**MATHEMATICS (ICSE 2016)**

**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)** Using remainder theorem, find the value of if on dividing by , leaves a remainder of . **[3]**

**(b)** Given and and . Find . **[4]**

**(c)** The mean of the following numbers is . Find the value of . Hence estimate the medium. **[3]**

**Answers:**

**(a)** Let

By remainder theorem, when is divided by means , , then the remainder is .

Therefore

**(b)** Given and and .

LHS

RHS

Since LHS = RHS

Also

Hence the value of

**(c) **

Now arrange the numbers in ascending order we get

Number of terms (odd number of terms)

Median terms

term

(median)

**Question 2**

**(a) **The slope of a line joining . Find: i) ii) Midpoint of using the value of found in (i). **[3]**

**(b)** Without using trigonometrical tables evaluate: **[4]**

**(c)** A certain number of metallic cones, each of radius and height are melted and, recast into a solid, sphere of radius . Find, the number of cones. **[3]**

**Answers:**

**(a) **i) Let

Given: Slope of

Formula for slope of

ii) Let

Given: Slope of

Formula for slope of

**(b)** Given:

**(c)** Let number of cones

Volume of cones = Volume of sphere [Provide Formulas]

Therefore the number of cones needed is

**Question 3**

**(a)** Solve the following inequation, write the solution set and represent it on number line. **[3]**

**(b)** In the given figure below, is the diameter. is the center of the circle. is parallel to and . Find: i) ii) iii) **[4]**

**(c)** If . **[3]**

**Answers:**

**(a)**

Therefore we have:

… … … … … i)

Also we have

… … … … … i)

Combining i) and ii) we get

Solution Set

**(b)** Given

Therefore (alternate angles)

Since (radius of the same circle)

Therefore (angles opposite to equal side of the triangle are equal)

(angle at the center is twice that subtended at the circumference)

In

(sum of the angles of a triangle is )

(angle in the same segment)

Therefore

**(c)**

Cross multiplying

Or

**Question 4**

**(a)** A game of number has cards marked with A card is drawn at random. Find the probability that the number on the card drawn is: i) A perfect square ii) Divisible by **[3]**

**(b)** Use graph paper for, this question. (Take unit along both ). Plot the point . **[4]**

Reflect points on the y axis and name them respectively. i) Write down their coordinates. ii) Name the figure iii) State the line of symmetry of this figure

**(c)** Mr. Lalit invested at a certain rate of interest, compounded annually for two years. At the end of first year it amounts . Calculate: i) The rate of interest. ii) The amount at the end of second year, to the nearest rupee. **[3]**

**Answers:**

**(a)** Total number of all possible outcomes

Formula used:

i) The cards with perfect squares are:

The number of favorable outcomes

ii) The cards with numbers divisible by are::

Therefore the number of favorable outcomes

**(b)** i)

ii) Arrow Head

iii) is the line of symmetry

**(c)** Given: Principal , Time , After one year amount =

i) We know that Amount

For

ii) Amount (A) at the end of

Formula for compound interest:

Given

Therefore

Hence Amount at the end of 2 years

**SECTION B [40 Marks]**

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

**Question 5**

**(a)** Solve the quadratic equation . Give answer correct to three significant figures. **[3]**

**(b)** A page form the saving bank account of Mrs. Ravi is given below:

Date | Particulars | Withdrawal (Rs.) | Deposit (Rs.) | Balance (Rs.) |

April 3^{rd} 2006 |
B / F | – | – | 6,000 |

April 7^{th} |
By Cash | – | 2,300 | 8,300 |

April 15^{th} |
By Cheque | – | 3,500 | 11,800 |

May 20^{th} |
To Self | 4,200 | – | 7,600 |

June 10^{th} |
By Cash | – | 5,800 | 13,400 |

June 15^{th} |
To Self | 3,100 | – | 10,300 |

August 13^{th} |
By Cheque | – | 1,000 | 11,300 |

August 25^{th} |
To Self | 7,400 | – | 3,900 |

September 6^{th} 2006 |
By Cash | – | 2,000 | 5,900 |

She closed the account an . Calculate the interest Mrs. Ravi earned, at the end of per annunm interest. Hence find the amount she receives on closing the account. **[4]**

