### Compulsory Mathematics Chapter wise model Question Class 10

**SET**

**For question No. (1)**

1. In an examination, 80 students passed in mathematics, 70 failed in Mathematics, 90 failed in science & 20 failed in both

- Illustrate the given information in the Vann diagram.
- How many students passed science?
- How many students passed in both
- How many students passed in both mathematics & Science?

2. 75 students in a class like Cristiano Ronaldo or Lionel Messi or both. Out of the 10 like both players. The ratio of the number of students who like Cristiano Ronaldo to those who like Lionel Messi is 2:3.

- Find the number of students who like Cristiano Ronaldo
- Find the number of students who like Lionel Messi only.
- Represent the result in a Venn diagram.

3. A survey conducted in a group of students showed that 50 liked singing, 40 liked dancing, 35 liked acting, 20 liked singing as well as dancing, 12 liked dancing as well as acting, 18 liked acting as well as singing, and 7 liked all three activities. If every student liked at least one activity, answer the following questions by drawing a Venn diagram.

- How many students were asked this question?
- How many students liked dancing only?
- How many students liked acting only?
- How many students liked only one activity?

4. There are 400 students in a school. They are allowed to cast votes either only for Ajay or for Ohaswi as their school prefect. 50 of them cast votes for both Ajay and Ojaswi and 24 did not cast the vote. The candidate Ojaswi won the election with a majority of 56 more votes than Ajay.

- How many students cast the vote?
- How many valid votes were received by Ajay?
- How many votes were valid?
- Show the result in the Venn diagram.

5. In a group of students, 20 study Economics. 18 study History, 21 study Science, 7 study Economics only, 10 study Science only, 6 study Economics and Science only, and 3 study Science and History only.

- Represent the above information in Venn diagram
- How many students study all the subjects?
- How many students are there altogether?

**Arithmetic **

**For question No. (2) **

1. Minakshi invested Rs. 85,000 for 1 year in Goodwill Finance at the rate of 8% per annum.

- How much interest will she receive if it is compounded each year?
- How much interest will she receive if it is compounded in each 6 months?
- How much interest will she receive if it is compounded in each 3 months?

2. Mr. Bantawa borrowed a sum of Rs. 1,00,000 from Agricultural Development Bank for 3 years to upgrade his poultry farming. the bank charged 5% interest compounded annually for the first year and the rate of interest was gradually increased by 1% every year. How much interest did he pay at the end of third year?

3. The compound interest of a sum of money at 8% p.a. for 2 years is more than the simple interest on the same sum at the same rate for the same time by Rs. 76.80. find

- The sum
- The interest compounded annually.

4. A man borrowed Rs. 1,00,000 from a bank. If he paid a compound interest of Rs.33,100 at the end of 3 years, find the rate of interest compounded annually charged by the bank.

5. If a sum becomes Rs.6,655 in 3 years and Rs 7,320.6=50 in 4 years interest being compounded annually, find:

- the rate of interest
- the sum

6. Gita borrowed Rs. 85,000 from a bank at the rate of 12% p.a. compounded semi-annually for 2 years. After one year, the bank changed its policy to charge the interest compounded quarterly at the same rate.

- How much interest did she pay in the first year?
- How much interest did she pay in the second year?
- How much less interest would she pay if the bank had not changed the policy?

**For Question No. (3)**

- After the depreciation at the rate of 15% p.a. the value of a vehicle became Rs.13,26,000 after one year. find the original value of the vehicle.
- The value of a machine is depreciated every year by 10% what will be the value of the machine worth Rs.1,80,000 after 3 years
- A mobile costing Rs. 6,000 is depreciated per year, and after 2 years its price becomes Rs. 5,415. Find the rate of depreciation.
- Mr. Chhetri bought a scanner machine 3 years before for Rs. 40,000. If the value of the machine is depreciated by 5%, 8%, and 10% in the first, the second, and the third year respectively. At what price did he sell the machine at the end of 3 years?
- At the annual rate of compound depreciation, if the value of a computer becomes Rs. 40,500 in 2 years and Rs.36,450 in 3 years, find the rate of depreciation. Also, find its original cost.
- 3 years ago, the Population of a village was 16,000. The population growth rate of that village is 5%. What is the population at present?
- In how many years will the population of a town be 2,09,475 from 1,90,000 at a growth rate of 5% per annum?
- The population of a village increases every year by 5%. At the end of two years, if 460 people migrated to other villages and the population of the village remained at 26,000, what was the population of the village in the beginning?
- After two years the population of a town will be 33,620 at the population growth rate of 2.5% p.a. Find the present population of the town.
- The population of a village increased from 10,000 to 11,000 in one year. Find the rate of growth of the population.

