Mathematics

1

According to a survey of 205 students who are studying in class ten of a school, the number of participated students in Nepali language handwriting competition is 140 and the number of participated students in English language handwriting competition is 100. The number of students who participated in other language except these two language handwriting is 15.

[6 marks]

a.

In the above context, 'N' and 'E' denote the sets of students participated in Nepali language and English language handwriting competition respectively. Write the cardinality notation of the set of students who participated in other language except these two languages.

[1 mark]

b.

Present the given information in a Venn-diagram.

[1 mark]

c.

Find the number of students participating in exactly one language handwriting competition.

[3 marks]

d.

Compare between the number of students participated in at least one and at most one language handwriting.

[1 mark]

2

Ram borrowed a sum of Rs.50,000 from Shyam at the rate of 10% per annum compound interest compounded annually for two years and immediately he lent the sum to Ghanshyam at the same rate of interest for the same period of time compounded semi-annually.

[5 marks]

a.

Write the formula to find the annual compound interest.

[1 mark]

b.

How much total compound interest should be paid by Ram? Find it.

[2 marks]

c.

How much was gained or lost to Ram in this transaction? Find it.

[2 marks]

3

In the beginning of 2075 B.S. the population of a city was 4,00,000. The population of the same city was 5,32,400 in the starting of 2078 B.S.

[4 marks]

a.

What was the annual population growth rate of the city? Find it.

[2 marks]

b.

If the population is increased by the same rate, in which year the population of the city was 6,44,204? Find using formula.

[2 marks]

4

In a certain day, the buying and selling rate of one pound sterling are NRs.196.32 and NRs.197.12 respectively. After certain time Nepali currency was devaluated by 2%.

[4 marks]

a.

What is exchange rate of currency? Write it.

[1 mark]

b.

How many pound sterling can be exchanged with NRs.49,280? Find it.

[1 mark]

c.

How much Nepalese rupees can be exchanged with 500 pound sterling after devaluation? Find it.

[1 mark]

d.

Compare the buying rate and selling rate after devaluation.

[1 mark]

5

The perimeter of a square base pyramid is 24 meter and its height is 4 meter.

[4 marks]

a.

Write the formula to find the volume of square based pyramid.

[1 mark]

b.

What is the volume of pyramid? Find it.

[1 mark]

c.

Find the total surface area of the pyramid.

[2 marks]

6

A water tank with circular base built to distribute drinking water for 40 household in a rural municipality is given in the figure. Which is made with the combination of cylinder and hemi-sphere. The height of the cylinder is 4.2 meter and diameter of the base is 7 meter.

[4 marks]

a.

Write the formula to find the base area of that tank.

[1 mark]

b.

How much liter of water is required to fill the whole tank? Find it.

[2 marks]

c.

If equal amount of water is being distributed to all the household everyday then one fourth water is consumed of full tank. How much amount should a family pay for consuming water in a month at the rate of 4 paisa per liter? Find it.

[1 mark]

7

The volume and height of a square based room are 243 cubic meter and 3 meter respectively. The area occupied by a door and two windows in the room is 12 sq. meter.

[5 marks]

a.

What is the total cost of plastering the four walls at the rate of Rs.250 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.

[2 marks]

8

The number of shoes sold by a shop in the first three days are given below in the table. The shop is selling double number of shoes each day compared to the previous day.

Days	$1^{\text{st}}$	$2^{\text{nd}}$	$3^{\text{rd}}$
Pair of shoes	4	8	16

[5 marks]

a.

The number of geometric means between the first term ( $a$ ) and last term ( $b$ ) is ( $n$ ), write the formula to find common ratio ( $r$ ).

[1 mark]

b.

How many pair of shoes will be sold up to 8 days as given in the above table? Calculate it by using formula.

[2 marks]

c.

In how many days the shop sells 4092 pair of shoes? Find it.

[2 marks]

9

The sum of the present ages of two sisters is 40 and the product is 391 numerically.

[5 marks]

a.

If the present age of elder sister is $x$ years, then express the age of younger sister in terms of $x$ .

[1 mark]

b.

Find the present ages of both sisters.

[2 marks]

c.

After how many years, if the square of that number of years is added to the present age, will the new ratio become 13:11?

[2 marks]

10

[5 marks]

a.

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

[2 marks]

b.

Solve: $2^{x-2} + 2^{3-x} = 3$

[3 marks]

11

Triangle QRT and Parallelogram PQRS are formed on the same base QR and between the same parallel lines QR and PT.

[4 marks]

a.

Write the relationship between the area of parallelogram PQRS and area of triangle QRT.

[1 mark]

b.

In the triangle QRT, the length of a side QT is 16 cm. and the length of perpendicular drawn from the vertex R to the side QT is 6 cm. Find the area of parallelogram PQRS.

[2 marks]

c.

What would be the relation between the area of triangle QTU and parallelogram PQRS if the base of the triangle QTU was double of the base of the parallelogram PQRS? Where the points Q, R and U lie on the same straight line. Write with logic.

[1 mark]

12

In the given figure, O is the centre of the circle and $\angle SPQ = 25^\circ$ .

[4 marks]

a.

What is the value of $\angle PSQ$ ? Write it.

[1 mark]

b.

Find the value of $\angle PRS$ .

[1 mark]

c.

Verify experimentally the relation of $\angle PRS$ and $\angle PQS$ . (Two circles of different measurement having radii at least 3 cm are necessary).

[2 marks]

13

[5 marks]

a.

Construct a triangle PQR in which PQ = 5.5 cm, QR = 7 cm and $\angle PQR = 60^\circ$ . Also construct a parallelogram having one side 6.5 cm and whose area is equal to the area of the triangle.

[3 marks]

b.

In the figure, O is the centre of a circle and AB is a diameter. The chords AC and BD are intersected at a point E. Prove that:

$\angle AOD + \angle BOC = 2\angle AED$

[2 marks]

14

The radius of a circular pond is 21 meter. A pole is fixed at the center of the pond. A man 1.4 meter tall observed the top of the pole from the edge of the pond and found the angle of elevation to be $45^\circ$ .

[4 marks]

a.

Define the angle of elevation.

[1 mark]

b.

Sketch the figure from the above context.

[1 mark]

c.

What was the height of the pole above the water level? Find it.

[1 mark]

d.

By how much should the height of the pole above the water surface be reduced so that the angle of elevation becomes $30^\circ$ ? Give the logic with calculation.

[1 mark]

15

In an examination, the marks obtained by the students in mathematics subject are given below in the table.

Marks obtained	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50
No. of Students	2	3	4	p	6

[6 marks]

a.

If the median of the given data is 34, then write the median class.

[1 mark]

b.

Find the value of p.

[2 marks]

c.

Calculate the mean of the given data.

[2 marks]

d.

Among all the participants in the exams, what percentage of students obtained mark below the median class? Find it.

[1 mark]

16

A box contains 3 red and 4 black balls of same shape and size. Two balls are drawn one after another without replacement.

[5 marks]

a.

What type of events are mentioned above? Write it.

[1 mark]

b.

Show the above experiment in a tree diagram.

[2 marks]

c.

Find the probability of getting first ball red and second ball black.

[1 mark]

d.

If 2 red balls are added in the box and two balls are drawn randomly one after another from the box without replacement then what is the probability of getting both balls of red colour? Find it.

[1 mark]