Other Solved Mathematics Board Papers

**MATHEMATICS (ICSE – Class X Board Paper 2010)**

**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) **Solve the following inequation and. represent the, solution set on the number line.

** [3]**

**(b) **Tarun bought an article for and, spent for transportation. He marked the article at and, sold it to a customer. If the customer had. To pay sales tax, find

(i) the customer’s price

(ii) Tarun’s profit percent** [3]**

**(c) **Mr. Gupta opened a recurring deposit account in a bank. He deposited per month for two years. At the time of maturity he got . Find. :

(i) the total interest earned by Mr Gupta.

(ii) the rate of interest per annum** [4]**

**Answers:**

**(a) **

or

or

or

Therefore

**(b) **Cost Price

Overheads

Listed Price

Sales Tax rate

(i) Customer Price

(ii) Profit

**(c) **

**Question 2:**

**(a) **Given , , and . Find .** [3]**

**(b) **Nikita invests for two years at a certain rate of interest compounded annually, At the end of first year it amounts to . Calculate;

(i) The rate of interest;

(ii) The amount at the end of the second year.** [3]**

**(c) ** are two points on the respectively. is the mid-point of . Find the

(i) Co-ordinates of

(ii) Slope of line

(iii) Equation of line ** [4]**

**Answers:**

**(a) **

**(b) **Compound Interest for 1 year

Given

Amount at the end of second year

**(c) **Let and

is the mid point

(i) Therefore

Hence and

(ii) Slope of

(iii) Equation of

**Question 3:**

**(a) **Cards marked with numbers 1, 2, 3, 4 … 20 are well shuffled and a card is drawn at random. What is the probability that the number of the cards is

(i) a prime number

(ii) divisible by 3

(iii) a perfect square** [3]**

**(b) **Without using trigonometric tables evaluate :

** [3]**

**(c) **Use graph paper for this question are the vertices of .

(i) Plot the reflection of in the , and write its co-ordinates.

(ii) Give the geometrical name of the figure .

(iii) Write the equation of the line of symmetry of the line ** [4]**

**Answers:**

**(a) **Given: Cards marked with numbers 1,2, … , 20

(i) Prime Numbers

P (Prime number) = P(A) =

(ii) No. divided by

P (no. divided by 3) = P(A) =

(iii) No. perfect square

P (Perfect square) = P(A) =

**(b) **Given:

**(c) **

(i) Plotted above

(ii) The coordinates of

(iii) Arrow Head

(iv) is the equation of line of symmetry

**Question 4:**

**(a) **When divided by the polynomials and leave the same remainder. Find the value of .** [3]**

**(b) **In the figure, given below, and are two parallel chords and is the center. If the radius of the circle is , find the distance between the two chords of lengths and respectively. ** [3]**

**(c) **The distribution given below shows the marks obtained by students in an aptitude test. Find the mean, median and mode of the distribution.** [4]**

Marks obtained | 5 | 6 | 7 | 8 | 9 | 10 |

No. of students | 3 | 9 | 6 | 4 | 2 | 1 |

**Answers:**

**(a) **When

Remainder1

Remainder2

Given Remainder1 = Remainder 2

**(b) **We know that perpendicular drawn from the center of the circle will bisect the chord.

Therefore

Hence

**(c) **

5 | 3 | 15 | 3 |

6 | 9 | 54 | 12 |

7 | 6 | 42 | 18 |

8 | 4 | 32 | 22 |

9 | 2 | 18 | 24 |

10 | 1 | 10 | 25 |

Mean =

Since is odd,

Median =

Mode (maximum frequency)

**Section B [40 Marks]**

Answer any **four** questions in this section.

**Question 5:**

**(a) **Without solving the following quadratic equation, find the value of ‘p’ for which the roots are equal.** [3]**

**(b) **Rohit borrows from Arun for years at per annum simple interest. He immediately lends his money to Akshay at compounded interest annually for the same period. Calculate Rohits profit at the end of two years.** [3]**

**(c) **Mrs. Kapoor opened a Saving Bank Account in State Bank of India on 9^{th} January 2008. Her passbook entries for the year 2008 are given below:

Date | Particulars | Withdrawals (Rs.) | Deposits (Rs.) | Balance (Rs.) |

Jan. 9, 2008 | By Cash | – | 10,000 | 10,000 |

Feb. 12, 2008 | By Cash | – | 15,500 | 25,500 |

April 6, 2008 | To Cheque | 3,500 | – | 22,000 |

April 30, 2008 | To Self | 2,000 | – | 20,000 |

July 16, 2008 | By Cheque | – | 6,500 | 26,500 |

Aug. 4, 2008 | To Self | 5,500 | – | 21,000 |

Aug. 20, 2008 | To Cheque | 1,200 | – | 19,800 |

Dec. 12, 2008 | By Cash | – | 1,700 | 21,500 |

Mrs. Kapoor closed the account on 31^{st} December 2008. If the bank pays interest at 4% per annum, find the interest he receives on closing the account. Give your answer correct to the nearest rupee.** [4]**

**Answers:**

**(a) **Comparing with , we get

For roots to be equal, we should have

**(b) **Simple Interest for 2 years

Compound Interest for 2 years

Gain

**(c) **Qualifying principal for various months:

Month | Principal (Rs.) |

January | 10000 |

February | 10000 |

March | 25500 |

April | 20000 |

May | 20000 |

June | 20000 |

July | 20000 |

August | 19800 |

September | 19800 |

October | 19800 |

November | 19800 |

Total | 204700 |

**Question 6:**

**(a) **A manufacturer marks an article at . He sells this article to a wholesaler at a discount of on the marked price and the wholesaler sells it to a retailer at a discount of on its marked price. If the retailer sells the article without any discount and at each stage the sales-tax is , calculate the amount of VAT paid by:

i. The Wholesaler

ii. The Retailer** [3]**

**(b)** In the following figure O is the center of the circle and AB is a tangent to it at point B. . Find .** [3]**

**(c) **A doorway is decorated as shown in the figure. There are four semi-circles. , the diameter of the larger Semi-circles. the diameter of the larger semi-circle is of length . Center of the three equal semi-circles lie on . is an isosceles triangle with . If , find the area of the shaded, region.** [4]**

**Answers:**

**(a) **Market Price of the article for wholesaler

Cost Price of the article for wholesaler

Amount of tax paid by the wholesaler

Cost Price of the article for retailer

Amount of tax paid by the retailer

Selling price of the article for retailer

Amount of tax paid by the end customer

Vat paid by retailer

Vat paid by wholesaler

**(b) ** is a tangent to the circle.

(given)

(angle at the center)

**(c) **Let

As angle in semi-circle is

In right angled , by Pythagoras theorem, we get

Also, Area of

Diameter of semicircle

Area of semi-circle

Diameter of each (three equal) semi-circles

Radius of the 3 equal semi-circles

Therefore Area of three equal semi circles

Area of shaded region = Area of semi-circles + Area of three equal circles – Area of

**Question 7:**

**(a) **Use ruler and compasses only for this question :

(i) Construct , where and .

(ii) Construct the locus of points inside the triangle which are equidistant from and .

(iii) Construct the locus of points inside the triangle which are equidistant from and .

(iv) Mark the point which is equidistant from and also equidistant from and . Measure and record the length of .** [3]**

**(b) **The equation of a line is . Find:

(i) Slope of the line.

(ii) The equation of a line perpendicular to the given line and passing through the intersection of the lines and ** [3]**

**(c) **The mean of the following distribution is 52 and the frequency of class interval 30-40 is . Find . ** [4]**

Class Interval | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |

Frequency | 5 | 3 | 7 | 2 | 6 | 13 |

**Answers:**

**(a) **

(i) Step 1: Draw a line . The with as the vertex, draw an angle of using a compass.

Step 2: Draw of length . Join . That gives is the .

(ii) Points which are equidistant from and are on the angle bisector of . Therefore draw angle bisector of

(iii) Points which are on the perpendicular bisector of are equidistant from point and . Draw a perpendicular bisector of .

(iv) The point where the angle bisector and the perpendicular bisector intersect is the point that is equidistant from and but also equidistant form points and .

