Class 10 Mathematics - Questions Bank 🔥
Chapters 1-5 | Board Exam Level | Previous Year Questions | Scoring Practice
⚠️ DIFFICULTY WARNING
This question bank contains ONLY HARD QUESTIONS - the type that separates 90%+ scorers from average students!
What's Inside:
- ✅ Previous Year Board Questions (PYQs)
- ✅ Sample Paper Questions (SPQs)
- ✅ High-Order Thinking Questions
- ✅ 5-Mark Questions Only
- ✅ NO Easy/Medium Questions
Why These Questions? These are the questions that:
- Appear in Section C & D of board papers
- Require deep conceptual understanding
- Combine multiple concepts
- Test your problem-solving skills
- Decide whether you score 35/40 or 25/40
How to Use:
- ⚠️ Attempt ONLY if you've studied the chapter thoroughly
- ⚠️ Time yourself: 7-8 minutes per question
- ⚠️ If stuck, check our detailed notes at ncert-nation.blogspot.com
- ⚠️ Review solutions carefully - understand the approach
- ⚠️ Retry wrong questions after 2 days
📚 CHAPTER 1: REAL NUMBERS (20 Questions)
Q1. [PYQ 2023] Prove that √2 + √3 is irrational. Hence show that 5 - √2 - √3 is also irrational.
Q2. [SPQ] Use Euclid's division lemma to show that cube of any positive integer is of the form 9m, 9m+1, or 9m+8 for some integer m.
Q3. [PYQ 2022] The HCF of two numbers is 16 and their product is 3072. If one number is 64, find the other number. Also find their LCM.
Q4. [SPQ] Prove that one and only one out of n, n+2, and n+4 is divisible by 3, where n is any positive integer.
Q5. [PYQ 2024] Find the largest number which divides 245 and 1029 leaving remainder 5 in each case. Verify your answer.
Q6. [SPQ] Show that the square of any positive integer cannot be of the form 5m + 2 or 5m + 3 for any integer m.
Q7. [PYQ 2023] Prove that if x and y are both odd positive integers, then x² + y² is even but not divisible by 4.
Q8. [SPQ] Three bells toll at intervals of 9, 12, and 15 minutes respectively. If they start tolling together, after what time will they toll together again? If they toll together at 11:00 AM, at what time will they toll together for the 5th time?
Q9. [PYQ 2022] Explain why 7×11×13+13 and 7×6×5×4×3×2×1+5 are composite numbers.
Q10. [SPQ] Prove that √5 is irrational. Hence prove that 2√5 - 3 is also irrational.
Q11. [PYQ 2024] Show that any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer. Hence prove that square of any positive integer is of the form 3m or 3m+1.
Q12. [SPQ] Find HCF and LCM of 306 and 657 by prime factorization method. Verify that HCF × LCM = Product of two numbers.
Q13. [PYQ 2023] On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm, and 45 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
Q14. [SPQ] Prove that the product of three consecutive positive integers is divisible by 6.
Q15. [PYQ 2022] Express 0.6̄3̄4̄ (0.634634634...) as a rational number in the form p/q where p and q are integers and q ≠ 0.
Q16. [SPQ] Show that every positive even integer is of the form 2q, and every positive odd integer is of the form 2q+1, where q is some integer. Using this, prove that the square of any positive integer is of the form 4m or 4m+1 for some integer m.
Q17. [PYQ 2024] Two tankers contain 850 litres and 680 litres of petrol. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times.
Q18. [SPQ] A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number of barfis, and they take up the least area of the tray. What is the number of barfis that can be placed in each stack for this purpose?
Q19. [PYQ 2023] Prove that if p is a prime number and p divides a², then p divides a, where a is a positive integer. Using this theorem, prove that √7 is irrational.
Q20. [SPQ] Find the largest number that will divide 398, 436, and 542 leaving remainders 7, 11, and 15 respectively.
📚 CHAPTER 2: POLYNOMIALS (20 Questions)
Q21. [PYQ 2023] If α and β are zeros of polynomial x² - 5x + k such that α - β = 1, find the value of k.
