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Question Paper Solutions - Mathematics 2016 - 2017 ICSE Class 10

SubjectMathematics
Year2016 - 2017 (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

If b is the mean proportion between a and c, show that `(a^4 + a^2b^2 + b^4)/(b^4 + b^2c^2 + c^4) = a^2/c^2`

Chapter: [2.04] Ratio and Proportion
Concept: Proportions
[3]1.2

Solve the equation 4x2 – 5x – 3 = 0 and give your answer correct to two decimal places

Chapter: [2.03] Quadratic Equations
Concept: Quadratic Equations
[4]1.3

AB and CD are two parallel chords of a circle such that AB = 24 cm and CD = 10 cm. If the
radius of the circle is 13 cm. find the distance between the two chords.

Chapter: [3.04] Circles
Concept: Chord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof)
[10]2
[3]2.1

Evaluate without using trigonometric tables, 

`sin^2 28^@ + sin^2 62^@ + tan^2 38^@ - cot^2 52^@ + 1/4 sec^2 30^@`

Chapter: [5] Trigonometry
Concept: Trigonometry Problems and Solutions
[3]2.2

if A = `[(1,3),(3,4)]` and B = `[(-2,1),(-3,2)]` and A2 - 5B2 = 5C. Find matrix C, where C is a 2 by 2

Chapter: [2.06] Matrices
Concept: Matrices Examples
[4]2.3

Jaya borrowed Rs. 50,000 for 2 years. The rates of interest for two successive years are 12% and 15% respectively. She repays 33,000 at the end of the first year. Find the amount she must pay at the end of the second year to clear her debt.

Chapter: [1.01] Compound Interest
Concept: Compound Interest as a Repeated Simple Interest Computation with a Growing Principal
[10]3
[3]3.1

The catalogue price of a computer set is Rs. 42,000. The shopkeeper gives a discount of
10% on the listed price. He further gives an off-season discount of 5% on the discounted
price. However, sales tax at 8% is charged on the remaining price after the two successive
discounts. Find

1)the amount of sales tax a customer has to pay

2)the total price to be paid by the customer for the computer set.

Chapter: [1.02] Sales Tax and Value Added Tax
Concept: Introduction to Sales Tax and Value Added Tax
[3]3.2

P(1, -2) is a point on the line segment A(3, -6) and B(x, y) such that AP : PB is equal to 2 : 3. Find the coordinates of B.

Chapter: [2.07] Co-ordinate Geometry
Concept: Co-ordinates Expressed as (x,y)
[4]3.3

The marks of 10 students of a class in an examination arranged in ascending order is as follows:

13, 35, 43, x, x + 4, 55, 61, 71, 80

If the median marks is 48, find the value of x. Hence find the mode of the given data.

Chapter: [6] Statistics
Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data
[10]4
[3]4.1

What must be subtracted from 16x3 – 8x2 + 4x + 7 so that the resulting expression has
2x + 1 as a factor?

Chapter: [2.02] Quadratic Equations
Concept: Solutions of Quadratic Equations by Factorization
[3]4.2

In the given figure ABCD is a rectangle. It consists of a circle and two semi-circles each of
which are of radius 5 cm. Find the area of the shaded region. Give your answer correct to
three significant figures

Chapter: [2.07] Co-ordinate Geometry
Concept: Simple Applications of All Co-ordinate Geometry.
[4]4.3

Solve the following inequation and represent the solution set on a number line.

`-8 1/2 < -1/2 -4x <= 7 1/2`, x ∈ 1

Chapter: [2.07] Co-ordinate Geometry
Concept: Equation of a Line
[10]5
[3]5.1

Given matrix B = `[(1,1),(8,3)]`, find the matrix X if X = B2 - 4B. Hence solve for a and b given `X[(a),(b)] = [(5),(50)]`

Chapter: [2.06] Matrices
Concept: Matrices Examples
[3]5.2

How much should a man invest in Rs. 50 shares selling at Rs. 60 to obtain an income of Rs. 450, if the rate of dividend declared is 10%. Also find his yield percent, to the nearest whole number.

Chapter: [1.04] Shares and Dividends
Concept: Shares and Dividends Examples
[4]5.3

Sixteen cards are labelled as a, b, c, ……………. m, n, o, p. They are put in a box and shuffled.a boy is asked to draw a card from the box. What is the probability that the card drawn is:

(1) a vowel

(2) a consonant

(3) none of the letters of the word median

Chapter: [5.02] Probability
Concept: Simple Problems on Single Events
[10]6
[3]6.1

Using a ruler and a compass construct a triangle ABC in which AB = 7 cm, CAB = 60o and
AC = 5 cm. Construct the locus of

1) points equidistant from AB and AC

2) points equidistant from BA and BC

Hence construct a circle touching the three sides of the triangle internally.

Chapter: [3.05] Constructions
Concept: Circumscribing and Inscribing a Circle on a Triangle
[3]6.2

A conical tent is to accommodate 77 persons. Each person must have 16 m3 of air to breathe. Given the radius of the tent as 7 m, find the height of the tent and also its curved surface area.

