Other Solved Mathematics Board Papers

**MATHEMATICS (ICSE – Class X Board Paper 2013)**

**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:**

(b) At what rate p.a. will a sum of yield as compound interest in years? [3]

(c) The median of the following observation arranged in ascending order is Find the value of and hence find the mean. [4]

**Answers:**

Substituting these values in the given expression we get,

(b) Given: Principle

Amount

Time

(c) Given observation are and mediam

Since which is odd, therefore

**Question 2:**

(a) What number must be added to each of the number to make them proportional? [3]

(b) If is a factor of the expression and, when the expression is divided by , it leaves a remainder , find the values of [3]

(c) Draw a histogram from the following frequency distribution and find the made from the graph: [4]

Class | 0-5 | 5-10 | 10-15 | 15-20 | 20-25 | 25-30 |

Frequency | 2 | 5 | 18 | 14 | 8 | 5 |

**Answers:**

(a) Let the number that must be added be , then

The new number

Since they are proportional,

(b) Let is a factor of the given expression;

Since

In the given expression, we substitute we get

When given expression is divided by

Similarly, in the given expression, we substitute we get

Solving equation (i) and (ii),

(c)

**Question 3:**

(a) Without using tables evaluate: [3]

(b) In the given feature,

Prove:

(i) AC is the diameter of the circle

(ii) Find [3]

(c) is a diameter of a circle with center , If , Find;

(i) The length of radius

(ii) The Coordinates of [4]

**Answers:**

(a)

(b) Given:

(i) Since is a cyclic quadrilateral

In

( sum property of a triangle)

Now from ,

Hence makes right angle belongs in semi-circle therefore is a diameter of the circle.

(ii) (Angles in the same segment of a circle)

Therefore

(c)

(i) Length of the radius

(ii) Let the point be

Given is the mid-point of Therefore

Hence, the co-ordinate of

**Question 4:**

(a) Solve the following equation and calculate the answer correct to two decimal places. [3]

(b) In the given figure, are perpendicular to

(i) Prove that

(ii) If , Calculate

(iii) Find the ratio of the area of [3]

**(c)** Using graph paper, plot the point

(i) Reflect in the origin to get the images

(ii) Write the co-ordinate of

(iii) State the geometrical name for the figure

(iv) Find its perimeter [4]

**Answers:**

(a) Given :

Comparing this expression with

**(b)**

(i) From

Given

And

(ii) In

Since

Given:

(iii) Since

(c)

(i) Please see graph

(ii) Reflection of in the origin

(iii) The geometrical name for the figure is a parallelogram

(iv) From the graph,

In

Therefore since is a parallelogram

Perimeter of

**SECTION B [40 Marks]**

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

**Question: 5**

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

(b) Mr. Britto deposits a certain sum of money each month in a Recurring Deposit Account of a bank. If the rate of interest is of 8% per annum and Mr. Britto gets Rs. 8088 from the bank after 3 years, find the value of his monthly installment. [3]

(c) Salman buys 50 shares of face value Rs. 100 available at Rs. 132.

(i) What is his investment?

(ii) If the dividend is 7.5% what will be his annual income?

(iii) If he wants to increase his annual income by Rs. 150. How many extra shares should he buy? [4]

**Answers:**

Taking L.C.M. of 3, 2 and 6 is 6.

(b) Let the monthly installment

Given: Maturity amount

Actual sum deposited

Maturity amount = Interest +Actual sum deposited

Hence the monthly installment be

(c) Number of shares

Face value of each share

Market value of each shares

Total face value

(i) Total investment

(ii) Rate of dividend

(iii) Let extra share should he buy be

Then total number of shares

Total face value

Hence the extra shares should be buy

**Question 6:**

[3]

(b) In the given circle with center is a tangent to the circle at Find [3]

(c) Given below are the entries in a Saving Bank A/C pass book

Date | Particular | Withdrawal | Deposit | Balance |

Feb. 8
Feb .18 April. 12 June.15 July. 8 |
B/F
To Self By Cash To Self By Cash |
–
Rs. 4000 – Rs. 5000 – |
–
– Rs. 2230 – Rs. 6000 |
Rs. 8500
– – – – |

Calculate the interest for six months from February to July at [4]

**Answers:**

(b) Given:

We know that,

(sum of the opposite angles in a cyclic quadrilateral is )

Join , we have a isosceles

(Radius of the same circle)

(angle in a semi circle)

In

Since (in a triangle, angles opposite to equal sides are equal)

Hence,

Since

The tangent at a point to circle is perpendicular to the radius through the point to contact.