Q22. [SPQ] Find a quadratic polynomial whose zeros are 2+√3 and 2-√3. Verify the relationship between zeros and coefficients.
Q23. [PYQ 2024] If α and β are zeros of polynomial 6x² + x - 2, find the value of: (α/β) + (β/α)
Q24. [SPQ] If one zero of polynomial p(x) = (k²+4)x² + 13x + 4k is reciprocal of the other, find k.
Q25. [PYQ 2022] Divide 3x⁴ + 5x³ - 7x² + 2x + 2 by x² + 3x + 1 and verify the division algorithm.
Q26. [SPQ] If α and β are zeros of polynomial 2x² - 4x + 5, find the value of: (i) α² + β² (ii) α³ + β³ (iii) 1/α + 1/β
Q27. [PYQ 2023] Find all zeros of polynomial 2x⁴ - 9x³ + 5x² + 3x - 1 if two of its zeros are (2 + √3) and (2 - √3).
Q28. [SPQ] If α and β are zeros of polynomial x² - p(x+1) - c, show that (α+1)(β+1) = 1 - c.
Q29. [PYQ 2024] If α, β are zeros of polynomial x² + x - 2, find a polynomial whose zeros are 2α + 1 and 2β + 1.
Q30. [SPQ] Divide polynomial x⁴ - 3x² + 4x + 5 by x² + 1 - x. Find the quotient and remainder. Verify division algorithm.
Q31. [PYQ 2023] If the sum of zeros of polynomial 3x² - kx + 6 is 3, find k. Also find the product of its zeros.
Q32. [SPQ] Find a cubic polynomial whose zeros are 2, 3, and 4. Verify the relationship between zeros and coefficients.
Q33. [PYQ 2022] If α and β are zeros of polynomial f(x) = x² - 5x + k such that α² + β² = 19, find the value of k.
Q34. [SPQ] If α and β are zeros of polynomial 4x² + 4x + 1, form a quadratic polynomial whose zeros are 2α and 2β.
Q35. [PYQ 2024] Find all zeros of polynomial p(x) = x³ - 5x² - 16x + 80, if its two zeros are equal in magnitude but opposite in sign.
Q36. [SPQ] If α and β are zeros of polynomial x² - x - 6, find the value of (1/α) + (1/β) - 2αβ.
Q37. [PYQ 2023] Check whether polynomial g(x) = x³ - 3x² + 4x - 12 is a multiple of x - 3. If yes, find all its zeros.
Q38. [SPQ] If the product of zeros of polynomial ax² - 6x - 6 is 4, find a. Also find the sum of zeros.
Q39. [PYQ 2022] Verify division algorithm for polynomials by dividing f(x) = 6x³ + 11x² - 39x - 65 by g(x) = x² + x - 1.
Q40. [SPQ] If α, β, γ are zeros of cubic polynomial ax³ + bx² + cx + d, prove that: (i) α + β + γ = -b/a (ii) αβ + βγ + γα = c/a (iii) αβγ = -d/a
📚 CHAPTER 3: PAIR OF LINEAR EQUATIONS IN TWO VARIABLES (20 Questions)
Q41. [PYQ 2024] Five years ago, a man was seven times as old as his son. Five years hence, the man will be three times as old as his son. Find their present ages.
Q42. [SPQ] A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.
Q43. [PYQ 2023] The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits differ by 2, find the number. How many such numbers are there?
Q44. [SPQ] A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, it would have taken 3 hours more. Find the original speed of the train.
Q45. [PYQ 2022] The area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.
Q46. [SPQ] Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?
Q47. [PYQ 2024] A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. Find the fraction.
Q48. [SPQ] For what value of k will the equations x + 2y = 5 and 3x + ky + 15 = 0 have: (i) A unique solution (ii) No solution (iii) Infinitely many solutions
Q49. [PYQ 2023] The difference between two numbers is 26 and one number is three times the other. Find the numbers. If one of these numbers is represented by x, find the two equations in one variable and their solution.
