### MATHEMATICS CLASS 10 SEE HIGH PRIORITY QUESTIONS

**Group A **

## 1 A.

- If the initial population of any place is P and the annual growth rate is R% pa then what is the population of that place after T years? Write it.
- If the initial price of an article is Rs Y and the annual rate of compound depreciation is R% then what will be the price of an article after T years? Write it
- If the marked price of an article is Rs x, the discount amount is Rs y and the VAT amount is Rs. z then what is the selling price of the article including VAT? Find it.
- Write the formula for finding the rate of discount when the discounted amount and marked price are given.
- Write down the definition of value-added tax (VAT)?

## B.

- What is the area of an equilateral triangle whose side is ‘a’ unit?
- If the three sides of a triangle are a,b, and c respectively, what is the semi-perimeter of the triangle? Write it.
- The base of an isosceles triangle is b cm and equal sides are a cm. What is the area of the triangle?
- What is the total surface area of a square-based pyramid having a side of base p cm and slant height q cm write it.
- If the area of a triangle is $S\sqrt{(s-a)(s-b)(s-c)}$ then what is the value of s? Write it.

## 2 A.

- In which condition $\sqrt[m]{P}$ and $\sqrt[m]{q}$Are they like surds? Write it
- What does n represent in the surd $\sqrt[n]{p}$? Write it.
- Simplify: $x^{a-b}xx^{b-a}$
- Find out the value of x if $\sqrt{x}= 9$
- Define LCM

## 2 B.

1. In the following data in which series does median line?

x |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |

c.f |
16 |
36 |
61 |
77 |
80 |

2. Write the formula to find the value of lower. Quartile $Q_{1}$ of a continuous data.

3. If the lowers limit o the median class of any data is ⋉frequency of that class is f, class interval 1, sum of the frequencies N and the cumulative frequency at pre-median class is cf then write the formula for finding the median (Md)

4. Write the formula to find the value of the upper quartile $Q_{3}$ of continuous data.

5. Write the formula for finding the median of a continuous series.

## 3 A.

- What is the relation between the area of a parallelogram and triangle standing on the same base and between the same parallel? Write it.
- Write the relationship between the area of a rectangle and a triangle standing on the same base and between the same parallel lines.
- In the given figure write the relation between areas of triangle ABC and square ABPQ

4. From the given figure write the relation between areas of $\Delta$ ABC and $\Delta$ABD

5. Write the relationship between the parallelogram ABCD and MNOP standing on the same base and between the same parallel lines.

## 3 B.

1. In the given figure O is the center of the circle. What is the value of x?

2. In the adjoining figure, O is the center of a circle. If ∢NMP = $x^{o}$, find the value of ∢NOP.

3. In the given figure, O is the center of the circle then what is the value of ∢XYZ? Write it

4. In the given figure, O is the center of the circle If ∢ABC = $120^{o}$ what is the size of the reflex ∢AOC? Write it.

5. In the given figure find the value of x

**GROUP B**

# For question number 4(a)

1. According to the money exchange rate of Nepal Rastra bank, the purchasing and selling rate of 1 American dollar is NRs 103. 72 and NRs 104.32 respectively,

- How many American dollars can be exchanged with NR 57, 376?
- How many Nepali rupees can you exchange for $500?

2. After a 10% devaluation of Nepali money, $2000 can be exchanged for Rs 231000. Find the initial exchange rate.

3. If $100 = £68 and Rs 155 = £1, how many dollars. will be equal to Rs 3,16,200?

4. If a tourist paid Rs 5085 for a carved window made up of wood with a discount of Rs 500 including 13%, value-added tax (VAT), how much does he get while leaving Nepal? Find it.

5. How many times Rs 1500 is valued including 15% VAT on Rs 15,000?

# For question number 4(b)

- Samundra deposits Rs 850 at NBL at the yearly rate of 8% for 1 year. How much interest will he get?
- Due to the use of medicine, the number of bacteria decreases at the rate of 20% per day so that the number of bacteria after 3 days is ux107, what was the number of bacteria before 3 days?
- The growth rate of Covid patients is 5% per hour. If the number of covid patients in the morning at 8:00 am is 44,100 what was the number before 2 hours?
- The present number of students in a school is 1000. If per 5 students carry I, student, every year, find the number of students increased after 2 years.
- The value of a machine is depreciated from Rs 3200 to Rs 3040 in a year. Find the rate of depreciation in percent.

