Mathematics 2015-2016 ICSE Class 10 Question Paper Solution

Mathematics
Date & Time: 14th March 2016, 11:00 am
Duration: 2h30m

Question 1 to Question 4 is Compulsory

Attempt any four questions from Question 5 to Question 11

[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

Concept: Remainder Theorem
Chapter: [0.0203] Factorization
[4]1.2

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

Concept: Matrices Examples
Chapter: [0.0208] Matrices
[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.

Concept: Median of Grouped Data
Chapter: [0.06] Statistics
[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).

Concept: Slope of a Line
Chapter: [0.021] Co-ordinate Geometry Equation 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^@

Concept: Trigonometry
Chapter: [0.05] Trigonometry
[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.

Concept: Surface Area of a Sphere
Chapter: [0.04] Mensuration [0.04] Mensuration
[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

Concept: Representation of Solution on the Number Line
Chapter: [0.020499999999999997] Linear Inequations
[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  angle OBD

Concept: Number of Tangents from a Point on a Circle
Chapter: [0.032] Circles
[3]3.3

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

Concept: Concept of Ratio
Chapter: [0.0204] Ratio and Proportion [0.0204] Ratio and Proportion
[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

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

Concept: Concept of Lines Symmetry
Chapter: [0.034] 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.

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

Concept: Bank
Chapter: [0.013000000000000001] Banking
[3]5.3

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

Concept: Concept of Compound Interest - Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years
Chapter: [0.011000000000000001] Compound Interest
[10]6
[3]6.1

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

Concept: Circumscribing and Inscribing a Circle on a Triangle
Chapter: [0.033] Constructions
[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.

Concept: Cyclic Properties
Chapter: [0.032] Circles
[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’.

Concept: Matrices Examples
Chapter: [0.0208] Matrices
[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

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

Concept: Finding the Median, upper quartile, lower quartile from the Ogive
Chapter: [0.06] Statistics
[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)

Concept: Trigonometric Identities
Chapter: [0.05] Trigonometry
[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.

Concept: Construction of Tangents to a Circle
Chapter: [0.033] Constructions
[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.

Concept: Equation of a Line
Chapter: [0.021] Co-ordinate Geometry 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

Concept: Sales Tax, Value Added Tax, and Good and Services Tax
Chapter: [0.013999999999999999] Gst (Goods and Services 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:

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

Concept: Surface Area of a Sphere
Chapter: [0.04] Mensuration [0.04] Mensuration
[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:
(i) the monthly instalment
(ii) the amount of maturity.

Concept: Bank
Chapter: [0.013000000000000001] Banking
[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

Concept: Graphical Representation of Data as Histograms
Chapter: [0.06] Statistics [0.06] Statistics
[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’.

Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables
Chapter: [0.05] Trigonometry
[10]11
[3]11.1

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

Concept: Trigonometric Identities
Chapter: [0.05] Trigonometry
[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.

Concept: Circumscribing and Inscribing a Circle on a Triangle
Chapter: [0.033] Constructions

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.

Concept: Loci Examples
Chapter: [0.031] Loci

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.

Concept: Loci Examples
Chapter: [0.031] Loci

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.

Concept: Loci Examples
Chapter: [0.031] Loci
[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

Concept: Shares and Dividends
Chapter: [0.012] Shares and Dividends

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