MATHEMATICS (ICSE Paper 2014)
Two and Half Hour. Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during the first 15 minutes. This time is to be spent in reading the question paper.
The time given at the head of this Paper is the time allowed for writing the answers. Attempt all questions form Section A and any four questions from Section B. All working, including rough work, must be clearly shown and must be done on the same sheet as the rest of the Answer. Omission of essential working will result in the loss of marks.
The intended marks for questions or parts of questions are given in brackets [ ].
Mathematical tables are provided.
SECTION A [40 Marks]
(Answer all questions from this Section.)
Question 1:
(a) Given . Find the matrix such that . [3]
(b) At what rate p.a. will a sum of yield as compound interest in years? [3]
(c) The mediam of the following observation arranged in ascending order is . Find the value of and hence find the mean. [4]
Answers:
(a) Given
Substituting these values in the given expression we get,
(b) Given: Principle
Amount
Time
We know that;
Therefore,
(c) Given observation are and mediam
Since which is odd, therefore
Therefore,
Now, Mean
Question 2:
(a) What number must be added to each of the number to make them proportional? [3]
(b) If is a factor of the expression and, when the expression is divided by , it leaves a remainder , find the values of . [3]
(c) Draw a histogram from the following frequency distribution and find the made from the graph: [4]
Class  05  510  1015  1520  2025  2530 
Frequency  2  5  18  14  8  5 
Answers:
(a) Let the number that must be added be , then
The new number
Since they are proportional,
(b) Let is a factor of the given expression;
Since
In the given expression, we substitute we get
… … … … … (i)
When given expression is divided by
Similarly, in the given expression, we substitute we get
… … … … … (ii)
Solving equation (i) and (ii),
(c)
Question 3:
(a) Without using tables evaluate: [3]
(b) In the given feature,
, ,
Prove:
(i) AC is the diameter of the circle
(ii) Find [3]
(c) is a diameter of a circle with center , If , Find;
(i) The length of radius
(ii) The Coordinates of [4]
Answers:
(a)
(b) Given:
(i) Since is a cyclic quadrilateral
In
( sum property of a triangle)
Now from ,
Hence makes right angle belongs in semicircle therefore is a diameter of the circle.
(ii) (Angles in the same segment of a circle)
Therefore
(c) (i) Length of the radius
(ii) Let the point be
Given is the midpoint of . Therefore
Hence, the coordinate of
Question 4:
(a) Solve the following equation and calculate the answer correct to two decimal places. . [3]
(b) In the given figure, and are perpendicular to
(i) Prove that
(ii) If , Calculate ,
(iii) Find the ratio of the area of [3]
(c) Using graph paper, plot the point and .
(i) Reflect and in the origin to get the images and .
(ii) Write the coordinate of and
(iii) State the geometrical name for the figure
(iv) Find its perimeter [4]
Answers:
(a) Given :
Comparing this expression with , we get
(b) (i) From
Givem
And
(ii) In
Since
Given: ,
(iii) Since $latex \triangle ABC \sim \triangle DEC,
(c) (i) Please see graph
(ii) Reflection of and in the origin
(iii) The geometrical name for the figure is a parallelogram
(iv) From the graph,
In
Therefore since is a parallelogram
Perimeter of
SECTION B [40 Marks]
(Answer any four questions in this Section.)
Question: 5
(a) Solve the following inequation, write the solution set and represent it on the number line: [3]
(b) Mr. Britto deposits a certain sum of money each month in a Recurring Deposit Account of a bank. If the rate of interest is of per annum and Mr. Britto gets from the bank after years, find the value of his monthly installment. [3]
(c) Salman buys shares of face value available at .
(i) What is his investment?
(ii) If the dividend is what will be his annual income?
(iii) If he wants to increase his annual income by . How many extra shares should he buy? [4]
Answers:
(a) Given;
Taking L.C.M. of 3, 2 and 6 is 6.
(b) Let the monthly installment
Given: Maturity amount
Interest
Actual sum deposited
Maturity amount = Interest +Actual sum deposited
Hence the monthly installment be
(c) Number of shares
Face value of each share
Market value of each shares
Total face value
(i) Total investment
(ii) Rate of dividend
Annual income
(iii) Let extra share should he buy be
Then total number of shares
Total face value
Annual income
Hence the extra shares should be buy
Question 6:
(a) Show that [3]
(b) In the given circle with center and is a tangent to the circle at . Find . [3]
(c) Given below are the entries in a Saving Bank A/C pass book
Date  Particular  Withdrawal  Deposit  Balance 
Feb. 8
Feb .18 April. 12 June.15 July. 8 
B/F
To Self By Cash To Self By Cash

–
Rs.4000 – Rs.5000 – 
–
– Rs.2230 – Rs.6000 
Rs.8500
– – – – 
Calculate the interest for six months from February to July at . [4]
Answers:
(a) LHS
= RHS. Hence Proved.
(b) Given:
We know that,
(sum of the opposite angles in a cyclic quadrilateral is )
Join , we have a isosceles
(Radius of the same circle)
(angle in a semi circle)
In
Since (in a triangle, angles opposite to equal sides are equal)
Hence,
Since
The tangent at a point to circle is perpendicular to the radius through the point to contact.
