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Mathematics 2014-2015 ICSE Class 10 Question Paper Solution

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Mathematics
Marks: 80Date: 2014-2015 March

Question 1 to Question 4 is Compulsory

Attempt Any Four Quesion from Question 5 to Question 11


[10]1
[3]1.1

A shopkeeper bought an article for Rs. 3,450. He marks the price of the article 16% above the cost price. The rate of sales tax charged in the article is 10%

Find the:

1) market price of the article.

2) price paid by a customer who buys the article

Concept: Introduction to Sales Tax and Value Added Tax
Chapter: [0.013999999999999999] Gst (Goods and Services Tax)
[3]1.2

Solve the following inequation and write the solution set:

13x – 5 < 15x + 4 < 7x + 12, x ε R

Represent the solution on a real number line.

Concept: Representation of Solution on the Number Line
Chapter: [0.020499999999999997] Linear Inequations
[4]1.3

Without using trigonometric tables evaluate:

`(sin 65^@)/(cos 25^@) + (cos 32^@)/(sin 58^@) - sin 28^2. sec 62^@ + cosec^2 30^@`

Concept: Trigonometric Ratios of Complementary Angles
Chapter: [0.05] Trigonometry
[10]2
[3]2.1

if `A = [(3,x),(0,1)], B = [(9,16),(0,-y)]`, Find x and y  where `A^2 = B`

Concept: Matrices Examples
Chapter: [0.0208] Matrices
[3]2.2

The present population of the town is 2,00,000. The population is increased by 10% in the first year and 15% in the second year. Find the population of the town at the end of two years.

Concept: Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years
Chapter: [0.011000000000000001] Compound Interest
[4]2.3

Three vertices of parallelogram ABCD taken in order are A(3, 6), B(5, 10) and C(3, 2)

1) the coordinate of the fourth vertex D

2) length of diagonal BD

3) equation of the side AD of the parallelogram ABCD

Concept: Simple Applications of All Co-ordinate Geometry.
Chapter: [0.021] Co-ordinate Geometry Equation of a Line
[10]3
[3]3.1

In the given figure, ABCD is the square of side 21 cm. AC and BD are two diagonals of the square. Two semicircles are drawn with AD and BC as diameters. Find the area of the shaded region. (Take `pi = 22/7`)

Concept: Circle - Direct Application Problems Including Inner and Outer Area
Chapter: [0.04] Mensuration
[3]3.2

The marks obtained by 30 students in a class assignment of 5 marks are given below.

Marks 0 1 2 3 4 5
No. of
Students
1 3 6 10 5 5

Calculate the mean, median and mode of the above distribution

Concept: Median of Grouped Data
Chapter: [0.06] Statistics
[4]3.3

In the figure given below, O is the centre of the circle and SP is a tangent. If ∠SRT = 65°,
find the value of x, y and z.

Concept: Construction of Tangents to a Circle
Chapter: [0.033] Constructions
[10]4
[3]4.1

Katrina opened a recurring deposit account with a Nationalised Bank for a period of 2 years. If the bank pays interest at the rate 6% per annum and the monthly instalment is Rs. 1,000, find the:

1) Interest earned in 2 years.

2) Matured value

Concept: Computation of Interest
Chapter: [0.013000000000000001] Banking
[3]4.2

Find the value of ‘K’ for which x = 3 is a solution of the quadratic equation

`(K + 2)x^2 - kx + 6 = 0`

Concept: Quadratic Equations
Chapter: [0.0202] Quadratic Equations
[4]4.3

Construct a regular hexagon of side 5 cm. Construct a circle circumscribing the hexagon. All traces of construction must be clearly shown.

Concept: Circumscribing and Inscribing a Circle on a Regular Hexagon
Chapter: [0.033] Constructions
[10]5
[5]5.1

Use a graph paper for this question taking 1 cm = 1 unit along both the x and y-axis :

1) Plot the points A(0, 5), B(2, 5), C(5, 2), D(5, -2), E(2, -5) and F(0, -5).

2) Reflect the points B, C, D and E on the y-axis and name them respectively as B’, C’, D’ and E’.

3) Write the coordinates of B’, C’, D’ and E’.

4) Name the figure formed by B C D E E’ D’ C’ B’.

5) Name a line of symmetry for the figure formed.

Concept: Lines of Symmetry
Chapter: [0.034] Symmetry
[5]5.2

Virat opened a Savings Bank account in a bank on 16th April 2010. His pass book shows the following entries:

Date Particulars

Withdrawal

(Rs.)

Deposit (Rs.) Balance (Rs.)
April 16, 2010 , By cash - 2500 2500
April 28th By cheque - 3000 5500
May 9th To cheque 850 - 4650
May 15th By cash   1600 6250
May 24th To cash 1000 - 5250
June 4th To cash 500 - 4750
June 30th To cheque - 2400 7150
July 3rd By cash - 1800 8950

Calculate the interest Virat earned at the end of 31st July 2010 at 4% per annum interest. What sum of money will he receive if he closed the account on 1st August 2010?