**(c)** In what time will yield as compound interest at per annum compounded annually ? **[3]**

**Answers:**

**(a)** Given

Comparing with , we get

Since

Therefore

Solving we get

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

Month | Qualifying Principal (Rs.) |

April | 8300 |

May | 7600 |

June | 10300 |

July | 10300 |

August | 3900 |

Total | 40400 |

Amount received on 30th September (on closing the account)

**(c)** Given

**Question 6**

**(a)** Construct a regular hexagon of side . Hence construct all its lines of symmetry and name them. **[3]**

**(b)** In the given figure is a cyclic quadrilateral produced meet at point . i) Prove ii) Find. iii) Find area, of quadrilateral if area of **[4]**

**(c) ** Given matrix and

If, : i) Write the order of matrix . ii) Find the matrix . **[3]**

**Answers:**

**(a) ** Step 1: Given side of the hexagon is . Construct the hexagon as follows:

- First draw a line using a ruler of length 5 cm. Mark it AB.
- The using a compass, make an arc of
- Using compass, draw 120 degree angle and cut the line into 5 cm lengths using the compass. Continue this until the hexagon is completed
- One the hexagon is completed, draw perpendicular bisectors of each of the arms of the hexagon. You will get mid point for each of the arms of hexagon as marked
- Now join the mid points of the opposite sides to get lines
- Now draw the diagonals passing through the center to get lines

The six lines of symmetry

**(b)** (i) Consider and

Since exterior angle of cyclic quadrilateral is equal to the interior opposite angle, we get

is common

Therefore

(ii) Since

(iii) Since

Therefore Area of quadrilateral

**(c)** Given matrix and

If, :

(i) From the given matrix, order of and order of

We know

or

Hence

Therefore the order of

(ii) Let

Therefore

… … … … … i)

… … … … … ii)

Solving i) and ii) we get

Therefore

**Question 7**

**(a)** An aero plane is at a height of meters finds that two ships are sailing towards it in the same direction. The angle of depression as observed from the aero plane are respectively. Find the distance between the two ships. **[4]**

**(b)** The table shows the distribution of the scores obtained by shooters in a shooting competition. Use a graph sheet and draw an ogive of the distribution.

(Take scores on the and shooters on the ).

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

No of shooters | 9 | 13 | 20 | 26 | 30 | 22 | 15 | 10 | 8 | 7 |

Use your graph to estimate the following: i) The median. ii) The interquartile range. iii) The number of shooters who obtained a score of more than **[6]**

**Answer:**

**(a) ** In the adjacent diagram, let be the air plane and be the two ships.

Given:

Consider

Consider

Therefore .

Hence, distance between two ships .

**(b) **

Scores | No of Shooters | Cumulative Frequency (c.f.) |

0-10 | 9 | 9 |

10-20 | 13 | 22 |

20-30 | 20 | 42 |

30-40 | 26 | 68 |

40-50 | 30 | 98 |

50-60 | 22 | 120 |

60-70 | 15 | 135 |

70-80 | 10 | 145 |

80-90 | 8 | 153 |

90-100 | 7 | 160 |

(i) (which is even)

Median

From the graph, Median

(ii) Lower Quartile

Upper Quartile

Inter Quartile range

(iii) The number of shooter who obtained a score of more (using the graph)

**Question 8**

**(a)** If show that **[3]**

**(b)** Draw, a line . Mark a point such that . Using a ruler and compass only, construct:

(i) A circle of radius , passing through .

(ii) Construct two tangents to the circle from the external point . Measure and record the length of the tangents. **[4]**

**(c)** A line meets the at and at . divides in the ratio

(i) Find the coordinates of .

(ii) Find the equation of the line through and perpendicular to . **[3]**

Answers:

**(a)** Let

Therefore

LHS

Since

Substituting

$latex \frac{x}{a} \frac{y}{b} \frac{z}{c} = &s=2$ RHS Hence proved.

**(b) **

- First draw line . This you can just draw by using a ruler.
- From cut . This can be done by using a compass. Set the compass at 3 cm and then make this arc cutting AB at C.
- Draw perpendicular bisector of .
- Taking as center and radius, draw an arc to meet the perpendicular bisector of .
- Taking as the center and as radius, draw a circle, which passing through
- Join .
- Draw right bisector of , which cuts .
- Taking as center and as radius, draw a circle which cuts old Circle at and .
- Join .

These are required tangents.

(ii) Length of tangent .

