HALF-YEARLY EXAMINATION
MATHEMATICS
Class - IX
Session: 2026–27
Time: 3 Hours
Full Marks: 90
General Instructions:
- All questions are compulsory unless otherwise specified.
- The question paper consists of two sections — Section A and Section B.
- Section A contains 45 objective-type questions carrying 1 mark each.
- Section B contains descriptive-type questions carrying a total of 45 marks.
- Show all necessary steps in descriptive questions.
- Use mathematical symbols and notation correctly.
SECTION A — MULTIPLE CHOICE QUESTIONS (45 Marks)
1. If 30% of a number is 150, what is the number?
যদি এটা সংখ্যাৰ 30% ৰ মান 150 হয়, তেন্তে সংখ্যাটো কি?
| (A) 300 | (B) 400 | (C) 500 | (D) 600 |
2. What is the simplest form of the ratio 108 : 144?
108 : 144 অনুপাতটোৰ আটাইতকৈ সৰল ৰূপ কি?
| (A) 3:4 | (B) 4:3 | (C) 2:3 | (D) 3:5 |
3. Assertion (A): The simple interest on Rs 2000 for 3 years at 5% per annum is Rs 300.
Reason (R): Simple Interest = (Principal × Rate × Time) / 100.
Reason (R): Simple Interest = (Principal × Rate × Time) / 100.
দাবী (A): 2000 টকাৰ 5% বাৰ্ষিক সুতৰ হাৰত 3 বছৰৰ সৰল সুত 300 টকা।
কাৰণ (R): সৰল সুত = (মূলধন × সুতৰ হাৰ × সময়) / 100।
কাৰণ (R): সৰল সুত = (মূলধন × সুতৰ হাৰ × সময়) / 100।
| (A) Both A and R are true and R is correct explanation of A. | (B) Both A and R are true but R is NOT the correct explanation. |
| (C) A is true but R is false. | (D) A is false but R is true. |
4. If x - 7 = 12, then the value of x is:
যদি x - 7 = 12, তেন্তে x ৰ মান হ'ব:
| (A) 5 | (B) 19 | (C) 15 | (D) 14 |
5. Which of the following is equal to 4³?
তলৰ কোনটো 4³ ৰ সমান?
| (A) 16 | (B) 64 | (C) 128 | (D) 256 |
6. Every rational number is a:
প্ৰতিটো পৰিমেয় সংখ্যা হৈছে এটা:
| (A) Natural number | (B) Integer | (C) Real number | (D) Whole number |
7. Assertion (A): √3 is an irrational number.
Reason (R): The decimal expansion of an irrational number is non-terminating and non-recurring.
Reason (R): The decimal expansion of an irrational number is non-terminating and non-recurring.
দাবী (A): √3 এটা অপৰিমেয় সংখ্যা।
কাৰণ (R): অপৰিমেয় সংখ্যাৰ দশমিক প্ৰসাৰণ অসীমিত আৰু অনাবৃত্ত।
কাৰণ (R): অপৰিমেয় সংখ্যাৰ দশমিক প্ৰসাৰণ অসীমিত আৰু অনাবৃত্ত।
| (A) Both A and R are true and R is correct explanation. | (B) Both A and R are true but R is not the correct explanation. |
| (C) A is true but R is false. | (D) A is false but R is true. |
8. The value of (81)1/2 is:
(81)1/2 ৰ মান হৈছে:
| (A) 3 | (B) 9 | (C) 27 | (D) 81 |
9. Which of the following is the rationalising factor of 1/(√5 - √3)?
1/(√5 - √3) ৰ হৰৰ পৰিমেয়কৰণ উৎপাদক কোনটো?
| (A) √5 + √3 | (B) √5 - √3 | (C) √3 - √5 | (D) None of these |
10. Statement I: Between two distinct rational numbers, there are infinitely many rational numbers.
Statement II: 0 is not a rational number.
Statement II: 0 is not a rational number.
