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# Question Paper Solutions - Mathematics 2015 - 2016 ICSE Class 10

SubjectMathematics
Year2015 - 2016 (March)

Question 1 to Question 4 is Compulsory

Attempt any four questions from Question 5 to Question 11

Marks: 80
[10]1
[3]1.1

Using remainder theorem, find the value of k if on dividing 2x3 + 3x2 - kx + 5 by x - 2, leaves a remainder 7

Chapter: [2.05] Factorization
Concept: Remainder Theorem
[4]1.2

Given A = [(2,0),(-1,7)] and I = [(1,0),(0,1)] and A^2 = 9A+ ml. Find m

Chapter: [2.06] Matrices
Concept: Matrices Examples
[3]1.3

The mean of following numbers is 68. Find the value of ‘x’.

45, 52, 60, x, 69, 70, 26, 81 and 94

Hence estimate the median.

Chapter: [5] Statistics
Concept: Median of Grouped Data
[10]2
[3]2.1

The slope of a line joining P(6, k) and Q (1-3k, 3) is 1/2 Find

1) k

2) A midpoint of PQ, using the value of ‘k’ found in (1).

Chapter: [2.07] Co-ordinate Geometry
Concept: Slope of a Line
[4]2.2

Without using trigonometrical tables, evaluate:

cosec^2 57^@ - tan^2 33^@ + cos 44^@ cosec 46^@ - sqrt2 cos 45^@ -  tan^2 60^@

Chapter: [5] Trigonometry
Concept: Trigonometry Problems and Solutions
[3]2.3

A certain number of metallic cones, each of radius 2 cm and height 3 cm are melted and recast into a solid sphere of radius 6 cm. Find the number of cones.

Chapter: [4] Mensuration
Concept: Area and Volume of Solids - Sphere
[10]3
[3]3.1

Solve the following inequation, write the solution set and represent it on the number line.

-3(x - 7) >= 15 - 7x > (x+1)/3, x ∉ R

Chapter: [2.01] Linear Inequations
Concept: Representation of Solution on the Number Line
[4]3.2

In the figure given below, an AD is a diameter. O is the centre of the circle

AD is parallel to BC and angle CBD = 32°. Find

1) angle OBD

2) angle AOB

3) angle BED

Chapter: [3.05] Constructions
Concept: Circumscribing and Inscribing a Circle on a Triangle
[3]3.3

If (3a + 2b) : (5a + 3b) = 18 : 29. Find a : b

Chapter: [2.04] Ratio and Proportion
Concept: Ratio and Proportion Example
[10]4
[3]4.1

A game of numbers has cards marked with 11, 12, 13, … 40. A card is drawn at random. Find the probability that the number on the card drawn is :

1) A perfect square

2) Divisible by 7

Chapter: [7] Probability
Concept: Simple Problems on Single Events
[4]4.2

Use graph paper for this question. (Take 2 cm = 1 unit along both x and y-axis.) Plot the points O(0, 0), A(-4, 4), B(-3, 0) and C(0, -3)

1) Reflect points A and B on the y-axis and name them A’ and B’ respectively. Write down their coordinates.

2) Name the figure OABCB’A’.

3) State the line of symmetry of this figure

Chapter: [3.01] Symmetry
Concept: Lines of Symmetry
[3]4.3

Mr Lalit invested Rs. 5000 at a certain rate of interest, compounded annually for two years. At the end of the first year, it amounts to Rs. 5325. Calculate

1) The rate of interest

2) The amount at the end of the second year, to the nearest rupee.

Chapter: [1.01] Compound Interest
Concept: Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years
[10]5
[3]5.1

Solve the quadratic equation x2 -  3(x + 3) = 0; Give your answer correct two significant figures

[4]5.2

A page from the savings bank account of Mrs Ravi is given below.

 Date Particulars Withdrawal(Rs.) Deposit (Rs.) Balance (Rs.) April 3rd 2006 B/F 6000 April 7th By cash 2300 8300 April 15th By cheque 3500 11800 May 20th To self 4200 7600 June 10th By cash 5800 13400 June 15th To self 3100 10300 August 13th By cheque 1000 11300 August 25th To self 7400 3900 September 6th2006 By cash 2000 5900

She closed the account on 30th September 2006. Calculate the interest Mrs Ravi earned
at the end of 30th September 2006 at 4.5% per annum interest. Hence, find the
amount she receives on closing the account.

Chapter: [1.03] Banking
Concept: Types of Accounts
[3]5.3

In what time will Rs. 1500 yield Rs. 496.50 as compound interest at 10% per annum compounded annually?

Chapter: [1.01] Compound Interest
Concept: Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years
[10]6
[3]6.1

Construct a regular hexagon of side 5 cm. Hence construct all its lines of symmetry and name them.

Chapter: [3.01] Symmetry
Concept: Lines of Symmetry
[4]6.2

In the given figure PQRS is a cyclic quadrilateral PQ and SR produced meet at T

1) Prove ΔTPS ~ ΔTRQ.

2) Find SP if TP = 18 cm, RQ = 4 cm and TR = 6 cm

3) Find the area of quadrilateral PQRS if the area of ΔPTS = 27 cm2.

