Class 10-2004

X-2004-math
SECTION A

Question 1

Solve the following system of linear equations:

6(ax + by) = 3a + 2b, 6(bx – ay) = 3b – 2a

Question 2

Using quadratic formula, solve the following quadratic equation for x: p2x2 + (p2 – q2)x – q2 = 0

Question 3

If (x – 3) (x + 2) is the GCD of the polynomials

P(x) = (x2 – 2x – 3) (2×2 + ax – 2) and Q(x) = (x2 + x – 2) (3×2 + bx – 3) find the value of a and b.

Question 4

4.If P= x³+y³/(x-y)² + 3xy  and R= x²-y²/xy

Express [P × Q ¸ R] as a rational expression in the lowest form.

Question 5

The 8th term of an arithmetic progression (A.P.) is 37 and its 12th term is 57. Find the A.P.

Question 6

Find the sum of the first 25 terms of an A.P. whose nth term is given by tn = 7 – 5n.

Question 7

An electric fan is available for Rs. 600 cash or for Rs. 250 cash down payment followed by 3 monthly installments of Rs. 125 each. Find the rate of interest charged under the installment plan.

Question 8

A loan of Rs. 4,200 is to be returned in two equal annual installments. If the rate of interest is 10% per annum, compounded annually, calculate the amount of each installment.

Question 9

In figure 1, ABCD is a trapezium in which AB ||DC. The diagonals AC and BD intersect at O. Prove that

Question 10

In figure 2, AB is a diameter of a circle, with centre O. If ‹ABC = 70°, ‹CAD = 30° and <BAE = 60°. Find <BAC, <ACD and <ABE.

SECTION B

Question 11

Solve the following system of linear equations graphically

5x – 6y + 30 = 0, 5x + 4y – 20 = 0. Also find the vertices of the triangle formed by the above two lines and X-axis.

Question 12

Solve for x:

Question 13

A hemispherical bowl of internal diameter 30cm contains some liquid. This liquid is to be filled into cylindrical shaped bottles each of diameter 5cm and height 6cm. Find the number of bottles necessary to empty the bowl.

Question 14

Prove that 

Question 16

Show that the points A(2, -2), B(14, 10), C(11,13) and D(-1,1) are the vertices of a rectangle.

Question 17

Prove that the coordinates of the centroid of a DABC, with vertices

A(x1, y1) B(x2, y2), C(x3, y3) and are given by 


Question 18

The mean of the following frequency distribution is 57.6 and the sum of the observations is 50. Find the missing frequencies f1 and f2.

Question 19

A bag contains 4 red, 5 black and 6 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is (i) white (ii) red (iii) not black (iv) red or white.

Question 20

The following data shows the expenditure of an individual over various items:

_

Represent the above data with the help of a pie-chart.

SECTION C

Question 21

Prove that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides. Using the above, prove the following: In DABC, ÐA = 90° and AD ^ BC, prove that  AB2 + CD2 = BD2 + AC2.

Question 22

Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Using the above, do the following:

Prove that the angle formed by a chord in the major segment is acute.

Question 23

If the radii of the circular ends of a bucket, 45 cm high, are 28 cm and 7 cm (as shown in Fig. 4), find the capacity of the bucket.

Fig.4   _

Question 24

A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 600 and angle of depression of the base of the hill as 450 .  Find the distance of the hill from the ship and height of the hill.

Question 25

Mohan Lal has an annual income of Rs.2,50,000 (excluding HRA). He contributes Rs. 4,000 per month in his Provident Fund and pays an annual premium of Rs. 15,000 for his LIC policy. How much should he invest in N.S.Cs so as to get maximum rebate? He pays Rs. 2,500 per month for 11 months towards income tax. How much income tax has he to pay in the last month of the year? Assume the following for calculating income tax:

(a) Standard Deduction
Gross income Deduction
(i) Upto Rs.75,000
(ii) From Rs.75,001 to Rs.5,00,000
(iii) More than Rs,5,00,000
40% of gross income
Rs.30,000
Rs.20,000
(b) Rates of Income Taxes
Income Income tax
(i) Upto Rs. 50,000
(ii) From Rs. 50,001 to Rs. 60,000
(iii) From Rs. 60,001 to Rs. 1,50,000

(iv) Rs. 1.50,000 and above

No tax
10% of the amount exceeding Rs.50,000
Rs.1,000+20% of the amount by which taxable income exceeds Rs.60,000
Rs.19,000+30% of the amount by which taxable income exceeds Rs.1,50,000
(c) Rebate in Tax (i) 20% of the savings subject to a maximum of Rs. 14,000 if the gross income is upto Rs. 1,50,000.
(ii)15% of the savings subject to a maximum of Rs. 10,500 if the gross income is from Rs. 1,50,001 to Rs. 5,00,000.

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