Q50. [SPQ] A lending library has fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹27 for a book kept for seven days, while Susy paid ₹21 for a book kept for five days. Find the fixed charge and the charge for each extra day.
Q51. [PYQ 2022] A man earns ₹600 per month more than his wife. One-tenth of the man's salary and one-sixth of the wife's salary amount to ₹1500, which is saved every month. Find their incomes.
Q52. [SPQ] The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.
Q53. [PYQ 2024] A shopkeeper sells a saree at 8% profit and a sweater at 10% discount, thereby getting a sum of ₹1008. If she had sold the saree at 10% profit and the sweater at 8% discount, she would have got ₹1028. Find the cost price of the saree and the list price of the sweater.
Q54. [SPQ] Solve for x and y: (a-b)x + (a+b)y = a² - 2ab - b² and (a+b)(x+y) = a² + b²
Q55. [PYQ 2023] In a competitive examination, one mark is awarded for each correct answer while 1/2 mark is deducted for every wrong answer. Jayanti answered 120 questions and got 90 marks. How many questions did she answer correctly?
Q56. [SPQ] A two-digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. Find the number.
Q57. [PYQ 2022] The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
Q58. [SPQ] Points A and B are 70 km apart on a highway. A car starts from A and another car starts from B at the same time. If they travel in the same direction, they meet in 7 hours, but if they travel towards each other, they meet in 1 hour. Find the speed of each car.
Q59. [PYQ 2024] A man sold a chair and a table together for ₹1520, thereby making a profit of 25% on the chair and 10% on the table. By selling them together for ₹1535, he would have made a profit of 10% on the chair and 25% on the table. Find the cost price of each.
Q60. [SPQ] Solve: x/(a-b) + y/(a+b) = 2a/(a²-b²) and x/(a+b) - y/(a-b) = 4ab/(a²-b²)
📚 CHAPTER 4: QUADRATIC EQUATIONS (20 Questions)
Q61. [PYQ 2024] Find the nature of roots of quadratic equation 2x² - 4x + 3 = 0. Also find the roots if they exist.
Q62. [SPQ] Find the values of k for which the quadratic equation (3k+1)x² + 2(k+1)x + 1 = 0 has equal roots. Also find the roots.
Q63. [PYQ 2023] A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Q64. [SPQ] The sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares.
Q65. [PYQ 2022] If the roots of the equation (a²+b²)x² - 2(ac+bd)x + (c²+d²) = 0 are equal, prove that a/b = c/d.
Q66. [SPQ] A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square metres more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 m. Find the length and breadth of the rectangular park.
Q67. [PYQ 2024] Find the value of p for which the roots of the equation px(x-2) + 6 = 0 are equal.
Q68. [SPQ] Two water taps together can fill a tank in 9 3/8 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Q69. [PYQ 2023] A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
Q70. [SPQ] If α and β are roots of equation x² - px + q = 0, find the value of: (i) α² + β² (ii) α³ + β³ (iii) α⁴ + β⁴
Q71. [PYQ 2022] The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
Q72. [SPQ] For what value of k, does the equation 12x² + kx + 5 = 0 have roots in the ratio 2:3?
Q73. [PYQ 2024] The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
Q74. [SPQ] A piece of cloth costs ₹200. If the piece was 5 m longer and each metre of cloth costs ₹2 less, the cost of the piece would have remained unchanged. How long is the piece and what is its original rate per metre?
Q75. [PYQ 2023] Find the values of k for which the equation x² + 5kx + 16 = 0 has no real roots.
Q76. [SPQ] An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.
Q77. [PYQ 2022] If the equation (1+m²)x² + 2mcx + (c²-a²) = 0 has equal roots, prove that c² = a²(1+m²).
Q78. [SPQ] In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/h and the time of flight increased by 30 minutes. Find the original duration of the flight.
Q79. [PYQ 2024] A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was ₹750. Find the number of toys produced on that day.