# For questions 5(a), (b) and 8(c)

- The edges of a triangular land are in the ratio of 12:17:25 and the perimeter is 540 feet. What is the area of land? Find it.
- The curved surface area of a cylinder is one-third of its T.S.A and the height is 5 cm. Find the radius.
- If a metallic sphere having $45πcm^{2}$ is melted to form a cylinder of height 5 cm. Find the radius.
- Find the volume of a given solid cone.
- If the area of the rectangular surface and height of a triangular prism is 660$cm^{2}$ and 22 cm respectively, find the perimeter of the base.

6. If the area of an equilateral triangle is $36\sqrt{3}cm^{2}$, what is the perimeter of the base?

7. If 8 solid iron spheres with radius ‘a’ cm each are melted to form a sphere with radius ‘b’ cm. Find a:b.

8. A cynical tent has a 3cm slant height and the radius of its base is 2.8cm. Find the cost of cloth required to make the tent at the rate of Rs300 per square meter.

9. A solid metallic cone of diameter 14cm, height 24 cm, and slant height 25 cm is cut into two halves. Find the T.S.A of two halves of cones so formed.

10. If a prism is 15 cm in height with its base a triangle sides 8cm, 9 cm, and 10cm. Find the area of rectangular surfaces.

11. If T.A.S of a solid hemisphere is 675$\pi cm^{2}$What is its volume?

12. The volume of a given triangular prism is 360$cm^{2}$ then, find its length.

13. What is the area of an isosceles triangle whose equal sides are 5cm and base side is 6cm?

14. Three metal hemispheres of radius 1cm, 6cm, and 8cm are melted to form a new hemisphere, finding the radius of the new hemisphere.

15. The perimeter and length of two sides of a triangle are 36cm, 8cm, and 12cm respectively. Find the area of the triangle.

16. The volume of a cylindrical can is 1.54liter and are of its base is 77$cm^{2}$, find the height.

# For question number 6(a)

1. Find the LCM of

$x^{2}y-xy^{3},(x^{2}-xy)^{2}$

2. If $a = b^{s}=c^{i}$and c= $a^{x}$ prove that sin x - 1=0

3. Simplify

$\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{5}}$+$\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{5}}$

4. express the given surd into the complete surds

$\sqrt{\frac{2m-3n}{2m+3n}}$

# For question number 6(b)

1. If x = $\frac{1}{2-\sqrt{3}}$, show that x² - 4x + 1 = 0

2. Simplify $\frac{a-b}{ab}+\frac{b-c}{bc}+\frac{c-a}{ca}$

3. If A =($4x-\frac{1}{x}$) then find ($A-\frac{1}{A}$)

4. Simplify: $x\sqrt[3]{y-1}\times y\sqrt{x^{-1}}$

5. simplify: $\frac{7^{x+2}-28\times 7^{x-1}}{7^{x}\times 45}x$

# For Questions 7(a), (b), (c)

- Simplify = $\frac{1}{1-p+p^{2}}+\frac{1}{1+p+p^{2}}+\frac{1}{1-p^{2}+p^{4}}$
- If apaq= $a^{p}a^{q} = (a^{p})^{q}$, prove that: p(q-2)+q(p-2) = 0
- If 4 is subtracted from the square of a number the result is 21. Find the number.
- Solve: $\sqrt{x^{2}+5}-1 =x$
- Solve for ‘z’: $\frac{2(2+3)}{\sqrt{2}+ 1}= \frac{2}{3}(\sqrt{2}-1)$
- Find three consecutive numbers such that twice the first, three times the second, and four times the third together make 191.
- Find L.C.M of $e^{4}+4$ and $2e^{3}-4e^{2}+4e$
- Simplify $\frac{3\times 2^{y}-7\times 2^{y-2}}{5\times 2^{y-2}}$
- Prove that: $(1-\frac{2}{y})(1-\frac{2}{y-2})(1+\frac{2}{y-3})=1$
- If a = $6^{x}$,b = $6^{y}$ and $a^{y}\times b^{x}$ = 36 , then prove that xy = 1

# For 8(a), (b), and (c)

1. In the given figure, AE‖DF and b are mid-point of AF. if the area of $\Delta$AEF is 30 $cm^{2}$then, find the area of parallelogram ABCD.