**For Question No. (4)**

- The buying and selling rates of 1 Canadian dollar in Annapurna Money Exchange Centre are NRs. 95.96 and NRs. 96.40. How Many Canadian dollars should it buy and sell to make a profit of NRs 4,400?
- If the exchange rate of £ 1 is Rs. 147.00 and the exchange rate of US\$1 is Rs. 113.00, how many dollars can be exchanged for 100 sterling pounds?
- Mrs. Magar wants to buy a book online. She finds a publisher in London selling the book for £ 15. This publisher is offering free transportation on the product. She then finds the same book from a publisher in New York for \$17 With a transportation fee of \$2. Which publisher should she buy the book from? (Exchange rate: £1 = NPR 159.88, \$1 = NPR 132.46)
- Mr. Gurung bought 10 Tola gold in Hong Kong for HKD\$ 3,900 per Tola at the rate of HKD\$ 1 = NPR 16.88. and bought in Nepal.

- Find the cost of gold in NPR if 25% customs duty was charged.

- A businessman arrived in Nepal from Kuwait with some Kuwaiti Dinars and he booked a ticket immediately for China. He exchanged 111,000 Kuwaiti dinars for Chinese Yuan. (Given exchange rates are 1 KWD = NPR 420 and 1 CNY = NPR 18.50)

- How many Chinese Yuan did he exchange?
- Had the Nepali rupee been devalued by 2.5% relative to the Dinar and the exchange rate of the Chinese Yuan been unchanged, how many Yuan would have been exchanged?

**Mensuration**

**For Question No. (5) **

- The length of the side of a square-based pyramid is 16 cm and its lateral surface area is 448 cm2. Find the length of its slant height.
- From the following square-based pyramids find the length of slant height (l)

3. The nets of a square-based pyramid are given below. find the area of triangular faces (L.S.A) and the total surface area of the pyramids.

4. 40 tourists came to visit Mt. Annapurna from Canada. They planned to stay at Annapurna base camp for 4 days. For this purpose. They ordered some square-based pyramid tents in Nepal. A tent can hold 8 people and each person has 6 ft. ×3 ft space on the ground with 48 cu. ft of air to breathe.

- Find the length of the side of the base of each tent.
- Find the total cost of all tents at the rate of Rs. 450 per sq. ft.

5. The diameter of the circular base of a right cone is 10cm and its total surface area is 90 𝝅 cm2. find its

- Slant height
- Vertical height
- Volume

**For Question No. (6)**

6. Find the volume of the following right circular cones.

7. The following combined solids are made up of the pyramid and the prism (cuboid) having a common base. find?

- The lateral surface area of the uppermost pyramid.
- The lateral surface area of the lowermost prism (cuboid)
- The total surface area of the given combined solid.

8. In the given combined solid. a right cone is placed on a hemisphere of equal radii. write the formula to find.

- C.S.A of the hemispherical portion
- C.S.A of the conical portion
- The surface area of combined solid
- Volume of conical portion

9. A tent is cylindrical up to a height of 9 ft. and conical above it. The diameter of the base is 16ft. and the total height of the tent is 24 ft.

- What is the height of the conical part?
- What is the slant height of the conical part?
- How much canvas is required to make the tent?

**For Question No. (7) **

10. Find the cost of pencil shaped solid object at the rate of Rs 10 per $cm^2$

11. Find the volume, T.S.A and C.S.A

12. Find T.S.A and volume.

13. Find the total cost of painting the solid object at the rate of Rs 20 per $cm^2$.

14. Find the volume of the given figure

15. Find T.S.A and volume

**Algebra **

**For Question No. (8)**

1. What is the arithmetic mean between 5 and 7?

2. Find the 2nd term of the following arithmetic sequences. 1st term = -5 and 3rd term = -55

3. There are n arithmetic means between 7 and 27. If the second mean is 15, find:

- The common difference
- The value of n
- The remaining means

4. In an A.P. a few arithmetic means are inserted between 75 and 105. if the ratio of the first mean to the last mean is 4:5, find:

- The common difference
- The number of means
- The means

5. The population of a municipality increased by 500 in the year B.S. 2079. the rate of population growth is expected to decrease by a number every year so that there will be an increase of only 400 people in B.S. 2084.

- What is the expected decrease rate in population growth per year?
- What will be the increased number of population in each year from B.S. 2080 to B.S. 2083?

6. find the sum of the following series. 2+4+6+......to 40 terms

7. If the sum of the first 55 terms of an AP adds up to 6600; find the 28th term.

8. The fifth term and the twelfth term of an arithmetic progression are 30 and 65 respectively. find:

- The first term and common difference
- The sum of the first 26 terms.

