MATHEMATICS (ICSE 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) Ranbir borrows at per annum compound interest. If he repays at the end of the first year and at the end of the second year, find the amount of loan outstanding at the beginning of the third year. [3]
(b) Find the value of , which satisfy the in equation . Graph the solution set on the number line. [3]
(c) A die has 6 faces marked by the given numbers as shown below: . The die is thrown once. What is the probability of getting?
(i) A positive integer
(ii) An integer greater than
(iii) The smallest integer [4]
Answer.
(a) Given: Principal for the first year
We known that [Reference Link]
Amount after the 1st year
Money repaid at the end of 1st year
Principle for the 2nd year
Amount after 2nd year
Money repaid at the end of the second year
The loan amount outstanding at the beginning of the third year
(b) Given
Multiplying throughout by 6
Therefore
Hence the solution set is
Therefore the values of are
The graph of the solution set is shown by dots on the number line.
(c) No. of sample space
A positive integer
No. of favorable cases
Probability
An integer greater than
No. of favorable cases
Probability
Smallest integer
Probability of smallest integer
Question 2:
(a) Find if [3]
(b) Sharukh opened a Recurring Deposit Account in a bank and deposited Rs. 800 per month for years. If he received Rs. 15,084 at the time of maturity. Find the rate of interest per annum. [3]
(c) Calculate the ratio in which the line joining is divided by point Also find (i) (ii) Length of . [4]
Answer:
(a) Given
(b) Here, = money deposited per month
Time for which the money is deposited =
Let the rate of interest be per annum, then
Total money deposited
Since money deposited +Interest = Maturity value
Hence rate of interest
(c) Let divide the line segment joining the points and in the ratio
Coordinate of is: [Reference Link]
But Coordinate of
The required ratio is , (Divides Internally)
(i) Therefore
Substituting
(ii) Coordinate of is
Length of
Question 3:
(a) Without using trigonometric tables, evaluate
[3]
(b) Using the remainder and factor theorem, factor the following polynomial: [3]
(c) In the figure given below is a rectangle . From the rectangle a quarter circle and a semicircle are removed Calculate the area of the remaining piece of the rectangle (Take ) [4]
Answers:
(a) Given
(b) Let
Putting , we get
By factor theorem is actor of
On dividing by , we get as the quotient and remainder
Therefore the other factor of are the factor of
Now,
Hence
(c) Area of rectangle
Area of quarter circle
Area of semicircle
Area of remaining piece of rectangle
Question 4:
(a) The number and are arranged in an ascending order. If the mean of the observations is equal to the median, find the value of [3]
(b) In the figure, is a diameter of the circle. Calculate: [3]
(i)
(ii)
(iii)
(c) Using graph paper to answer the following questions. (Take unit on both axis)
(i) Plot the points and
(ii) is the image of when reflected in the . Plot it on the graph paper and write the coordinates of
(iii) is the image of when reflected in the line write the coordinates of
(iv) Write the geometric name of the figure
(v) Name a line of symmetry of the figure formed. [4]
Answers:
(a) Arrange numbers in ascending order are
Mean
No. of terms
Median
Median
According to given condition
or
(b) In
(i) (angle in the semi circle)
(ii) (cyclic quadrilateral)
(iii) (angles in the same segment)
(c) As shown in the graph below:
(i) Coordinate of
(ii) Coordinate of
(iii) Geometric name
(iv) is the symmetric line.
SECTION B [40 Marks]
(Answer any four questions in this Section.)
Question 5:
(a) A shopkeeper bought a washing machine at a discount of from a wholesaler, the printed price of the washing machine being . The shopkeeper sells it to a consumer at a discount of on the printed price. If the rate of sales tax is find:
(i) the VAT paid by the shopkeeper
(ii) the total amount that the consumer pays for the washing machine. [3]
(b) If , then find the value of
(i)
(ii) [3]
(c) In
(i) Prove that is similar to
(ii) Find
(iii) Find area of [4]
Answers:
(a) Given: Printed price of washing machine
Rate of discount
(i) Amount of discount to shopkeeper
Shopkeeper’s Price
Sales Tax paid by shopkeeper
Discount for consumer
Price of consumer
Tax charged by the shopkeeper
Since, Tax paid by the shopkeeper
VAT paid by the shopkeeper=Tax charged – Tax Paid
(ii) Total amount paid by the consumer for washing machine
(b) Given:
(i) Applying componendo and dividend
(ii) Cubing both sides we get
Applying componendo and dividend
(c) (i) In and
(common angle)
(given)
Therefore (AAA Postulate)
(ii) Since
(corresponding sides are proportional)
and
(iii) Since
Question 6:
(a) Find the value of for which the following points and are collinear. Hence find the equation of the line. [3]
(b) Salman invests a sum of money in shares, paying dividend quoted at premium. If his annual dividend is , calculate;
(i) the number of shares he bought
(ii) his total investment
(iii) the rate of return on his investment [3]
(c) The surface area of a solid metallic sphere is . it is melted and recast into solid right circular cones of radius and height . calculate:
(i) the radius of the sphere.
(ii) the number of cones recast. (Take ) [4]
Answers:
(a) Given: and are collinear.