2. In the given figure, ABCD is a cyclic quadrilateral. If ∠ADE =$75^{o}$ then finds the value of ∠ADC and ∠ABC.

3. In the given circle, AB is a tangent, BC is diameter, and ∠BAO =$32^{o}$ , find the value of ∠ABO.

4. In the given figure, BD‖AD,∠BCD =$90^{o}$, BC = 4cm, CD = 12 cm and area of $\Delta$BOD is 36$cm^{2}$ then, find the area of quadrilateral ABCD.

5. Figure D is the center of the circle. If ∠ACB = $30^{o}$, find the area of x.

6. In the given circle, o is the center of the circle. AC is the tangent and A is the point of contact. If OA = 5 cm and CM = 8 cm. Find the length of AC.

7. In the given figure, BE DC, DC =12 cm, and BE =10cm, then find the area of $\Delta$FBC.

8. In the given figure, o is the center of the circle if ∠ABD = $25^{o}$, find the value of ∠DCF.

9. In the given figure. A is the center of the circle, CD is a tangent, BC= 4 cm, and CD=8cm. Find the length of AB.

10. In the given figure, WXYZ is a parallelogram. If ∠WMZ=90°, WM = 6cm, and MZ = 4cm, Find the area of ㅁWXYZ.

11. In the given figure, X is the center of the circle, ∠ACB = 70°, and find the value of ∠ADC.

12. In the given figure, P is the center of the circle, and QR is tangent to the circle. If ∠PRQ= 30°, find the value of ∠QPR.

13. In the given figure; BC丄 AE, BA丄CD, BE丄CE, and AX 丄CD. IF AB=12cm and area of quadrilateral BCEF = 84.sq.cm Find the length of Ax,

14. In the given figure, O is the center of The circle. If ∠BDA = 30°, find the measurement of ∠BCD.

15. In the figure, P is the center of the circle and TO is tangent to the circle. If TO = 12 cm and PO = 5 cm then what is the length of TR? Find it.

**For Question 9(a) (b)**

1. In the given figure AB=8cm, Bc=6 cm, and the area of $\Delta$ABC is 12$cm^{2}$ .Find the value of angle ABC.

2. In a continuous series, the average weight of some students is 75 Kg and the sum of their weight is 600 kg. Find the number of students.

**For Question **10(a), (b)

1. If a device is rolled and a coin is tossed at the same time, find the probability of occurring prime number on the dice and head on the coin.

2. From a bag containing 10 red and 6 black balls of the same shape and size. Two balls are taken randomly in succession without replacement.Show the probabilities of all outcomes in a tree diagram.

Group - C

**Question no - 11**

1. In a group of 300 people, 240 people like to play cricket and 170 people like to play football. If 50 people like to play none of the games.

- Show the above information in the Venn- diagram.
- How many people like cricket only?

2. In a group of 200 students, 80 students obtained an ‘A’ only in mathematics. 70 obtained ‘A’ only in science and 30 of them got the other grades in both subjects. Find using a Venn - diagram.

- The number of students who obtained an ‘A’ in both subjects.
- The number of students who obtained an ‘A’ in mathematics.
- The number of students who got an ‘A’ in science.

3. In a survey it was found that 75% of people liked to visit Pokhara, 60% liked to visit Lumbini and 20% did not like to visit both places. If 198 people liked to visit both places, how many people took part in the survey? Find it by using a Venn - diagram.

4. In a group of 300 people the ratio of people is 6:5. Out of which, 60 people like both drinks and 10% of them didn't like both drinks then,

- Show the above information in Venn - diagram
- Find the total number of people who liked only one drink.

5. A survey was conducted among some people. The survey showed 120 like to listen to the radio, 80 don't like to listen to the radio. If 50 people like to listen to the radio but do not like to watch television and 110 don't like to watch television.

- Illustrate the information in Venn - diagram.
- Find the number of people who like to watch television only.
- find the total number of people who took part in the survey.

6. 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?

7. In a survey of 70 people it was found that 50 liked football only and 10 liked both football and volleyball. The number of people who liked football is 4 times the number of people who liked volleyball y using the Venn diagram find the number of people who liked volleyball only and who don’t like any of the games

8. In a survey, one-third of people like only milk, and 28 don’t like milk. Half of them like curd but 10 don’t like both. Show the above information in Venn diagrams and find how many people like both of these drinks.

