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# ICSE Board Exam 2005 : Mathematics

8 pages, 42 questions, 41 questions with responses, 106 total responses
 ICSE Indian Certificate of Secondary Education (ICSE), New Delhi
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MATHEMATICS -2005 (Two hours and a half) Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during the first 15 minutes. This time is to be spent in reading the question paper. The time given at the head of this Paper is the time allowed for writing the answers. Attempt all questions from Section A and any four questions from Section B. All working, i ncluding rough work, must be clearly shown and must be done on the same sheet as the rest of the answer. Omission of essential working will result in the loss of marks. The intended marks for questions or parts of questions are given in brackets [ ]. Mathematical tables are provided. S ECTION A (40 Marks) Attempt all questions from this Section. Question 1 (a) (x 2) is a factor of the expression x3 + ax2 + bx + 6. When this expression is divided by (x 3), it leaves the remainder 3. Find the vlaues of a and b. (Type in the answers separated by commas. Leave no spaces around the commas.) [3] (b) What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional? [3] (c) If the interest is compounded half yearly, calculate the amount when the Principal is Rs 7,400, the rate of interest is 5% per annum and the duration is one year. [4] Question 2 (a) M r. R.K. Nair gets Rs 6,455 at the end of one year at the rate of 14% per annum in a recurring deposit account. Find the monthly instalment. [3]

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[3] (b) A = {x : 11x 5 > 7x + 3, x R} and B={x : 18x 9 l5+12x, x R}. Find the range of Set A B and represent it on a number line. (c) In the given figure, AB and DE are perpendicular to BC. If AB = 9 cm, DE = 3 cm and AC = 24 cm, calculate AD. [10] Question 3 [10] (a) Use a graph paper for this question. (Take 10 small divisions = 1 unit on both axes). P and Q have co-ordinates (0, 5) and ( 2, 4). (i) P is invariant when reflected in an axis. Name the axis. (ii) Find the image of Q on reflection in the axis found in (i). (iii) (0, k) on reflection in the origin is invariant. Write the value of k. (iv) Write the co-ordinates of the image of Q, obtained by reflecting it in the origin followed by reflection in x-axis. (Type in the answers separated by commas. Leave no spaces around the commas.) (b) If the mean of the following distribution is 7.5, find the missing frequency 'f' : Variable : 5 6 7 8 9 10 11 12 Frequency : 20 17 f 10 8 6 7 6 (c) In the figure given below, OACB is a quadrant of a circle. The radius OA = 3.5 cm, OD = 2 cm. Calculate the area of the shaded portion. Question 4 (a) Draw a histogram to represent the following data : Pocket money in Rs. No. of S tudents [3]

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150 - 200 10 200 - 250 5 250 - 300 7 300 - 350 4 350 - 400 3 (b) Prove that (1 + tan A)2 + (1 tan A)2 = 2 sec2A. [3] (c) The catalogue price of a computer set is Rs 45,000. The shopkeeper gives a discount of 7% on the listed price. He gives a further off-season discount of 4% on the balance. However, sales tax at 8% is charged on the remaining amount. Find : (i) The amount of sales tax a customer has to pay, (ii) The final price he has to pay for the computer set. (Type in the answers separated by commas. Leave no spaces around the commas.) [2] (ii) The final price he has to pay for the computer set. [2] S ECTION B (40 Marks) Attempt any four questions from this Section. Question 5 (a) Solve the following equation and give your answer up to two decimal places : x2 5x 10 = 0 (Type in the answers separated by commas. Leave no spaces around the commas.) [3] (b) PQR is a right-angled triangle with PQ = 3 cm and QR = 4 cm. A circle which touches all the sides of the triangle is inscribed in the triangle. Calculate the radius of the circle. [3] (c) In the given figure, write (i) the co-ordinates of A, B and C. (Type in the answers separated by commas. Leave no spaces around the commas.) [2]

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(ii) the equation of the line through A and // to BC. [2] Question 6 (a) In the figure, PM is a tangent to the circle and PA = AM . Prove that : (i) PM B is isosceles. [1 1/2] (ii) PA PB = M B2. [1 1/2] (b) Find the value of x given that A 2 = B A= [3] 2 12 4x B= 01 01 (c) Write down the relation denoted by the arrow diagram, by listing the ordered pairs. State the domain, co-domain and the range of the relation. ls the relation a function? If so, state its type. [4] Question 7 (a) M r. Tiwari invested Rs 29,040 in 15% Rs 100 shares quoted at a premium of 20%. Calculate : [1]

