**MATHEMATICS **(ICSE 2015)

**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)** A shopkeeper bought an article for Rs. 3450. He marks the price of the article 16% above the cost price. The rate of sales tax charged on the article is 10%. Find the:

(i) marked price of the article

(ii) price paid by a customer who buys the article. **[3]**

**(b)** Solve the following in equation and write the solution set:

Represent the solution on a real number line.** [3]**

**(c)** Without using trigonometric tables evaluate:-

**[4]**

**Answers:**

**(a) **(i) Cost price of the article

Marked Price

(ii) Price paid by the customer

**(b) **Given

Solution:

**(c) **

**Question 2**

**(a)** If and . **[3]**

**(b)** The present population of a town is . It population increases by in the first year and in the second year. Find the population of the town at the end of the two years. **[3]**

**(c)** Three verticals of a parallelogram taken in order are find:

(i) the coordinates of the fourth vertex .

(ii) length of diagonal

(iii) equation of side of the parallelogram . **[4]**

**Answers:**

**(a)** and

Therefore

And

**(b) **Given:

Population after first year

Principle for the second year

Population at the end of second year

**(c)** Let the coordinate of . In a parallelogram, the diagonals bisect each other.

Mid point of

Mid point of

(i)

Coordinates of

(ii)

(iii) Equation of AB:

**Question 3 **

**(a) **In the given figure, ABCD is a square of side 21 cm, AC and BD are two diagonals of the square. Two semi circle are drawn with AD and BC as diameters. Find the area of the shaded region. (Taken ) **[3]**

**(b)** The marks obtained by 30 students in a class assessment of 5 marks is given below:

Marks |
0 | 1 | 2 | 3 | 4 | 5 |

No. of Students |
1 | 3 | 6 | 10 | 5 | 5 |

Calculate the mean, mediam and mode of the above distribution. **[3]**

**(c) **In the figure given below O is the center of the circle and SP is a tangent. If **[4]**

**Answer:**

**(a)** Given: Side

Let Diagonal of the square

Area of

Area of semicircle

Area of shaded region = Area of 2 semicircles + Area of +

**(b)** Below

0 | 1 | 0 | 1 |

1 | 3 | 3 | 4 |

2 | 6 | 12 | 10 |

3 | 10 | 30 | 20 |

4 | 5 | 20 | 25 |

5 | 5 | 25 | 30 |

Mean

Median

Mode = 3 marks (as highest frequency is 10)

**(c) **In (Radius is always perpendicular to the tangent)

In

**Question 4**

**(a)** Katrina opened a recurring deposit account with a Nationalized Bank for a period of 2 years. If the bank pays interest at the rate of 6% per annum and the monthly installment is Rs. 1000 find the:** [3]**

(i) interest earned in 2 years

(ii) matured value

**(b)** Find the value of for which is a solution of the quadratic equation, . Thus find the root of the equation. [3]

**(c)** Construct a regular hexogon of side 5 cm. Construct a circle circumscribing the hexagon. All traces of construction must be clearly shown. **[4] **

**Answer:**

**(a)** (i) Given

(ii) Total money deposit in 24 months =

** (b)**

in given equation:

Putting in given equation:

is the other root of the given equation.

**(c)** Steps of Construction:

- First draw a regular hexagon. The length of one side is 5 cm. You will get hexagon ABCDEF.
- Take any two adjacent sides and draw perpendicular bisectors.
- The point where these two bisectors intersect, is the center of the circle.
- With O as the center, draw a circle which will pass through all the vertices of the hexagon.

**SECTION B [40 Marks]**

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

**Question 5**

**(a) **Use a graph paper for this question taking unit along both the :

(i) Plot the points

(ii) Reflect the points on the and name them respectively as .

(iv) Name the figure formed by .