9. In a group of 50 students 20 like only math and 15 like only science. If the number of students who do not like any of the two subjects is double the number of students who like both subjects. Find the number of students who like at most one subject by using a Venn diagram.

10. 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.

**Question 12**

1. A tourist paid Rs.5085 for a Nepali Dhaka made of cotton fabric with a discount of 15% including 13% value-added tax(VAT).

- How much does he get back while leaving Nepal
- What was his discount amount? Find it.

2. Hari Bahadur sold a watch after allowing a 20% discount and then adding 13% VAT if the discount amount is Rs. 192 more than the VAT amount then find the marked price and selling price of the watch.

3. A businessman bought some American dollars of NRs. 4,40,000 according to the exchange rate of the American dollar $1 = NRs. 110. After two days in comparison to the American dollar, the Nepali currency was devalued by 10%. What was the profit or loss of the businessman when he exchanged the American dollars he had into Nepali rupees again on that day? Find it.

4. After allowing a 10% discount on the M.P. of radio 13% VAT was levied and sold it. If the difference between S.P with VAT and S.P after discount Rs. 1170 find the M.P of radio.

5. An article is sold for Rs. 4158.40 after allowing an 8% discount on the marked price and adding 13% value-added tax. That is the discount amount found.

6. An article is sold for Rs. 3051 after allowing Rs. 300 discount on the marked price and adding 13% VAT. What is the discount percent allowed on the marked price of the article? Find it.

7. The marked price of an article was 25% above the cost price. If it was sold allowing a 10% discount on the marked price, find the profit percentage.

8. A foreigner bought an article at a discount of 15% and paid Rs. 17289. If 13% VAT was levied on it By what VAT amount paid by him is less than the discount received by him?

9. Laligurash suppliers sold construction materials for Rs. 2,00,000 to Himchuli supplies making 20% profit and adding 13% VAT. Himchulis suppliers spent Rs. 6,000 for transportation costs and levied a local tax of Rs.5,000 and made a 10% profit on their total outlay and sold to construction. What VAT amount should be paid by the constructor?

10. A man buys an article at a discount of 13% and pays 16% VAT. if he pays Rs. 261 for VAT find the marked price of the article and also the amount paid by him to buy the article.

**For question number 13 (menstruation) **

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

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

3. Find T.S.A and volume.

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

5. Find the volume of the given figure

6. Find T.S.A and volume

7. An umbrella is made up of 8 isosceles triangular pieces of clothes. The measurement of the base of each triangular piece is 28cm and two equal sides are 50 cm each. If the rate of cost of cloth is Rs. 0.50, find the cost of the cloth required to make the umbrella.

8. The area of curved surface of a solid cylinder is equal to $\frac{2}{3}$ of the total surface area of the same cylinder. If T.S.A is 924$cm^{2}$ find the volume.

9. In the given figure, the solid pyramid is of the square based pyramid. The total surface area of the pyramid is 896$cm^{2}$and the side of the square base is 14cm. Find the straight height (OP) of the pyramid.

Find C.S.A and volume

10. The perimeter of a right-angled triangle is 12cm and its area is 6$cm^{2}$. Find the sides of the triangle.

**for question numbers 14 and 15**

1. Find LCM of: $a^{4}= 6a^{2}-7-8x-x^{2},a^{4}+2a^{2}x+x^{2}-1$

2. Solve $\frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}-\sqrt{a}}= 4\sqrt{2}+\frac{\sqrt{x}-\sqrt{a}}{\sqrt{x}+\sqrt{a}}$

3. $7^{m}+\frac{343}{7^{m}}=56$

4. $\frac{3x-1}{9x^{2}-3x+1}$-$\frac{3x+1}{9x^{2}+3x+1}$+$\frac{54x^{3}}{81x^{4}-9x^{2}+1}$

5. The sum of the digit number is 9 and if the tens place digit is increased by 3, the number formally will be equal to $1\frac{2}{3}$ times the original number. What is the original number?

