Other Solved Mathematics Board Papers

**MATHEMATICS (ICSE – Class X Board Paper 2019)**

**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 in-equation and write down the solution set:

(b) A man invests Rs. 4500 in shares of a company which is paying 7.5% dividend. If Rs. 100 shares are available at a discount of 10%. Find:

(i) Number of shares he purchases.

(ii) His annual income.

(c) In a class of 40 students, marks obtained by the students in a class test (out of 10) are given below: Calculate the following for the given distribution:

Marks | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Number of Students | 1 | 2 | 3 | 3 | 6 | 10 | 5 | 4 | 3 | 3 |

(i) Median

(ii) Mode

**Answer:**

(a) Given

Case 1:

Case 2:

(b) Rs. 100 shares at a discount of 10% will cost

(c)

Marks |
No. of Students |
Cumulative Frequency |

1 |
1 | 1 |

2 |
2 |
3 |

3 |
3 |
6 |

4 |
3 |
9 |

5 |
6 |
15 |

6 |
10 |
25 |

7 |
5 |
30 |

8 |
4 |
34 |

9 |
3 |
37 |

10 |
3 |
40 |

Total |
40 |

i) Total number of Students which is even

Which is between . Therefore Median

ii) Mode frequency of is the highest. Therefore Mode

**Question 2: **

(a) Using the factor theorem, show that is a factor of . Hence factorize the polynomial completely.

(b) Prove that:

(c) In an Arithmetic Progression (A.P.) the fourth and sixth terms are respectively. Find the:

(i) first term

(ii) common difference

(iii) sum of the first terms.

**Answer:**

(a)

Let

Therefore is a factor of

Hence

(b) To prove:

LHS

RHS. Hence proved.

(c) Let the first term of the sequence is and the common difference is .

… … … … … i)

… … … … … ii)

Solving i) and ii) we get , and

Therefore

i) First term

ii) Common difference

**Question 3: **

(b) are two points on the axis and axis respectively. divides the line segment in the ratio . Find:

(i) the coordinates of

(ii) slope of the line .

(c) A solid metallic sphere of radius is melted and made into a solid cylinder of height . Find the:

(i) radius of the cylinder

(ii) curved surface area of the cylinder [Take ]

**Answer:**

(b) Let coordinates of is is . Point divides in ratio

Therefore i) The coordinates of is

(c) Let the radius of the sphere is and radius of cylinder is and the height of the cylinder is .

Therefore Volume of sphere Volume of cylinder

Therefore Radius of the cylinder is

Curved surface area

Question 4:

(a) The following numbers, are in proportion. Find .

(b) Solve for the quadratic equation . Give your answer correct to three significant figures

(c) Use ruler and compass only for answering this question.

Draw a circle of radius . Mark the center as . Mark a point outside the circle at a distance of from the center. Construct two tangents to the circle from the external point .

Measure and write down the length of any one tangent.

**Answer:**

(a) Given are in proportion.

(b) Given

Comparing the above equation by

(c)

i. Draw a line segment

With center and radius , draw a circle.

iii. Draw the mid point of .

With center and diameter , draw a circle which intersect the circle at .

Joint

are the required tangent on measuring the length of

**SECTION B (40 Marks)**

Attempt any four questions from this Section

**Question 5:**

(a) There are discs numbered to . They are put in a closed box and shaken thoroughly. A disc is drawn at random from the box. Find the probability that the number on the disc is:

(i) an odd number

(ii) divisible by both.

(iii) a number less than .

(b) Rekha opened a recurring deposit account for months. The rate of interest is per annum and Rekha receives Rs. as interest at the time of maturity. Find the amount Rekha deposited each month.

(c) Use a graph sheet for this question. Take unit along both axis.

(i) Plot the following points:

(ii) Reflect the points on the y axis and name them as respectively.

(iii) Write down the coordinates of .

(iv) Join the points in order and give a name to the closed figure .

**Answer:**

(a) Total number of cases

(i) an odd number

ii) Divisible by both

Since months and

Therefore Interest

Interest Rs.

Rs.

(c) Please refer to the graph below for answers. the shape of the figure is “Arrow Head”.

**Question 6:**

(a) In the given figure, and .

(i) Prove that .

(ii) Find Area of Area of quadrilateral .

(b) The first and last term of a Geometrical Progression (G.P.) are respectively. If the common ratio is , find:

(i) the number of terms of the G.P.

(ii) Sum of the terms.

(c) A hemispherical and a conical hole is scooped out of a solid wooden cylinder. Find the volume of the remaining solid where the measurements are as follows:

The height of the solid cylinder is , radius of each of hemisphere, cone and cylinder is . Height of cone is . Give your answer correct to the nearest whole number. Take .

**Answer:**

(a)

i) To prove

Consider

(Given)

is common

(By AA criterion)

Taking the reciprocals on both sides

Now deducting both sides by

(b) Given first term and last term

i)

We know

(c)

**Question 7:**

(a) In the given figure is a tangent to the circle with center . If find . Give reasons for your answers.