Chapter: [3.04] Circles
Concept: Cyclic Properties
[3]6.3

Given matrix A = [(4sin30^@, cos0^@),(cos0^@, 4sin30^@)]and B = [(4),(5)]

if AX = B

1) Write the order of matrix X.

2) Find the matrix ‘X’.

Chapter: [2.06] Matrices
Concept: Matrices Examples
[10]7
[4]7.1

An aeroplane at an altitude of 1500 metres, finds that two ships are sailing towards it in the same direction. The angles of depression as observed from the aeroplane are 45° and 30° respectively. Find the distance between the two ships

Chapter: [5] Trigonometry
Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables
[6]7.2

The table shows the distribution of the scores obtained by 160 shooters in a shooting competition. Use a graph sheet and draw an ogive for the distribution. (Take 2 cm = 10 scores on the X-axis and 2 cm = 20 shooters on the Y-axis).

 Scores 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 No. ofshooters 9 13 20 26 30 22 15 10 8 7

Use your graph to estimate the following:

1) The median

2) The interquartile range.

3) The number of shooters who obtained a score of more than 85%.

Chapter: [6] Statistics
Concept: Median from the Ogive
[10]8
[3]8.1

If x/a=y/b = z/c show that x^3/a^3 + y^3/b^3 + z^3/c^3 = (3xyz)/(abc)

Chapter: [5] Trigonometry
Concept: Trigonometric Identities
[4]8.2

Draw a line AB = 5 cm. Mark a point C on AB such that AC = 3 cm. Using a ruler and a compass only, construct :

1) A circle of radius 2.5 cm, passing through A and C.

2) Construct two tangents to the circle from the external point B. Measure and record the length of the tangents.

Chapter: [3.03] Constructions
Concept: Construction of Tangents to a Circle
[3]8.3

A line AB meets X – axis at A and Y –axis at B. P (4, -1) divides AB in the ratio 1 : 2.

1) Find the coordinates of A and B.

2) Find the equation of the line through P and perpendicular to AB.

Chapter: [2.07] Co-ordinate Geometry
Concept: Equation of a Line
[10]9
[3]9.1

A dealer buys an article at a discount of 30% from the wholesaler, the marked price being Rs. 6000. The dealer sells it to a shopkeeper at a discount of 10% on the marked price. If the rate of the VAT is 6%, find

1) The price paid by the shopkeeper including the tax.

2) The VAT paid by the dealer

Chapter: [1.02] Sales Tax and Value Added Tax
Concept: Introduction to Sales Tax and Value Added Tax
[4]9.2

The given figure represents a kite with a circular and a semicircular motifs stuck on it.
The radius of a circle is 2.5 cm and the semicircle is 2 cm. If diagonals AC and BD are
of lengths 12 cm and 8 cm respectively, find the area of the:

Chapter: [2.07] Co-ordinate Geometry
Concept: Simple Applications of All Co-ordinate Geometry.
[3]9.3

A model of a ship is made to a scale 1: 300

1) The length of the model of the ship is 2 m. Calculate the lengths of the ship.

2) The area of the deck ship is 180,000 m2. Calculate the area of the deck of the model.

3) The volume of the model in 6.5 m3. Calculate the volume of the ship.

Chapter: [4] Mensuration
Concept: Area and Volume of Solids - Sphere
[10]10
[3]10.1

Mohan has a recurring deposit account in a bank for 2 years at 6% p.a. simple interest. If he gets Rs. 1200 as interest at the time of maturity find:

1) the monthly instalment

2) the amount of maturity

Chapter: [1.03] Banking
Concept: Types of Accounts
[4]10.2

The histogram below represents the scores obtained by 25 students in a mathematics mental test. Use the data to :

1) Frame a frequency distribution table

2) To calculate mean

3) To determine the Modal class

Chapter: [5] Statistics
Concept: Graphical Representation of Histograms
[3]10.3

A bus covers a distance of 240 km at a uniform speed. Due to heavy rain, its speed gets reduced by 10 km/h and as such it takes two hrs longer to cover the total distance. Assuming the uniform speed to be ‘x’ km/h, form an equation and solve it to evaluate ‘x’.

Chapter: [5] Trigonometry
Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables
[10]11
[3]11.1

Prove that cosA/(1+sinA) + tan A =  secA

Chapter: [5] Trigonometry
Concept: Trigonometric Identities
[4]11.2

Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown

Construct a  ABC in which BC = 6.5 cm,  ABC = 60°, AB = 5 cm.

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

Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown.

Construct the locus of points at a distance of 3.5 cm from A.

Chapter: [3.03] Loci
Concept: Loci Examples

Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown

Construct the locus of points equidistant from AC and BC.

Chapter: [3.03] Loci
Concept: Loci Examples

Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown

Mark 2 points X and Y which are a distance of 3.5 cm from A and also equidistant from AC and BC. Measure XY.

Chapter: [3.03] Loci
Concept: Loci Examples
[3]11.3

Ashok invested Rs. 26,400 on 12%, Rs. 25 shares of a company. If he receives a dividend of Rs. 2,475. Find the :

1) number of shares he bought

2) The market value of each share

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