Other Solved Mathematics Board Papers

**MATHEMATICS (ICSE – Class X Board Paper 2012)**

**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)** Given . Find . **[3]**

**(b) **The monthly pocket money of Ravi and Sanjeev are in the ratio . There expenditures are in the ratio . If each saves every month, find their monthly pocket money. **[3]**

**(c)** Using the Reminder Theorem factories completely the following polynomial: **[4]**

Answers:

**(a)** Given

**(b) **Let monthly pocket money be Ravi is and Sanjeev is .

They both save per month.

Therefore, their expenditure would be and respectively.

Hence

Ravi’s pocket money

Sanjeev’s pocket money

**(c)** Given

Try

, therefore (x-1) is not a factor of the given function.

Try

Therefore is a factor of

To factories

Hence,

**Question 2:**

**(****a)** On what sum of money will the difference between the compound interest and simple interest for years be equal if the rate of interest charged for both is p.a.? **[3]**

**(b) ** is an is isosceles right-angled triangle with . A semi-circle is drawn with as the diameter. If find the area of the shaded region. . **[3]**

**(c)** Given a line segment joining the points and Find:

(i) the ratio in which is divided by the

(ii) find the coordinates of the point of intersection.

(iii) the length of . **[4]**

Answers:

**(a)** Let the sum be

Simple Interest for 2 years

Amount Compound Interest

Given difference = 25 Rs.

Therefore

**(b)** is a right angled triangle. Therefore

Area of semi circle

Area of

Area of the shaded region = Area of the semi circle – Area of

**(c)** Let the required ratio be and the point of intersection be

Since

Therefore the point intersection is

Length of .

**Question 3:**

**(a) **In the given figure is the central of the circle and is a tangent at . If and . Calculate the radius of the circle. **[3]**

**(b)** Evaluate without using trigonometric tables:

**[3]**

**(c)** Marks obtained by students in a short assessment is given below, where are two missing data:

Marks | 5 | 6 | 7 | 8 | 9 |

No. of Students | 6 | A | 16 | 13 | B |

If the mean of the distribution is find . **[4]**

Answers:

**(a) ** Let the radius of the circle

Here we apply intercept theorem. Therefore:

**(b) **

=

=

**(c) **Given, the total number of students

Therefore

… … … (i)

Given mean

Therefore

… … … (ii)

Solving (i) and (ii) we get and

**Question 4:**

**(a)** Kiran deposited per month for months in a Bank’s recurring deposit account. If the bank pays interest at the rate of per annum, find the amount she gets on maturity. **[3]**

**(b)** Two coins are tossed once; Find the probability of getting:

(i) heads

(ii) At least tail **[3]**

**(c)** Using graph paper and taking along both and ;

(i) Plot the points and

(ii) Reflect in the origin to get the images respectively.

(iii) Write down the co-ordinates of .

(iv) Give the geometrical name for the figure .

(v) Draw and name its lines of symmetry. ** [4]**

Answers:

**(a)**

**(b)** Let Heads – and Tails –

If two coins are tossed once, then the total number of possibilities would be as shown: Sample Space

(i.e. there are 4 possible outcomes)

(i) Event: getting two heads

Hence the probability

(ii) Events : At least one tail

Hence the probability

**(c)** (i) Please refer to the graph shown below

(ii) Please refer to the graph shown below

(iii) Reflection of in the origin are respectively.

(iv) Name of the geometrical figure in the graph show is Rhombus

(v) Two lines of symmetry: Both diagonal

**SECTION B [40 Marks]**

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

**Question 5:**

**(a)** In the given figure, is the diameter of a circle with center . . Find (i) (ii) ** [3]**

**(b)** Given Write (i) the order of the matrix (ii) the matrix . ** [3]**

**(c)** A page from the savings Bank Account of Mr. Prateek is given below: ** [4]**

Date | Particular | Withdrawal | Deposit | Balance |

Jan. 1^{st} 2006 |
B/F | – | – | 1,270 |

Jan. 7^{th} 2006 |
By Cheque | – | 2310 | 3580 |

March 9^{th} 2006 |
To Self | 2000 | – | 1580 |

March 26^{th} 2006 |
By Cash | 6200 | 7780 | |

June 10^{th} 2006 |
To Cheque | 4500 | – | 3280 |

July 15^{th} 2006 |
By Clearing | – | 2630 | 5910 |

October 18^{th} 2006 |
To Cheque | 530 | – | 5380 |

October 27^{th} 2006 |
To Self | 2690 | – | 2690 |

November 3^{rd} 2006 |
By Cash | – | 1500 | 4190 |

December 6^{th} 2006 |
To Cheque | 950 | – | 3240 |

December 23^{rd} 2006 |
By Transfer | – | 2920 | 6160 |

If he receives as interest on 1^{st} January, 2007. Find the rate of interest paid by the bank.

Answers:

**(a)** (i) is a cyclic quadrilateral)

(ii) In

**(b)**

Therefore . Hence the order of Matrix is

Let

Therefore

Therefore

and

Solving we get

Hence

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

Month | Principal (Rs.) |

January | 3580 |

February | 3580 |

March | 1580 |

April | 7780 |

May | 7780 |

June | 3280 |

July | 3280 |

August | 5910 |

September | 5910 |

October | 2690 |

November | 4190 |

December | 3240 |

Total | 52800 |

Therefore Rate

**Question 6:**

**(a)** The principal price of an article is . The wholesaler allows a discount of to the shopkeeper. The shopkeeper sells the article to the customer at the printed price. Sales tax (under VAT) is charged at the rate of at every stage. **[4]**

Find:

(i) The cost of the shopkeeper inclusive of tax

(ii) VAT paid by the shopkeeper to the Government.