Rejecting
(b) Nominal value of share
Dividend on 1 share
Total dividend of Salman
(i) No. of shares Salman bought
(ii) Premium on 1 share
Market value of 1 share
Total investment for shares
(iii) Rate of return=
(c) (i) Let the radius sphere
Surface area of sphere
(ii) Volume of sphere
Volume of cone
No. of cones recast
Question 7:
(a) Calculate the mean of the distribution given below using the short cut method; [3]
Marks  1120  2130  3140  4150  5160  6170  7180 
No. of students  2  6  10  12  9  7  4 
(b) In the figure given below, diameter of a circle meet at is a tangent to the circle at find:
(i)
(ii) the length of tangent . [3]
(c) Let , and . Find . [3]
Answers:
(a) Table as follows

Mean Value




1120 2130 3140 4150 5160 6170 7180 
2
6 10 12 9 7 4 
155
255 355 455 555 655 755 
30
20 10 0 10 20 30 
60 120 100 0 90 140 120 
(b) (i) Since chord and tangent at point intersect each other at .
… … … … (i)
Since chord and tangent at point interested each other at ,
… … … … (ii)
From (i) (ii)
Given;
Putting these values in eq. (3)
Hence,
(ii) From (i),
Length of tangent
(c)
Question 8:
(a) The compound interest, calculate yearly, on a certain sum of money for the second year is and for the third year is . Calculate the rate of interest and the original sum of money. [3]
(b) Construct a with , Construct the in circle of the triangle measure and record the radius of the in circle. [3]
(c) Use a graph paper for this question. The daily pocket expenses of students in a school are given below;
Pocket expenses (in Rs.)  05  510  1015  1520  2025  2530  3035  3540 
Number of students (frequency)  10  14  28  42  50  30  14  12 
Draw a histogram representing the above distribution and estimate the mode from the graph. [4]
Answers:
(a) Compound Interest for the third year
Compound Interest for the second year
Simple Interest on for one year
Rate of interest
Let the original money be
Amount after 2 year – amount after one year = Compound Interest for second year.
Rate of interest
Original sum of money
(b) Steps of construction:
 Construct a with the given data:
 Draw a line BC of 6.5 cm length using a ruler
 The make an arc of 5.5 cm from B and similarly, make an arc of 5 cm from C
 The place where the two arcs intersect, join that point to B and C and complete the triangle.
 Draw the internal bisectors of . Let these bisectors cut at
 Draw an arc from point B so that it cuts the two sides of the angle ABC
 From the point of intersections, draw two arcs
 Join the point B and the point of intersection and draw a line. This line bisects the angle ABC.
 Repeat the above 3 steps for point C.
 The two bisectors will intersect at the point O which is the center of the incircle.
 Taking as center and touching the side of the circle as the radius, Draw a incircle which touches all the sides of the
 From draw a perpendicular to side which cut at .
 Measure which is required radius of the in circle. .
(c)
Question: 9
(a) If is the duplicate ratio of , find the value of . [3]
(b) Solve for using the quadratic formula. Write your answer correct to two significant figures. [3]
(c) A page from the saving bank account of Priyanka is given below: [4]
Date  Particular  Amount
Withdrawal 
Amount
Deposited 
Balance 
03/04/2006
05/04/2006 18/04/2006 25/05/2006 30/05/2006 20/07/2006 10/09/2006 19/09/2006 
B/F
By cash By Cheque To Cheque By Cash By Self By Cash To Cheque 
–
– – 5,000.00 – 4,000.00 – 1,000.00 
–
2,000.00 6,000.00 – 3,000.00 – 2,000.00 – 
4,000.00
6,000.00 12,000.00 7,000.00 10,000.00 6,000.00 8,000.00 7,000.00 
If the interest earned by Priyanka for the period of ending September, 2006 is , find the rate of interest.
Answers:
(a) Given: is the duplicate ratio of
(b) Given:
Comparing with , we get
and
and
(c) Qualifying principal for various months:
Month  Principal (Rs.) 
April  6000 
May  7000 
June  10000 
July  6000 
August  6000 
September  7000 
Total for 1 month  42000 
Now,
Interest
Question: 10
(a) A two digit number is such that the product of its digits is 6, If 9 is added to the number, the digits interchange their places. Find the number. [4]
(b) The number obtained by 100 students in a Mathematics test are given below; [6]
Marks  010  1020  2030  3040  4050  5060  6070  7080  8090  90100 
No. of Students  3  7  12  17  23  14  9  6  5  4 
Draw an ogive for the distribution on a graph sheet. (Use a scale of 2 cm = 10 units on both axis)
Use the ogive to estimate the:
(i) median
(ii) lower quartile
(iii) number of students who obtained more than 85% marks in the test.
(iv) number of students who did not pass in the if the pass percentage was 35
Answers:
(a) Let the required two digit number be
Given: and
Since (given)
(not possible)
When
The required two digit number
(b)
Marks  No. of Students  Cumulative Frequency (c.f) 
010
1020 2030 3040 4050 5060 6070 7080 8090 90100 
8
7 12 17 23 14 9 6 5 4 
3
10 22 39 62 76 85 91 96 100 
On the graph paper, we plot the following points:
(i) Medium
From the graph
(ii) Lower quartile
From the graph
(iii) The number of students who obtained more than marks in test students
(iv) The number of students who did not pass in the test if test if the pass percentage was 35 students.
Question 11:
(a) In the figure given below, is the center of the circle, and are two chords of the circle. and .
Find the:
(i) radius of the circle
(ii) length of chord [3]
(b) Prove the identity:
[3]
(c) An airplane at an altitude of observed of depression of two boats on the opposite banks of a river to be and respectively. Find the answer correct to the nearest whole number. [4]
Answers:
(a) Given
is mid point of
(i) Let radius of circle
From
(ii) Now from
As is mid point of
(b) To prove:
LHS
Hence Proved.
(c) Let height of airplane
Two boats are at .
Let as shown in the diagram.
From
From
Width of river