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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.

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SECTION A [40 Marks]

(Answer all questions from this Section.)

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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

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SECTION B [40 Marks]

(Answer any four questions in this Section.)

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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 3rd 2006	B / F	–	–	6,000
April 7th	By Cash	–	2,300	8,300
April 15th	By Cheque	–	3,500	11,800
May 20th	To Self	4,200	–	7,600
June 10th	By Cash	–	5,800	13,400
June 15th	To Self	3,100	–	10,300
August 13th	By Cheque	–	1,000	11,300
August 25th	To Self	7,400	–	3,900
September 6th 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