উক্তি I: দুটা বেলেগ বেলেগ পৰিমেয় সংখ্যাৰ মাজত অসীম সংখ্যক পৰিমেয় সংখ্যা থাকে।
উক্তি II: 0 এটা পৰিমেয় সংখ্যা নহয়।
উক্তি II: 0 এটা পৰিমেয় সংখ্যা নহয়।
| (A) Only Statement I is correct. | (B) Only Statement II is correct. |
| (C) Both are correct. | (D) Both are incorrect. |
11. Find the equivalent fraction of 0.111....
0.111... ৰ সমতুল্য ভগ্নাংশ উলিওৱা।
| (A) 1/9 | (B) 1/10 | (C) 1/11 | (D) 1/99 |
12. The degree of the polynomial 5x&sup5; - 3x³ + 2x + 1 is:
5x&sup5; - 3x³ + 2x + 1 বহুপদটোৰ মাত্ৰা কিমান?
| (A) 2 | (B) 3 | (C) 5 | (D) 1 |
13. What is the zero of the polynomial p(x) = 3x - 2?
p(x) = 3x - 2 বহুপদটোৰ শূন্য কি?
| (A) 2/3 | (B) -2/3 | (C) 3/2 | (D) -3/2 |
14. Assertion (A): The polynomial x² - 4 has exactly two real zeroes, 2 and -2.
Reason (R): A quadratic polynomial can have at most two real zeroes.
Reason (R): A quadratic polynomial can have at most two real zeroes.
দাবী (A): x² - 4 বহুপদটোৰ ঠিক দুটা বাস্তৱ শূন্য আছে, 2 আৰু -2।
কাৰণ (R): এটা দ্বিঘাত বহুপদৰ সৰ্বাধিক দুটা বাস্তৱ শূন্য থাকিব পাৰে।
কাৰণ (R): এটা দ্বিঘাত বহুপদৰ সৰ্বাধিক দুটা বাস্তৱ শূন্য থাকিব পাৰে।
| (A) Both A and R are true and R is correct explanation. | (B) Both A and R are true but R is not the correct explanation. |
| (C) A is true but R is false. | (D) A is false but R is true. |
15. The coefficient of x² in (x + 3)(x - 2) is:
(x + 3)(x - 2) ত x² ৰ সহগ কিমান?
| (A) -6 | (B) 1 | (C) 3 | (D) -2 |
16. Match the polynomials with their types:
| Column I | Column II |
|---|---|
| (a) x²-x | (i) Linear |
| (b) 2x³+1 | (ii) Quadratic |
| (c) 5x-3 | (iii) Cubic |
বহুপদবোৰক সিহঁতৰ প্ৰকাৰৰ সৈতে মিলোৱা:
| (A) (a)-(ii), (b)-(iii), (c)-(i) | (B) (a)-(iii), (b)-(ii), (c)-(i) |
| (C) (a)-(i), (b)-(iii), (c)-(ii) | (D) (a)-(ii), (b)-(i), (c)-(iii) |
17. If p(x) = x² - 3x + 2, then p(-1) is equal to:
যদি p(x) = x² - 3x + 2, তেন্তে p(-1) ৰ মান কিমান হ'ব?
| (A) 0 | (B) 4 | (C) 6 | (D) 2 |
18. Which of the following is a trinomial?
তলৰ কোনটো ত্ৰিপদ ৰাশি?
| (A) 3x² | (B) x² + x | (C) x² + 2x + 1 | (D) x³ - 1 |
19. The horizontal axis in the Cartesian plane is called:
কাৰ্টেছীয় সমতলত অনুভূমিক অক্ষডালক কি বোলে?
| (A) y-axis | (B) x-axis | (C) Origin | (D) Quadrant |
20. In which quadrant does the point (-2, -5) lie?
(-2, -5) বিন্দুটো কোনটো চৌকাত অৱস্থিত?
| (A) First | (B) Second | (C) Third | (D) Fourth |
21. Assertion (A): The point (3, 0) lies on the x-axis.
Reason (R): For any point on the x-axis, its y-coordinate is 0.
Reason (R): For any point on the x-axis, its y-coordinate is 0.
