Other Solved Mathematics Board Papers

**MATHEMATICS (ICSE – Class X Board Paper 2014)**

**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) Ranbir borrows Rs. 20,000 at 12% per annum compound interest. If he repays Rs. 8400 at the end of the first year and Rs. 9680 at the end of the second year, find the amount of loan outstanding at the beginning of the third year. [3]

(b) Find the value of , which satisfy the in equation Graph the solution set on the number line. [3]

(c) A die has 6 faces marked by the given numbers as shown below: 1, 2, 3, -1, -2, -3. The die is thrown once. What is the probability of getting?

(i) A positive integer

(ii) An integer greater than -3

(iii) The smallest integer [4]

**Answer.**

(a) Given: Principal for the first year

Money repaid at the end of 1st year

Principle for the 2nd year

Money repaid at the end of the second year

The loan amount outstanding at the beginning of the third year

Multiplying throughout by 6

Therefore the values of

The graph of the solution set is shown by dots on the number line.

(c) No. of sample space

A positive integer

No. of favorable cases

An integer greater than

No. of favorable cases

Smallest integer

**Question 2:**

(b) Sharukh opened a Recurring Deposit Account in a bank and deposited Rs. 800 per month for 1.5 years. If he received Rs. 15,084 at the time of maturity. Find the rate of interest per annum. [3]

(c) Calculate the ratio in which the line joining is divided by point Also find (i) (ii) Length of [4]

**Answer:**

(b) Here, Principal (P) = money deposited per month = Rs. 800

Let the rate of interest be per annum, then

Total money deposited

Since money deposited +Interest = Maturity value

Hence rate of interest

(c) Let divide the line segment joining the points in the ratio

Coordinate of is: *[Reference Link]*

But Coordinate of

Length of

**Question 3:**

(a) Without using trigonometric tables, evaluate

[3]

(b) Using the remainder and factor theorem, factor the following polynomial: [3]

(c) In the figure given below is a rectangle From the rectangle a quarter circle and a semicircle are removed Calculate the area of the remaining piece of the rectangle (Take ) [4]

**Answers:**

(a) Given

(b) Let

Putting , we get

By factor theorem is actor of

On dividing by , we get as the quotient and remainder

Therefore the other factor of are the factor of

Now,

Hence

(c) Area of rectangle

**Question 4: **

(a) The number are arranged in an ascending order. If the mean of the observations is equal to the median, find the value of [3]

(b) In the figure, is a diameter of the circle. Calculate: [3]

(i)

(ii)

(iii)

(c) Using graph paper to answer the following questions. (Take unit on both axis)

(i) Plot the points

(ii) is the image of when reflected in the Plot it on the graph paper and write the coordinates of

(iii) is the image of when reflected in the line write the coordinates of

(iv) Write the geometric name of the figure

(v) Name a line of symmetry of the figure formed. [4]

**Answers:**

(a) Arrange numbers in ascending order are

No. of terms

According to given condition

or

(b) In

(i) (angle in the semi circle)

(ii) (cyclic quadrilateral)

(iii) (angles in the same segment)

(c) As shown in the graph below:

(i) Coordinate of

(ii) Coordinate of

(iii) Geometric name

(iv) is the symmetric line.

**SECTION B [40 Marks]**

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

**Question 5:**

(a) A shopkeeper bought a washing machine at a discount of 20% from a wholesaler, the printed price of the washing machine being Rs. 18,000. The shopkeeper sells it to a consumer at a discount of 10% on the printed price. If the rate of sales tax is 8% find:

(i) the VAT paid by the shopkeeper

(ii) the total amount that the consumer pays for the washing machine. [3]

(i)

(c) In

(i) Prove that is similar to

(ii) Find

(iii) Find area of [4]

**Answers:**

(a) Given: Printed price of washing machine

Rate of discount

Shopkeeper’s Price

Price of consumer

Since, Tax paid by the shopkeeper

VAT paid by the shopkeeper=Tax charged – Tax Paid

(ii) Total amount paid by the consumer for washing machine

(i) Applying componendo and dividend

(ii) Cubing both sides we get

Applying componendo and dividend

(c)

(i) In

(common angle)

(given)

Therefore (AAA Postulate)

(ii) Since

(corresponding sides are proportional)

(iii) Since

**Question 6:**

(a) Find the value of for which the following points are collinear. Hence find the equation of the line. [3]

(b) Salman invests a sum of money in Rs. 50 shares, paying 15% dividend quoted at 20% premium. If his annual dividend is Rs. 600 , calculate;

(i) the number of shares he bought

(ii) his total investment

(iii) the rate of return on his investment [3]

(c) The surface area of a solid metallic sphere is it is melted and recast into solid right circular cones of radius 3.5 cm and height 7 cm. calculate:

(i) the radius of the sphere.

