Mathematics

1

In a survey conducted on the students studying in class ten of a school to determine suitable place for educational tours to Janakpur and Birgunj. It was found that 60 students considered Janakpur, 40 students considered Birgunj and 20 students considered both places suitable but 10 students did not consider any of the two places suitable.

[6 marks]

a.

If 'J' and 'B' denote the set of students who prefer Janakpur and Birgunj respectively. Write the set of students who considered as both places suitable in cardinality notation.

[1 mark]

b.

Present the above information in a Venn diagram.

[1 mark]

c.

How many students are studying in class ten in the school? Find it.

[3 marks]

d.

If 10 students who did not consider any of the two places suitable had said Janakpur as a suitable place, then what would be the ratio of students who consider only Janakpur and only Birgunj as the suitable place? Find it.

[1 mark]

2

Anil deposited Rs. 2,00,000 in a Bank at the rate of 10% p.a. compound interest for 2 years.

[5 marks]

a.

Write the formula for finding quarterly compound interest.

[1 mark]

b.

How much annual compound interest did Anil receive in 2 years? Find it.

[2 marks]

c.

If the bank provides the semiannual compound interest instead of the annual compound interest at the same rate of interest, how much more interest would be received by Anil? Find it.

[2 marks]

3

The present population of a municipality is 20,000 and the annual population growth rate is 3%.

[3 marks]

a.

Write the formula to find the population $P_T$ after T years if the initial population is P and rate of annual growth is R.

[1 mark]

b.

What will be the population of municipality after 2 years? Find it.

[1 mark]

c.

Is the calculation process of population growth and compound interest same? Give your opinion.

[1 mark]

4

According to the money exchange rate of certain time, one American dollar ($1) was equal to NRs. 136.04.

[5 marks]

a.

How many American dollars ($) can be exchanged with NRs. 20,406? Find it.

[1 mark]

b.

How many Nepali rupees can be exchanged with American dollars 2,500 when Nepali currency is devaluated by 2%? Find it.

[2 marks]

c.

If the Nepali currency was revaluated up by 2% instead of devaluation, how much less or more Nepali rupees can be obtained while exchanging 2500 American dollars? Find it.

[2 marks]

5

The vertical height of a square based pyramid is 24 cm and length of the base side is 14 cm.

[5 marks]

a.

Write the formula to find the area of a triangular surface of the pyramid.

[1 mark]

b.

Find the volume of the pyramid.

[2 marks]

c.

Find the total surface area of the pyramid.

[2 marks]

6

A solid object made up of a cone and a cylinder is given in the figure.

[4 marks]

a.

How many curved surfaces are there in the given solid object? Write it.

[1 mark]

b.

Find the height of cone.

[1 mark]

c.

Compare the volume of cone and cylinder.

[2 marks]

7

The length, breadth and height of a rectangular classroom are 18 ft, 14 ft and 10 ft respectively. In the classroom, there are two windows with size 6 ft × 4 ft and two doors with size 6 ft × 3 ft.

[4 marks]

a.

How much does it cost to paint four walls and ceiling of the classroom excluding doors and windows at the rate of Rs. 40 per square feet? Find it.

[3 marks]

b.

If a painter paints 202 square feet in a day, how many days will two painters take to paint the classroom? Find it.

[1 mark]

8

The first and last terms of an arithmetic series are 5 and 329 respectively. The sum of all terms is 4676.

[5 marks]

a.

What is called mean in arithmetic series? Write it.

[1 mark]

b.

Find the common difference of the series.

[2 marks]

c.

How many terms should be removed from last in the series to make the sum 51? Find it by calculation.

[2 marks]

9

In a positive number of two digits, the product of two digits is 27. When 54 is subtracted from the number, the places of the digits are interchanged.

[5 marks]

a.

If the two digits number be $10x + y$ , then write the number obtained by interchanging its digits.

[1 mark]

b.

Make a quadratic equation from the given verbal problem.

[2 marks]

c.

Find the number.

[2 marks]

10

[5 marks]

a.

Simplify: $\frac{a^3 + 1}{a^2 - a + 1} + \frac{a^3 - 1}{a^2 + a + 1}$

[2 marks]

b.

Solve: $5^x + \frac{1}{5^x} = 5\frac{1}{5}$

[3 marks]

11

In the adjoining figure, $\Delta PQR$ , parallelograms PQRS and PQTU are standing on the same base PQ and between the same parallel lines PQ and UR.

[3 marks]

a.

Write the relation between the area of parallelograms PQRS and PQTU.

[1 mark]

b.

Prove that the area of $\Delta PQR$ is half of the area of parallelogram PQTU.

[2 marks]

12

In the given figure, O is the centre of the circle. Where, central angle AOC is equal $140^\circ$ .

[4 marks]

a.

Write the relation between the inscribed angle ADC and the arc ABC.

[1 mark]

b.

Find the value of $\angle ADC$ from the given figure.

[1 mark]

c.

Experimentally verify that the opposite angles ABC and ADC of cyclic quadrilateral ABCD are supplementary. (Two circles with at least 3 cm radii are necessary.)

[2 marks]

13

[6 marks]

a.

Construct a triangle CAT having sides AT = 4.4 cm, AC = 5.5 cm and $\angle CAT = 60^\circ$ . Construct another triangle BAT whose area is equal to the area of the given triangle, where AB = 6.2 cm.

[3 marks]

b.

Why the area of $\Delta CAT$ and $\Delta BAT$ are equal? Give a reason.

[1 mark]

c.

In the parallelogram ROSE, if P and Q are any points of sides ES and ER respectively, prove that: $\Delta ROP = \Delta SOQ$ .

[2 marks]

14

In the given figure, height of the electric pole (PQ) is 18 meter and height of a man (RS) is 1.5 meter. SQ represents the distance between electric pole and man, where $\angle PRT = 30^\circ$ .

[4 marks]

a.

Define the angle of elevation.

[1 mark]

b.

Find the value of PT.

[1 mark]

c.

Find the distance between the electric pole and the man.

[1 mark]

d.

By how many degrees will the angle of elevation less or more when PT and TR are equal? Find it.

[1 mark]

15

The marks obtained by 20 students in an examination with full marks 50 are given in the following table.

Marks obtained	0-10	10-20	20-30	30-40	40-50
No. of students	2	3	4	7	4

[6 marks]

a.

Write the modal class of the given data.

[1 mark]

b.

Find the median from the given data.

[2 marks]

c.

Calculate the average mark from the given data.

[2 marks]

d.

How many maximum number of students could be there who obtained the marks less than the average mark? Find it.

[1 mark]

16

Two cards are drawn randomly one after another without replacement from a well shuffled deck of 52 cards.

[5 marks]

a.

If $P(A \cap B) = P(A) \times P(B)$ , what type of events are A and B? Write it.

[1 mark]

b.

Show the probability of all possible outcomes of getting and not getting king cards in a tree diagram.

[2 marks]

c.

Find the probability of getting both king cards.

[1 mark]

d.

Is the probability of getting both ace of diamond possible? Give reason.

[1 mark]