(v) Name a line of symmetry for, the figure formed. **[5]**

**(b) **Virat opened a Saving Bank account in a bank on 16th April 2010. His pass book shows the following entries:

Date | Particulars | Withdrawals (Rs.) | Deposits (Rs.) | Balance (Rs.) |

April 16, 2010 | By Cash | – | 2500 | 2500 |

April 28^{th} |
By Cheque | – | 3000 | 5500 |

May 9^{th} |
To Cheque | 850 | – | 4650 |

May 15^{th} |
By Cash | – | 1600 | 6250 |

May 24^{th} |
To Cash | 1000 | – | 5250 |

June 4^{th} |
To Cash | 500 | – | 4750 |

June 30^{th} |
By Cheque | – | 2400 | 7150 |

July 3^{rd} |
To Cash | – | 1800 | 8950 |

Calculate the interest Virat earned at the end, of 31st July, 2010 at 4% per annum interest. What sum of money will he receives if he closes the account on 1st August, 2010? **[5]**

**Answer:**

**(a) ** (i) Shown in the diagram.

(ii) Shown in then diagram.

(iii) Coordinates :

(iv) Octagon

(v) .

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

Month |
Principal (Rs.) |

April | 0 |

May | 4650 |

June | 4750 |

July | 8950 |

Total | 18350 |

Amount

**Question 6**

**(a)** If a, b, c are in continued proportion, prove that **[3]**

**(b)** In the given figure is a triangle and , is parallel to the . and intersects the at and respectively.

(i) Write the coordinates of .

(ii) Find, the length of .

(iii) Find the ratio in which divides .

(iv) Find the equation, of the line **[4]**

**(c)** Calculate the mean of the following distribution. **[3]**

Class Interval | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |

Frequency | 8 | 5 | 12 | 35 | 24 | 16 |

**Answer:**

**(a) ** To prove:

Given are in continued proportion

Therefore

This also implies

LHS

RHS

Hence LHS = RHS

Therefore Proved

**(b)** (i) Coordinates of

(ii) Coordinates of

Note: *The distance between any two points and is *

Length

Coordinates of

Length

(iii) Let the required ratio be

Let the coordinate of , since it lies on the y axis.

Since

(iv) Equation of through Coordinates of

Required equation of the line:

**(c)**

Class | Mid Value | ||

0-10 | 5 | 8 | 40 |

10-20 | 15 | 5 | 75 |

20-30 | 25 | 12 | 300 |

30-40 | 35 | 35 | 1225 |

40-50 | 45 | 24 | 1080 |

50-60 | 55 | 16 | 880 |

Total |

Mean

**Question 7**

**(a) **The solid spheres of radii 2 cm and 4 cm are melted and recast into a cone of height 8 cm. Find the radius of the cone so formed. **[3]**

**(b) **Find ‘a’ if the two polynomials , leaves the same remainder when divided by .** [3]**

**(c)** Prove that **[4]**

**Answers:**

**(a)** Volume of two spheres = Volume of cone

**(b)** Let and

Putting in both the expressions

Given

Therefore

**(c) ** Given

LHS

RHS

**Question 8**

**(a) ** are two chords of a circle intersecting at . Prove that . **[3]**

**(b)** A bag contain 5 white balls, 6 red balls and 9 green balls. A ball is drawn at random from the bag, Find the probability that the ball drawn is:

(i) a green ball

(ii) a white or a red ball

(iii) is neither a green ball nor a white ball. **[3]**

**(c) **Rohit invested Rs. 9600 on Rs. 100 shares at Rs. 20 premium paying 8% dividend. Rohit sold the shares when the price rose to Rs. 160. He invested the proceeds (excluding dividend) in 10% Rs. 50 shares at Rs. 40. Find the:

(i) original Number of shares

(ii) sale proceeds

(iii) new number of shares

(iv) change in the two dividends. **[4]**

**Answers:**

**(a)** To prove:

Construction : Join AD and CB.

