heights of the light-house and the building.
(ii) distance between the light-house and the building.
Ans. (20 m; 34.64 m)
Or
*The height of a house subtends a right angle at the opposite window. The angle of elevation of the window from the base of the house is 60°. If the width of the road is 6 m, find the height of the house.
Ans. (13.856 m)