Concept: Types of Accounts
Chapter: [0.013000000000000001] Banking
[10]6
[3]6.1

If a, b, c are in continued proportion, prove that (a + b + c) (a – b + c) = a2 + b2 + c2

Concept: Conept of Compound Interest - Compound Interest as a Repeated Simple Interest Computation with a Growing Principal
Chapter: [0.011000000000000001] Compound Interest
[4]6.2

In the given figure ABC is a triangle and BC is parallel to the y-axis. AB and AC intersect
the y-axis at P and Q respectively.

1) Write the coordinates of A.

2) Find the length of AB and AC

3) Find the ratio in which Q divides AC.

4) Find the equation of the line AC

Concept: Geometric Understanding of c as the y-intercept Or the Ordinate of the Point Where the Line Intercepts the y Axis Or the Point on the Line Where x=0
Chapter: [0.021] Co-ordinate Geometry Equation of a Line
[3]6.3

Calculate the mean of the following distribution :

Class Interval 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 8 5 12 35 24 16
Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data
Chapter: [0.06] Statistics
[10]7
[3]7.1

Two solid spheres of radii 2 cm and 4 cm are melted and recast into a cone of height 8 cm. Find the radius of the cone so formed.

Concept: Area and Volume of Solids - Sphere
Chapter: [0.04] Mensuration
[3]7.2

Find 'a' of the two polynomials ax3+3x2-9 and 2x3+4x+a, leaves the same remainder when divided by x+3.

Concept: Remainder Theorem
Chapter: [0.0203] Factorization
[4]7.3

Prove that `(sin theta)/(1-cottheta) + (cos theta)/(1 - tan theta) = cos theta + sin theta`

Concept: Trigonometric Identities
Chapter: [0.05] Trigonometry
[10]8
[3]8.1

AB and CD are two chords of a circle intersecting at P. Prove that AP x PB = CP x PD

Concept: Arc and Chord Properties - If Two Chords Intersect Internally Or Externally Then the Product of the Lengths of the Segments Are Equal
Chapter: [0.032] Circles
[3]8.2

A bag contains 5 white balls, 6 red balls and 9 green balls. A ball is drawn at random from the bag. Find the probability that the ball is drawn is:

1) a green ball

2) a white or a red ball

3) is neither a green ball nor a white ball.

Concept: Simple Problems on Single Events
Chapter: [0.07] Probability
[4]8.3

Rohit invested Rs. 9,600 on Rs. 100 shares at Rs. 20 premium paying 8% dividend. Rohit sold the shares when the price rose to Rs 160. He invested the proceeds (excluding dividend) in 10% Rs. 50 shares at Rs. 40. Find the:

1) original number of shares

2) sale proceeds

3) new number of shares

4) change in the two dividends.

Concept: Shares and Dividends
Chapter: [0.012] Shares and Dividends
[10]9
[4]9.1

The horizontal distance between two towers is 120 m. The angle of elevation of the top and angle of depression of the bottom of the first tower as observed from the second tower is 30° and 24° respectively.

Find the height of the two towers. Give your answer correct to 3 significant figures

Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables
Chapter: [0.05] Trigonometry
[6]9.2

The weight of 50 workers is given below:

Weight in Kg 50-60 60-70 70-80 80-90 90-100 100-110 110-120
No. of Workers 4 7 11 14 6 5 3

Draw an ogive of the given distribution using a graph sheet. Take 2 cm = 10 kg on one axis and 2 cm = 5 workers along the other axis. Use a graph to estimate the following:

1) The upper and lower quartiles.

2) If weighing 95 kg and above is considered overweight, find the number of workers who are overweight.

Concept: Ogives (Cumulative Frequency Graphs)
Chapter: [0.06] Statistics
[10]10
[3]10.1

A wholesaler buys a TV from the manufacturer for Rs. 25,000. He marks the price of TV 20% above his cost price and sells it to a retailer at a 10% discount on the market price. If the rate of the VAT is 8%, find the :

1) Market price

2) Retailer’s cost price inclusive of tax.

3) VAT paid by the wholesaler.

Concept: Introduction to Sales Tax and Value Added Tax
Chapter: [0.013999999999999999] Gst (Goods and Services Tax)
[3]10.2

if `A [(3,7),(2,4)], B = [(0,2),(5,3)]`  and `C = [(1,-5),(-4,6)]` Find AB - 5C

Concept: Matrices Examples
Chapter: [0.0208] Matrices
[4]10.3

ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that:

1) ΔADE ~ ΔACB

2) If AC = 13 cm, BC = 5 cm and AE = 4 cm. Find DE and AD.

3) Find. Area of ΔADE: area of quadrilateral BCED

Concept: Similarity of Triangles
Chapter: [0.035] Similarity
[10]11
[3]11.1

Sum of two natural numbers is 8 and the difference of their reciprocal is `2/15`. Find the numbers.

Concept: Quadratic Equations
Chapter: [0.0202] Quadratic Equations
[3]11.2

Given `(x^3 + 12x)/(6x^2 + 8) = (y^3+ 27y)/(9y^2 + 27)`. Using componendo and dividendo find x : y.

Concept: Componendo and Dividendo Properties
Chapter: [0.0204] Ratio and Proportion
[4]11.3

Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 105°

Hence:

1) Construct the locus of points equidistant from BA and BC

2) Construct the locus of points equidistant from B and C.

3) Mark the point which satisfies the above two loci as P. Measure and write the length of PC.

Concept: Constructions Under Loci
Chapter: [0.031] Loci

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