6. The sum of the ages of a father and his son is 40 years. If they both live on till the sons become as old. As the father is now, the sum of their ages will be 96 years. Find their present age

7. Simplify $\frac{x-1}{x-2}-\frac{x+1}{x+2}-\frac{2x}{x^{2}+4}+\frac{16x}{x^{4}+16}$

8. $\sqrt[bc]{\frac{x^{\frac{b}{c}}}{x^{\frac{c}{b}}}}$$\times\sqrt[ac]{\frac{x^{\frac{c}{a}}}{x^{\frac{a}{c}}}}$$\times\sqrt[ab]{\frac{x^{\frac{a}{b}}}{x^{\frac{b}{a}}}}$

9. Of $x^{a}=y^{b}=z^{c}$ and $y^{2}=xz$show that $\frac{2}{b}=\frac{1}{a}+\frac{1}{c}$

10. A year hence, a father will be 5 times as old as his son. Two years ago, he was 3 times as old as his son will be 4 years hence. Find their present ages.

11. When a two digit number is divided by the sum of its digits, then the quotient is 8 and the remainder is 2. If the digits are reversed and the resulting number is divided by the sum of its digit, the quotient is 2 and the remainder is 8. Find the digits.

12. X = $a^{q+r}\times b^{q},y = a^{r+p}\times b^{q}$ and z = $a^{p+q}\times b^{r}$, prove that $x^{q-r}\times x^{r-p}\times z^{p-q}=1$.

13. $\frac{2a-6}{a^{2}-9a+20}-\frac{a-1}{a^{2}-7a+12}-\frac{a-2}{a^{2}-8a+15}$

14. The present age of the father is twice as old as the age of the son 3 years hence. If 2 years hence, the age of the father will be thrice as old as the age of the son, 1 year age. Find their present ages.

15. If 2xyz =1 and $x^{3}+y^{3}+z^{3}$ = 1, prove that: $m^{x^{2}-y-z-1}\times m^{y^{2}-x-z-1}\times m^{z^{2}-x-y}= m^{2}$

16. Find HCF of $6x^{6}+6x^{4}+6x^{2}$ and $4x^{6}+4x^{3}$

17. $\frac{7\sqrt{3}}{\sqrt{10}+\sqrt{3}}+\frac{2\sqrt{5}}{\sqrt{6}+\sqrt{5}}+\frac{3\sqrt{2}}{\sqrt{15}+3\sqrt{3}}$

18. Four years ago, the sum of the ages of his father and son was 27 years. After 5 years, the ratio of the ages of the son and father will be 2:7. Find the present ages of the father and son.

19. Five times the sum of the digits of a two digit number is 9 less than the number formed by reversing its digits. If four times the value of the digit at once place is equal to half of the place value of the digit at tens place. Find the number.

20. If $\sqrt[p]{x}= \sqrt[q]{y}= \sqrt[r]{z}$ and xyz = 1, prove that p+q+r = 0

**Theorems for question 16 **

1. Prove that the area of parallelogram PQRS and PQMN standing on some base and same parallels are equal.

2. A triangle PMN and a parallelogram RMNQ are constructed on the same base MN and between the same parallel lines MN and PQ prove that: area or $\Delta$PMN = $\frac{1}{2}$ロRMNQ area of RMNQ

3. In the given figure, AE॥BC, AB॥DC, and BF॥CE then prove that.

- ABF= $\Delta$ DCE
- ロABCD = ロFBCE

4. Prove that the areas of $\Delta$PQR and $\Delta$PQS. standing on the same base PQ and between the same parallel lines PQ and SR are equal.

5. Prove that diagonals of parallelograms bisect each other.

**For question number 17 **

### Geometry construction

1. Construct a parallelogram MINA in which MI = 6cm IU = 4.5cm and <AMI = 600. Also, construct a triangle MUD equal in area to the parallelogram MINA and having a side UD = 7.5cm

2. Construct a quadrilateral ABCD having AB = 6cm, BC = 5.5cm AC = 6.5cm CD= 8cm, and AD = 7cm construct a $\Delta$ ABE is equal in area to the quadrilateral ABCD.

3. 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 $\Delta$ ABC and <DBC = $75^{o}$.

4. Construct a parallelogram PQRS in which PQ = 5cm, diagonal PR= 6cm and diagonal QS = 8cm. Construct a triangle PSA whose area is equal to the area of the parallelogram.

5. Construct a triangle xyz in which xy = 6.3cm, <x = $30^{o}$ and <y= $45^{o}$ construct another triangle wxy equal in area to $\Delta$XYZ and a side wy = 7.5cm

6. Construct a quadrilateral PQRS having PQ = QR = 5.9cm, RS = PS = 6.1cm and <QPS = $75^{o}$ then construct a $\Delta$PST which is equal in area to the given quadrilateral

7. 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.

