Mathematics

1

In a survey conducted among 120 students studying in class Ten of a secondary school, it was found that 60 students liked cricket game, 55 students liked basketball game and 20 students did not like any of these games.

[6 marks]

a.

If C and B denote the sets of students who liked cricket and basketball game respectively, write the cardinality of the students who did not like any of these games.

[1 mark]

b.

Present the above information in a Venn diagram.

[1 mark]

c.

Find the number of students who liked cricket game only.

[3 marks]

d.

Compare the number of students who liked cricket game only and who liked basketball game only.

[1 mark]

2

Aatmik wants to deposit Rs. 4,00,000 in a bank for 2 years. The bank offers 10% per annum compound interest to Aatmik with three alternates (annual compound interest, semi-annual compound interest and quarterly compound interest).

[5 marks]

a.

Which option among the above three alternates, Aatmik has to use to get more interest? Write it.

[1 mark]

b.

How much compound interest does he receive at the end of 2 years compounded semi-annually? Find it.

[2 marks]

c.

Is semi-annual compound interest received by Aatmik in 2 years double than the quarterly compound interest received in 1 year? Justify with calculation.

[2 marks]

3

A photocopy machine is purchased for Rs. 80,000. After using it for 2 years, only Rs. 30,000 is earned. The price of machine depreciates annually at the rate of 20% and the machine is sold after 2 years.

[4 marks]

a.

The initial price of a machine is $V_0$ , annual rate of compound depreciation is R and the price of machine after T years is $V_T$ , express $V_T$ in terms of $V_0$ , R and T.

[1 mark]

b.

Find the total profit or loss amount on selling the machine.

[2 marks]

c.

If he had sold the machine after using it one year more, then by how much is the selling price less or more than the purchased price? Compare it.

[1 mark]

4

A businessman exchanged Australian dollars with NRs. 1,29,090 at the exchange rate of Australian dollar 1 = NRs. 86.06. After some days, Nepali currency was revaluated up by 2% in comparison to Australian dollar and on that day, he exchanged the Australian dollars into Nepali currency again.

[4 marks]

a.

How many Australian dollars did the businessman exchange? Find it.

[1 mark]

b.

How many Nepali rupees did the businessman receive when he exchanged Australian dollar after revaluation in Nepali currency? Find it.

[2 marks]

c.

What profit or loss percent did the businessman make in that transaction? Find it.

[1 mark]

5

The vertical height of the square based pyramid is 24 cm and the length of base side is 20 cm.

[3 marks]

a.

Write the formula to find the volume of the pyramid.

[1 mark]

b.

Find the total surface area of the pyramid.

[2 marks]

6

In the figure, a metallic solid made of hemisphere and cone is given, where the height of cone is 24 cm and diameter of base is 14 cm. [Figure: A cone on top of a hemisphere. The total diameter is 14 cm, and the height of the cone part is 24 cm.]

[5 marks]

a.

Write the formula to find the slant height of cone when vertical height and radius of base are given.

[1 mark]

b.

Find the volume of the solid object.

[2 marks]

c.

If the solid object is melted and turned into a cylindrical object of radius 7 cm, what is the height of the cylinder? Calculate it.

[2 marks]

7

The volume and height of a square based room are 75 cubic meter and 3 meter respectively. The area occupied by a door and two windows in the room is 6 square meter.

[4 marks]

a.

What is the total cost of plastering the four walls without door and windows at the rate of Rs. 200 per square meter? Find it.

[3 marks]

b.

If the rate of plastering per square meter is increased by one-fourth, then what will be the increment in the total cost of plastering the walls? Find it.

[1 mark]

8

Hira collected the following sum of money in the first 5 days of the month of Baishakh.

Baishakh-1	Baishakh-2	Baishakh-3	Baishakh-4	Baishakh-5
Rs. 10	Rs. 20	Rs. 40	Rs. 80	Rs. 160

[5 marks]

a.

What is the mean value of the amounts collected on the 2nd Baishakh and the 4th Baishakh? Write it.

[1 mark]

b.

How much money will be collected by the 10th days? Find using formula.

[2 marks]

c.

Up to how many days of Baisakh can Rs. 1,63,830 be collected? Find it.

[2 marks]

9

The length of rectangular field is twice of its breadth and its area is 200 square meter.

[5 marks]

a.

Write the standard form of quadratic equation.

[1 mark]

b.

Find the length and breadth of the rectangular field.

[2 marks]

c.

How many maximum numbers of pieces having the size 5 m × 4 m can be made in the field? Also present it in a diagram.

[2 marks]

10

[5 marks]

a.

Simplify: $\frac{1}{x-y} - \frac{1}{x+y}$

[2 marks]

b.

If $x^2 = 3^{2/3} + 3^{-2/3} - 2$ , prove that: $3x^3 + 9x = 8$

[3 marks]

11

In the given figure, square ABCD and parallelogram EBCF are on the same base BC and between the same parallel lines AF and BC. [Figure: Square ABCD and Parallelogram EBCF sharing base BC]

Square ABCD and Parallelogram EBCF sharing base BC

Parallelogram PQRS with midpoint M on TR

[5 marks]

a.

Write the relation between the areas of parallelograms standing on the same base and between the same parallel lines.

[1 mark]

b.

Prove that: Area of parallelogram EBCF = Area of square ABCD

[2 marks]

c.

In the given figure, PQRS is a parallelogram and M is the mid-point of TR. Prove that: $\Delta TQM = \frac{1}{2}(\Delta PQT + \Delta SRT)$

[2 marks]

12

[4 marks]

a.

Construct a triangle ABC having BC = 6.4 cm, AB = 5.6 cm and AC = 6 cm. Also construct a triangle DAB having one side 7 cm equal in area to $\Delta ABC$ .

[3 marks]

b.

Why are the areas of $\Delta ABC$ and $\Delta DAB$ equal? Give reason.

[1 mark]

13

Central angle AOB and inscribed angles ADB and ACB are standing on the same arc AB in a circle with center O.

[4 marks]

a.

Write the relation between the inscribed angles standing on the same arc.

[1 mark]

b.

Experimentally verify that the central angle AOB is double of the inscribed angle ACB. (Two circles with radii at least 3 cm are necessary.)

[2 marks]

c.

The measure of central angle is $(5x)^\circ$ and the measure of inscribed angle is $(2x + 10)^\circ$ standing on the same arc in a circle, find the value of x.

[1 mark]

14

A tree x meter high is broken by the wind, at the height 6 meter from the ground so that its top touches the ground and makes an angle $30^\circ$ with the ground.

[4 marks]

a.

What is called the angle of elevation? Write it.

[1 mark]

b.

Express the length of the broken part of the tree in terms of x.

[1 mark]

c.

What was the height of the tree before broken? Find it.

[1 mark]

d.

At what height should the tree be broken so that its top makes an angle of $45^\circ$ with ground? Find it.

[1 mark]

15

The first quartile of the given data is 35.

Obtained Marks	0-20	20-40	40-60	60-80	80-100
Number of students	2	x	8	5	1

[6 marks]

a.

Illustrate the class where the first quartile lies.

[1 mark]

b.

Find the value of x.

[2 marks]

c.

Find the mode from the given data.

[2 marks]

d.

Find the ratio of students who are above and below the first quartile class.

[1 mark]

16

A married couple has given birth to two children in the interval of five years.

[5 marks]

a.

Define independent events.

[1 mark]

b.

Show all the possible outcomes in a tree diagram.

[2 marks]

c.

Find the probability of having both daughters.

[1 mark]

d.

By how much is the probability of getting both children son less or more than the maximum probability? Calculate it.

[1 mark]