Other Solved Mathematics Board Papers

**MATHEMATICS (ICSE – Class X Board Paper 2017)**

**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) If is the mean proportion between and , show that; ** [3]**

(b) Solve the equation and give your answer correct to two decimal. ** [4]**

(c) and are two parallel chords of a circle such that and . If the radius of the circle is , find the distance between the two chords. ** [3]**

Answer:

(a)

Given

Now, LHS

RHS.

Hence Proved.

(b) The equation is

We know the roots of a quadratic equation are:

Here

Substituting, we get

(c)

**Question: 2 **

(a) Evaluate without using trigonometric tables,

** [3]**

(b) If and and Find matrix where is a matrix ** [4]**

(c) Jaya borrowed for years . The rates of interest for two successive years are and respectively. She repays at the end of the first year. Find the amount she must pay at the end of the second year to clear her debt. ** [3]**

Answers:

(a)

(b)

(c) We know

is the amount and is the principal.

For first year:

For second year:

**Question: 3**

(a) The catalog price of a computer set is . The Shopkeeper gives a discount of on the listed price. He further gives an off-season discount of on the discounted price. However, sales tax at is charged on the remaining price after the two successive discounts. Find:

(i) The amount of sales tax a customer has to pay

(ii) The total price to be paid by the customer for the computer set. ** [3]**

(b) is a point on the line segment and such that is equal to . Find the coordinates of . ** [4]**

(c) The marks of 10 students of a class in an examination arranged in ascending order is as follows: . If the median marks is 48, find the value of . Find the mode of the given data. ** [3]**

Answers:

(a) Cost Price

1st Discount

Price after discount

2nd Discount

Price after discount

(i) Sales tax

(ii) Price to be paid by the customer

(b) and let

We know the ratio formula:

Therefore

Similarly

Hence the coordinate of

(c)

Given: and

Median for even

Hence

Hence the and the

Looking at all the terms, we see that is repeated twice and hence the mode is

Question 4:

(a) What must be subtracted from the resulting expressing has as a factor? ** [3]**

(b) In the given figure is a rectangle. It consists of a circle and two semi circles each of which are of radius . Find the area of the shaded region. Give your answer correct to three significant figures. ** [4]**

(c) Solve the following in equation and represent the solution set on a number line.

** [3]**

Answers:

(a)

Given is a factor

Therefore has to be subtracted from the given polynomial.

(b) Radius of the circle

Therefore, as shown in the diagram: Breadth of the rectangle and Length of the rectangle

Area of

Taking

Area of the circles and semi circles

Therefore the shaded area

(c) Given

Therefore we have two equations:

Hence

Since , the values of

**Section B (40 Marks)**

**Attempt any four questions from this Section:**

**Question: 5**

(a) Given matrix . Find the matrix if . Hence solve for given that [4]

(b) How much should a man invest in shares selling at to obtain an income of , if the rate of dividend declared is . Also find his yield percent, to the nearest whole number. ** [3]**

(c) Sixteen cards are labeled as . They are put in a box and shuffled. A boy is asked to draw a card from the box. What is the probability that the card draw is:

(i) A Vowel

(ii) A Consonant

(iii) None of the letters of the word median** [3]**

Answers:

(a)

Now

(b) Face Value

Market Value

Dividend

(i) Let the number of shares bought

Dividend Amount

(ii)

Investment

Yield

Therefore yield nearest to a whole number is

(c) No of cards

Cards:

(i) No of vowels:

Probability (vowels)

(ii) No of consonants:

Probability (consonants)

(iii) Letters other than ‘median’

Probability (other than median)

**Question: 6**

(a) Using a ruler and a compass construct a in which , and . Construct the locus of:

(i) Points equidistant from and

(ii) Point equidistant from and

Hence construct a circle touching the three sides of the triangles internally. ** [4]**

(b) A conical tent is to accommodate persons. Each person must have of air to breathe. Given the radius of the tent as find the height of the tent and also its curved surface area. ** [3]**

(c) If , use the properties of proportion to find:

(i)

(ii) ** [3]**

Answers:

(a)

(b) Radius of the cone

No of people

Volume per person

Therefore the volume of the tent

(i) Volume of the tent

(ii) Curved surface area of a cone

Therefore curved surface area

(c)

(i) Applying componendo and dividendo

(ii)

**Question: 7**

(a) A page from a Saving Bank Account Passbook is given below:

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

Jan 07, 2016 | B/F | – | – | 3000.00 |

Jan 10, 2016 | By Cheque | – | 2600.00 | 5600.00 |

Feb 08, 2016 | To Self | 1500.00 | – | 4100.00 |

Apr 06, 2016 | By Cheque | 2100.00 | – | 2000.00 |

May 04, 2016 | By Cash | – | 6500.00 | 8500.00 |

May 27, 2016 | By Cheque | – | 1500.00 | 10,000.00 |

(i) Calculate the interest for the 6 months from January to June 2016, at per annum.

(ii) If the account is closed on 1^{st} July 2016, find the amount received by the account holder. [5]

(b) Use a graph for this question (Take unit on both axis)

(i) Plot the following points : and

(ii) Reflect points on the and write down their coordinates. Name the images as respectively.

