Mathematics

1

In the survey conducted on 120 households of a village, 70 households are doing homestay business, 50 households are doing agriculture business and 30 households do other work. The sets of households doing homestay and agriculture business are denoted by H and A respectively.

[6 marks]

a.

Write the cardinality notation of the number of households who do not do any of the homestay and agriculture business.

[1 mark]

b.

Present the above information in a venn-diagram.

[1 mark]

c.

Find the number of households doing only one business.

[3 marks]

d.

If 10 households who do other work start home stay business, find the ratio of homestay and agriculture business households.

[1 mark]

2

Ramesh borrowed the loan Rs. 2,00,000 from the bank at annual compound interest rate of 7% per annum. After some time, he repaid Rs. 2,28,980 including the principal and interest to the bank.

[4 marks]

a.

How much interest has Ramesh paid? Find it.

[1 mark]

b.

For how many years has Ramesh used the loan? Find it.

[2 marks]

c.

If the interest rate is reduced by 1%, how much amount would be paid by Ramesh? Find it.

[1 mark]

3

The present population of village 'A' is 4500 and village B is 5000. The annual population growth rate of village A is 2%.

[4 marks]

a.

What does P denote in the population after T years $P_T = P\left(1 + \frac{R}{100}\right)^T$ ? Write it.

[1 mark]

b.

If 200 people are added by migration in the village 'A' after 1 year, what will be the population after 1 year? Find it.

[1 mark]

c.

If the population of village B decreases by the same growth rate of village A, what will be the population of village B after 2 years? Find it.

[2 marks]

4

When Rita came to Nepal from America, she brought 2500 American dollars ( $). When she came to Nepal, the exchange rate of American dollar was as follows.$ 1 = NRs. 127.35 buying rate and $1 = NRs. 127.95 selling rate

[5 marks]

a.

Which exchange rate is used to exchange Nepali currency by Rita from the bank? Write it.

[1 mark]

b.

What is the profit or loss for Rita if she exchanges the American dollars into Nepali currency and then exchanges Nepali currency into American dollars after two days? Find it.

[2 marks]

c.

On the same day, the exchange rate of $1 = NRs. 127.35 and NRs. 160 = Indian Rs. 100. How many dollars can be exchanged with Indian Rs. 1,20,000? Calculate it.

[2 marks]

5

The diagram alongside is of a square based pyramid. Its each side of base is 6 meter and vertical height is 4 meter.

[4 marks]

a.

Write the formula for finding the area of all triangular surfaces of pyramid.

[1 mark]

b.

Find the volume of the pyramid.

[1 mark]

c.

Find the total surface area of the pyramid.

[2 marks]

6

Given solid object is made up of a cone and hemisphere. The diameters of the base of a cone and hemisphere are equal and the total height of object is 17 cm and diameter of base is 10 cm.

Solid object made of a cone and hemisphere

[5 marks]

a.

Write the relation between height and radius of hemisphere.

[1 mark]

b.

Find the total surface area of solid object.

[3 marks]

c.

Will the amount of Rs. 150 be sufficient to color the surface of the solid object at the rate of 40 paisa per square cm? Give reason with calculation.

[1 mark]

7

A rectangular room has length 12 ft, breadth 8 ft and height 9 ft. There are two windows of size 2.5 ft × 3ft and two doors of size 6ft × 2ft.

[4 marks]

a.

Find the total area of four walls, floor and ceiling.

[2 marks]

b.

How much less or more is the cost of coloring the four walls excluding the doors and windows at the rate of Rs. 175 per square feet than Rs. 50,000? Find it.

[2 marks]

8

A shop sells shoes and slippers (in pairs) during winter season according to the following table.

Day	1st day	2nd day	3rd day	4th day	5th day
No. of shoes	2	4	8	16	..........
No. of slippers	3	6	9	12	..........

[5 marks]

a.

Write the relationship between arithmetic mean and geometric mean.