দাবী (A): (3, 0) বিন্দুটো x-অক্ষত অৱস্থিত।
কাৰণ (R): x-অক্ষৰ ওপৰত থকা যিকোনো বিন্দুৰ y-স্থানাংক 0 হয়।
কাৰণ (R): x-অক্ষৰ ওপৰত থকা যিকোনো বিন্দুৰ y-স্থানাংক 0 হয়।
| (A) Both A and R are true and R is correct explanation. | (B) Both A and R are true but R is not the correct explanation. |
| (C) A is true but R is false. | (D) A is false but R is true. |
22. The perpendicular distance of the point P(-3, 6) from the x-axis is:
P(-3, 6) বিন্দুৰ পৰা x-অক্ষলৈ লম্ব দূৰত্ব কিমান?
| (A) -3 units | (B) 3 units | (C) 6 units | (D) -6 units |
23. The ordinate of all points on the x-axis is:
x-অক্ষৰ ওপৰত থকা সকলো বিন্দুৰ কোটি কিমান?
| (A) 0 | (B) 1 | (C) -1 | (D) Any number |
24. A point with a positive x-coordinate and negative y-coordinate lies in:
যিটো বিন্দুৰ x-স্থানাংক ধনাত্মক আৰু y-স্থানাংক ঋণাত্মক, সেইটো কোনটো চৌকাত থাকিব?
| (A) Quadrant I | (B) Quadrant II | (C) Quadrant III | (D) Quadrant IV |
25. The linear equation 2x + 5 = 15 in one variable has:
2x + 5 = 15 এক চলকযুক্ত ৰৈখিক সমীকৰণটোৰ:
| (A) Unique solution | (B) Two solutions | (C) No solution | (D) Infinitely many solutions |
26. Which of the following is a solution of 3x - y = 6?
তলৰ কোনটো 3x - y = 6 ৰ এটা সমাধান?
| (A) (2, 0) | (B) (0, 2) | (C) (3, 1) | (D) (1, 3) |
27. Assertion (A): The graph of y = -2 is a line parallel to the x-axis.
Reason (R): The equation of a line parallel to the x-axis is of the form y = k.
Reason (R): The equation of a line parallel to the x-axis is of the form y = k.
দাবী (A): y = -2 ৰ লেখডাল x-অক্ষৰ সমান্তৰাল।
কাৰণ (R): x-অক্ষৰ সমান্তৰাল ৰেখাৰ সমীকৰণৰ আৰ্হি হৈছে y = k।
কাৰণ (R): x-অক্ষৰ সমান্তৰাল ৰেখাৰ সমীকৰণৰ আৰ্হি হৈছে y = k।
| (A) Both A and R are true and R is correct explanation. | (B) Both A and R are true but R is not the correct explanation. |
| (C) A is true but R is false. | (D) A is false but R is true. |
28. The equation of the y-axis is:
y-অক্ষৰ সমীকৰণটো কি?
| (A) x = 0 | (B) y = 0 | (C) x = y | (D) x + y = 0 |
29. If (2, 3) is a solution of 3x + ky = 12, find k.
যদি (2, 3) বিন্দুটো 3x + ky = 12 ৰ সমাধান হয়, তেন্তে k ৰ মান নিৰ্ণয় কৰা।
| (A) 1 | (B) 2 | (C) 3 | (D) 4 |
30. Any point on the line y = -x is of the form:
y = -x ৰেখাডালত থকা যিকোনো বিন্দুৰ আৰ্হি হ'ল:
| (A) (a, a) | (B) (a, -a) | (C) (0, a) | (D) (-a, -a) |
31. According to Euclid, a line has:
ইউক্লিডৰ মতে, এডাল ৰেখাৰ থাকে:
| (A) No breadth | (B) Length and breadth | (C) Only thickness | (D) All three dimensions |
32. Assertion (A): If equals are added to equals, the wholes are equal.
Reason (R): This is one of Euclid's axioms.
Reason (R): This is one of Euclid's axioms.
