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ICSE CLASS 10 MATHS PREVIOUS YEAR PAPER 2016

ICSE

Class 10th Mathematics

Previous Year Question Paper 

2016

(Two and a half hours)

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 from 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.

The intended marks for questions or parts of questions are given in brackets [ ].

Mathematical tables are provided.

 

SECTION A (40 Marks)

 

Attempt all questions from this Section.

 

Question 1

a) Using remainder theorem, find the value of k if on dividing $  2 x^{3}+3 x^{2}-k x+5 \text { by } x-2  $, leaves a remainder 7.                 [3]

b) $\text { Given } A=\left(\begin{array}{cc}2 & 0 \\-1 & 7\end{array}\right) \text { and } I=\left(\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right) \text { and } A^{2}=9 \mathrm{~A}+\mathrm{ml} \text { Find } \mathrm{m} \text { . }$   [4]

c) The mean of following numbers is 68. Find the value of ‘x’. 45, 52, 60, x, 69, 70, 26, 81 and 94. Hence estimate the median.  [3]

Question 2

a) $\text { The slope of a line joining } \mathrm{P}(6, \mathrm{k}) \text { and } \mathrm{Q}(1-3 \mathrm{k}, 3) \text { is } \frac{1}{2}$. Find:

  1. K

  2. Midpoint of PQ, using the value of ‘k’ found in (i).

b) Without using trigonometrical tables, evaluate:                                                                         [4]

 

$\operatorname{cosec}^{2} 57^{\circ}-\tan ^{2} 33^{\circ}+\cos 44^{\circ} \operatorname{cosec} 46^{\circ}-\sqrt{2} \cos 45^{\circ}-\tan ^{2} 60^{\circ}$

 

c) A certain number of metallic cones, each of radius 2 cm and height 3 cm are melted and recast into a solid sphere of radius 6 cm. Find the number of cones. [3]                                                                                                                                              

 

Question 3

  1. Solve the following inequation, write the solution set and represent it on the number line. $ -3(x-7) \geq 15-7 x>\frac{x+1}{3}, x \in R$                                                                                                                                                                 [3]

 

  1. In the figure given below, AD is a diameter. O is the centre of the circle.  AD is parallel to   

         $\mathrm{BC} \text { and } \angle \mathrm{CBD}=32^{\circ} . \text { Find: }$ 

 

$\text { i) } \angle O B D$                                                                                                                     

$\text { i) } \angle \mathrm{AOB}$ A picture containing night sky

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$\text { i) } \angle B E D$

  1. If (3a + 2b): (5a + 3b) = 18: 29. Find a: b                                                                                           [3]



Question 4

 

  1. A game of numbers has cards marked with 11, 12, 13, … 40. A card is drawn at random. Find the probability that the number on the card drawn is:                                              [3]

  1. A perfect square

  2. Divisible by 7



  1. Use graph paper for this question. (Take 2 cm = 1 unit along both x and y axis.) Plot the points O (0, 0), A (-4, 4), B (-3, 0) and C (0, -3)                                                                             [4]

    1. Reflect points A and B on the y-axis and name them A’ and B’ respectively. Write down their coordinates.

    2. Name the figure OABCB’A’.

    3. State the line of symmetry of this figure

  2. Mr. Lalit invested Rs. 5000 at a certain rate of interest, compounded annually for two years. At the end of first year, it amounts to Rs. 5325. Calculate                                                                 [3]

    1. The rate of interest

    2. The amount at the end of second year, to the nearest rupee.            




SECTION B (40 Marks)

Attempt any four questions from this Section

Question 5

 

  1. solve the quadratic equation $  x^{2}-3(x+3)=0 $; Give your answer correct two  significant figures.                                                                                                                                                                [3]                                                                                                                           

        b)    A page from the savings bank account of Mrs. Ravi is given below.

Date

 

Particulars

Deposit (Rs.)

Balance (Rs.)

April 3rd 2006

 

B/F

 

6000

April 7th

 

By cash

2300

8300

April 15th

 

By cheque

3500

11800

May 20th

 

To self

 

7600

June 10th

By cash

 

5800

13400

June 15th

To self

3100

 

10300

August 13th

By cheque

 

1000

11300

August 25th

To self

7400

 

3900

September 6th 2006

By cash

 

2000

5900

 

She closed the account on 30th September, 2006. Calculate the interest Mrs. Ravi earned at the end of 30th September, 2006 at 4.5% per annum interest. Hence, find the amount she receives on closing the account.                                                                                                                             [4]

 

  1. In what time will Rs. 1500 yield Rs. 496.50 as compound interest at 10% per annum compounded annually?                                                                                                                                       [3]

 

Question 6

  1. Construct a regular hexagon of side 5 cm. Hence construct all its lines of symmetry and name them.                                                                                                                                                          [3]

  2. In the given figure PQRS is a cyclic quadrilateral PQ and SR produced meet at T.                   [4]

      i. $\text { Prove } \Delta \mathrm{TPS} \sim \Delta \mathrm{TRQ}$

      ii. Find SP if TP = 18 cm, RQ = 4 cm and TR = 6 cm.

      iii. $\text { Find area of quadrilateral } \mathrm{PQRS} \text { if area of } \Delta \mathrm{PTS}=27 \mathrm{~cm}^{2} \text { . }$


Shape

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$\text { c) Given matrix } A=\left(\begin{array}{cc}4 \sin 30^{\circ} & \cos 0^{\circ} \\\cos 0^{\circ} & 4 \sin 30^{\circ}\end{array}\right) \text { and } B=\left[\begin{array}{l}4 \\5\end{array}\right]$

If AX = B                                                                                                                                                          [3]

            i. Write the order of matrix X.

            ii. Find the matrix ‘X’.