(c)

Date | Particular | Withdrawal | Deposit | Balance |

Feb. 8
Feb .18 April. 12 June.15 July. 8 |
B/F
To Self By Cash To Self By Cash |
–
Rs. 4000 – Rs. 5000 – |
–
– Rs. 2230 – Rs. 6000 |
Rs. 8500
Rs. 4500 Rs. 6730 Rs. 1730 Rs. 7730 |

Principle for the month of Feb = Rs. 4500

Principle for the month of March = Rs. 4500

Principle for the month of April = Rs. 4500

Principle for the month of May = Rs. 6730

Principle for the month of June = Rs. 1730

Principle for the month of July = Rs. 7730

Total principle from the month of Feb. to July = Rs. 29690

**Question 7:**

(a) In Find the equation of the equation of the median through A. [3]

(b) A shopkeeper sells an article at the listed price of Rs. 1500 and the rate of VAT is 12% at each stage of sale. If the shopkeeper pays a VAT of Rs. 36 to the Government. What was the price, inclusive of Tax, at which the shopkeeper purchased the article from the wholesaler? [3]

(c) In the figure given, from the top of a building high, the angles of depression of the top and bottom of a vertical lamp post are observed to be respectively. [4]

Find:

(i) The horizontal distance between

(ii) The height of the lamp paid.

**Answers:**

(a) Since

Now equation of mediam

Here,

(b) Given: List price of an article , Rate of

Let C.P. of this article be , then

If the shopkeeper pays a

Therefore the price at which the shopkeeper purchased the article inclusive of sales tax

(c) Given; AB=60 m

Since

(alternate angles, AP and BC are parallel)

(i) Now in

Hence the horizontal distance between

(ii) Let and proved above

Therefore

Since

Hence, the height of the lamp post

**Question 8:**

(b) A solid sphere of radius 15 cm is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate the number of comes recast. [3]

(c) Without solving the following quadratic equation, find the value of for which the given equation has real and equal roots; [4]

**Answers:**

Therefore

and

Hence

(b) Radius of a solid sphere,

Now, radius of right circular come

Height

(c) Given equation

Since roots are real and equal,

Comparing the coefficient of with equation , we get

Substituting the values in (i) we get

**Question 9: **

(a) In the figure along side is a quadrant of a circle, The radius Calculate the area of the shaded portion.

(b) A box contain some black balls and 30 white balls. If the probability of drawing a black ball is two-fifths of a white ball, find the number of black balls in the box. [3]

(c) Find the mean of the following distribution by step deviation method: [4]

Class Internal | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |

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

**Answers:**

(a) Radius of quadrant

(b) Let the number of black balls be , then

Total number of balls

Hence, the number of black balls

(c)

C.I | Frequency | Mid-Value | ||

20-30
30-40 40-50 50-60 60-70 70-80 |
106
8 12 5 9 |
25
35 45 55 65 75 |
-2
-1 0 1 2 3 |
-20
-6 0 12 10 27 |

Here,

**Question:10**

(a) Using a ruler and compasses only:

(i) Construct a with the following data:

(ii) In the some diagram draw a circle with as diameter. Find a point on the circumstance of the circle which is equidistant from

(iii) Measure [4]

(b) The mark obtained by 120 students in a test are given below;

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

Students | 5 | 9 | 16 | 22 | 26 | 18 | 11 | 6 | 4 | 3 |

Draw an ogive for the given distribution on a graph sheet; Using suitable scale for ogive to estimate the following:

(i) The mediam

(ii) The number of students who obtained more than 75% marks in the test.

(iii) The number of students who did not pass the test if minimum marks required to pass is 40. [6]

**Answers:**

(a) Steps of Construction:

Using a ruler draw a line

With the help of the point , draw You could do it by drawing an arc with B as the center. Then cut the arc twice using the same radius as set in the compass.

Given that the length of Taking radius cut This gives you point A.

Now join to with the help of a ruler.

We now need to draw a perpendicular bisector of BC. This can be done by taking a certain length in the compass and make arcs as shown in the diagram keeping point B and C as the center. Make sure that you keep the width of the compass same for all the four arcs.

The join the two points of intersection. This gives the perpendicular bisector. Draw perpendicular bisector of

Draw a circle as center and as radius.

Now draw angle bisector of which intersects circle at

Join

Now,

(b)

Marks | No. of students | Cumulative Frequency |

0-10
10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 |
5 9 16 22 26 18 11 6 4 3 |
5 14 30 52 78 96 107 113 117 120 |

N=120 |

On the graph paper we plot the following points:

From the graph 60th term

(ii) The number of students who obtained more than marks in test

(iii) The number of students who did not pass the test if the minimum pass marks

**Question 11:**

(a) In the figure given below the segment meets at at The point on divides it in the ration find the coordinates of [3]

(c) A shopkeeper purchases a certain number of books for Rs. 960. If the cost per book was Rs. 8 less, the number of books that could be purchased for Rs. 960 would be 4 more. Write an equation, taking the original cost of each book to be , and solve it to find the original cost of the books. [4]

**Answers:**

(a) Let the co-ordinates of

Since the co-ordinates of a point divides it in the ratio it implies that

By using section formula, we get

Hence, the co-ordinate of

By using componendo and dividendo, we get:

Taking square root on both sides, we get

(c) Given the original cost of each book be

Total cost

According to question,

(not possible)

Hence the cost of the original book is