Mathematics

1

Below is a survey of 200 students studying in class ten asking which subject do you prefer between Mathematics and English.

• 120 students like Mathematics.

• 110 students like English.

• 30 students like none of these two subjects.

[6 marks]

a.

Write the set of students who like none of these two subjects in cardinal notation by supposing the set of students who like mathematics by 'M' and English by 'E'.

[1 mark]

b.

Present the above information in a Venn-diagram.

[1 mark]

c.

Find the number of students, who like exactly one subjects.

[3 marks]

d.

Ram said, "The number of students who like both the subjects is double of the number of students who like none of these subjects." Do you agree with Ram's statement? Justify it.

[1 mark]

2

A man took a loan of Rs.1,50,000 for two years at the rate of compound interest 10% per annum compounded annually from his close relative for the purpose of business. He paid Rs.85,000 at the end of the first year to reduce the interest and some loan.

[5 marks]

a.

Write the formula for finding annual compound interest.

[1 mark]

b.

How much money did he return as the loan at the end of one year? Find it.

[2 marks]

c.

How much total money did he pay to clear the debt on time? Find it.

[2 marks]

3

A bus owner bought a luxury bus for Rs.20,00,000 and used it in the Kathmandu - Pokhara route for 3 years. He earned Rs.7,43,124 only in 3 years. The price of the bus is depreciated at the rate of 8% per year.

[4 marks]

a.

Define the compound depreciation.

[1 mark]

b.

What will be the selling price of the bus after 3 years? Find it.

[1 mark]

c.

How much profit or loss percent will bus owner make in the total transaction? Find it.

[2 marks]

4

A student exchanged some American dollar for Nepali rupees 13,00,000 to go to America for his/her higher study. On that day, the selling rate and buying rate of 1 American dollar ($) were NRs. 143.75 and NRs. 143.15 respectively.

[4 marks]

a.

How many American dollars was exchanged with NRS. 13,00,000 by the student? Find it.

[1 mark]

b.

Due to health problem of the student, the student could not go to America and after a few days he/she exchanged American dollars into Nepali rupees. On that day, Nepalese currency was revaluated by 0.5%. How many Nepalese rupees did the student get on that day? Find it.

[2 marks]

c.

Write the reason of gain or loss in exchanging the currency for the student.

[1 mark]

5

The length of side of the base of a square based pyramid is 24 cm and height is 9 cm.

[4 marks]

a.

Write the formula for finding the volume of a square based pyramid.

[1 mark]

b.

Find the volume of the given pyramid.

[1 mark]

c.

Find the area of triangular surfaces of the pyramid.

[2 marks]

6

The given figure is of a toy made with the combination of a cone and hemi-sphere. The common radius of the toy is 7 cm and total height is 31 cm.

[6 marks]

a.

Write the relation among the radius of the base of a cone ( $r$ ), vertical height ( $h$ ) and slant height ( $l$ ).

[1 mark]

b.

Find the curved surface area of the conical part of the toy.

[3 marks]

c.

By how much the volume of conical part is less or more than the volume of hemi-spherical part? Compare with calculation.

[2 marks]

7

The parking area outside of a building is in geometric shape as shown in the given figure.

[3 marks]

a.

Find the area of the parking area.

[2 marks]

b.

Find the total cost of leveling the parking area at the rate of Rs.2500 per 100 sq. ft.

[1 mark]

8

The monthly salary of an employee is Rs.45,000 at present. He receives a grade of Rs.1,500 on his monthly salary every year.

[6 marks]

a.

Write the name of sequence formed from above context.

[1 mark]

b.

What will be his total income in 4 years when the number of grades increases yearly at the same rate? Calculate it.

[3 marks]

c.

In how many years, his total income will be Rs.35,10,000 when the number of grades increases yearly by the same rate? Find it.

[2 marks]

9

The sum of the present ages of two brothers is 28 years and the product of their ages is 187 numerically.

[5 marks]

a.

In the quadratic equation of $ax^2 + bx + c = 0$ , write the formula to find the value of $x$ .

[1 mark]

b.

Find their present ages.

[2 marks]

c.

After how many years will the product of their ages be 247? Find it.

[2 marks]

10

[4 marks]

a.

Simplify: $\frac{1}{m+n} - \frac{1}{m-n}$

[2 marks]

b.

Solve: $3^{y+1} + 3^y = 36$

[2 marks]

11

In the given figure, NE // AW, AN // ME and SN // WE.

[5 marks]

a.

Write the relation between parallelogram NEWS and triangle NWS.

[1 mark]

b.

Prove that the areas of parallelograms NAME and NEWS are equal.

[2 marks]

c.

If 'S' is the midpoint of AM, Prove that, $\Delta ANS = \frac{1}{4}\square NEWS$ .

[2 marks]

12

In the given figure, PARM is a cyclic quadrilateral in circle with centre O.

[4 marks]

a.

Write the relation between $\angle PAR$ and $\angle PMR$ .

[1 mark]

b.

Find the value of $x$ from the given figure.

[1 mark]

c.

Verify experimentally that in the cyclic quadrilateral PARM, $\angle APM + \angle ARM = 180^\circ$ (Two circles having radii at least 3 cm are necessary.)

[2 marks]

13

In a quadrilateral PQRS, PQ = 5 cm, QR = 5.5 cm, RS = 5.7 cm, PR = 7.1 cm and PS = 6.1 cm are given.

[4 marks]

a.

Construct quadrilateral PQRS. Also construct a triangle equal in area to the area of quadrilateral PQRS.

[3 marks]

b.

From above construction, why are the area of quad. PQRS and area of triangle equal? Give reason.

[1 mark]

14

A man 1.3 meter tall is observing at the top of a tower 41.3 meter high and found an angle of elevation of $45^\circ$ .

[4 marks]

a.

Define the angle of elevation.

[1 mark]

b.

How high is the tower than the man? Find it.

[1 mark]

c.

Find the distance between the tower and man.

[1 mark]

d.

How many meter should the man walk backward from that place to find the angle of elevation of the top of the tower $30^\circ$ ? Find it.

[1 mark]

15

The marks obtained by 30 students are presented in the table below.

Marks obtained	0 - 10	0 - 20	0 - 30	0 - 40	0 - 50
No. of students	12	17	20	22	30

[6 marks]

a.

What does 'L' represent for in formula to find the median $(M_d) = L + \frac{\frac{N}{2} - c.f.}{f} \times i$ ? Write it.

[1 mark]

b.

Find the median of the given data.

[2 marks]

c.

Calculate the average marks from the data.

[2 marks]

d.

If the student securing (0 - 30) marks are avoided from the given data, what difference shows in the average marks? Find it.

[1 mark]

16

Two cards are drawn randomly one after another with replacement from a well shuffled deck of 52 cards.

[5 marks]

a.

What are independent events? Write it.

[1 mark]

b.

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

[2 marks]

c.

What is the probability of getting both faced cards? Find it.

[1 mark]

d.

What is the difference between the probabilities of both cards being faced cards if two cards are drawn one after another with replacement and without replacement? Find it.

[1 mark]