9. The sum of the first seven terms of an arithmetic series is 14 and the sum of the first ten terms is 125. find.

- The first term and common difference
- The sum of the first 25 terms.

10. Identify the second mean in the geometric sequence 1, 3, 9, 27, 81, ?

11. If 8, x, y, 27 are in G.P. find

- The common ratio
- The values of x and y.

There are n geometric means between 1 and 64 If the ratio of the first mean to the last mean is 1:16 find

- The common ratio
- The value of n
- The means

13. The first term of a geometric series is 4 and the common ratio is 2, find the sum of its first 8 terms.

14. In a G.P., the second term is 48 and its fifth term is 6. find:

- The first term and common ratio
- The fourth term
- the sum of the first 6 terms.

15. A man borrows Rs.18,900 without interest and repays the loan in 6 installments each installment double the preceding one. find:

- The first installment
- The last installment

**For Question No. (9)**

1. Solve these equations by factorization method:

- $2\sqrt{3x^2}-8x+2\sqrt{3} = 0$
- $\frac{x}{2x+1} = \frac{x}{4x-1}$
- $\frac{3}{x-1}+\frac{2}{x-3} = \frac{2}{x}$

2. Solve each equation by completing the square:

- $x^2+3x-28 = 0$
- $3x^2 = 10x-3$
- $\frac{x+3}{x+2} = \frac{3x-7}{2x-3}$

3. Solve each equation by using the formula:

- $2x^2-6x = 0$
- $3x^2-5x+6 = 2(x^2+3) $
- $4x^2 = \frac{4}{15}x+3$

4. Solve these equations by factorization method:

- $\frac{x}{x+1}+\frac{x+1}{x} = \frac{13}{6}$
- $\frac{1}{x-2}+\frac{2}{x-1} = \frac{6}{x}$

5. If 8 is subtracted from thrice the square of a natural number, the difference is 40. find the number.

6. If the sum of a number and 21 times its reciprocal is 10, find the number.

7. A number exceeds another number by 3 and their product is 40. Find the numbers.

8. The present ages of the two brothers are 15 years and 22 years respectively. After how many years will the product of the number of their ages be 408?

9. One year hence, a father's age will be 5 times the age of his son. the product of the present age of the father and his son is 145. find their present ages.

**For question No (9)**

**1. Simplify: **

- $\frac{n^2}{7n-49}+\frac{49}{7n-n^2}$
- $\frac{x+y}{x-y}+\frac{x-y}{x+y}-\frac{2(x^2+y^2)}{x^2-y^2} $
- $\frac{a+2}{1+a+a^2}-\frac{a-2}{1-a+a^2} -\frac{2a^2}{1+a^2+a^4}$

2. Prove that

- $\frac{1}{1-x^2}+\frac{x}{1+x^3} = \frac{1}{(1-x)(a+x^2)} $
- $\frac{1}{a-1}-\frac{a}{a^2-1}-\frac{a^2}{a^4-1} = \frac{1}{a^4-1}$

**For question No. (10) **

1. Solve the following exponential Equations:

- $(\frac{4}{3})^x = \frac{64}{27}$
- $4^x-2×2^{x+1}+27= 0$
- $7^x+\frac{343}{7^x} = 56$

2. Mr. Kamal bought a plot of 5 Aana land in a municipality. Now, he has built a house covering the space of $4^{2-x}$ Aana and the kitchen garden covers the remaining $4^{x-1}$ Aana of his plot.

- Make the exponential equation.
- Solve equation for x.
- Find the area of the kitchen garden.
- Find the area of plot occupied by the house.

3. If $a^x$ = $b^y$ = $c^z$ and $b^3$ = ac, Prove that $\frac{3}{y} = \frac{1}{x}+\frac{1}{z}$

4. If $x^p = yz, y^q$ = zx and $z^r$= xy, prove that pqr = p+q+r+2.

**Geometry **

**For Question No. (11)**

- Construct a parallelogram MINA in which MI = 6cm IU = 4.5cm and <AMI = $60^0$. Also, construct a triangle MUD equal in area to the parallelogram MINA and having a side UD = 7.5cm
- Construct a quadrilateral ABCD having AB = 6cm, BC = 5.5cm AC = 6.5cm CD= 8cm, and AD = 7cm construct a ABE is equal in area to the quadrilateral ABCD.
- Construct a triangle ABC in which a = 7.8cm, b = 7.2cm and c = 6.3cm. Then construct a parallelogram DBEF equal in area to ABC and <DBC = $75^0$
- Construct a parallelogram PQRS in which PQ = 5 cm, diagonal PR= 6cm and diagonal QS = 8cm. Construct a triangle PSA whose area is equal to the area of the parallelogram.