দাবী (A): সমানৰ লগত সমান যোগ কৰিলে, যোগফলবোৰ সমান হয়।
কাৰণ (R): এইটো ইউক্লিডৰ স্বতঃসিদ্ধসমূহৰ এটা।
কাৰণ (R): এইটো ইউক্লিডৰ স্বতঃসিদ্ধসমূহৰ এটা।
| (A) Both A and R are true and R is correct explanation. | (B) Both A and R are true but R is not the correct explanation. |
| (C) A is true but R is false. | (D) A is false but R is true. |
33. How many dimensions does a point have?
এটা বিন্দুৰ মাত্ৰা কিমান?
| (A) 1 | (B) 2 | (C) 3 | (D) 0 |
34. Euclid's fifth postulate is related to:
ইউক্লিডৰ পঞ্চম স্বীকাৰ্য কিহৰ লগত জড়িত?
| (A) Intersecting lines | (B) Parallel lines | (C) Right angles | (D) Circles |
35. The boundaries of solid bodies are:
গোটা বস্তুৰ সীমাবোৰ হৈছে:
| (A) Points | (B) Lines | (C) Surfaces | (D) Curves |
36. Which of the following does not need a proof?
তলৰ কোনটোৰ বাবে প্ৰমাণৰ প্ৰয়োজন নাই?
| (A) Theorem | (B) Axiom | (C) Corollary | (D) Lemma |
37. If two adjacent angles are supplementary, they form a:
যদি দুটা সন্নিহিত কোণ সম্পূৰক হয়, তেন্তে সিহঁতে গঠন কৰে এটা:
| (A) Right angle | (B) Linear pair | (C) Vertically opposite angle | (D) Reflex angle |
38. The sum of the exterior angles of a triangle is:
ত্ৰিভুজৰ বহিঃকোণবোৰৰ সমষ্টি কিমান?
| (A) 180° | (B) 270° | (C) 360° | (D) 540° |
39. Assertion (A): The sum of two interior opposite angles is equal to the exterior angle of a triangle.
Reason (R): This is known as the Exterior Angle Theorem.
Reason (R): This is known as the Exterior Angle Theorem.
দাবী (A): দুটা অন্তঃস্থ বিপৰীত কোণৰ সমষ্টি ত্ৰিভুজটোৰ বহিঃকোণৰ সমান।
কাৰণ (R): ইয়াক বহিঃকোণৰ উপপাদ্য বুলি জনা যায়।
কাৰণ (R): ইয়াক বহিঃকোণৰ উপপাদ্য বুলি জনা যায়।
| (A) Both A and R are true and R is correct explanation. | (B) Both A and R are true but R is not the correct explanation. |
| (C) A is true but R is false. | (D) A is false but R is true. |
40. If one angle of a linear pair is acute, then its other angle will be:
ৰৈখিক যোৰৰ এটা কোণ সূক্ষ্মকোণ হ'লে, আনটো কোণ হ'ব:
| (A) Acute | (B) Obtuse | (C) Right | (D) Straight |
41. Statement I: If two lines are parallel to the same line, they are parallel to each other.
Statement II: Vertically opposite angles are supplementary.
Statement II: Vertically opposite angles are supplementary.
উক্তি I: একেডাল ৰেখাৰ সমান্তৰাল হোৱা দুডাল ৰেখা পৰস্পৰ সমান্তৰাল হয়।
উক্তি II: বিপ্ৰতীপ কোণবোৰ সম্পূৰক।
উক্তি II: বিপ্ৰতীপ কোণবোৰ সম্পূৰক।
| (A) Only Statement I is true. | (B) Only Statement II is true. |
| (C) Both are true. | (D) Both are false. |
42. The measure of an angle which is twice its supplement is:
যিটো কোণ তাৰ সম্পূৰক কোণৰ দুগুণ, তাৰ মাপ কিমান?
| (A) 60° | (B) 120° | (C) 90° | (D) 150° |
43. In ΔABC, if AB = AC and ∠A = 80°, then ∠B is:
ΔABC ত, যদি AB = AC আৰু ∠A = 80°, তেন্তে ∠B হ'ল:
| (A) 50° | (B) 80° | (C) 100° | (D) 40° |
44. Which congruence rule involves a right angle, hypotenuse, and a side?
কোনটো সৰ্বসমতাৰ বিধিত সমকোণ, অতিভুজ আৰু এটা বাহু অন্তৰ্ভুক্ত থাকে?
| (A) SAS | (B) ASA | (C) SSS | (D) RHS |
45. Assertion (A): Equilateral triangles are always isosceles.
Reason (R): An equilateral triangle has all three sides equal.
