Other Solved Mathematics Board Papers

**MATHEMATICS (ICSE – Class X Board Paper 2020)**

**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 quadratic equation:

Give your answer correct to two decimal places. **[3]**

(b) Given

If , where is the identity matrix of Order , find and . **[3]**

(c) Using ruler and compass construct a triangle where cm, cm and . Hence construct a circle circumscribing the triangle . Measure and write down the radius of the circle. **[4]**

Answers:

(a) Given

Comparing it with we get , and

We know,

Therefore or

Hence roots are and

(b) Given: and

Comparing we get

Similarly,

(c)

cm and cm

Radius of the circle cm

**Question 2:**

(a) Use factor theorem to factorize completely. **[3]**

(b) Solve the following in-equation and represent the solution set on the number line.

**[3]**

(c) Draw a histogram for the given data using graph paper. **[4]**

Weekly Wages (in Rs) | No. of People |

3000-4000 | 4 |

4000-5000 | 9 |

5000-6000 | 18 |

6000-7000 | 6 |

7000-8000 | 7 |

8000-9000 | 2 |

9000-10000 | 4 |

Answers:

(a) Let

By trial and error, let

Therefore

Hence we can say that is a factor of

Now,

Now factorize

Therefore complete factorization is

(b)

First solve:

Also solve

Hence the solution set

(c)

**Question 3:**

(a) In the figure given below, is the center of the circle and is a diameter, if and , find ** [3]**

i)

ii)

iii)

(b) Prove that

**[3]**

(c) In what ratio is the line joining and is divided by y-axis. Also find the coordinates of the point of intersection. **[4]**

Answers:

(a)

i) We know, The angle which an arc of a circle subtends at the center is double that which it subtends on any point of the part of the circumference.

Therefore

ii) Given

The chord of the same length subtend the same angle

iii) (semi circular angles are )

In

(b)

LHS

RHS Hence proved.

(c) Let the point of intersection be

Let the ratio of in which the line segment gets divided be

Using section formula

and

Hence the ratio is

Therefore

Therefore the coordinates of the point of intersection is

**Question 4:**

(a) A solid spherical ball of radius cm is melted and recasted in identical spherical marbles. Find the radius of each marble. ** [3]**

(b) Each of the letters of the word is written on identical circular discs and put in a bag. They are well shuffled. If a disc is drawn at random from the bag, what is the probability that the letter is

i) a vowel

ii) One of the first letters of the English alphabet which appears in a given word

iii) One of the last letters of the English alphabet which appears in a given word **[3]**

(c) Mr. Bedi visits the market and buys the following articles:

Medicines costing Rs. , GST @

A pair of shoes costing Rs. , GST @

A laptop bag costing Rs with a discount or , GST @

i) Calculate the total amount of GST paid

ii) The total bill amount including GST paid by Mr. Bedi **[4]**

Answers:

(a) Let be the radius of spherical ball and be the radius of the spherical marble

Volume of spherical ball

Volume of spherical marble

cm

(b) First 9 letters:

Last 9 letters:

Total outcomes

i) No. of vowels

Therefore probability

ii) There are probable outcomes

Therefore probability

iii) There are probable outcomes

Therefore probability

(c)

Article | Cost ( Rs.) | Final Cost (Rs.) | GST Rate | GST ( Rs.) | Final Price (Rs.) |

Medicines | 950 | 950 | 5% | 47.50 | 997.50 |

Shoes | 3000 | 3000 | 18% | 540 | 3540 |

Laptop Bag | 1000 @ 30% discount | 700 | 18% | 126 | 829 |

Total (Rs.) | 713.50 | 5363.50 |

Therefore

i) Calculate the total amount of GST paid Rs.

ii) The total bill amount including GST paid by Mr. Bedi Rs.

**SECTION B [40 Marks]**

*(Attempt any four questions from this Section.)*

**Question 5:**

(a) A company with shares of nominal value Rs. declares and annual dividend of . Calculate:

i) the total amount of dividend paid by the company

ii) annual income of Mr. Sharma who holds shares of the company.

If the return percent of Mr. Sharma for his shares is , find the market value of each share. ** [3]**

(b) The mean of the following data is . Calculate the value of . ** [3]**

Marks | 5 | 10 | 15 | 20 | 25 |

No. of Students | 3 | 7 | 9 | 6 |

(c) The and the last term of a geometric progression are and respectively. If the common ratio is positive, find the first term, common ration and the number of terms of the series. ** [4]**

Answers:

(a) Number of shares

Nominal Value Rs.

Dividend

i) Dividend Rs.

ii) shares

Dividend Rs.

Return

Rs.

Therefore Market Value Rs.

