Mathematics

1

In a survey of a group of 360 students, 100 students like basketball game only, 60 like cricket game only and 100 do not like any of the two games.

[6 marks]

a.

If 'B' and 'C' denote the set of students who like basketball and cricket game respectively, then what is the value of $n(B \cup C)$ ? Write it.

[1 mark]

b.

Present the above information in a venn-diagram.

[1 mark]

c.

Find the number of students who like both the games.

[3 marks]

d.

If everyone who is not interested in any game liked cricket game in the second survey and found others to be same then, what would be the number of students who like at least one game? Find it.

[1 mark]

2

Rajan borrowed a loan of Rs. 10,000 from Ram for 2 years at the rate of 10% simple interest. Immediately, Rajan lent the same sum for same time and same rate of interest compounded annually to Shyam.

[5 marks]

a.

According to the given context, which interest is more among simple interest and compound interest for 2 years? Write it.

[1 mark]

b.

How much profit did Rajan get during the transaction of 2 years? Find it.

[2 marks]

c.

How much more interest should Shyam need to pay to Rajan if Rajan had lent the amount at semi-annual compound interest? Find it.

[2 marks]

3

The population of a village is 20,000. The population increases by 2% annually in the village.

[4 marks]

a.

If the initial population is $P$ , growth rate is $R$ per annum and population after $T$ years is $P_T$ then write the formula to find $P_T$ .

[1 mark]

b.

After how many years the population of the village will be 20,808? Find it.

[2 marks]

c.

If the population increases at the rate of 3% per annum, by what number will the population of the village be increased in 2 years? Find it.

[1 mark]

4

According to the currency exchange rate, the buying rate of 1 American dollar was NRs. 136.13 and selling rate was NRs. 137.25 in a certain day.

[4 marks]

a.

Which rate buying or selling is used when you exchange American dollar into Nepali rupees? Write it.

[1 mark]

b.

How many Nepali rupees can American tourist get by exchanging 1000 dollars? Find it.

[1 mark]

c.

The American tourist spent NRs. 1,01,817.50 while staying in Nepal, how many American dollars can he/she exchange from remaining Nepalese rupees, while returning back to own country? Find it.

[2 marks]

5

A group of students constructed a square based pyramid shaped tent having length of base side 24 meter and vertical height 5 meter.

[4 marks]

a.

How many triangular surfaces are there in the square based pyramid? Write it.

[1 mark]

b.

Find the slant height of the above square based tent.

[1 mark]

c.

What is the total cost of cloths required to make triangular surfaces at the rate of Rs. 125 per square metre? Find it.

[2 marks]

6

In the given figure, wooden cylinder and cone having equal base are shown. [Figure: A wooden cylinder with height 24m and diameter 24m, and a cone with height 16m and equal base diameter.]

[5 marks]

a.

Write the formula to find the volume of a cone.

[1 mark]

b.

Find the volume of the cone in the given objects.

[2 marks]

c.

If given wooden cylinder is drilled out in the given conical shape, what will be the volume of remaining wood in cylinder? Find it.

[2 marks]

7

The length of a wall is 10 m, width is 0.5 m and height is 2 m. Bricks of size 25 cm $\times$ 12 cm $\times$ 8 cm are used to build the wall. Also, $\frac{1}{10}$ part of the wall is occupied by the clay joints.

[4 marks]

a.

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

[3 marks]

b.

Estimate the cost of bricks used in the wall at the rate of Rs. 14500 per 1000 bricks.

[1 mark]

8

Ramesh deposits the amount in a co-operative for 7 days by increasing the amount every day double than the previous day. He deposited Rs. 10 on the first day, Rs. 20 on the second day, Rs. 40 on the third day and so on till the $7^{th}$ day.

[5 marks]

a.

What type of series is formed from the deposit amount according to above context? Write it.

[1 mark]

b.

How much amount will Ramesh deposit by the end of 7 days? Find it using formula.

[2 marks]

c.

If Ramesh withdraws the amount deposited by 4 days, how much will he receive at the end of the $7^{th}$ days? Find it.

[2 marks]

9

The longer side of a rectangular field is 40 m more than the shorter side and its diagonal is 40 m more than its longer side. [Figure: A rectangle showing length l, breadth b, and diagonal d.]

Rectangle showing length l, breadth b, and diagonal d

[5 marks]

a.