(b) The model of a building is constructed with the scale factor .

(i) If the height of the model is , find the actual height of the building in meters.

(ii) If the actual volume of a tank at the top of the building is , find the volume of the tank on the top of the model.

(i) State the order of matrix .

(ii) Find the matrix .

**Answer:**

(a) In

[Sum of angles in a triangle]

Also

In

(b)

i) If the scale factor is , then actual height will be times the height of the model.

The height of the model is

Therefore actual height

ii) Now, actual volume of a tank will be times the volume of a tank in the model.

Therefore Volume of tank in model Actual volume of a tank

(c) Given

i)

Therefore has the order of

ii)

Similarly,

Now,

also,

**Question 8:**

(a) The sum of the first three terms of an Arithmetic Progression (A.P.) is and the product of the first and third term is . Find the first term and the common difference.

(b) The vertices of a are . Find:

(i) Slope of .

(ii) Equation of a line perpendicular to and passing through .

(c) Using ruler and a compass only construct a semi-circle with diameter . Locate a point on the circumference of the semicircle such that is equidistant from . Complete the cyclic quadrilateral , such that is equidistant from . Measure and write it down.

**Answer:**

(a) Let the three times terms of an A.P. be

Sum:

Also

When

When

Hence the two APs are

(b) Given

Therefore required line is

Let us assume

(c)

Draw a line segment

Taking mid point of as center , draw a semi-circle with radius

iii. Now, the semicircle circumscribes the

Draw angle bisector of and make it intersect the semi-circle at .

Measure the angle which comes out to be

**Question 9:**

(a) The data on the number of patients attending a hospital in a month are given below. Find the average (mean) number of patients attending the hospital in a month by using the shortcut method. Take the assumed mean as . Give your answer correct to decimal places.

Number of Patients | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |

Number of Days | 5 | 2 | 7 | 9 | 2 | 5 |

(c) Sachin invests Rs. in , Rs. shares at Rs. . He sells the shares when the price of each share rises by Rs. . He invests the proceeds in Rs. shares at Rs. . Find:

(i) the sale proceeds.

(ii) the number of Rs. shares he buys.

(iii) the change in his annual income.

**Answer:**

(a)

Number of Patients | Number of Days | Mid Value | Assumed Mean | |

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

20-30 | 2 | 25 | -20 | -40 |

30-40 | 7 | 35 | -10 | -70 |

40-50 | 9 | 45 | 0 | 0 |

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

60-70 | 5 | 65 | 20 | 100 |

Total | 30 | -140 |

Using componendo and dividendo on both sides

On squaring both sides

(c)

i) Market value of shares Rs.

Total face value of shares

Selling price of the shares

Sale proceed

ii) Market value of the shares bought

Total face value of shares

iii) Income from Rs. shares

Therefore increase in income

**Question 10: **

(a) Use graph paper for this question. The marks obtained by students in an English test are given below:

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

Number of Students | 5 | 6 | 16 | 22 | 26 | 18 | 11 | 6 | 4 | 3 |

Draw the ogive and hence, estimate:

(i) the median marks.

(ii) the number of students who did not pass the test if the pass percentage was .

(iii) the upper quartile marks.

(b) A man observes the angle of elevation of the top of the tower to be . He walks towards it in a horizontal line through its base. On covering m the angle of elevation changes to . Find the height of the tower correct to significant figures.

**Answer:**

(a)

Class Interval | Frequency | Cumulative Frequency |

0-10 | 5 | 5 |

10-20 | 9 | 14 |

20-30 | 16 | 30 |

30-40 | 22 | 52 |

40-50 | 26 | 78 |

50-60 | 18 | 96 |

60-70 | 11 | 107 |

70-80 | 6 | 113 |

80-90 | 4 | 117 |

90-100 | 3 | 120 |

i) Median

ii) Number of students who failed

(b) Let the height of the tower be m.

… … … … … i)

… … … … … ii)

Using i) and ii), we get

m

Therefore Height of the tower m

**Question 11:**

(a) Using the Remainder Theorem find the remainders obtained when is divided by . Hence find if the sum of the two remainders is .

(b) The product of two consecutive natural numbers which are multiples of is equal to . Find the two numbers.

(c) In the given figure, is a pentagon inscribed in a circle such that is a diameter and side . If , find giving reasons:

(i)

(ii)

(iii)

Hence prove that is also a diameter.

**Answer:**

(a) Remainder theorem:

Dividend Divisors Quotient Remainder

Dividing by gives remainder as

Also

Sum of remainder

(b) Let the numbers be

Therefore

But , because must be natural number.

Therefore

Therefore the numbers are

(c) Let

We know Angle in a semi-circle

i) In

ii) We know

Alternate interior angles since is transversal to parallel lines

Also

iii)

Therefore in

Also in

We know, if an angle of a triangle in a circle is then, the hypotenuse must be the diameter of the circle. Hence, is the diameter of the circle.