Chapter: [5] Trigonometry
Concept: Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables
[4]6.3

if `(7m + 2n)/(7m - 2n) = 5/3`use properties of proportion to find

1) m:n

2)`(m^2 + n^2)/(m^2 - n^2)`

Chapter: [2.04] Ratio and Proportion
Concept: Componendo and Dividendo Properties
[10]7
[4]7.1

A page from a savings bank account passbook is given below:

Date Particulars Amount
withdrawn (Rs.)
Amount
Deposited (Rs.)
Balance
(Rs.)
Jan 7, 2016 B/F     3,000.00
Jan 10, 2016 By Cheque   2600.00 5600.00
Feb 8, 2016 To Self 1500.00   4100.00
Apr 6, 2016 By Cheque 2100.00   2000.00
May 4, 2016 By Cash   6500.00 8500.00
May 27, 2016 By Cheque   1500.00 10000.00

1) Calculate the interest for the 6 months from January to June 2016, at 6% per annum.

2) If the account is closed on 1st July 2016, find the amount received by the account holder.

Chapter: [1.03] Banking
Concept: Computation of Interest
[6]7.2

Use a graph paper for this question (Take 2 cms = 1 unit on both x and y-axis)

1) Plot the following points:A(0, 4), B(2, 3), C(1, 1) and D(2, 0)

2) Reflect points B, C, D on the y-axis and write down their coordinates. Name the images as B’, C’, D’ respectively

3) Join the points A, B, C, D, D’, C’, B’ and A in order, so as to form a closed figure. Write
down the equation of the line of symmetry of the figure formed

 

Chapter: [3.01] Symmetry
Concept: Lines of Symmetry
[10]8
[3]8.1

Calculate the mean of the following distribution using step deviation method.

Marks 0-10 10-20 20-30 30-40 40-50 50-60
Number of
students
10 9 25 0 16 10
Chapter: [6] Statistics
Concept: Measures of Central Tendency - Mean, Median, Mode for Raw and Arrayed Data
[3]8.2

In the given figure PQ is a tangent to the circle at A, AB and AD are bisectors of `angleCAQ` and `angle PAC`. if `angleBAQ = 30^@. prove that:

1) BD is a diameter of the circle

2) ABC is an isosceles triangle

Chapter: [3.05] Constructions
Concept: Construction of Tangents to a Circle
[4]8.3

The printed price of an air conditioner is Rs. 45000/-. The wholesaler allows a discount of 10%
to the shopkeeper. The shopkeeper sells the article to the customer at a discount of 5% of the
marked price. Sales tax (under VAT) is charged at the rate of 12% at every stage. Find:

1) VAT paid by the shopkeeper to the government

2) The total amount paid by the customer inclusive of tax.

Chapter: [1.02] Sales Tax and Value Added Tax
Concept: Introduction to Sales Tax and Value Added Tax
[10]9
[3]9.1

In the figure given, O is the centre of the circle. `angle DAE = 70^@`, Find giving suitable reasons the measure of:

1) `angle BCD`

2) `angle BOD`

3) `angle OBD`

Chapter: [2] Circle
Concept: Cyclic Properties
[3]9.2

A(-1, 3), B(4, 2) and C(3, -2) are the vertices of a triangle.

1) Find the coordinates of the centroid G of the triangle

2) Find the equation of the line through G and parallel to AC

Chapter: [2.07] Co-ordinate Geometry
Concept: Equation of a Line
[4]9.3

Prove the following identities, where the angles involved are acute angles for which the expressions are defined

`(sin theta-2sin^3theta)/(2cos^3theta -costheta) = tan theta`

Chapter: [5] Trigonometry
Concept: Trigonometric Identities
[10]10
[4]10.1

The sum of the ages of Vivek and his younger brother Amit is 47 years. The product of their
ages in years is 550. Find their ages.

Chapter: [2.03] Quadratic Equations
Concept: Nature of Roots
[6]10.2

The daily wages of 80 workers in a project are given below.

Wages
(in Rs.)
400-450 450-500 500-550 550-600 600-650 650-700 700-750
No. of
Workers
2 6 12 18 24 13 5

Use a graph paper to draw an ogive for the above distribution. (Use a scale of 2 cm = Rs.
50 on x-axis and 2 cm = 10 workers on y-axis). Use your ogive to estimate:

1) the median wage of the workers

2) the lower quartile wage of workers

3) the numbers of workers who earn more than Rs. 625 daily

Chapter: [5] Statistics
Concept: Ogives (Cumulative Frequency Graphs)
[10]11
[3]11.1

The angles of depression of two ships A and B as observed from the top of a lighthouse 60 m high are 60° and 45° respectively. If the two ships are on the opposite sides of the lighthouse, find the distance between the two ships, Give your answer correct to the nearest whole number.

Chapter: [5] Trigonometry
Concept: Trigonometric Identities
[3]11.2

PQR is a triangle. S is a point on the side QR of ΔPQR such that `angle PSR = angle QPR`. Given QP = 8 cm,PR = 6 cm and SR = 3 cm.

1) Prove ΔPQR ~ ΔSPR

2) Find the length of QR and PS

3) `" area of ΔPQR"/"area of ΔSPR"`

Chapter: [3.02] Similarity
Concept: Axioms of Similarity of Triangles
[4]11.3

Mr Richard has a recurring deposit account in a bank for 3 years at 7.5% p. a. simple interest. If he gets Rs. 8325 as interest at the time of maturity, find

1) The monthly deposit

2) The maturity value

Chapter: [1.03] Banking
Concept: Types of Accounts
S