Write the relation among the length ( $l$ ), breadth ( $b$ ) and diagonal ( $d$ ) of the field according to the above context.

[1 mark]

b.

Find the length of the shorter side and longer side of the rectangular filed.

[2 marks]

c.

How many maximum numbers of plots of size 30 m $\times$ 20 m can be made from the rectangular filed? Find it.

[2 marks]

10

[5 marks]

a.

Simplify: $\frac{p+q}{pq} - \frac{q+r}{qr} - \frac{r+p}{rp}$

[2 marks]

b.

Solve: $3^y + 3^{-y} = 9\frac{1}{9}$

[3 marks]

11

In the given figure, $\triangle ABC$ and $\triangle BCD$ are standing on same base $BC$ and between same parallel lines $AD$ and $BC$ . From the point $B$ , a perpendicular $BP$ is drawn to the line $AC$ . [Figure: Quadrilateral ABCD with diagonals and perpendicular BP, and a trapezium PQRS with midpoints M and N.]

Quadrilateral ABCD with diagonals and perpendicular BP

[5 marks]

a.

Write the name of triangle whose area is equal to area of $\triangle BAD$ in the given figure.

[1 mark]

b.

If $AC = 9\text{ cm}$ and $BP = 6\text{ cm}$ , find the area of triangle $BCD$ .

[2 marks]

c.

In the given figure, $PQRS$ is a trapezium, where $PQ \parallel SR$ . $M$ and $N$ are the mid points of the diagonals $PR$ and $QS$ respectively. Prove that: $\triangle MSR = \triangle NSR$ .

[2 marks]

12

In a triangle $PQR$ , $\angle PQR = 60^{\circ}$ , $QR = 8\text{ cm}$ and $PQ = 6\text{ cm}$ are given.

[4 marks]

a.

Construct a $\triangle PQR$ according to above measurements and also construct a rectangle $RITA$ equal in area to the triangle.

[3 marks]

b.

Why the areas of triangle and rectangle so formed are equal? Write reason.

[1 mark]

13

$O$ is the centre of the given circle. Inscribed angles $PAQ$ and $PBQ$ are standing on the same arc $PQ$ . [Figure: A circle with center O, central angle POQ, and inscribed angles PAQ and PBQ.]

Circle with center O, central angle POQ, and inscribed angles PAQ and PBQ

[4 marks]

a.

Write the relation between the circumference angles $PAQ$ and $PBQ$ .

[1 mark]

b.

If the measures of central angle $POQ$ is $(12x + 4)^{\circ}$ and the measures of inscribed angle $PAQ$ is $(3x + 20)^{\circ}$ , find the value of $x$ .

[1 mark]

c.

Verify experimentally that central angle is double of the inscribed angle formed on same arc. (Two circles having radii more than 3 cm are necessary.)

[2 marks]

14

In the given figure, height of the tower $AB$ is 24.5 meter and height of a house $CD$ is 4.5 meter. $BC$ denotes the distance between tower and house. [Figure: A tower AB and house CD standing on horizontal ground BC. A horizontal line ED is drawn from D to AB.]

Tower AB and house CD standing on horizontal ground BC

[4 marks]

a.

Define the angle of elevation.

[1 mark]

b.

Find the value of $AE$ .

[1 mark]

c.

If $\angle ADE = 30^{\circ}$ , find the distance between the tower and the house.

[1 mark]

d.

By how many degrees is the angle of elevation less or more when $AE$ and $ED$ are equal? Compare it.

[1 mark]

15

The marks obtained by the students in an exam of mathematics of 75 full marks are given in the following table.

Obtained Marks	0-15	15-30	30-45	45-60	60-75
Number of students	2	5	4	6	3

[6 marks]

a.

Illustrate the modal class from the above data.

[1 mark]

b.

Find the median from the above table.

[2 marks]

c.

Find the mean from the above table.

[2 marks]

d.

Among all the participants in the exam, what percentage of students obtained marks below the modal class? Find it.

[1 mark]

16

A box contains 6 white and 10 black balls of same shape and size. Two balls are drawn at random one after another with replacement.

[5 marks]

a.

If A and B are two independent events, write the multiplication law of probability.

[1 mark]

b.

Show the probability of all the possible outcomes in a tree diagram.

[2 marks]

c.

Find the probability of getting both balls of same color.

[1 mark]

d.

By how much the probability of getting both balls of different color is less or more than probability of getting both balls of white color? Find it.

[1 mark]