Reason (R): An equilateral triangle has all three sides equal.
দাবী (A): সমবাহু ত্ৰিভুজবোৰ সদায় সমদ্বিবাহু হয়।
কাৰণ (R): সমবাহু ত্ৰিভুজৰ তিনিওটা বাহু সমান।
কাৰণ (R): সমবাহু ত্ৰিভুজৰ তিনিওটা বাহু সমান।
| (A) Both A and R are true and R is correct explanation. | (B) Both A and R are true but R is not the correct explanation. |
| (C) A is true but R is false. | (D) A is false but R is true. |
46. In ΔLMN, if LM > LN, then:
ΔLMN ত, যদি LM > LN, তেন্তে:
| (A) ∠N > ∠M | (B) ∠M > ∠N | (C) ∠L > ∠M | (D) ∠N = ∠M |
47. If ΔABC ≅ ΔDEF, then by CPCT which is true?
যদি ΔABC ≅ ΔDEF, তেন্তে CPCT অনুসৰি কোনটো সত্য?
| (A) ∠A = ∠D | (B) ∠B = ∠F | (C) ∠C = ∠E | (D) AB = EF |
SECTION B — DESCRIPTIVE QUESTIONS (45 Marks)
[2 Marks]
46. Simplify by rationalising the denominator: 1 / (√7 + √3).
হৰৰ পৰিমেয়কৰণ কৰি সৰল কৰা: 1 / (√7 + √3)।
[2 Marks]
47. Find the value of k, if x - 1 is a factor of p(x) = kx² - √3x + 1.
যদি p(x) = kx² - √3x + 1 ৰ এটা উৎপাদক x - 1 হয়, তেন্তে k ৰ মান নিৰ্ণয় কৰা।
[2 Marks]
48. Write the coordinates of the points where the line 3x + 4y = 12 intersects the x-axis and y-axis.
3x + 4y = 12 ৰেখাডালে x-অক্ষ আৰু y-অক্ষক ছেদ কৰা বিন্দুবোৰৰ স্থানাংক লিখা।
[2 Marks]
49. If A, B and C are three points on a line, and B lies between A and C, then prove that AB + BC = AC.
যদি এডাল ৰেখাত থকা A, B আৰু C তিনিটা বিন্দু হয়, আৰু B বিন্দুটো A আৰু C ৰ মাজত থাকে, তেন্তে প্ৰমাণ কৰা যে AB + BC = AC।
[2 Marks]
50. In a triangle, if the angles are in the ratio 1:2:3, find the measure of each angle.
এটা ত্ৰিভুজৰ কোণবোৰৰ অনুপাত 1:2:3 হ'লে, প্ৰতিটো কোণৰ মাপ উলিওৱা।
[2 Marks]
51. Find four different solutions of the equation x + 3y = 9.
x + 3y = 9 সমীকৰণটোৰ চাৰিটা বেলেগ বেলেগ সমাধান উলিওৱা।
[3 Marks]
52. Locate √5 on the number line.
সংখ্যা ৰেখাত √5 ৰ অৱস্থান নিৰ্ণয় কৰা।
[3 Marks]
53. Factorise: 3x² + 8x + 5 using splitting the middle term.
মধ্যপদ বিভাজন পদ্ধতিৰে উৎপাদকত বিশ্লেষণ কৰা: 3x² + 8x + 5।
[3 Marks]
54. Plot the points (x, y) given in the following table on the plane, choosing suitable units of distance on the axes:
| x | -3 | -1 | 0 | 2 | 4 |
|---|---|---|---|---|---|
| y | 7 | 5 | -2.5 | 4 | -3 |
উপযুক্ত একক লৈ ওপৰৰ তালিকাত দিয়া (x, y) বিন্দুবোৰ সমতলত স্থাপন কৰা।
[3 Marks]
55. In the given figure, if lines AB and CD intersect at O such that ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.
