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. in shares of a company which is paying dividend. If Rs. shares are available at a discount of . Find:

(i) Number of shares he purchases.

(ii) His annual income.

(c) In a class of students, marks obtained by the students in a class test (out of ) 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. shares at a discount of will cost Rs.

i) Therefore Number of shares

ii) His annual income at dividend Rs.

(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

Median

term term

term

term

Which is between and . 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 and 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

iii) Sum of the first terms

Question 3:

(a) Simplify:

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

(i) the coordinates of and

(ii) slope of the line .

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

(i) radius of the cylinder

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

Answer:

(a) Given

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

and

Therefore i) The coordinates of is and

ii) Slope of line

(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 cm

Curved surface area

Question 4:

(a) The following numbers, and 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 cm. Mark the center as . Mark a point outside the circle at a distance of cm 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 and are in proportion.

(b) Given

Comparing the above equation by

or

(c) i. Draw a line segment cm

ii. With center and radius cm, draw a circle.

iii. Draw the mid point of .

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

v. Joint and

and are the required tangent on measuring the length of cm

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 and 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 cm unit along both and axis.

(i) Plot the following points: and

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

(iii) Write down the coordinates of and .

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

Answer:

(a) Total number of cases

(i) an odd number

Therefore Probability of an odd number

ii) Divisible by and both

Probability of the number divisible by and both

iii) Probability a number less than

(b) Let the monthly installment i.e Rs.

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, cm and cm.

(i) Prove that .

(ii) Find Area of Area of quadrilateral .

(b) The first and last term of a Geometrical Progression (G.P.) are and 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 cm, radius of each of hemisphere, cone and cylinder is cm. Height of cone is cm. Give your answer correct to the nearest whole number. Take .

Answer:

(a)

i) To prove

Consider and

(Given)

is common

(By AA criterion)

ii)

Taking the reciprocals on both sides

Now deducting both sides by

(b) Given first term and last term

i)

We know

ii) Sum of the terms

(c) Required volume Volume of cylinder Volume of hemisphere Volume of cone

Volume of cone

Volume of hemisphere

Volume of cylinder

Therefore required volume

Question 7:

(a) In the given figure is a tangent to the circle with center . If , find and . 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 cm, 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.

(c) Given , where is a matrix and is unit matrix of order .

(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 cm

Therefore actual height cm cm

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

Therefore volume of tank

(c) Given

i)

Therefore has the order of

ii) Let us assume

=

=

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 and . 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 cm. Locate a point on the circumference of the semicircle such that is equidistant from and . Complete the cyclic quadrilateral , such that is equidistant from and . 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 and

(b) Given

i) Slope of BC

ii) Slope of line perpendicular to

Therefore required line is

(c)

i. Draw a line segment cm

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

iii. Now, the semicircle circumscribes the

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

v. 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 |

(b) Using properties of proportion solve for , given

(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 |

Mean

(b) Given

Using componendo and dividendo on both sides

On squaring both sides

(c)

i) Market value of shares Rs.

Therefore the number of shares bought

Total face value of shares Rs.

Income form these shares Rs.

Selling price of the shares Rs.

Sale proceed Rs.

ii) Market value of the shares bought Rs.

Therefore number of Rs. shares bought

Total face value of shares Rs.

iii) Income from Rs. shares Rs.

Therefore increase in income Rs.

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 |

Here , therefore

i) Median

ii) Number of students who failed

iii) The upper Quartile marks term term

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

In

… … … … … i)

In

… … … … … 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 and . 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

or

But , because must be natural number.

Therefore

Therefore the numbers are

(c) Let and

We know Angle in a semi-circle

i) In

ii) We know

Alternate interior angles since is transversal to parallel lines and

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.