**For Question No. (12) **

- Construct a triangle xyz in which xy = 6.3cm, <x = $30^0$ and <y= $45^0$ construct another triangle wxy equal in area to XYZ and a side wy = 7.5cm
- Construct a quadrilateral PQRS having PQ = QR = 5.9cm, RS = PS = 6.1cm and <QPS = $75^0$ then construct a PST which is equal in area to the given quadrilateral
- Construct a quadrilateral ABCD in which AB = 5.4cm, BC = 5.1cm, CD = 4.9cm, AD = 6.1cm and diagonal BD = 5.7cm also construct a triangle equal in area to the quadrilateral ABCD.
- Construct a triangle having an angle $60^0$ and whose area equals to the rectangle having length 6 cm and breadth 4.5cm

**For question No. (13) **

- Construct a MNQ equal in area to the quadrilateral MNOP having NO=OP= 5.5cm, PM = MN = 4.5cm and <MNO = $75^0$
- Construct a triangle having an angle $75^0$ and whose area is equal to the area of a parallelogram ABCD having base side 6cm diagonal 7 cm and angle made by diagonal with base side.
- In the given figure O is the center of the circle. If AOB=95°, find the degree measures of arc AB and arc ACB.

**Statistics and Probability **

**For question No. (14) **

1. The mean of the given data is 28 find the value of k

Class interval |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |

Frequency |
12 |
18 |
27 |
k |
17 |
6 |

2. The following marks are obtained by students in mathematics in an examination.

51,20,60,34,48,61,79,25,39,52,43,46,37,40,67,76,71,33,44,55

- Make a frequency table of class interval
- Find the median.

3. Find the median from the following.

Marks obtained |
0-10 |
0-20 |
-0-30 |
0-40 |
0-50 |

No. of frequency |
4 |
12 |
24 |
44 |
62 |

4. According to the given data, find the maximum marks obtained by below 75% of students.

C.I |
0-5 |
5-10 |
10-15 |
15-20 |
20-25 |

f |
6 |
4 |
7 |
5 |
8 |

5. If the first quartile of the data is 30.625, find the value of a

C.I |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |

f |
8 |
a |
5 |
4 |
3 |

6. Find $Q_1, Q_2, Q_3$

Class |
Less than 12 |
Less than 24 |
Less than 36 |
Less than 48 |
Less than 60 |

f |
3 |
9 |
19 |
22 |
24 |

7. The $Q_3$ of the given data = 60 find the value of P.

C.I |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |
70-80 |

f |
3 |
5 |
4 |
5 |
4 |
p |
3 |

For Question No. (15)

1. Two events Q and R are mutually exclusive with P(Q) = $\frac{3}{5}$ and P(R) = $\frac{1}{5} , Find the probabilities of the following events.

- P(QUR)
- P$(\bar{QUR)}$

2. A basket contains 5 yellow, 3 blue, and 2 green balls. If a ball is drawn randomly from the basket, find the probability of not getting a blue ball.

3. A box contains 5 black, 7 blue, and 4 yellow balls. A ball is drawn at random and it is replaced, then another ball is drawn. Find the probability that

- The first is blue and the second is black,
- Both of them are yellow
- Both of them are of the same color

4. The probability of solving a mathematical problem by two students A and B are $\frac{1}{2}$ and $\frac{1}{5}$ respectively. If the problem is given to both students, find the probability of solving the problem.

5. Two children were born to a married couple. Find the probability of having at least one son by drawing a tree diagram.

6. A bag contains 3 red and 5 white balls. A ball is drawn at random and replaced, then another ball is drawn. Draw a probability tree diagram and find the probability that both of them are not of the same color.

**Trigonometry **

**For question No. (16)**

1. In the given figure, AB = height of a tree, A = the position of an eagle, C = position of a snake, BC = distance between the snake and the tree,<DAC = angle made by the line of sight with horizontal ground when the eagle looks down the snake.

- What is the <DAC called?
- What is the measure of <ACB?

2. A circular pond has a pole standing vertically at its centre. The top of the pole is 30 m above the water surface and the angle of elevation of it from a point on the circumference is $60^0$. Find the length of the pond.

3. From the top of a building 20 m high, a 1.7 m tall man observes the elevation of the top of a tower and finds it $45^o$.If the distance between the building and the tower is 50m, find the height of the tower.

4. Two men are on the opposite side of a tower 40m high on the same horizontal line. They observed the angle of elevation of the top of the tower and found it to be $45^0$ and $30^o$. Find the distance between them.