চিত্ৰত, AB আৰু CD ৰেখাই O বিন্দুত কটাকটি কৰে যাতে ∠AOC + ∠BOE = 70° আৰু ∠BOD = 40°। ∠BOE আৰু প্ৰবিদ্ধ ∠COE নিৰ্ণয় কৰা।
[3 Marks]
56. Show that the angles of an equilateral triangle are 60° each.
দেখুওৱা যে সমবাহু ত্ৰিভুজৰ প্ৰতিটো কোণৰ মাপ 60°।
[3 Marks]
57. Evaluate using suitable identities: (998)³.
উপযুক্ত অভেদ ব্যৱহাৰ কৰি মান নিৰ্ণয় কৰা: (998)³।
[3 Marks]
58. If a transversal intersects two parallel lines, then prove that each pair of alternate interior angles is equal.
এডাল তিৰ্যকে দুডাল সমান্তৰাল ৰেখাক ছেদ কৰিলে, প্ৰমাণ কৰা যে প্ৰতিযোৰ একান্তৰ অন্তৰ্কোণ সমান।
[4 Marks]
59. Represent √7.2 on the number line.
সংখ্যা ৰেখাত √7.2 ক উপস্থাপন কৰা।
[4 Marks]
60. State and prove the Angle Sum Property of a Triangle.
ত্ৰিভুজৰ কোণৰ সমষ্টিৰ ধৰ্মটো লিখা আৰু প্ৰমাণ কৰা।
[4 Marks]
61. In an isosceles triangle ABC with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that:
(i) OB = OC
(ii) AO bisects ∠A.
(i) OB = OC
(ii) AO bisects ∠A.
AB = AC হোৱা এটা সমদ্বিবাহু ত্ৰিভুজ ABC ৰ ∠B আৰু ∠C ৰ সমদ্বিখণ্ডক দুডালে পৰস্পৰ O বিন্দুত কটাকটি কৰে। A আৰু O সংযোগ কৰা হ'ল। দেখুওৱা যে:
(i) OB = OC
(ii) AO ৰেখাডালে ∠A ক সমদ্বিখণ্ডিত কৰে।
(i) OB = OC
(ii) AO ৰেখাডালে ∠A ক সমদ্বিখণ্ডিত কৰে।
This question paper prepared by © Sikshalens.in
Staring at a textbook for hours is rarely enough to guarantee top grades. The real secret to exam success lies in active recall and practical application, which is exactly where question papers come into play. Whether they are previous year board papers, mock tests, or chapter-wise practice sheets, question papers serve as a crucial bridge between passive learning and actual exam performance. By integrating these tools into a daily study routine, students can drastically improve their academic outcomes and overall confidence.
Decoding Exam Patterns and Trends
One of the most significant advantages of analyzing past question papers is the direct insight they provide into the examiner's mind. Textbooks contain vast amounts of information, but not all of it carries equal weight in a final assessment.
- Highlighting high-yield topics: Consistent practice quickly reveals which chapters, theories, or mathematical concepts are tested repeatedly year after year.
- Understanding question formats: Students become familiar with the specific phrasing of objective MCQs, short-answer questions, and long-form descriptive prompts, eliminating surprises on exam day.
- Mastering marking schemes: Reviewing question papers helps students understand exactly how marks are distributed, allowing them to prioritize their revision efforts efficiently.
Building Essential Time Management Skills
Knowing the right answer is only half the battle; writing it down clearly within a strict time limit is the other. Many students struggle with completing their exams, often leaving high-scoring sections unattempted because they spent too long on earlier questions. Solving question papers under timed, simulated exam conditions teaches students how to allocate their minutes wisely. They learn to quickly navigate past mental roadblocks, ensuring they secure marks for the concepts they know well before dedicating time to the most challenging problems.
Pinpointing Weaknesses and Reducing Anxiety
A practice question paper acts as an honest, unfiltered diagnostic tool. It strips away the illusion of competence that often happens when a student is simply reading through highlighted notes. When a student attempts a paper from scratch, they immediately see where their memory falters, which definitions are blurry, and which formulas require more practice. This targeted feedback allows them to spend their remaining study hours fixing actual knowledge gaps rather than reviewing material they have already mastered.
Furthermore, the familiarity bred by regular practice is the ultimate antidote to exam anxiety. Walking into an examination hall having already solved dozens of similar papers transforms a high-stress event into a routine, familiar task. The brain shifts from panic mode to execution mode, allowing the student to perform with maximum clarity and precision.