(b)

Marks | No. of Students | |

5 | 3 | 15 |

10 | 7 | 70 |

15 | ||

20 | 9 | 180 |

25 | 6 | 150 |

Given: Mean

We know

(c) term term Last term

We know,

… … … … … i)

Similarly, … … … … … ii)

Dividing ii) by i) we get

Using i) we get

We know that

**Question 6:**

(a) If and . Find: ** [3]**

(b) In the given figure cm, cm and cm

Chord and intersect at .

i) Prove that

ii) Find the length of

iii) Find area of area of ** [3]**

(c) From the top of a cliff, the angle of depression of the top and bottom of a tower are observed to be and respectively. If the height of the tower is m, Find:

i) the height of the cliff

ii) the distance between the cliff and the tower. **[4]**

Answers:

(a) Given

and

(b)

i) Consider and

( Vertically opposite angles)

(angles subtended by an arc on the circumference are equal)

(By AA similarity criterion)

ii) Since

cm

iii)

c)

In

… … … … … i)

In

… … … … … ii)

Substituting

m

m

i) the height of the cliff m

ii) the distance between the cliff and the tower m

**Question 7:**

(a) Find the value of is the lines and are perpendicular to each other. Hence find the equation of a line passing through and parallel to . **[3]**

(b) Using properties of proportion find given **[3]**

(c) In the given figure, and are two tangents to the circle with center , touching at and respectively. If and , find:

i) and

ii)

iii) **[4]**

Answers:

(a) Given

Therefore slope

Similarly,

Therefore slope

We know

Therefore slope

Therefore equation of line

(b)

Applying componendo and dividendo

Applying componendo and dividendo

(c)

i)

Since is an isosceles triangle

Similarly, is an isosceles triangle

ii)

Angle subtended by an arc at the center is twice that subtended on the circumference.

iii) is a quadrilateral. We know that the sum of all the internal angles of a quadrilateral is

Hence,

**Question 8:**

(a) What must be added to the polynomial , so that it leaves a remainder when divided by ? **[3]**

(b) Mr. Sonu has a recurring deposit account and deposits Rs. per month for years. If he gets Rs. at the time of maturity, find the rate of interest. **[3]**

(c) Use a graph paper for this. ** [4]**

Take cm unit on both x and y axes.

i) Plot the following points on your graph sheet and

ii) Reflect point and on the x axis and name then and respectively.

iii) Join the points and in order. Name the closed figure formed.

Answers:

(a) Given :

When divided by

(b) Given Rs. years months Rs.

Interest Rs.

Interest

(c)

The figure formed is a nonagon (nine edges). It is a FISH.

**Question 9:**

(a) students enter for a game of shot put competition. The distance thrown ( in meters) is recorded below. ** [6]**

Distance in m | 12-13 | 13-14 | 14-15 | 15-16 | 16-17 | 17-18 | 18-19 |

Number of Students | 3 | 9 | 12 | 9 | 4 | 2 | 1 |

Use a graph paper to draw an ogive for the above distribution.

Use a scale of cm m on x-axis and cm students on the other axis.

Hence using your graph paper find:

i) the median

ii) Upper Quartile

iii) Number of students who cover a distance which is above m

(b) If , prove that **[4]**

Answers:

(a)

i) the median

ii) Upper Quartile

iii) Number of students who cover a distance which is above m students

(b)

Applying componendo and dividendo

Squaring both sides

Applying componendo and dividendo

. Hence proved.

**Question 10:**

(a) If the terms of an AP is equal to four time its first term and the sum of first six terms is , find the first term and the common difference. **[3] **

(b) The difference of two natural numbers us and their product is . Find the numbers. ** [3]**

(c) Use ruler and compass for this question. Construct a circle of radius cm. Draw a chord cm.

i) Find the locus of points equidistant from and . Mark the point where it meets the circle as .

ii) Join and find the locus of points which are equidistant from and . mark the point where it meets the circle as .

Join and . Measure and write down the length of side of quadrilateral . ** [4]**

Answers:

(a) and

We know that

… … … … … i)

Also

… … … … … ii)

Solving i) and ii)

Substituting in i)

Therefore first term and common difference

(b) Let the two natural numbers be and

Therefore

Also

Substituting

( not a natural number).

Therefore . Substituting we get

(c)

Step 1:Draw a circle of radius 3.5 cm.

Step 2: Then take a point A and draw an arc 6 cm long to intersect the circle at B. Join AB. That is the chord

Step 3: Locus of equidistant points for AB is the perpendicular bisector. Draw perpendicular bisector.

Step 4: Mark point D. Join AD.

Step 5: Draw an angle bisector of angle DAB. This is the locus for equidistant point for point on line AD and AB.

Step 6: Measure CD.

**Question 11:**

(a) A model of a high rise building is made to a scale of .

i) if the height of the model is m, find the height of the actual building.

ii) If the floor area of a flat in the building is , find the floor are of that in the model.

(b) From a solid wooden cylinder of height cm and diameter cm. Two conical cavities are hollowed out. The diameters of the cone is also cm and height is cm.

Taking , find the volume of the remaining solid.

(c) Prove the identity:

Answers:

(a) Scale

m

i)

m

ii)

or

(b) Volume of a cylinder

Volume of a cone

Given : cm cm cm

Volume of cylinder

Volume of cones

Therefore Volume of remaining solid

(c) LHS

RHS. Hence proved.