**MATHEMATICS (ICSE Paper 2011)**

**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)** Find the value of if is a factor of . Hence determine whether is also a factor. ** [3]**

**(b) **If and , is the product possible ? Give a reason. If yes, find . **[3]**

**(c)** Mr. Kumar borrowed for two years. The rate of interest for the two successive years are and respectively. If the repays at the end of the first year, find the outstanding amount at the end of the second year. **[4]**

**Answer:**

**(a) ** Let .

Since given that is a factor

Substituting the value of in the above function we get:

For to be a factor

Substituting the value of in the above function we get:

Hence ) is a factor of

**(b)** The order of matrix and the order of matrix .

Since the number of columns in is equal to the number of rows in , the product is possible.

**(c)** Principal

The rate of interest for the two successive years are and respectively.

Formula:

Therefore Amount after year

Principal at the start of year after repayment

Amount outstanding at the end of second year

**Question 2: **

**(a)** From a pack of 52 playing cards all cards whose numbers are multiples of 3 are removed. A card is now drawn at random.What is the probability that the card drawn is;

(i) a face card (King, Jack, or Queen)

(i) an even number red card **[3]**

**(b)** Solve the following equation: Give your answer correct to two significant figures. ** [3]**

**(c)** In the given figure is the center of the circle Tangent of and meet at if , find (i) (ii) (iii) ** [4]**

**Answer:**

**(a)** Total number of cards

Number of cards which are multiples of

Total number of cards left

(i) Number of face cards

Probability (of a face card)

(ii) Even numbered red cards

Probability (of a even number red card)

**(b)** Given

Simplifying:

Compare with equation , we get and

We know,

Therefore

Answer correct to two significant figures:

**(c)** Consider and

is common

(two tangents drawn from a point on a circle are of equal lengths)

Therefore (RHS postulate)

(i)

(ii)

(iii) (chord subtends twice the angle at the center than that it subtends on the circumference)

**Question 3:**

**(a) **Ahmed has a recurring deposit account in a bank. He deposits per month for years. If he gets at the time of maturity, find;

(i) The interest paid by the bank

(ii) The rate of interest ** [3]**

(b) Calculate the area of the shaded region, if the diameter of the semi circle is equal to . (Take ) **[3]**

**(c)** is a triangle and is the central of the triangle. If and find and find the length of side . ** [4]**

**Answer:**

**(a)**

**(b)** Area of shaded portion = Total area – area of the two quadrants

**(c) ** Since is the centroid

Therefore

Therefore units.

**Question 4:**

**(a) **Solve the following in equation and represent the solution set on the number line:

**[3]**

**(b)** Evaluate without using trigonometric tables:

** [3]**

**(c) **A mathematics aptitude test of 50 students was recorded as follows:

Marks | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 |

No. of students | 4 | 8 | 14 | 19 | 5 |

Draw a histogram for the above data using a graph paper and locate the mode. **[4]**

**Answers:**

**(a) **

or or

or or

Therefore

**(b) **

**(c) **

**SECTION B [40 Marks]**

*(Answer any four questions in this Section.)*

**Question 5:**

**(a)** A manufacturer sells a washing machine to a wholesaler for . The wholesaler sells it to a trader at a profit of and the trader in turn sells it to a consumer at a profit of . If the rate of VAT is find:

(i) The amount of VAT received by the state government on the sale of this machine from the manufacture and the wholesaler.

(ii) The amount that the consumer pays for the machine. **[3]**

**(b) **A solid cone of radius and height is melted and made into small spheres of radius . Find the number of sphere formed. **[3]**

**(c)** is a parallelogram where and Find

(i) Coordinates of

(ii) Equation of diagonal **[4]**

**Answer:**

**(a)** (i) Tax received by the manufacturer

For the trader the price

Tax paid by the trader

Therefore VAT received from wholesaler

Price for the consumer

(ii) Tax paid by the consumer

Hence the total price paid by the consumer

**(b)** Cone: Radius : and Height

Sphere:

Number of sphere

**(c) ** (i) Mid point of

Therefore we have and

is the mid point of as well (diagonals of a parallelogram bisect each other)

Hence

and

Hence

(ii) Equation of

**Question 6:**

**(a)** Use a graph paper to answer the following (Take on both axes)

(i) Plot and the vertices of a

(ii) Reflect on the and name it as

(iii) Write the coordinates of the image and

(iv) Give a geometrical name for the figure

(v) Identify the line of symmetry of **[5]**

**(b)** Choudhury opened a Saving account at State Bank of India an 1^{st} April 2007, The entries of one year as shows in his pass book are given below:the bank pays interest at the rate of 5% per annum, find the interest paid on 1^{st} April 2008. Give your answer correct to the nearest rupees. **[5]**

Date | Particulars | Withdrawals (Rs) | Deposit (Rs.) | Balance (in Rs.) |

1^{st} April 2007 |
By cash | – | 8550 | 8550 |

12^{th} April 2007 |
To Self | 1200 | – | 7350 |

24^{th} April 2007 |
By cash | – | 4550 | 11900 |

8^{th} July 2007 |
By cheque | – | 1500 | 13400 |

10^{th} Sep.2007 |
By cheque | – | 3500 | 16900 |

17^{th} Sep. 2007 |
By cheque | 2500 | – | 14400 |

11^{th} Oct.2007 |
By cash | – | 800 | 15200 |

6^{th} Jan 2008 |
To Self | 2000 | – | 13200 |

9^{th} March 2008 |
By cheque | – | 950 | 14150 |

**Answer:**

**(a) **(i) & (ii) Shown in the graph below.

(iii) and

(iv) The shape is that of a hexagon.