[4]

Answers:

Rejecting

(b) Nominal value of share

Total dividend of Salman

Market value of 1 share

Total investment for 80 shares

**(c)**

(i) Let the radius sphere

Surface area of sphere

**Question 7:**

(a) Calculate the mean of the distribution given below using the short cut method; [3]

Marks | 11-20 | 21-30 | 31-40 | 41-50 | 51-60 | 61-70 | 71-80 |

No. of students | 2 | 6 | 10 | 12 | 9 | 7 | 4 |

(b) In the figure given below, diameter of a circle meet at is a tangent to the circle at find:

(i)

(ii) the length of tangent [3]

**Answers:**

(a) Table as follows

Mean Value | ||||

11-20
21-30 31-40 41-50 51-60 61-70 71-80 |
2
6 10 12 9 7 4 |
15-5
25-5 35-5 45-5 55-5 65-5 75-5 |
-30
-20 -10 0 10 20 30 |
-60
-120 -100 0 90 140 120 |

(b)

(i) Since chord and tangent at point intersect each other at

……….. (i)

Since chord and tangent at point interested each other at ,

……….. (ii)

From (i) (ii)

Given;

Putting these values in eq. (3)

Hence,

(ii) From (i),

Length of tangent

(c)

**Question 8:**

(a) The compound interest, calculate yearly, on a certain sum of money for the second year is Rs. 1320 and for the third year is Rs. 1452. Calculate the rate of interest and the original sum of money. [3]

(b) Construct a with , Construct the in circle of the triangle measure and record the radius of the in circle. [3]

(c) Use a graph paper for this question. The daily pocket expenses of 200 students in a school are given below;

Pocket expenses (in Rs.) | 0-5 | 5-10 | 10-15 | 15-20 | 20-25 | 25-30 | 30-35 | 35-40 |

Number of students (frequency) | 10 | 14 | 28 | 42 | 50 | 30 | 14 | 12 |

Draw a histogram representing the above distribution and estimate the mode from the graph. [4]

**Answers:**

(a) Compound Interest for the third year

Compound Interest for the second year

Simple Interest on for one year

Let the original money be

Amount after 2 year – amount after one year = Compound Interest for second year.

Rate of interest

Original sum of money

(b) Steps of construction:

- Construct a with the given data:
- Draw a line BC of 6.5 cm length using a ruler
- The make an arc of 5.5 cm from B and similarly, make an arc of 5 cm from C
- The place where the two arcs intersect, join that point to B and C and complete the triangle.

- Draw the internal bisectors of Let these bisectors cut at
- Draw an arc from point B so that it cuts the two sides of the angle ABC
- From the point of intersections, draw two arcs
- Join the point B and the point of intersection and draw a line. This line bisects the angle ABC.
- Repeat the above 3 steps for point C.
- The two bisectors will intersect at the point O which is the center of the in-circle.

- Taking as center and touching the side of the circle as the radius, Draw a in-circle which touches all the sides of the
- From draw a perpendicular to side which cut at
- Measure which is required radius of the in circle.

(c)

**Question: 9**

(a) If is the duplicate ratio of , find the value of [3]

(b) Solve for using the quadratic formula. Write your answer correct to two significant figures. [3]

(c) A page from the saving bank account of Priyanka is given below: [4]

Date |
Particular |
Amount Withdrawal |
Amount Deposited |
Balance |

03/04/2006
05/04/2006 18/04/2006 25/05/2006 30/05/2006 20/07/2006 10/09/2006 19/09/2006 |
B/F By cash By Cheque To Cheque By Cash By Self By Cash To Cheque |
– – – 5,000.00 – 4,000.00 – 1,000.00 |
– 2,000.00 6,000.00 – 3,000.00 – 2,000.00 – |
4,000.00 6,000.00 12,000.00 7,000.00 10,000.00 6,000.00 8,000.00 7,000.00 |

If the interest earned by Priyanka for the period of ending September, 2006 is Rs.175, find the rate of interest.

**Answers:**

(a) Given: is the duplicate ratio of

(b) Given:

Comparing with , we get

(c) Qualifying principal for various months:

Month | Principal (Rs.) |

April | 6000 |

May | 7000 |

June | 10000 |

July | 6000 |

August | 6000 |

September | 7000 |

Total for 1 month | 42000 |

**Question: 10**

(a) A two digit number is such that the product of its digits is 6, If 9 is added to the number, the digits interchange their places. Find the number. [4]

(b) Marks obtained by 100 students in a Mathematics test are given below; [6]

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

No. of Students | 3 | 7 | 12 | 17 | 23 | 14 | 9 | 6 | 5 | 4 |

Draw an ogive for the distribution on a graph sheet. (Use a scale of 2 cm = 10 units on both axis)

Use the ogive to estimate the:

(i) median

(ii) lower quartile

(iii) number of students who obtained more than 85% marks in the test.

(iv) number of students who did not pass in the if the pass percentage was 35

**Answers:**

(a) Let the required two digit number be

Given:

Since (given)

(not possible)

When

The required two digit number

(b)

Marks | No. of Students | Cumulative Frequency (c.f) |

0-10
10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 |
8
7 12 17 23 14 9 6 5 4 |
3
10 22 39 62 76 85 91 96 100 |

On the graph paper, we plot the following points:

From the graph

From the graph

(iii) The number of students who obtained more than 85% marks in test students

(iv) The number of students who did not pass in the test if test if the pass percentage was 35 students.

**Question 11:**

(a) In the figure given below, is the center of the circle, are two chords of the circle.

Find the:

(i) radius of the circle

(ii) length of chord [3]

(b) Prove the identity:

[3]

(c) An airplane at an altitude of observed of depression of two boats on the opposite banks of a river to be respectively. Find the answer correct to the nearest whole number. [4]

**Answers:**

(a) Given

is mid point of

(i) Let radius of circle

From

(ii) Now from

As is mid point of

(b) To prove:

LHS

Hence Proved.

(c) Let height of airplane

Two boats are at

Let as shown in the diagram.

Width of river