**Sample paper of mathematics for class 10th**

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Q1. A loan has to be returned in two equal annual installments. If the rate of interest is 16%per annum compounded annually and each installment is of Rs.1682, find the sum Borrowed and the total interest paid

Q2. Solve the following system of equations:

15x + 4y = 61

4x + 15y = 72

Q3. From the top of a tower 60m high, the angles of depression of the top and bottom of a building whose base is in the same straight line with the base of the tower are observed to be 30° and 60° respectively. Find the height of the building.

Q4. Find graphically, the vertices of the triangle formed by the x-axes and the lines

2x – y + 8 = 0……….. Eq(1)

8x + 3y – 24 = 0………..Eq(2)

Q5. A suit is available for Rs.1500 cash or for Rs.500 cash down payment followed by 3monthly installments of Rs.345 each. Find the rate of interest charged under the installment scheme.

Q6. If (x — 2) is a factor of x2 + ax + b and a + b = 1, find the values of a and b.

Q7. Derive the formula for the sum of first n terms of an A.P. whose first term is ‘a’ and the common difference is’d’.

Q8. The sum of the squares of two positive integers is 208. If the square of the larger number is 18 times the smaller, find the numbers.

Q9. Which term of the A.P. 3, 15, 27, 39….is 132 more than its 54th term?

Q10. Find the sum of the following arithmetic progression

1+3+5+7+…………….+199

Q11. An unbiased dice is tossed

i) Write the sample space of the experiment

ii) Find the probability of getting a number greater than 4

iii) Find the probability of getting a prime number.

Q12. Show that a line drawn parallel to the parallel sides of a trapezium divides the non-parallel Sides proportionally.

Q13. Construct a triangle ABC in which BC = 13cm, CA = 5cm and AB = 12cm. Draw its

In circle and measure its radius.

Q14. The total surface area of a closed right circular cylinder is 6512 cm², and the circumference of its base is 88 cm. Find the volume of the cylinder (use π =22/7)

Q15. Find the ratio in which the point (-3, p) divides the line segment joining the points (-5, -4) and (-2, 3). Hence find the value of p.

Q16. Prove the identity:

(1 + Cotθ – Cosecθ) (1 + tanθ + secθ) = 2.

Q17. Show that the points (7, 10), (-2, 5) and (3, -4) are the vertices of an isosceles right triangle.

Q18. Using distance formula, show that the points (-1, -1), (2, 3) and (8, 11) are collinear.

Q19. Prove that the angle subtended by an arc of a circle at its center is double the angle subtended by it at any point on the remaining part of the circle.

Using the above result prove that the angle in a major segment is acute.

Q20. A solid toy is in the form of a hemisphere surmounted by a right circular cone. If the height of the cone is 4 cm and diameter of the base is 6 cm calculate:

i) The volume of the toy

ii) Surface area of the toy (use π = 3.14)

Q21. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the Squares of their corresponding sides.

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

Q22.A bucket of height 8cm. and made up of copper sheet is in the form of frustum of aright circular cone with radii of its lower and upper ends as 3 cm and 9 cm respectively.

Calculate:

i) The height of the cone of which the bucket is a part

ii) The volume of water which can be filled in the bucket.

iii) The area of copper sheet required to make the bucket (Leave the answer in Terms of π)

Q23.An airplane flying horizontally at a height of 1.5km above the ground is observed at a certain point on earth to subtend an angle of 60°. After 15 seconds, its angle of elevation at the same point is observed to be 30°. Calculate the speed of the airplane in km/h.

Q24.A bag contains 5 red balls, 8 white balls, 4 green balls and 7 black balls. A ball is drawn at random from the bag. Find the probability that it is.

(i) Black

(ii) Not green

Q25. Anil’s total annual salary excluding HRA is Rs.1,96,000. He contributes Rs.5000 per month in his G.P.F. How much he should invest in N.S.C. to get maximum rebate? After getting maximum rebate he wants to pay income tax in equal monthly installments.

Find the amount which should be deducted per month towards tax from his salary.

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.100, 000.

(ii) Rs.30, 000/- in case the total annual income is from Rs. 100,001 to Rs.500, 000.

b) Rate of income Tax:

Slab Income Tax

i) Up to 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.1000 + 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 income tax:

i) 20% of the amount of saving subject to maximum

Rs.14,000/-, if gross income is up to Rs.1,50,000

ii) 15% of the amount of saving subject to a maximum of Rs.10,500/-if gross income is above Rs.1,50,000 but not exceeding Rs.500,000

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