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Question Paper Solutions - Mathematics 2009 - 2010 ICSE Class 10

Year2009 - 2010 (March)

Attempt all questions form Question 1 to Question 4

Attempt any four questions From Question 5 to Quesation 11

Marks: 80

Solve the following in equation and represent the solution set on the number line.

`R - 3 < -1/2 - (2x)/3 <= 5/6, x ∈ R`

Chapter: [2.01] Linear Inequations
Concept: Representation of Solution on the Number Line

Tarun bought and article for Rs. 8000 and spent Rs. 1000 for transportation. He marked the article Rs. 11,700 and sold it to a customer. If the customer had to pay 10% sales tax, find:
(1) the customer’s price
(2) Tarun’s profit percent.

Chapter: [1.02] Sales Tax and Value Added Tax
Concept: Computation of Tax

Mr. Gupta opened a recurring deposit account in a bank. He deposited Rs. 2500 per month for two years. At the time of maturity he got Rs. 67,500. Find:

1) the total interest earned by Mr Gupta.

2) the rate of interest per annum.

Chapter: [1.03] Banking
Concept: Types of Accounts

Given `A = [(3,-2),(-1,4)] B = [(6),(1)], C = [(-4),(5)] and D = [(2),(2)]` Find AB + 2C - 4D

Chapter: [2.06] Matrices
Concept: Matrices Examples

Nikita invests Rs. 6000 for two years at a certain rate of interest compounded annually. At the end of the first year, it amounts to Rs. 6720. Calculate:

1) the rate of interest.

2) the amount at the end of the second year.

Chapter: [1.01] Compound Interest
Concept: Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years

A and B are two points on the x-axis and y-axis respectively. P (2, −3) is the midpoint of AB. Find the:

(1) coordinates of A and B
(2) slope of line AB.
(3) an equation of line AB.

Chapter: [3] Co-ordinate Geometry
Concept: Slope of a Line

Cards marked with numbers 1, 2, 3, 4… 20 are well shuffled and a card is drawn at random. What is the probability that the number on the card is:
(1) A prime number,
(2) A number divisible by 3,
(3) A perfect square?

Chapter: [5.02] Probability
Concept: Simple Problems on Single Events

Without using trigonometric tables evaluate

`(sin 35^@ cos 55^@ + cos 35^@ sin 55^@)/(cosec^2 10^@ - tan^2 80^@)`

Chapter: [5] Trigonometry
Concept: Trigonometric Identities

Use graph paper for this question) (4)
A(0, 3), B(3, −2) and O(0, 0) are the vertices of triangle ABO.

(1) Plot the triangle on a graph sheet taking 2 cm = 1 unit on both the axes.1
(2) Plot D the reflection of B in the Y-axis, and write its coordinates.
(3) Give the geometrical name of the figure ABOD.
(4) Write the equation of the line of symmetry of the figure ABOD.

Chapter: [3.01] Symmetry
Concept: Lines of Symmetry

When divided by x – 3 the polynomials x3 – px2 + x + 6 and 2x3 – x2 – (p + 3) x – 6 leave the same remainder. Find the value of ‘p’.

Chapter: [2.05] Factorization
Concept: Factorising a Polynomial Completely After Obtaining One Factor by Factor Theorem

In the figure given below AB and CD are two parallel chords and O is the centre. If the radius of the circle is 15 cm, find the distance MN between the two chords of length 24 cm and 18 cm respectively.

Chapter: [3.04] Circles
Concept: Chord Properties - Chords Equidistant from the Center Are Equal (Without Proof)

The distribution given below shows the marks obtained by 25 students in an aptitude test. Find the mean, median and mode of the distribution.

Marks obtained 5 6 7 8 9 10
No. of students 3 9 6 4 2 1
Chapter: [6] Statistics
Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data

Without solving the following quadratic equation, find the value of ‘p’ for which the roots are equal.

px2 – 4x + 3 = 0.

Chapter: [2.03] Quadratic Equations
Concept: Quadratic Equations

Rohit borrows Rs. 86,000 from Arun for two years at 5% per annum simple interest. He immediately lends out this money to Akshay at 5% compound interest compounded annually for the same period. Calculate Rohit’s profit in the transaction at the end of two years.

Chapter: [1.01] Compound Interest
Concept: Compound Interest as a Repeated Simple Interest Computation with a Growing Principal

Mrs Kapoor opened a Savings Bank Account in State Bank of India on 9th January 2008. Her pass book entries for the year 2008 are given below:

Date Particulars Withdrawals (in Rs.) Deposits (in Rs.) Balance (in Rs.)
Jan 9, 2008 By Cash - 10000 10000
Feb 12, 2008 By Cash - 15500 25500
April 6, 2008 To Cheque 3500 - 22000
April 30, 2008 To Self 2000 - 20000
July 16, 2008 By Cheque - 6500 26500
August 4, 2008 To Self 5500 - 21000
August 20, 2008 To Cheque 1200 - 19800
Dec. 12, 2008 By Cash - 1700 21500

Mrs Kapoor closes the account on 31st December 2008. If the bank pays interest at 4% per annum, find the interest Mrs Kapoor receives on closing the account. Give your answer correct to the nearest rupee.