**(b) **(i) Slope

(ii) Slope of perpendicular

For point of intersection solve and

and

Therefore intersection

Therefore equation of line

**(c) **

Interval | Frequency | |||

10-20 | 5 | 15 | -30 | -150 |

20-30 | 3 | 25 | -20 | -60 |

30-40 | F | 35 | -10 | -10f |

40-50 | 7 | 45 (A) | 0 | 0 |

50-60 | 2 | 55 | 10 | 20 |

60-70 | 6 | 65 | 20 | 120 |

70-80 | 13 | 75 | 30 | 390 |

**Question 8:**

**(a) **Use the remainder theorem to factorize the following expression: .** [3]**

**(b) **If are in continued proportion, prove that: . ** [3]**

**(c) **From the top of a light house high the angles of depression of two ships on opposite sides of it are and respectively. Find the distance between the two ships to the nearest meter.** [4]**

**Answers:**

**(a) **Let

Remainder

Hence is a factor of

Hence

**(b) **If are in continued proportion, then

Applying componendo and dividendo

Squaring both sides

Substituting

**(c) **In

In

Therefore

**Question 9:**

**(a) **Evaluate ** [3]**

**(b) **In the given figure, is a triangle with . Prove that . If and area of . Calculate the:

(i) length of

(ii) area of ** [3]**

**(c) **Vivek invests in , shares at . He sells the shares when the price rises to , and invests the proceeds in shares at . Calculate; i) The sale proceeds ii) The number of shares he buys; iii) The change in his annual income from dividend.** [4]**

**Answers:**

**(a) **

**(b) **Consider

(given)

(common)

Therefore

(AAA postulate)

(i) Given

(ii)

Area of

**(c) **First Investment

Let the amount invested

Nominal Value of the share

Market Value of the share

Dividend earned

Number of shares bought

Sale Proceed

Dividend earned

Second Investment

Therefore the amount invested

Nominal Value of the share

Market Value of the share

Dividend earned

Number of shares bought

Dividend earned

Hence the change in income

**Question 10:**

**(a) **A positive number is divided into two parts such that the sum of the squares of the two parts is 20. The square of the larger part is 8 times the smaller part. Taking x as the smaller part of the two parts, find the number.** [4]**

**(b) **The monthly income of a group of 320 employees in a company is given below:

Monthly Income | No. of Employees |

6000-7000 | 20 |

7000-8000 | 45 |

8000-9000 | 65 |

9000-10000 | 95 |

10000-11000 | 60 |

11000-12000 | 30 |

12000-13000 | 5 |

Draw an ogive of the given. distribution on a graph sheet taking on one axis and on the other axis. From the graph determine:

(i) the median wage

(ii) the number of employees whose income is below

(iii) If the salary of a senior employee is above find the number of senior employees in the company.

(iv) The upper quartile** [6]**

**Answers:**

**(a) **Let the two parts be

Given

Also

Substituting it back

(ignore this as the number is positive)

Therefore the larger part is

Hence the number is

**(b) **

Monthly Income | No. of Employees | Cumulative Frequency |

6000-7000 | 20 | 20 |

7000-8000 | 45 | 65 |

8000-9000 | 65 | 130 |

9000-10000 | 95 | 225 |

10000-11000 | 60 | 285 |

11000-12000 | 30 | 315 |

12000-13000 | 5 | 320 |

Here n (no. of employees) (even)

(i) Median

Required median (from graph)

(ii) Number of employees whose income is below approx

(iii) Number of senior employees in the company

(iv) Upper Quartile term

Upper Quartile

**Question 11:**

**(a) **Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon. ** [3]**

**(b)** A hemispherical bowl of diameter 7.2 cm is filled completely with chocolate sauce. This sauce is poured into an inverted cone of radius 4.8 cm. Find the height of the cone.** [3]**

**(c) **Given . Use componendo and dividendo to prove that: .** [4]**

**Answers:**

**(a) **Given side of the hexagon is . Construct the hexagon as follows:

- First draw a line using a ruler of length 5 cm. Mark it AB.
- The using a compass, make an arc of
- Using compass, draw 120 degree angle and cut the line into 5 cm lengths using the compass. Continue this until the hexagon is completed
- One the hexagon is completed, draw perpendicular bisectors of each of the arms of the hexagon. You will get mid point for each of the arms of hexagon as marked
- Now join the mid points of the opposite sides to get lines
- Now draw the diagonals passing through the center to get lines

The six lines of symmetry

**(b) **Given: Diameter of hemispherical bowl

Radius of hemispherical bowl

Volume of hemispherical bowl

Volume of cone

Volume of cone $latex = Volume of hemi-sperical bowl

**(c) **Given

Applying componendo and dividendo

Simplifying

Square both sides

Applying componendo and dividendo

Simplifying