8. Construct a triangle having an angle $60^{o}$ and whose area equals to the rectangle having length 6cm and breadth 4.5cm

9. Construct a $\Delta$MNQ equal in area to the quadrilateral MNOP having NO=OP= 5.5cm, PM = MN = 4.5cm and <MNO = $75^{o}$

10. Construct a triangle having an angle $75^{o}$ and whose area is equal to the area of a parallelogram ABCD having base side 6cm diagonal 7cm and angle made by diagonal with base side.

**For question number 18 **

1. Verify experimeatally that angle AKB at the cerate is twice the inscribed angle ACB subtended by the same are AB (Two figures with at least 3cm radii are necessary)

2. Verify experimentally that the angles in the same segment of a circle are equal.

3. Explore experimentally the relationship between opposite angles of cyclic quadrilateral MNOP.

4. Prove theoriivally that the centre angle is twice the inscribed angle standing on same are.

5. Verify theoretically that inscribed angles standing on some are equal

6. Prove that the opposite angles of a cyclic quadrilateral are supplementary theoretically.

**Q.No. 19 **

1. A tower on the bank of river is of 20 m high and the angle of elevation of the top of the tower from the opposite bank is $30^{o}$. find the breath of the river.

2. A man 1.7m tall observe a bird sitting on the top of a free in front of him find the angle of elevation to be $60^{o}$. if the height of the tree is 53.7m find the distance between the man and the tree.

3. The distance between a tower and a house is one third of the height of the tower. If the height of the tower is 60 meter and the angle of depression from the top of the tower to the roof of the house is $45^{o}$. find the height of the house.

4. A man of height 1.6m is flying a kite from the roof of a house 25m high. If the length of the string of the kite is $80\sqrt{3}m$ and it makes the angle of $60^{o}$. with horizon find the height of the kite from the ground.

5. If the top of a tree which is broken by the wind makes an angle of $60^{o}$. with the ground at a distance of $15\sqrt{3}$ from the root of the tree find the height of the tree before it was broken.

6. The diameter of a circular pond is 90 meters and a pillar is fixed at the center of the pond. The height of the pole is 48 meters and the pond is 3 meters deep. Find the angle of elevation of the top of the pole from a point of the circumference.

7. Two vertical poles are fixed 60m apart. The angle of depression of the top of the first pole having height 115.36m as seen from the top of the second which is 150m height is Q. find the value of $0^{o}$.

8. An electric pole on the side of a road is tied with a wire to the ground. If the wire having length 30 meters makes an angle $60^{o}$ with the ground. Find the height of the pole from the ground up to the point where the wire is tied.

9. From the top of a tower 75 ft high. The angle of depression of roof of a house is $25\sqrt{3}$ ft. find the height of the house

10. A man 1.6m tall observes the angle of elevation of top of the house and found to be $60^{o}$. if the distance of man from the foot of the house is 100m, find the height of the house.

**Statistics (for Q.no. 20)**

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 10. 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. If the median of given table is 24, calculate the value of x

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

No. of students |
9 |
21 |
1 |
15 |
10 |

5. 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 |

6. 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 |

7. Find $Q_{1}$ and $Q_{3}$

Mid value |
6 |
18 |
30 |
42 |
54 |

f |
8 |
12 |
14 |
10 |
6 |

8. Find median

Marks |
10-19 |
20-29 |
30-39 |
40-49 |
50-59 |

f |
4 |
5 |
9 |
3 |
2 |

9. 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 |

10. The $Q_{1}$ 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 |

Group D

# Q.no:21

1. A sum of money is placed at simple interest for 3 years at 10% per annum and then the amount is invested for 2 years at the same rate at compound interest. If the total amount of 5 years became Rs 471,900. What was the sum? Ans: 300000

2. Binita deposited Rs 50,000 in a bank at the rate of 10% p.a compounded annually. At the end of the second year, she withdrew Rs 15000 and at the end of 3rd year, she deposited Rs 10,000 again. Find the total interest received by her in 4 years if the interest is compounded half-yearly in the last two years. Ans: 21330.53.