(v) Line of symmetry :

**(b)** Qualifying principal for various months:

Month | Principal (Rs.) |

April | 7350 |

May | 11900 |

June | 11900 |

July | 13400 |

August | 13400 |

September | 14400 |

October | 14400 |

November | 15200 |

December | 15200 |

January | 13200 |

February | 13200 |

March | 14150 |

Total | 157700 |

**Question 7:**

**(a)** Using component and dividend, find the value of x:

**[3]**

**(b)** If , and is the identity matrix of the same order and is transpose of matrix , find . **[3] **

**(c) **In the adjoining figure is a right angled triangle with .

(i) Prove

(ii) If

(iii) Find the ratio of the area of is to the area of ** [4]**

**Answer:**

**(a)**

Applying componendo and dividendo

Simplifying

Square both sides

Simplifying we get

**(b)**

**(c)** (i) Let

Therefore

Therefore (AAA postulate)

(ii)

Therefore

(iii)

**Question 8:**

**(a) **Using step division method, calculate the mean marks of the following distribution: State the modal class: **[5]**

Class Interval | 50-55 | 55-60 | 60-65 | 65-70 | 70-75 | 75-80 | 80-85 | 85-90 |

Frequency | 5 | 20 | 10 | 10 | 9 | 6 | 12 | 8 |

**(b) **Marks obtained by 200 students in an examination are given below:

Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 |

No. of students | 5 | 11 | 10 | 20 | 28 | 37 | 40 | 29 | 14 | 6 |

Draw an ogive for the given distribution taking marks on one axis and students on the other axis. Using the graph, determine.

(i) The median marks

(ii) The number of students who failed if minimum marks required to pass is .

(iii) If scoring and more marks is considered as grade one, find the number of students who secured grade on in the examination; **[5]**

**Answers:**

**(a) **

C.I | |||||

50-55 | 5 | 52.5 | -15 | -3 | -15 |

55-60 | 20 | 57.5 | -10 | -2 | -40 |

60-65 | 10 | 62.5 | -5 | -1 | -10 |

65-70 | 10 | 67.5 | 0 | 0 | 0 |

70-75 | 9 | 72.5 | 5 | 1 | 9 |

75-80 | 6 | 77.5 | 10 | 2 | 12 |

80-85 | 12 | 82.5 | 15 | 3 | 36 |

85-90 | 8 | 87.5 | 20 | 4 | 32 |

(i)

(ii) Modal class is (class with highest freq.)

**(b) **

Class Interval | Frequency | Cumulative Frequency |

0-10 | 5 | 5 |

10-20 | 11 | 16 |

20-30 | 10 | 20 |

30-40 | 20 | 46 |

40-50 | 28 | 74 |

50-60 | 37 | 111 |

60-70 | 40 | 151 |

70-80 | 29 | 180 |

80-90 | 14 | 194 |

90-100 | 6 | 200 |

(i) (even)

Median observation observation observation

(ii) Number of student who failed

(iii) Number of students who secured grade one

**Question 9:**

**(a)** Parekh invested on shares at a discount of paying dividend. At the end of one year he sells the shares at a premium of . Find;

(i) The annual dividend

(ii) The profit earned including hid dividend;

**(b)** Draw a circle of radius . mark a point outside the circle at a distance of . from the center. Construct two tangent from to the given circle. Measure and write down the length of one tangent.

**(c)** Prove that

Answer:

**(a) ** Nominal Value of the share

Market Value of the share

Number of shares bought

Dividend earned

Sale proceeds

Profit

**(b) **

Steps of construction (diagram not to scale):

- With the help of a ruler measure 3.5 cm in your compass and draw a circle of radius 3.5 cm.
- The draw a point P, 6 cm away from the circle.
- The next step is to bisect PP’
- The with this as the center, O, draw a circle that goes through point Q and R on the original circles.
- Join PQ and PR. These are the tangents.
- Length of the tangents is 4.9 cm

**(c) **

LHS

=

=

= RHS

Hence Proved.

**Question 10:**

**(a)** is the mean proportion two numbers and and is the third proportion of and . Find the numbers.

**(b)** In what period of time will yield as compound interest at per annum, if compounded on an yearly basis?

**(c)** A man observed the angle of elevation of the top of a building to be . He walks towards it in a horizontal line through its base. On covering the angle of evaluation changes to . Find the height of the building correct to the nearest meter.

Answer:

**(a) ** Given is the mean proportion between two numbers

Therefore … … … … … … i)

Also given is the third proportion to

Therefore … … … … … … ii)

Solving i) and ii)

Hence

Hence the numbers are .

**(b)** Given

**(c) **From

Similarly, from

Equating for we get

**Question 11:**

**(a)** with and (not drawn to scale) Three circle are drawn touching each other with the vertices as their centers. Find the radii of the three circles. ** [3] **

**(b) ** is divided equally among children. If the number of children were more than each would have got less. Find ** [3]**

**(c)** Give equation of line is

(i) Write the slope of line if is the bisector of angle .

(ii) Write the co-ordinates of point .

(iii) Find the equation of . ** [4]**

Answer:

**(a) **Let the radius of the three circles

Given: and

Therefore

… … … … … (i)

… … … … … (ii)

… … … … … (iii)

Adding (i), (ii) and (iii) we get

… … … … … (iv)

Using (i) and (iv) we get

Using (ii) and (iv) we get

Using (iii) and (iv) we get

**(b)** Let the number of children

Therefore

Hence the number of children is 20

**(c) **(i) Slope of

(ii) Equation of line . It passes through

is the point of intersection of and

Solving and we get and .

Hence .

(iii) Equation of