(Angles of the same segment)

(Angles of the same segment)

Therefore (AAA Postulate)

Hence Proved

**(b)** Number of White Balls:

Number of Red Balls:

Number of Green Balls :

Total number of Balls:

(i) Probability (Green)

(ii) Probability (Red or White)

(iii) Probability (neither a green ball nor a white ball}

**(c) ** First Investment

Nominal Value of the share

Market Value of the share

Dividend

(i) Number of shares bought

(ii) Sale Price

Sale Proceed

Second Investment

Nominal Value of the share

Market Value of the share

Dividend

(iii) Number of shares bought

Dividend on 1st Investment earned

Dividend on 2nd Investment earned

Change in Dividend income

**Question 9**

**(a)** The horizontal distance between two towers is 120 m. The angle of elevation of the top and angle of depression of the bottom of the first tower as observed from the second tower is and respectively.

Find the height of the two towers. Give your answer correct to 3 significant figures. ** [4]**

**(b)** The weight of 50 workers is given below:

Weight in kg | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 | 100-110 | 110-120 |

No. of workers | 4 | 7 | 11 | 14 | 6 | 5 | 3 |

Draw on give of the given distribution using a graph sheet. Take 2 cm =10 kg on one axis and 2 cm =5 workers along the other axis. Use a graph to estimate the following:

(i) the upper and lower quartiles

(ii) if weight 95 kg and above is considering find the number of workers who are overweight. **[6]**

**Answers:**

**(a) **Let AB and CD are two towers.

In the right

In right angle

Height of tower is and tower is .

**(b)** Below the table

Weight | ||

50-60 | 4 | 4 |

60-70 | 7 | 11 |

70-80 | 11 | 22 |

80-90 | 14 | 36 |

90-100 | 6 | 42 |

100-110 | 5 | 47 |

110-120 | 3 | 50 |

Total | 50 |

(i) Lower quartile=

Upper quartile =

(ii) No. of over weight workers

**Question 10**

**(a)** A wholesale buys a TV from the manufacture for Rs. 25000. He marks the price of the TV 20% above his cost price and sell it to a retailer at 10% discount on the marked price. If the rate of VAT is 8%, find the: **[3]**

(i) marked Price

(ii) retailer’s cost price inclusive of tax

(iii) VAT paid by the wholesaler.

**(b)** If , and . Find AB-5C. ** [3]**

**(c)** ABC is a right angled triangle with . D is any point on AB and DE is perpendicular to AC. Prove that:

(i)

(ii) If AC = 13 cm, BC = 5 cm and AE = 4 cm. Find DE and AD.

(iii) Find, area, of : area of quadrilateral BCED. **[4]**

**Answers:**

**(a) **(i) Cost price for wholesaler

Market Price

(ii) Discount

Cost Price for retailer

Cost Price inclusive Tax

(iii) Cost price for wholesaler

Sale price for wholesaler

Profit for wholesalers

VAT

**(b)** Given , ,

**(c) **To prove:

(i) In

(is common)

Therefore (AAA Postulate)

(ii)

Since

Therefore

Therefore

cm

cm

(iii)

**Question 11**

(a) Sum of two natural numbers is 8 and the difference of their reciprocal is find the numbers. ** [3]**

(b) Given using componendo and dividendo find **[3]**

(c) Construct the , and . Hence

(i) Construct the locus of points equidistant from BA and BC

(ii) Construct the locus of points equidistant from B and C

(iii) Mark the point which satisfy he above two loci as P. Measure and write the length of PC. **[4]**

**Answers:**

**(a)** Let 1st number be and the 2nd number be

Given

Therefore the first number is 3 and the second number is 5.

**(b)**

Applying componendo and dividendo

Again Applying componendo and dividendo

**(c)** Steps of Construction:

- First draw a line AB=5.5 cm.
- With the help of the point A, draw . This you can draw using a compass.
- Take radius 6 cm, cut AC=6 cm and join C to B. This way you get CA and BC. This completes the triangle.
- Then draw perpendicular bisector of BC and draw angle of bisector both intersecting at P.
- P is the required point PC=4.8 cm