ICSE Board Class 10 Math Sample Papers 2009

 Sample Paper 
 Class – X
 Mathematics

Section A
( Question 1 is compulsory and five other questions )

Q.1 ( Each sub question carries equal marks & total marks 30 )
1. Using Cramer's rule solve the following system of linear equations : 3x – y = 7 ; 2x + 5y = –1.
2. Find the equation to the pair of lines passing through ( 2 ,–3 ) and perpendicular to 3x² – 2xy – 4y² = 0.
3. Find the equation of the ellipse with its center at ( 1 , 2 ), focus at ( 6 , 2 ) and containing the point ( 4 , 6 ).
4. Solve the equation tan-1(2 + x) + tan-1(2 - x) = tan-1(2/3)
5. Find : .
6. Find the coefficient of correlation from the following regression lines :
x – 2y + 3 = 0 ; 4x – 5y + 1 = 0.
7. Evaluate : ò sec4x dx.
8. Show that ?n + ?2n is equal to 2 if n is a multiple of 3 and - 1 if n is not a multiple of 3.
9. Given : P(A) = 0.4, P(B) = 0.7 and P(B nA) = 0.2, find (i) P(A/B) (ii) P(A’/B) (iii) P(A’/B’).
10. Solve : (x2 – yx2) dy + (y2 + xy2) dx = 0

Q. 2. (a) Using properties of determent, prove that
x2  y2  z2
x3 y3  z3 = xyz (x – y) (y – z) ( z – x) (xy + yz + zx)
xyz xyz  xyz
(b) Solve by matrix method :
x – 2y + 3z = 6; x – 3y + 2z = 1; x + 4y + z = 12

Q. 3. (a) Draw a circuit for the expression ABC’+ AB’C + AB’C’. Also draw the circuit for the simplified form of the given expression.
(b) If , prove that .

Q. 4. (a) Taking the function f(x) = ( x – 3 ) logx, prove that there is at least one value of x in ( 1 , 3 ) which satisfies x logx =
3-x.
(b) ABC is a right-angled triangle of given area S. Find the sides of the triangle for which the area
of the circumscribed circle is least.

Q. 5 (a) Evaluate :
         (b) Evaluate :

Q. 6 (a) If the pair of lines ax² + 2pxy – ay² = 0, bx² + 2qxy – by² = 0 are such that each pair bisects the angle between the other pair, then show that ab + pq = 0.
(b) Find the area included between the curves y² = 4ax and x² = 4ay.

Q. 7 (a) The following table gives the two kinds of assessment of ten post-graduate students, performance :

Find Spearman's coefficient of rank correlation and intercept the result.
(b) Calculate the range and semi-interquartile range of wages :
Wages (Rs) 30-32 32-34 34-36 36-38 38-40 40-42 42-44
Labourers 12 18 16 14 12 8 6
(c) A pair of dice is thrown. If the two numbers appearing on them are different, find the probability that the sum of the numbers appearing is 6.

Q. 8 (a) A problem in Mathematics is given to four students A, B, C and D. Their chances of solving the problem respectively are 1/2, 1/3, 1/4 and 1/5. What is the probability that the problem will be solved?
(b) If z = x + iy, = 2 - iz / 2z - i, and | | = 1, find the locus of z and illustrate it in the complex plane.

Q. 9. (a) Using De Moivre's theorem, find the value of (1 + i 3)4 + (1 - i 3)4
(b) Solve : .

(Section B)
Answer any two questions

Q.10.(a) The price relative to a base 100, and weights of a set of commodities are given below.
Commodity A B C D E F
Price-Relative 250 200 120 154 100 114
Weight x x 2x y 2y 2
If the weights add up to 36 and the price index is 142, find the values of x and y.
(b) The profits of a soft drink firm in thousands of rupees during each month of a year were
Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. 1.2 0.8 1.4 1.6 2.0 2.4 3.6 4.8 3.4 1.8 0.8 1.2 Plot these on a graph. Calculate four-monthly moving averages and plot these on the same graph. Comment on the general trend.
Q.11.(a) Given the total cost function for x units of a commodity as . Find : (i) the marginal cost (ii) the average cost (iii) show that the marginal average cost is given by { x MC – C(x) }/ x².
(b) What is the average due date of the following installments payable under contract ?
June 1st ? Rs 3200; July 1st ? Rs 4000; August 1st ? Rs 1800.

Q.12.(a) A dealer advertises that a tape recorder is sold at Rs 450 cash down followed by two yearly installments of Rs 680 and Rs 590 at the end of the first and second year respectively. If the interest charged is 18% per annum compounded annually, find the cash price of the tape recorder.
(b) A bill of Rs 12750 was payable 60 days after sight. It was accepted on 4 March 1990 and discounted on 25 March 1990. What was the discounted value of the bill if the rate of interest was 7/2 % per annum ?