(c)
Date  Particular  Withdrawal  Deposit  Balance 
Feb. 8
Feb .18 April. 12 June.15 July. 8 
B/F
To Self By Cash To Self By Cash 
–
Rs. 4000 – Rs. 5000 – 
–
– Rs. 2230 – Rs. 6000 
Rs. 8500
Rs. 4500 Rs. 6730 Rs. 1730 Rs. 7730 
Principle for the month of Feb = Rs. 4500
Principle for the month of March = Rs. 4500
Principle for the month of April = Rs. 4500
Principle for the month of May = Rs. 6730
Principle for the month of June = Rs. 1730
Principle for the month of July = Rs. 7730
Total principle from the month of Feb. to July = Rs. 29690
Time
Rate of interest
Interest
Question 7:
(a) In . Find the equation of the equation of the median through A. [3]
(b) A shopkeeper sells an article at the listed price of and the rate of VAT is at each stage of sale. If the shopkeeper pays a VAT of to the Government. What was the price, inclusive of Tax, at which the shopkeeper purchased the article from the wholesaler? [3]
(c) In the figure given, from the top of a building high, the angles of depression of the top and bottom of a vertical lamp post are observed to be and respectively. [4]
Find:
(i) The horizontal distance between and
(ii) The height of the lamp paid.
Answers:
(a) Since is midpoint of
The coordinate of
Now equation of mediam ,
Here,
(b) Given: List price of an article , Rate of
VAT on the article
Let C.P. of this article be , then
If the shopkeeper pays a VAT =Rs.36
Then,
Therefore the price at which the shopkeeper purchased the article inclusive of sales tax
(c) Given; AB=60 m
Since
(alternate angles, AP and BC are parallel)
(i) Now in
Hence the horizontal distance between
(ii) Let and proved above
Therefore
In
Since
Hence, the height of the lamp post
Question 8:
(a) Find if [3]
(b) A solid sphere of radius 15 cm is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate the number of comes recast. [3]
(c) Without solving the following quadratic equation, find the value of for which the given equation has real and equal roots;
[4]
Answers:
(a) Given
Therefore
and
Hence
(b) Radius of a solid sphere,
Volume of solid sphere
Now, radius of right circular come
Height
Volume of right circular cone
Therefore The number of cones s=1$
(c) Given equation
Since roots are real and equal, … … … … … (i)
Comparing the coefficient of with equation , we get
Substituting the values in (i) we get
Question 9:
(a) In the figure along side is a quadrant of a circle, The radius , Calculate the area of the shaded portion. (Take ) [3]
(b) A box contain some black balls and 30 white balls. If the probability of drawing a black ball is twofifths of a white ball, find the number of black balls in the box. [3]
(c) Find the mean of the following distribution by step deviation method: [4]
Class Internal  2030  3040  4050  5060  6070  7080 
Frequency  10  6  8  12  5  9 
Answers:
(a) Radius of quadrant
Area of quadrant
Here,
and
Then area of
Area of shaded portion =Area of quadrant – Area of triangle
(b) Let the number of black balls be , then
Total number of balls
Thus, the probability of blackballs
And the probability of white balls
Given, Probability of black ball
Therefore
Hence, the number of black balls
(c)
C.I  Frequency  MidValue  
2030
3040 4050 5060 6070 7080 
10
6 8 12 5 9 
25
35 45 55 65 75 
2
1 0 1 2 3 
20
6 0 12 10 27 
Here, and
Mean
Question:10
(a) Using a ruler and compasses only:
(i) Construct a with the following data:
(ii) In the some diagram draw a circle with as diameter. Find a point on the circumstance of the circle which is equidistant from and .
(iii) Measure [4]
(b) The mark obtained by 120 students in a test are given below;
Marks  010  1020  2030  3040  4050  5060  6070  7080  8090  90100 
Students  5  9  16  22  26  18  11  6  4  3 
Draw an ogive for the given distribution on a graph sheet; Using suitable scale for ogive to estimate the following:
(i) The mediam
(ii) The number of students who obtained more than 75% marks in the test.
(iii) The number of students who did not pass the test if minimum marks required to pass is 40. [6]
Answers:
(a) Steps of Construction:
 Using a ruler draw a line
 With the help of the point , draw . You could do it by drawing an arc with B as the center. Then cut the arc twice using the same radius as set in the compass.
 Given that the length of AB = 3.5 cm. Taking radius cut . This gives you point A.
 Now join to with the help of a ruler.
 We now need to draw a perpendicular bisector of BC. This can be done by taking a certain length in the compass and make arcs as shown in the diagram keeping point B and C as the center. Make sure that you keep the width of the compass same for all the four arcs.
 The join the two points of intersection. This gives the perpendicular bisector. Draw perpendicular bisector of .
 Draw a circle as center and as radius.
 Now draw angle bisector of which intersects circle at
 Join and
 Now,
(b)
Marks  No. of students  Cumulative Frequency 
010
1020 2030 3040 4050 5060 6070 7080 8090 90100 
5
9 16 22 26 18 11 6 4 3 
5
14 30 52 78 96 107 113 117 120 
N=120 
On the graph paper we plot the following points:
(i) Mediam
From the graph 60^{th} term
(ii) The number of students who obtained more than marks in test
(iii) The number of students who did not pass the test if the minimum pass marks
Question 11:
(a) In the figure given below the segment meets at and at . The point on divides it in the ration find the coordinates of and . [3]
(b) Using the properties of proportion solve for , given [3]
(c) A shopkeeper purchases a certain number of books for . If the cost per book was less, the number of books that could be purchased for would be more. Write an equation, taking the original cost of each book to be , and solve it to find the original cost of the books. [4]
Answers:
(a) Let the coordinates of
Since the coordinates of a point divides it in the ratio it implies that
By using section formula, we get
$latex \frac{2 \times 0 + 3 \times x}{2+3} = \frac{3x}{5} &s=1$
Similarly,
Hence, the coordinate of
(b) Given
By using componendo and dividendo, we get:
Taking square root on both sides, we get
(c) Given the original cost of each book be
Total cost
Therefore the number of books for
If the cost per book was less, ) then
Number of books
According to question,
(not possible)
Hence the cost of the original book is
nice….
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