Chapter: [1.03] Banking
Concept: Computation of Interest

A manufacturer marks an article for Rs. 5000. He sells it to a wholesaler at a discount of 25% on the marked price and the wholesaler sells it to a retailer at a discount of 15% on the marked price. The retailer sells it to a consumer at the marked price and at each stage, the VAT is 8%. Calculate the amount of VAT received by the government from:
(1) the wholesaler,
(2) the retailer.

Chapter: [1.02] Sales Tax and Value Added Tax
Concept: Computation of Tax

In the following figure, O is the centre of the circle and AB is a tangent to it at point B. ∠BDC = 65°. Find ∠BAO.

Chapter: [3.04] Circles
Concept: Tangent to a Circle

A doorway is decorated as shown in the figure. There are four semi-circles. BC, the diameter of the larger semi-circle is of length 84 cm. Centres of the three equal semicircles lie on BC. ABC is an isosceles triangle with AB = AC. If BO = OC, find the area of the shaded region. (Take `pi = 22/7`)

Chapter: [3.01] Symmetry
Concept: Lines of Symmetry

Use ruler and compasses only for this question:

I. Construct  ABC, where AB = 3.5 cm, BC = 6 cm and ABC = 60o.
II. Construct the locus of points inside the triangle which are equidistant from BA and BC.
III. Construct the locus of points inside the triangle which are equidistant from B and C.
IV. Mark the point P which is equidistant from AB, BC and also equidistant from B and C. Measure and records the length of PB.

Chapter: [3.03] Loci
Concept: Constructions Under Loci

The equation of a line 3x + 4y − 7 = 0. Find

1) The slope of the line.

2) The equation of a line perpendicular to the given line and passing through the intersection of the lines x – y + 2 = 0 and 3x + y – 10 = 0.

Chapter: [2.07] Co-ordinate Geometry
Concept: Equation of a Line

The Mean of the following distribution is 52 and the frequency of class interval 30-40 is ‘f’. Find ‘f’.

10-20 20-30 30-40 40-50 50-60 60-70 70-80
Frequency 5 3 f 7 2 6 13
Chapter: [6] Statistics
Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data

Use the Remainder Theorem to factorise the following expression:]

`2x^3 + x^2 - 13x + 6`

Chapter: [2.01] Polynomials
Concept: Remainder Theorem

If x, y, z are in continued proportion, prove that `(x + y)^2/(y + z)^2 = x/z`

Chapter: [2.04] Ratio and Proportion
Concept: Proportions

From the top of a lighthouse, 100 m high the angles of depression of two ships on opposite sides of it are 48° and 36° respectively. Find the distance between the two ships to the nearest metre.

Chapter: [5] Trigonometry
Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables


`[(4sin 30^@,  2cos 60^@),(sin 90^@, 2 cos 0^@)][(4,5),(5,4)]`

Chapter: [2.06] Matrices
Concept: Matrices Examples

In the given figure ABC is a triangle with ∠EDB = ∠ACB.  Prove that Δ ABC ~ Δ EBD. If BE = 6 cm, EC = 4 cm, BD = 5 cm. And area of Δ BED = 9 cm2. Calculate the

(1) length of AB
(2) area of Δ ABC

Chapter: [1] Similarity
Concept: Similarity of Triangles

Vivek invests Rs 4500 in 8%. Rs. 10 shares at Rs. 15. He sells the shares when the price
rises to Rs. 30, and invests the proceeds in 12% Rs. 100 shares at Rs. 125. Calculate.
(1) the sale proceeds
(2) the number of Rs. 125 shares he buys.
(3) the change in his annual income from dividend.

Chapter: [1.04] Shares and Dividends
Concept: Shares and Dividends

A positive number is divided into two parts such that the sum of the squares of the two parts is 20. The square of the larger part is 8 times the smaller part. Taking x as the smaller part of the two parts, find the number.

Chapter: [2.03] Quadratic Equations
Concept: Quadratic Equations

The monthly income of a group of 320 employees in a company is given below:

Monthly Income No. of Employees
6000-7000 20
7000-8000 45
8000-9000 65
9000-10000 95
10000-11000 60
11000-12000 30
12000-13000 5

Draw an ogive the given distribution on a graph sheet taking 2 cm = Rs. 1000 on one axis and 2 cm = 50 employees on the other axis. From the graph determine:

(1) the median wage
(2) the number of employees whose income is below Rs. 8500.
(3) if the salary of a senior employee is above Rs. 11,500, find the number of senior employees in the company.
(4) the upper quartile.

Chapter: [6] Statistics
Concept: Graphical Representation of Ogives

Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon.

Chapter: [3.05] Constructions
Concept: Circumscribing and Inscribing a Circle on a Regular Hexagon

A hemispherical bowl of diameter 7.2 cm is filled completely with chocolate sauce. This sauce is poured into an inverted cone of radius 4.8 cm. Find the height of the cone.

Chapter: [4] Mensuration
Concept: Area and Volume of Solids - Cone

Given x = `(sqrt(a^2 + b^2) + sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) - sqrt(a^2 - b^2))`

Use componendo and dividendo to prove that `b^2 = (2a^2x)/(x^3 + 1)`

Chapter: [2.04] Ratio and Proportion
Concept: Componendo and Dividendo Properties