(iii) The cost of the customer inclusive of tax.

**(b)** Solve the following inequation and represent the solution set on the number line:

**[3]**

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

**[3]**

Answers:

**(a)** Printed Price

Discounted price

Price charged by the wholesaler

The cost to the shopkeeper inclusive of the tax

VAT paid by the shopkeeper

The cost to the customer inclusive of the tax

**(b)**

or or or

or or or

Therefore

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

For roots to be equal, we should have

**Question 7:**

**(a)** A hollow sphere of internal and external radii and respectively is melted and recast into small cones of base radius and height . Fins the number of cones. **[3]**

**(b)** Solve the following equation and give your answer correct to significant figures: **[3]**

**(c)** As observed from the top of a tall lighthouse, the angles of depression of two ships on the same side of the light house in horizontal line with its base are and Find the distance between the two ships. Give your answer correct to the nearest meter. **[4]**

Answers:

**(a)** Sphere: Internal radius , External radius

Cone: Radius , Height

**(b)** Given :

Comparing this expression with , we get

We know

Therefore or

**(c)** In

In

Therefore

**Question 8:**

**(a)** A man invests on shares at . If the company pays him dividend. Find:

(i) The number of shares he buys

(ii) His total dividend.

(iii) His percentage return on the shares. **[3]**

**(b)** In the given figure and are right angled at respectively. Given and .

(i) Prove

(ii) Find **[3]**

**(c)** If using properties of proportion show that **[4]**

Answers:

**(a)** Investment , Dividend

Nominal Value of the share

Market Value of the share

(i) Number of shares

(ii) Dividend

(iii)

**(b)** In

(common angle)

(right angles)

Therefore (AAA postulate)

Since

Given AC=10 cm, AP = 15 cm and PM= 12 cm

**(c)** Given

Applying componendo and dividendo

Simplify

Now square both sides

Simplifying

or

**Question 9:**

**(a)** The line through and is perpendicular to the line . Find the value of $latex b. **[3]**

**(b)** Prove that **[3]**

**(c)** A car covers a distance of 400 km at a certain speed. Had the speed been 12 km per hours more, the line taken for the journey would have been 1 hour and 40 minutes less. Find the original speed of the care. **[4]**

Answers:

**(a)** Slope of

Given equation is

Since they are perpendicular,

**(b)** RHS

LHS. Hence proved.

**(c)** Let the speed of the car be

Therefore

Therefore speed of the car is 48 km/hr.

**Question 10:**

**(a)** Construct a in which base and [3]

(i) Construct a circle circumscribing the

(ii) Draw a cyclic quadrilateral so that is equidistant from . **[4]**

**(b)** The following distribution represents the height of 160 students of a school

Height | 140-145 | 145-150 | 150-155 | 155-160 | 160-165 | 165-170 | 170-175 | 175-180 |

Students | 12 | 20 | 30 | 38 | 24 | 16 | 12 | 8 |

Draw a given for the given distribution taking 2 cm =5 cm of height on one axis and 2 cm=20 students on the other axis. Using the graph, determine:

(i) The medium height

(ii) The interquartile range

(iii) The number of students whose height is above 172 cm. ** [6]**

Answers:

**(a) **(i) Following Step to Constructions:

- First, using a ruler, draw a line segment
- Construct This you can do it by using a compass. Draw an arc with B as the center. The using the same arc and intersection point ‘x”, draw two arcs one at ‘y’ and then one at ‘z’. Craw a line through B and ‘z’. You will get 120^o angle.
- The cut from . This you can do by setting the compass for a width of 5.5 cm.
- Then join . So you get the \triangle ABC.
- Now construct perpendicular bisectors of and intersecting at . Join .
- Taking O as the center and as radius draw a circle, passing through , and

(ii)

- Extend the right bisector of intersecting the circle at .
- Join and
- is required cyclic quadrilateral.

**(b) **Following table:

Height |
F | c.f. |

140-145
145-150 150-155 155-160 160-165 165-170 170-175 175-180 |
12
20 30 38 24 16 12 8 |
12 32 62 100 124 140 152 160 |

(i) Mean

(ii) Interquartile range

(iii) No. of students above

**Question 11:**

**(a)** In and . Find the radius of the inscribed circle. **[3]**

**(b) **Find the mode and medium of the following frequency distribution: ** [3]**

X | 10 | 11 | 12 | 13 | 14 | 15 |

f | 1 | 4 | 7 | 5 | 9 | 3 |

**(c)** The line through intersects at . [4]

(i) Write the slope of the line

(ii) Write the equation of the line

(iii) Find the Co-ordinates of ** [4]**

Answers:

**(a)** Given is a right triangle. Therefore using Pythagoras theorem:

Draw as shown. They will be perpendicular to the sides of the triangle.

is a square since the angels are and (Since the tangent to a circle from an exterior point are equal in length).

Therefore

(Since the tangent to a circle from an exterior point are equal in length).

(Since the tangent to a circle from an exterior point are equal in length).

Since

**(b)** Mode is the value of the highest frequency.

Therefore Mode

For Median, first write the data in ascending order as follows:

.

Since Median is the middle most value.

Median

**(c)** Given points

Slope

Equation of line:

When

Hence the co-ordinates of