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(i) The number of shares bought by M r. Tiwari. (ii) M r. Tiwari's income from the investment. [1] (iii) The percentage return on his investment. [1] (b) From the top of a cliff 92 m high, the angle of depression of a buoy is 20 . Calculate to the nearest metre, the distance of the buoy from the foot of the cliff. [3] (c) A circle with centre O, diameter AB and a chord AD is drawn. Another circle is drawn with AO as diameter to cut AD at C. Prove that BD = 2OC. [4] Question 8 (a) M r. Rakesh Sharma receives his annual salary as given below : [3] Basic Salary : Rs. 6,000 per month Dearness Allowance : Rs. 5,000 per month Savings: Contribution towards Provident Fund : Rs. 13,200 per year Contribution towards LIC premium : Rs. 5,000 per year Donations : To Prime M inister's Relief Fund : Rs. 2,000 (eligible for 100% tax exemption) Calculate: (i) M r. Sharma's taxable income (ii) The tax M r. Sharma has to pay for the financial year. Tax slab:Upto Rs. 50,000 Rs. 50,001 to Rs. 60,000 : No tax. : 10% of the income exceeding Rs. 50,000. Rs. 60,001 to Rs. 1,50,000 : Rs. 1,000 + 20% of the income exceeding Rs. 60,000. [3]

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Above Rs. 1,50,000 Standard Deduction Rebate in tax CESS : : : : Rs. 19,000 + 30% of the income exceeding Rs. 1,50,000. Rs. 20,000. 20% of the total savings or Rs. 14,000, whichever is less. 2% of the tax payable after rebate. (b) A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. Find the number of cones thus obtained. [4] Question 9 (a) Use a graph paper for this question. The graph of a linear equation in x and y, passes through A( 1, 1) and B(2, 5). From your graph, find the values of h and k, if the line passes through (h, 4) and (1/2, k). [3] (b) In an isosceles triangle ABC, with AB = AC, BD is the perpendicular from B to the [3] side AC. Prove that BD 2 CD 2 = 2AD CD. (c) A page from the passbook of M rs. Rama Bhalla is given below : Withdrawals Deposits Balance Date S ignature Particulars Rs. P. Rs. P. Rs. P. YEAR 2004 January 1 B/F -- -- 2000.00 January 9 By cash -- 200.00 2200.00 500.00 -- 1700.00 February 10 February 24 July 29 To cheque By cheque -- 300.00 2000.00 200.00 -- 1800.00 November 7 By cash -- 300.00 2100.00 December 8 By cash -- 200.00 2300.00 To cheque [4]

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Calculate the interest due to M rs. Bhalla for the period from January 2004 to December 2004, at the rate of 5 % per annum. Question 10 (a) Using a ruler and compass only : (i) Construct a triangle ABC with BC = 6 cm, ABC = l20 and AB = 3.5 cm. [2] (ii) In the above figure, draw a circle with BC as diameter. Find a point 'P' on the circumference of the circle which is equidistant from AB and BC. M easure BCP. [2] (b) Using a graph paper, draw an Ogive for the tollowing distribution which shows a record of the weight in kilograms of 200 students. [3] Weight Frequency 40 - 45 5 45 - 50 17 50 - 55 22 55 - 60 45 60 - 65 51 65 - 70 31 70 - 75 20 75 - 80 9 Use your Ogive to estimate the following : (i) The percentage of students weighing 55 kg or more, (ii) The weight above which the heaviest 30% of the students fall, (iii) The number of students who are (l) under-weight and (2) overweight, if 55.70 kg is considered as standard weight. [3] Question 11 (a) In the given figure, O is the centre of the circle and AOC = 160 . Prove that 3 y 2 x = 140 [3]

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(b) Without using mathematical tables, find the value of x if cos x = cos 60 cos 30 + sin 60 sin 30 . [3] (c) By increasing the speed of a car by 10 km/hr, the time of journey for a distance of 72 km is reduced by 36 minutes. Find the original speed of the car. [4]

Additional Info : Solved ICSE Board exam paper study guide - ICSE 2005 : MATHEMATICS - I.C.S.E. Free Online Question Paper
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