3. According to yearly compound interest, the compound interest on a sum of money for two successive years is Rs 225 and 240 respectively. Find the rate of interest and the compound interest for next year (Ans; 20/3 % , Rs 256

4. Suvam trader borrows Rs 500 at 12% simple interest and invests the same amount at 14% compound interest. If the transaction is closed at the end of two years. What is his gain? Ans: (298)

5. At a certain rate of yearly compound interest , a sum of money amounts to Rs 60500 in 2 years and Rs 73205 in 4 years. Find the rate of interest and sum.

6. A man borrowed Rs 50,000 for 3 years at the rate of 12% . simple interest but he could pay only Rs 25,000 at the end of 3 years and agreed to clear his debt at the end of next two years at the rate of 10 % annual compound interest . Find how much he should pay to clear his debt? Also find the amount of interest he paid altogether.

7. Karan borrowed Rs 20000 for 3 years at the rate of 10% per annum compounded annually . After 3 years , he paid only half of the original principal. He cleared the debt of remaining principal and the interest of the first 3 years at the same rate after the next 2 years.

- How much did he pay for the last 2 years?
- How much total interest did he pay in 5 years?

8. A bank has fixed the rate of interest 10% per annum annually compound interest in account A and 9.5 % per annum semi-annually compound interest in account B. If you are going to deposit Rs 80,000 for 2 years in the same bank , in which account will you deposit and why? Give your reasons.

9. A man working in the UAE deposits to citizen international bank Nepal for 2 years in compound interest . Compounded annually at the rate of 11% p.a (Dirham=Rs32). After one year he deposits another 10,000 Dirham at the same interest rate in the same bank . But this time the Nepa;i currency is evaluated by 5 % in comparison to Dirham. Find the total compound interest paid by the bank to the man. What percent did he get more or less interest in the first year than in the second year?

10. Prakash lent altogether Rs 6000 to Anima agreed to pay simple interest at 10% p.a and Elina agreed to pay compound interest at rate of 8% p.a. If Elina paid Rs 50 more than Anima as the interest, how much did he lend to each.

# Q.No.22

1. A person bought a water cylindrical tank of circular base having the radius 1.05 metre and height 3.5 metre for the use of own house from the shop . If the upper part of the tank is hemispherical how many litres of water will be contained in the tank?

2. Binaya has fixed a water tank in the shape of a cylinder surmounted by a hemisphere on the root of his house. The tank has its internal radius 90cm and total height 3.8cm.

- Find the cost of filling the tank with water at the rate of 20 paisa per litre.
- In how many days will the water of the overhead water tank last if his family consumes 495 litres of water in a day? Find it.

3. In a ceremony Rahul has to make arrangements for accommodation of 200 people for which he plans to build a conical tent in such a way that each person has 0.5m2 of space on the ground and 2.5m3 air to breath. What is the area of clothes required to make the conical tent? Also find the total cost of cloth at the rate of Rs 50 per square metre.

4. A solid object is made with the combination of a hemisphere and a cone having the same radius of the base . The diameter of its hemispherical part is 28cm . The cost of painting the surface of the solid at Rs 150 per 200cm2 was 1831.50. Find the total height of the solid.

5. A solid is made up of a combination of pyramid and a square based prism with the height of 24 cm and length of the prism is 12 cm and length of the base is 10 cm. How much money is needed to paint the total surface area of the solid at the rate of Rs. 1.50 per cm2?

6. An iron pillar is formed with the combination of a cylinder in the lower part and cone above it . The radius of the base of each of the cylinder and cone is 8 cm , The cylindrical part is 240 cm high and the conical part is 240 cm high and the conical part is 36 cm high ,1 cubic cm of iron weighs 10 gm. Find the weight of the pilar in kilogram (Ans: 506.88Kg)

7. A cylindrical can having radius 12 cm and height 30 cm is full of ice -cream .If the ice cream is to be filled in the cones of height 12cm and radius 3cm by making hemispherical shape on the top, then how many cones are required , Find it.

8. A conical hole of equal radius and height is drilled out from a cylindrical wooden log. Find the curved surface area of the remaining part of wood . Also , find the total cost of pointing the curved surface at the rate of rs 7.50 per sq.cm.

9. A container shaped right circular cylinder having diameter 12cm and height 15 cm is full of ice cream. If ice cream is to be filled into cones of height 12cm and diameter 6cm having a hemispherical shape on the top find the number of such cones which can be filled with ice-cream, If the cost of each slope of an ice cream in a cone is Rs 50, what was the cost of the full container of ice cream?

