ICSE Board Sample Paper

Subject – Mathematics

Class - X

1. (a) Rohan Verma opened a bookshop with some initial investment. In the 1^st year he incurred a loss of 10%. However, during the 2^nd year he incurred a profit of 20%, which in 3^{rd year} increased to 25%. Calculate his net profit percent for the entire period of 3 years.                                       4

(b) If the points (a,1) , (1,2) and (0,b+1) are collinear show that 1/a+ 1/b =1.                                                   3

(c) Calculate the mean, median and mode of the following data: 9,0,3,2,8,5,5,2,7,1.                                        3

2. (a) Puja purchases an article for ` 3600 and sells it to Nilakshi for ` 4800. Nilakshi , in turn, sells the article to Shubham for ` 5500. If the VAT rate is 10% find the VAT levied on Puja and Nilakshi.                          3

(b) If A, B, C are the angles of a triangle, prove that :                                      3

                  sin {^(B+C)/₂} = cos ^A/₂

(c) (x -2) is a factor of the expression x³ + ax² + bx + 6 . When this expression is divided by (x- 3), it leaves the reminder 3. Find the values of “a” and “b” .

With this value of a and b, factorise the expression completely.                                               4

3. (a) Calculate the arithmetic mean, correct to one decimal place, for the following frequency distribution of marks obtained in a Geometry test.

Marks	0-10	10-20	20-30	30-40	40-50
No of students	7	13	15	12	3

(b) Without using a trigonometric table calculate:                                                                 3

     4 (sin32⁰/cos58⁰) + 5 (tan48⁰/cot42⁰) - 8 (sec72⁰/cosec18⁰)

(c) Solve the following inequation and graph the solution on the number line :                     3

                         2x – 5 ≤ 5x + 4 ≤ 11 , x R

4. (a) Without using trigonometric tables, evaluate: (2tan53⁰/cot37⁰) – cot80⁰/tan10⁰)                                 3

(b) Find the mean for the following frequency distribution:                                                                            4

Class intervals	84-90	90-96	96-102	102-108	108-114
Frequency	8	12	15	10	5

(c) A copper wire when bent in the form of an equilateral triangle has area 121√3 cm². If the same wire is bent into the form of a circle, find the area enclosed by the wire.                                          3

SECTION B (40 Marks)

Attempt any four questions from this Section

5. (a) A certain sum amounts to Rs 4840 in 2 years and to Rs 5324 in 3 years at compound interest. Find the rate of interest and the sum.                         5

(b) The angles of elevation of the top of a tower from two points P and Q at distances a and b respectively, from the base and in the same straight line with it, are complementary. Prove that the height of the tower is √(ab).                                          5

6. (a)  David opened a Recurring Deposit Account in a bank and deposited Rs 300 per month for two years. If the received Rs 7725 at the time of maturity, find the rate of interest per annum.                                        4

(b) With out using mathematical tables, find the value of x if cosx = cos 60⁰ cos 30⁰ + sin 60⁰ sin 30⁰.                                    3

(c) Solve the following quadratic equation for x and give your answer correct to three significant figures: 2 - 4^x – 3 = 0.              3

7. (a) Bosco wishes to start a 200 rectangular vegetable garden. Since he has only 50 m barbed wire, he fences three sides of the rectangular garden letting his house compound wall act as the fourth side of the fence. Find the dimensions of the garden.                                   3

(b) Prove that A(2, 1), B(0,3) and C(-2,1) are the three vertices of an isosceles right angled triangle. Hence find the coordinates of a point D, if ABCD is a square.                                       4

(c) Prove that 1 /(secx - tanx) + 1/( secx + tanx) = 2/cosx.                                                3

8. (a) The angle of elevation of an eroplane from a point P on the ground is 60.After 12 seconds from the same point P, the angle of elevation of the same plane changes to 30. If the plane is flying horizontally at a speed of 600 km / h, find the height at which the plane is flying.                             4

(b) The manufacturer sold a TV to a wholesaler for Rs.7000. The wholesaler sold it to a trader at a profit of Rs.1000. If the trader sold it to the customer at a profit of Rs.1500, find:

(i) the total VAT (value added tax) collected by the state government at the rate of 5% .

(ii) the amount that the customer pays for the TV.                                                                                   4

c) In a single throw of a die, find the probability of getting:

               i. An odd number.                ii. A prime number.                                                                           2

9. (a) By selling an article for ` 24 , a trader loses as much percent as the cost price of the article. Calculate the cost price.                 3

(b) If p : q = r : s then show that (mp + nq) : q = (mr + ns) : s                                              3

(c) The contents of 100 match boxes were checked to determine the number of matches they contained.

No of matches	35	36	37	38	39	40	41
No. of boxes	6	10	18	25	21	12	8

(i) Using step deviation method calculates to one decimal place, the mean of matches per box.

(ii) Determine, how many extra matches would have to be added to the total contents of the 100 boxes to bring the mean up to exactly 39 matches.            4

10. (a) What quantity must be added to each term of the ratio (m + n ) : (m – n) to make it equal to (m + n )² : (m – n)².                               3

(b) Solve 3x² – 8x +2 = 0 correct to 2 decimal places.                                               4

(c) ABCD is a cyclic quadrilateral in which ÐA = (x + y + 10)^o, ÐB = (y + 20)^o, ÐC = (x + y – 30)^o and ÐD = (x + y)^o. Find x and y.                            3