(iii) Join the points and in order, so as to form a closed figure, Write down the equation of the line of symmetry of the figure formed. [5]

Answers:

(a) Qualifying principal for various months:

Month | Principal (Rs.) |

January | 5600 |

February | 4100 |

March | 4100 |

April | 2000 |

May | 8500 |

June | 10000 |

Total | 34300 |

(i)

(ii) Amount received by the holder on 1st July

(b)

(i) Please refer to the diagram above.

(ii)

(iii) The enclosed figure is a kite. The line of symmetry is: or

**Question: 8 **

(a) Calculate the mean of the following distribution using step deviation method: [4]

Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |

No. of Students | 10 | 9 | 25 | 30 | 16 | 10 |

(b) In the given figure is a tangent to the circle at and are bisectors of and , if prove that:

- is a diameter of the circle
- is an isosceles triangle. [3]

(c) The printed price of an air conditioner is . The wholesaler allows a discount of to the shopkeeper. The shopkeeper sells the article to customer at a discount of of the marked price. Sales tax (under VAT) is changed at the rate of at every stage. Find:

- VAT paid by the shopkeeper to the government;
- The total amount paid by the customer inclusive of tax. [3]

Answers:

(a) Table as follows:

Marks | Mid Term | No. of Students | |||

0-10 | 5 | 10 | -20 | -2 | -20 |

10-20 | 15 | 9 | -10 | -1 | -9 |

20-30 | 25 | 25 | 0 | 0 | 0 |

30-40 | 35 | 30 | 10 | 1 | 30 |

40-50 | 45 | 16 | 20 | 2 | 32 |

50-60 | 55 | 10 | 30 | 3 | 30 |

Mean

(b) Consider the diagram as shown:

(Given)

(i) (since DB is the diameter)

(ii) (angles subtended by the cord CB on the circumference of the circle)

(iii) (angles subtended by the cord DC on the circumference of the circle)

When the chord bisects the tangent…

Also

is an isosceles triangle.

(c) Price

Wholesaler to Shopkeeper

Price

Discount

Therefore discount amount

Therefore discounted price

VAT

Therefore VAT paid by the wholesaler

Shopkeeper to Customer

Price

Discount

Therefore discount amount

Discounted price

VAT

Therefore VAT paid by the retailer

(i) VAT paid by the shopkeeper

(ii) Total price for the customer

**Question: 9 **

(a) In the figure given, is the center of the circle. . Find giving suitable reasons, the measure of: ** [4]**

(i)

(ii)

(iii)

(b) and are the vertices of a triangle.

(i) Find the coordinates of the centroid of the triangle.

(ii) Find the equation of the line through and parallel to .

(c) Prove that

** [3]**

Answers:

(a)

(radius of the same circles)

(i) ( is a cyclic quadrilateral)

(ii) (angle subtended by a chord at the center is twice that subtended on the circumference)

(iii) ( is isosceles)

(b) are the given coordinates of the three vertices of the triangle

(i) Let the centroid be

Therefore

and

Hence centroid is

(ii) Slope of

Therefore the slope of the line passing through and parallel to

Hence the equation of the line passing through and parallel to is:

or

(c)

LHS

RHS. Hence proved.

**Question: 10**

(a) The sum of the ages of Vivek and his younger brother Amit is 47 years. The product of their ages is a year is 550. Find their ages. ** [4]**

(b) The Daily wages of 80 workers in a project are given below. ** [6]**

Wages in Rs | 400-450 | 450-500 | 500-550 | 550-600 | 600-650 | 650-700 | 700-750 |

No. of workers | 2 | 6 | 12 | 18 | 24 | 13 | 5 |

Use a graph paper to draw an ogive for the above distribution. (Use a scale of on and workers on ). Use your ogive to estimate:

(i) The medium wage of the workers.

(ii) The lower quartile wage of workers.

(iii) The number of workers who can more than Rs.625 daily;

Answers:

(a) Let the age of Vivek

and Let the age of Amit

Given: and

Hence

When

When

Hence the age of the two is years.

(b)

Wages | No. workers | Cumulative Frequency (c.f) |

400-450
450-500 500-550 550-600 600-650 650-700 700-750 |
2
6 12 18 24 13 5 |
2
8 20 38 62 75 80 |

On the graph paper, we plot the following points:

(i) Median

From the graph

(ii) Lower quartile

From the graph

(iii) The number of workers earning more that 625 per day students

**Question: 11**

(a) The angles of depression of two ships and as observed from the top of a light house high are and respectively. If the two ships are on the opposite sides of the light house, find the distance between the two ships, Give your answer correct to the nearest whole number. ** [4]**

(b) is a triangle, is a point on the side of triangle such that Given , and

(i) Prove

(ii) Find the length of and

(iii) **[3]**

(c) Richard has a recurring deposit account in a bank for 3 years at p.a. simple interest. If he gets as interest at the time of maturity, find;

(i) The monthly deposit

(ii) The maturity Value **[3]**

Answers:

(a) In

In

Hence

Rounding off the the nearest whole number, we get

(b)

(i) Consider and

(given)

is common

is common.

Therefore by AAA postulate,

(ii) Since

(iii)

(c)

months

(i)

(ii) Maturity Value