**(c) **

(i) Let coordinates of be

The coordinate of a point divides it in the ratio

If a point divides two points and in the ratio , then the coordinates of the point at

Therefore

Therefore coordinates are

(ii) Slope of AB

Slope of perpendicular bisector

Required equation of the line:

**Question 9**

**(a) **A dealer buys an article at a discount of from the wholesaler, the marked price being . The dealer sells it to a shopkeeper at a discount of on the marked price. If the rate of VAT is , find:

(i) The price paid by the shopkeeper including the tax.

(ii) The VAT paid by the dealer. **[3]**

**(b) **The given figure represents a kite with a, circular and a Semicircular motifs stuck on it. The radius of circle is and the semicircle is . If diagonals are the lengths respectively, find the area of the :

(i) shaded, part. Give your answer correct to the nearest whole number.

(ii) unshaded part.** [4]**

**(c) **A model of a ship is made to a scale .

(i) The length of the model of the ship is . Calculate the length of the ship.

(ii) The area of the deck ship is . Calculate the area of the deck of the model.

(iii) The volume of the model is . Calculate the volume of the ship. **[3]**

**Answers**

**(a)** Given: Market Price Rate of discount

Therefore the cost price of the article for the dealer

Sales tax paid by the dealer to the wholesaler

(i) Discount rate for shop keeper

The cost price for the shopkeeper

The price paid by the shopkeeper including the tax

(ii) VAT paid by the dealer = Tax charged – Tax paid

**(b)**

(i) Given: Radius of circle Radius of semicircle .

Area of shaded part = Area of semicircle + Area of circle

(ii) Let kite be a quadrilateral.

Area of quad

Area of kite

Area of the unshaded part = Area of kite – Area of shaded part

**(c)**

Length of the ship

Area of the model

Volume of the ship

**Question 10**

**(a)** Mohan has a recurring deposit account in a bank for years at p.a. simple interest. If he gets as interest at the time of maturity, find:

(i) the monthly installment

(ii) the amount of maturity **[3]**

**(b)** The histogram below represents the scores obtained by students in a Mathematics mental test. Use the data to:

(i) Frame a frequency distribution, table.

(ii) To calculate mean.

(iii) To determine the Modal class. **[4]**

**(c)** A bus covers a distance of km at a uniform speed. Due to heavy rain its speed gets reduced by and as such it takes two hrs longer to cover the total distance. Assuming the uniform speed to be , form an equation and solve it to evaluate . **[3]**

**Answers**

**(a)** the number of months for which the money is deposited

(i) Let the monthly instalment be

Using the formula:

Therefore the monthly instalment

(ii) The amount at maturity = Total money deposited + interest

**(b) **(i) Frequency Distribution Table

C.I | Frequency |

0-10 | 2 |

10-20 | 5 |

20-30 | 8 |

30-40 | 4 |

40-50 | 6 |

(ii) Mean

C.I | |||

0-10 | 2 | 5 | 10 |

10-20 | 5 | 15 | 75 |

20-30 | 8 | 25 | 200 |

30-40 | 4 | 35 | 140 |

40-50 | 6 | 45 | 270 |

Mean

(iii) Model Class = 20-30 &s=0$

**(c) ** Let the speed of the bus for onward journey

Distance

Time taken for onward journey

Due to rain the return speed of the bus

Time taken for the return journey

Given that the return time is more, we have

Hence

**Question 11**

**(a)** Prove that [3]

**(b)** Use ruler and compasses only for the following question. All construction lines and arcs much be clearly shown.

(i) Construct a in which .

(ii) Construct the locus of points at a distance of from .

(iii) Construct the locus of points equidistant from .

(iv) Mark points which are at a distance of from and also equidistant from . Measure . [4]

**(c)** Ashok invested on shares of a company. If he receives a dividend of find the :

(i) number of shares he bought.

(ii) Market value of each share. **[3]**

**Answers**

**(a)** To prove

= RHS Hence Proved

**(b)**

- Construct . Draw a line BC which is 6.5 cm long. The using your compass, make an angle of at point B of line BC.
- The taking the compass, make a cut a 5 cm mark on the side of the angle to mark A. AB is 5 cm long.
- Taking as center and radius is , draw a circle. This has been told in the question. Which is required locus.
- Draw an angle bisector of . Angle bisector of cut above circle at two points i.e., . These points are required points.
- Measure the length of .

**(c)**

(i) Let number of shares be

Face Value of x shares

Dividend

Therefore

(ii) Market value