10. Two pillars of height 8 ft each where its four faces are shown in the gate of a stadium have one-one pyramids of height 4 ft, Each having the same base on their tops. The base of each pillar is 6ft х 6ft. If the pillars with pyramids are pointed at the rate of Rs 75 per square ft. What will be the total cost?

# Qno.23

1. simplify $\frac{(a^{2}-\frac{1}{b^{2}})^{2}(a-\frac{1}{b})^{b-a}}{(b^{2}-\frac{1}{a^{2}})^{b}(b+\frac{1}{a})^{a-b}}$

2. If a+b+c=0, prove that : $\frac{1}{1+x^{a}+x^{-b}}+\frac{1}{1+x^{b}+x^{-c}}+\frac{1}{1+x^{-c}+a^{-a}}=1$

3. If pqr=1, prove that $\frac{1}{1+p+q^{-1}}+\frac{1}{1+q+r^{-1}}+\frac{1}{1+r+p^{-1}}=1$

4. If $x^{2}-2=2^{\frac{2}{3}}+2^{-\frac{2}{3}}$, then prove that

x= $2^{\frac{1}{3}}+2^{\frac{-1}{3}}$

x= $2x^{3}-6x=5$

5. If a+b+c=m then prove that : $\frac{x^{2a}}{x^{2a}+x^{m-b}x^{m-c}}+\frac{x^{2b}}{x^{2b}+x^{m-c}x^{m-x}}+\frac{x^{2c}}{x^{2c}+x^{m-a}x^{m-b}}= 1$

6. The students studying in class 10 organised a picnic with a total budget of Rs 1,21,600. They decided to collect an equal amount for the picnic , But 4 students could not attend the picnic and each of the participating students paid Rs 80 more.

- How many students participated in the picnic?
- Calculate the amount paid by each of the participated students.

7. The perimeter of a rectangular ground is 100m. When its side is reduced then new length and breadth are$\frac{5}{6}$ and 75% of the previous length and breadth . If the new perimeter of ground is 80m. What was the length and breadth of the ground in the beginning? Find it.

8. The total cost of 5Kg rice and 5 Kg sugar is Rs 940. If the rate of rice increases by 20% and the rate of sugar decreases by 10% , the total cost of 4 Kg rice and 3 Kg sugar will be Rs 627. By what percent the cost of 1 Kg rice is more or less than the cost of 1 Kg sugar . Find it?

9. In a number of two digits, the ratio of the digit in the tenth place and the digit in the unit place is 3:1 . If 3 is added to the three times of the sum of the digits, then the opposite number is formed. Find the number.

10. Solve when (a+b+c=2): $\frac{\sqrt{x-1}-b-c}{a}+\frac{\sqrt{x-1}-c-a}{b}+\frac{\sqrt{x-1}-a-b}{c}= 3$

# Q.no. 24

1. In the figure, PQ=QR=RP, From O, perpendiculars OX, OY and OZ are drawn on PQ,QR and PR respectively and PAQR Prove that: PA= OX+OY+OZ

2. In the given figure PQ is a diameter at a circle with centre O. If PX=XS, prove that : OX//QS

3. In the given figure , AP= CP then prove that AD= BC

4. In the given figure WXY is an equilateral triangle and WPQ is an isoceles triangle then prove that:

i) PQ// XY

ii) ፈPWX=ፈQWY

5. In the given figure RT// OQ. Prove that PM is the bisector of ፈOMQ.

6. In a trapezium MNOP if MN//PO and A is the midpoint of NO then Prove that:

Area of MAP=12Area of trapezium MNOP

7. In the figure AP,BP,CR and DR are the bisectors of ፈA, ፈB , ፈC and ፈD respectively in cyclic quadrilateral ABCD . Prove that PQRS is a cyclic quadrilateral.

8. In ABC D,E,F and G are the midpoints of BC, AD, BE and CF respectively. Prove that : ABC= 8EFG

9. In the given figure AM and CN are bisectors of ፈDAB and ፈDCb respectively . Prove that the MN is a diameter of a circle.

10.In the given figure , O is the centre of a circle. Two chords XY and MN are produced to meet at A, then prove that:

∡XOM-∡YON= 2∡XAM