Mathematics

1

Among the 450 students present at the program, 144 students were found to like mathematics subject only, 216 students were found to like science subject only but 63 students were found who did not like any of these two subjects.

[6 marks]

a.

If the total number of present is represented by 'U' then write cardinality of 'U'.

[1 mark]

b.

Show the given information in a Venn-diagram.

[1 mark]

c.

Find the number of students who liked mathematics subject.

[3 marks]

d.

How much percent the number of student who liked both subjects is less than the number of students who did not like any of two subjects?

[1 mark]

2

Kumar deposited Rs.5,50,000 in a commercial bank at the rate of 10% p.a. compound interest compounded yearly for 2 years. After 1 year the bank changed its policy to pay the interest compound semi-annually at the same rate.

[5 marks]

a.

What is compound interest? Write it.

[1 mark]

b.

Find the sum of amount accumulated in Kumar's account at the end of 2 years.

[2 marks]

c.

Compare the interest of the first year and second year.

[2 marks]

3

The present population of a municipality is 1,26,075. The annual population growth rate of the municipality is 2.5% per year.

[4 marks]

a.

Write the formula to calculate population after T years.

[1 mark]

b.

What was the population of that municipality before 2 years? Find it.

[2 marks]

c.

If 1075 people migrate to other places due to different circumstances in this year then what will be the population of that municipality after 1 year? Find it.

[1 mark]

4

The exchange rate published by Nepal Rastra Bank on dated 2082/08/05 states the buying and selling rate for American dollar ($) 1 were NRs. 141.64 and NRs.142.24 respectively.

[4 marks]

a.

Which rate will the bank use to exchange Nepalese rupees into American dollar? Write it.

[1 mark]

b.

How much American dollar ($) can be exchanged with NRs.7,96,544? Find it.

[1 mark]

c.

If the Nepali rupees is devaluated by 0.3% after some time, what is the profit or loss made when the US dollar exchanged back in Nepali rupees? Find it.

[2 marks]

5

The length of a side of the base of a square based pyramid is 20 cm, and the length of slant edge is 26 cm.

[5 marks]

a.

How many total surfaces are there in a square base pyramid? Write it.

[1 mark]

b.

Find the total surface area of the pyramid?

[3 marks]

c.

Compare between the vertical height and slant height of the pyramid.

[1 mark]

6

In the figure, a solid object is formed with the combination of a cone and a hemi-sphere having the same base. Where, the total height of the solid object is 20 cm. The diameter of the hemi sphere is 14 cm.

Solid object combination of cone and hemisphere

[4 marks]

a.

Write the formula to find the volume of the cone.

[1 mark]

b.

Find the height of the conical part in the given solid object.

[1 mark]

c.

Find the volume of the given solid object.

[2 marks]

7

The length, breath and height of a wall are 20 m, 30 cm and 5 m respectively. The bricks of size 25 cm × 12 cm × 5 cm are used to construct the wall. Also 10% part of the wall is occupied by the joints of cement and sand.

[4 marks]

a.

How many bricks are used to construct the wall? Find it.

[2 marks]

b.

2,000 bricks are loaded in a trip and the rent of it is Rs.600. Estimate the cost of bricks used to construct the wall at the rate of Rs.13,500 per 1,000 bricks.

[2 marks]

8

Ramila opened account on behalf of her daughter in a finance and the amounts deposited monthly is increased in common difference are presented below in the table.

Months	$1^{\text{st}}$	$2^{\text{nd}}$	$3^{\text{rd}}$	$4^{\text{th}}$	$5^{\text{th}}$
Amount Rs.	2000	4000	6000	8000	10000

[5 marks]

a.

What type of sequence is formed on saving for daughter of Ramila? Write it.

[1 mark]

b.

How much total amount did Ramila deposit in 1 year? Find it.

[2 marks]

c.

How many months will Ramila require to deposit Rs.4,20,000 in the account of her daughter? Find it.

[2 marks]

9

In a two digit number, the product of digits is 20 and their difference is 1.

[5 marks]

a.

Write the standard form of a number of two digits.

[1 mark]

b.

Find the number.

[4 marks]

10

[6 marks]

a.

Simplify: $5^{\sqrt{x}} = 25$

[1 mark]

b.

Solve: $2^a + 2^{-a} = 2\frac{1}{2}$

[3 marks]

c.

Simplify: $\frac{1}{p^2 - pq + q^2} - \frac{p^2 + q^2}{p^4 + p^2q^2 + q^4}$

[2 marks]

11

In the given figure, MN // SP, SM // QN and MR // NP.

[5 marks]

a.

Write the relation between areas of parallelogram MNQS and parallelogram MNPR.

[1 mark]

b.

Prove that the area of triangle SMN is half of the area of parallelogram MNPR.

[2 marks]

c.

If MR and QN intersect at the point A, prove that area of quadrilateral SMAQ and area of quadrilateral PRAN are equal.

[2 marks]

12

In a circle with centre O, the central angle SOP and inscribe angles SRP and STP are standing on same arc SP.

[4 marks]

a.

Write the relationship between central angle and inscribe angle standing on the same arc.

[1 mark]

b.

The measurement of the central angle is $(x + 60)^\circ$ and the measurement of the inscribe angle is $(2x)^\circ$ . Find the value of x.

[1 mark]

c.

Verify experimentally that the inscribed angles STP and SRP are equal. (Two different circles with radii of at least 3 cm are required.)

[2 marks]

13

In a triangle MNS, MN = 6.5 cm, NS = 6 cm and MS = 5.5 cm are given.

[4 marks]

a.

Construct a $\Delta MNS$ on the basis of above measurements. Also, construct a parallelogram having one side is 6.8 cm with equal area of that triangle.

[3 marks]

b.

Why the area of $\Delta MNS$ and parallelogram are equal? Give reason.

[1 mark]

14

The diameter of a circular pond is 100 m and a pillar is fixed at the centre of the pond. A person finds the angle of elevation to the top of the pillar be $\theta$ from the bank of the pond. The depth of the pond is 1.5 m and total high of the pillar is 51.5 m.

[4 marks]

a.

Draw a figure representing the above context.

[1 mark]

b.

What is the height of the pole above the water level?

[1 mark]

c.

Find the value of angle of elevation $\theta$ .

[1 mark]

d.

If the angle of elevation was $30^\circ$ , what would be the radius of the pond? Find it.

[1 mark]

15

The marks obtained by students in an examination with full marks 100 are given in the following table.

Marks obtained	0 - 20	20 - 40	40 - 60	60 - 80	80 - 100
No. of students	2	3	x	5	4

[6 marks]

a.

If the mean of the given data is 56, find the value of x.

[2 marks]

b.

Write the model class of the given data.

[1 mark]

c.

Find the first quartile from the given data.

[2 marks]

d.

Compare between the number of students securing 40 or above 40 marks and less than 40 marks.

[1 mark]

16

There are 6 white and 4 black balls of same shape and size in a box. Two balls are drawn randomly one after another with replacement.

[5 marks]

a.

If A and B are mutually exclusive events, then write the principle of $P(A \cup B)$ .

[1 mark]

b.

Present the probability of all the possible outcomes of given event in a tree diagram.

[2 marks]

c.

Find the probability of getting same coloured ball?

[1 mark]

d.

Find the ratio of the probabilities of getting same coloured balls and different colored balls.

[1 mark]