Sample Question Paper for Class 8

 **PART 1: QUESTION PAPER**

**RAMAKRISHNA MISSION BOYS' HOME**

**Annual Examination - 2026**

**Class-VIII**

**Subject : Mathematics (English Medium)**

**Time : 3 Hours** | **Full Marks : 100**

*Instructions:*

*   *Handwriting should be neat and clean.*

*   *Maintain the word-limit strictly.*

*   *Marks will be deducted for spelling mistakes, and illegible hand-writing.*

*   *Figures in the margin indicate full marks.*

*   *All questions are compulsory. Draw neat and labelled diagrams wherever necessary.*

### **Section A: Very Short Answer**

**Answer all questions. Each question carries 1 mark.** (10 × 1 = 10)

1. If the price of sugar increases by 20%, by what percent must a family reduce its consumption so that their expenditure on sugar remains unchanged?

2. What is the H.C.F. of the algebraic expressions $x^2 - 5x + 6$ and $x^2 - 4x + 4$?

3. In $\triangle ABC$, the lengths of the sides are $AB = 5$ cm, $BC = 6$ cm, and $AC = 7$ cm. Which angle of the triangle is the largest?

4. A mixture contains milk and water in the ratio 5:2. If 14 liters of water is added to it, the ratio becomes 5:4. Find the original quantity of milk in the mixture.

5. Factorise: $4x^2 - 12xy + 9y^2 - 25z^2$.

6. How many straight lines can be drawn parallel to a given line at a fixed distance of 4 cm from it?

7. In an election, a candidate who gets 40% of the total votes polled is elected by a majority of 240 votes. Find the total number of votes polled.

8. Simplify: $\frac{x}{x-y} - \frac{y}{x-y} + \frac{x+y}{x^2-y^2}$.

9. If the angles of a triangle are in the ratio 2:3:4, find the difference between the measurement of the largest and the smallest angles.

10. 12 men can reap a field in 20 days. How many men will be required to reap the same field in 15 days?

 **Section B: Short Answer**

**Answer all questions. Each question carries 2 marks.** (10 × 2 = 20)

11. The population of a town is 50,000. It increases by 5% in the first year and decreases by 4% in the second year. What is the population of the town at the end of the second year?

12. Find the L.C.M. of the algebraic expressions: $a^2 - b^2$, $a^2 - ab$, and $a^2 + 2ab + b^2$.

13. Draw a line segment of length 8 cm and divide it into 5 equal parts using a compass and a ruler. Write the steps of construction only.

14. A chemical solution contains 30% acid. If 10 liters of water is added to 40 liters of this solution, what is the new percentage of acid in the resulting mixture?

15. If $x + \frac{1}{x} = 5$, find the value of $x^2 + \frac{1}{x^2}$.

16. In $\triangle PQR$, $PQ = 8$ cm, $PR = 6$ cm, and $QR = 10$ cm. Verify the relation between the lengths of the sides and the measurements of their opposite angles.

17. A man's income increases by 20% and his expenditure increases by 10%. If his original income was Rs. 5,000 and his original expenditure was Rs. 4,000, find the percentage increase in his savings.

18. Factorise the algebraic expression: $x^4 + x^2 + 1$.

19. Draw a line segment $AB$ of length 6 cm. Draw a line parallel to $AB$ at a distance of 3 cm from it, using a compass and ruler. Write the steps of construction only.

20. A sum of money is divided among A, B, and C in the ratio 3:4:8. If the share of C is Rs. 800 more than the combined share of A and B, find the total sum of money.

### **Section C: Long Answer**

**Answer all questions. Each question carries 4 marks.** (10 × 4 = 40)

21. In an examination, 35% of the total students failed in History, 45% failed in Geography, and 20% failed in both subjects. Find the percentage of students who passed in both subjects. If 400 students passed in both subjects, find the total number of students who appeared in the examination.

22. Simplify the algebraic expression: 

$$\frac{1}{x-y} - \frac{1}{x+y} + \frac{2y}{x^2+y^2} - \frac{4y^3}{x^4-y^4}$$

23. Prove that in a triangle, the side opposite to the greater angle is greater than the side opposite to the smaller angle.

24. A vessel contains a mixture of milk and water in the ratio 7:3. 20 liters of the mixture is drawn out and replaced by water. If the new ratio of milk to water becomes 7:5, find the original quantity of milk in the vessel.

25. Find the H.C.F. and L.C.M. of the following algebraic expressions: 

$x^2 - 5x + 6$, $x^2 - 4x + 3$, and $x^2 - 2x - 3$.

26. Construct a $\triangle ABC$ with $BC = 6$ cm, $\angle B = 60^\circ$, and $\angle C = 45^\circ$. Draw the perpendicular bisector of the side $BC$. Write the exact steps of construction.

27. 8 men or 12 women can do a piece of work in 25 days. In how many days will 6 men and 11 women do the same work?

28. Factorise: $(x^2 - 4x)(x^2 - 4x - 2) - 24$.

29. In $\triangle ABC$, $AB = AC$. The side $BA$ is produced to $D$ such that $AD = AB$. Prove that $\angle BCD = 90^\circ$.

30. The price of a machine depreciates by 10% every year. If its present value is Rs. 87,480, what was its value exactly 3 years ago?

### **Section D: Higher Order Thinking / Application**

**Answer all questions. Each question carries 10 marks.** (3 × 10 = 30)

31. **(Arithmetic - Mixture & Percentage)**

A container contains a mixture of two liquids A and B in the ratio 4:1. When 10 liters of the mixture is taken out and 10 liters of liquid B is poured into the container, the ratio becomes 2:3.

(a) Find the original quantity of liquid A in the container. **(5)**

(b) If 5 liters of the original mixture is replaced by 5 liters of liquid B, and then again 5 liters of the new mixture is replaced by 5 liters of liquid B, find the final ratio of A to B in the container. **(5)**

32. **(Algebra - Factorisation & Simplification)**

(a) If $a, b, c$ are real numbers such that $a + b + c = 0$, prove that $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} = 3$. **(5)**

(b) Factorise the expression: $x^4 - 2x^3 - x^2 + 2x + 1$. **(5)**

33. **(Geometry - Construction & Verification)**

(a) Draw a line segment $PQ$ of length 9 cm. Divide it into 3 equal parts using a compass and ruler. Through the first division point (from P), draw a line parallel to $PQ$. Write the exact steps of construction. **(6)**

(b) In $\triangle XYZ$, $\angle X = 50^\circ$ and $\angle Y = 60^\circ$. A point $W$ is taken on $YZ$ such that $XW$ is the bisector of $\angle YXZ$. Verify the relation between the sides $XY, XW, YW$ and $XZ, XW, ZW$ using the angle-side relationship theorem of a triangle. **(4)**