Question 7

  1. An Aeroplane at an altitude of 1500 meters, finds that two ships are sailing towards it in the same direction. The angles of depression as observed from the Aeroplan are 45o and 30o respectively. Find the distance between the two ships.                                                                    [4]

 

  1. The table shows the distribution of the scores obtained by 160 shooters in a shooting competition. Use a graph sheet and draw an ogive for the distribution. (Take 2 cm = 10 scores on the X-axis and 2 cm = 20 shooters on the Y-axis).                                                           [6]

$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|}\hline \text { Scores } & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 & 50-60 & 60-70 & 70-80 & 80-90 & 90-100 \\\hline \text { No. of shooters } & 9 & 13 & 20 & 26 & 30 & 22 & 15 &10 & 8 & 7 \\\hline\end{array}$

 

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 85%.

 

Question 8

 

     a) $\frac{x}{a}=\frac{x}{b}=\frac{x}{c} \text { show that } \frac{x^{3}}{a^{3}}=\frac{y^{3}}{b^{3}}=\frac{z^{3}}{c^{3}}=\frac{3 x y z}{a b c}$                                                                                  [3]

 

     b) Draw a line AB = 5 cm. Mark a point C on AB such that AC = 3 cm. Using a ruler and a compass only, construct:                                                                                                                                     [4]

     i. A circle of radius 2.5 cm, passing through A and C.

     ii. Construct two tangents to the circle from the external point B. Measure and record the length of the tangents.

 

c) A line AB meets X – axis at A and Y –axis at B. P (4, -1) divides AB in the ratio 1: 2.             [3]

                i. Find the coordinates of A and B.

                ii. Find the equation of the line through P and perpendicular to AB.



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Question 9

  1. A dealer buys an article at a discount of 30% from the wholesaler, the marked price being Rs. 6000. The dealer sells it to a shopkeeper at a discount of 10% on the marked price. If the rate of VAT is 6%, find                                                                                                                                  [3]

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

                ii. The VAT paid by the dealer

 

  1. The given figure represents a kite with a circular and a semicircular motif stuck on it. The radius of a circle is 2.5 cm and the semicircle is 2 cm. If diagonals AC and BD are of lengths 12 cm and 8 cm respectively, find the area of the:                                                                                [4]

 

         i. Shaded part. Give your answer correct to the nearest whole number.

         ii. Unshaded partIcon

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 c) A model of a ship is made to a scale 1: 300                                                                                   [3]

          i. The length of the model of ship is 2 m. Calculate the lengths of the ship.

          ii. The area of the deck ship is 180,000 m2. Calculate the area of the deck of the model.

          iii. The volume of the model in 6.5 m3. Calculate the volume of the ship.



Question 10

  1. Mohan has a recurring deposit account in a bank for 2 years at 6% p.a. simple interest. If he gets Rs. 1200 as interest at the time of maturity, find                                  [3]

           i. the monthly installment

           ii. the amount of maturity

 

  1. The histogram below represents the scores obtained by 25 students in a mathematics mental test. Use the data to:                                                                                                                                       [4]

            i. Frame a frequency distribution table

            ii. To calculate mean

 

            iii. To determine the Modal class

 

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  1. A bus covers a distance of 240 km at a uniform speed. Due to heavy rain its speed gets reduced by 10 km/h and as such it takes two hrs. longer to cover the total distance. Assuming the uniform speed to be ‘x’ km/h, form an equation and solve it to evaluate ‘x’. [3]

 

Question 11

  1. $\text { Prove that } \frac{\cos \mathrm{A}}{1+\sin \mathrm{A}}+\tan \mathrm{A}=\sec \mathrm{A}$                                                                                                        [3]

 

  1. Use ruler and compasses only for the following questions. All constructions lines and arcs must be clearly shown.                                                                                                                                  [4]

           i. $\text { Construct } a \Delta A B C \text { in which } B C=6.5 \mathrm{~cm}, \angle A B C=60^{\circ}, A B=5 \mathrm{~cm} \text { . }$

           ii. Construct the locus of points at a distance of 3.5 cm from A.

           iii. Construct the locus of points equidistant from AC and BC.

           iv. Mark 2 points X and Y which are a distance of 3.5 cm from A and also equidistant from AC and BC. Measure XY.

 

  1. Ashok invested Rs. 26,400 on 12%, Rs. 25 shares of a company. If he receives a dividend of Rs. 2,475. Find the:                                                                                                                                       [3]

             i. number of shares he bought

 

             ii. Market value of each share

 
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