[1 mark]

b.

How many slippers are sold up to 8th days? Find it.

[2 marks]

c.

Compare the total number of shoes and slippers sold up to 8th days.

[2 marks]

9

The area and perimeter of a rectangular field are 150 sq.m and 50 m respectively.

[5 marks]

a.

What is the length and breadth of the field? Find it by making quadratic equation.

[3 marks]

b.

How much equal part should be subtracted from the length and breadth of the land to get the area 84 square meter? Solve it.

[2 marks]

10

[5 marks]

a.

Write $\frac{1}{x^{-6}}$ in the positive index of $x$ .

[1 mark]

b.

Simplify: $\frac{a^2}{a-2b} + \frac{4b^2}{2b-a}$

[2 marks]

c.

Solve: $4^{x-2} = 0.25$

[2 marks]

11

In the given figure, AE // BD, ED // AC and BE // CD.

Figure showing parallel lines AE // BD, ED // AC and BE // CD

[4 marks]

a.

Write the relationship between the area of parallelogram and triangle standing on the same base and between the same parallel lines.

[1 mark]

b.

Prove that: $\triangle ABE = \triangle BCD$

[2 marks]

c.

Compare between the area of triangle ABE and trapezium ACDE.

[1 mark]

12

In the given figure, PQRS is a cyclic quadrilateral. The side PQ is produced to the point T.

Cyclic quadrilateral PQRS with side PQ produced to T

[5 marks]

a.

Write the relation of $\angle PSQ$ and $\angle PRQ$ .

[1 mark]

b.

Verify experimentally that $\angle SPR$ and $\angle SQR$ are equal. (Two circles with radii at least 3cm are necessary)

[2 marks]

c.

Prove that: $\angle RQT = \angle PSR$

[2 marks]

13

In a quadrilateral PQRS, PQ = 5 cm, QR = 4.5 cm, RS = SP = 6 cm and QS = 6.5 cm.

[4 marks]

a.

Construct a quadrilateral PQRS according to the above measurements and then construct a triangle which is equal to the quadrilateral in area.

[3 marks]

b.

Why the area of the quadrilateral and triangle are equal? Give reason.

[1 mark]

14

In the figure, the height of tower (CD) is $15\sqrt{3}$ meter and the height of house (AB) is $10\sqrt{3}$ meter. The angle from the top of house to the top of tower is $30^\circ$ . [Figure: Tower CD and house AB on horizontal ground AC. Horizontal line BE drawn from B to tower. Angle of elevation from B to D is 30°.]

Tower CD and house AB diagram with angle of elevation 30 degrees

[4 marks]

a.

What type of angle is formed when the top of the tower is observed from the roof of the house? Write it.

[1 mark]

b.

Find the height of part DE of the tower.

[1 mark]

c.

Calculate the distance between house and tower.

[1 mark]

d.

Find the angle of the top of the tower from the basement of the house.

[1 mark]

15

The marks obtained by 50 students of a class in an exam of maths with full mark 60 are given in the table.

Obtained marks	10-20	20-30	30-40	40-50	50-60
Number of students	7	13	15	10	5

[6 marks]

a.

Write the class of mode.

[1 mark]

b.

Find the value of first quartile ( $Q_1$ ) from the given data.

[2 marks]

c.

Calculate the average mark from the given data.

[2 marks]

d.

What is the average mark of the students who obtained 40 or more than 40 marks? Find it.

[1 mark]

16

In a box, 5 red and 3 white balls of same size and shape are kept.

[5 marks]

a.

Write the addition law of probability of mutually exclusive events.

[1 mark]

b.

Two balls are drawn one after another without replacement from the box. Show the probability of all the possible outcomes in a tree diagram.

[2 marks]

c.

Find the probability of getting both white balls from the tree diagram.

[1 mark]

d.

Find the difference between the probability of both balls being white if two balls are drawn one after another with replacement and without replacement.

[1 mark]