Q80. [SPQ] If the roots of the equation (b-c)x² + (c-a)x + (a-b) = 0 are equal, prove that 2b = a + c.
📚 CHAPTER 5: ARITHMETIC PROGRESSIONS (20 Questions)
Q81. [PYQ 2024] How many terms of the AP: 9, 17, 25,... must be taken to give a sum of 636?
Q82. [SPQ] The sum of the third and seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.
Q83. [PYQ 2023] Find the sum of all three-digit numbers which are divisible by 7.
Q84. [SPQ] The sum of first n terms of an AP is 3n² + 5n. Find the AP and its 20th term.
Q85. [PYQ 2022] If the pth term of an AP is q and the qth term is p, prove that its nth term is (p+q-n).
Q86. [SPQ] The sum of first p terms of an AP is q and the sum of first q terms is p. Find the sum of first (p+q) terms.
Q87. [PYQ 2024] How many terms of the AP: 24, 21, 18,... must be taken so that their sum is 78? Explain why there are two answers.
Q88. [SPQ] In an AP, if the sum of first m terms is n and the sum of first n terms is m, find the sum of first (m+n) terms.
Q89. [PYQ 2023] The ratio of the 11th term to the 18th term of an AP is 2:3. Find the ratio of the 5th term to the 21st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms.
Q90. [SPQ] Find the sum of all integers between 100 and 1000 which are divisible by 9.
Q91. [PYQ 2022] The sum of first six terms of an AP is 42. The ratio of its 10th term to its 30th term is 1:3. Calculate the first and thirteenth term of the AP.
Q92. [SPQ] The 4th term of an AP is zero. Prove that the 25th term of the AP is three times its 11th term.
Q93. [PYQ 2024] In an AP, if Sₙ = n²p and Sₘ = m²p, where Sₙ denotes the sum of first n terms and Sₘ denotes sum of first m terms, prove that Sₚ = p³.
Q94. [SPQ] The sum of three numbers in AP is 12 and the sum of their cubes is 288. Find the numbers.
Q95. [PYQ 2023] If the sum of first p terms of an AP is ap² + bp, find the common difference.
Q96. [SPQ] The sum of n terms of two APs are in the ratio (3n+8):(7n+15). Find the ratio of their 12th terms.
Q97. [PYQ 2022] Find the sum of all odd integers between 2 and 100 divisible by 3.
Q98. [SPQ] The 17th term of an AP exceeds its 10th term by 7. Find the common difference.
Q99. [PYQ 2024] Two APs have the same common difference. The first term of one of these is -1 and that of the other is -8. The difference between their 4th terms is same as the difference between their 8th terms, and is same as the difference between any two corresponding terms. Why?
Q100. [SPQ] In an AP, the sum of first ten terms is -150 and the sum of its next ten terms is -550. Find the AP.
🎯 HOW TO TACKLE THESE QUESTIONS
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STEP 1: Prerequisite Check Before attempting these:
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- ⚠️ If stuck after 3 minutes, mark it and move on
- 🔄 Return to marked questions after completing all
STEP 3: Self-Evaluation
- ✅ Correct: Understand WHY the approach worked
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- Solutions readily available = Students copy without understanding
- Our goal: Make you THINK and LEARN
- Drive to Quality Content
- Each question's solution is in our detailed notes
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- Struggling with problem = Brain working hard
- This struggle = Real learning happens
- Instant solutions = No real learning
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- In board exam, no solutions available!
- Practice attempting without help
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🎯 SCORING STRATEGY
How These Questions Help You Score 95%+:
Level 1: Basic Understanding (60-70%)
- Solve NCERT + Easy questions
- Get basic concepts clear
Level 2: Good Score (70-85%)
- Solve NCERT + Medium questions
- Practice sample papers
Level 3: Excellent Score (85-95%)
- Solve NCERT + Medium + These HARD questions
- Master all question types
Level 4: Outstanding (95%+)
- Solve ALL question types
- Can solve ANY variation
- Complete confidence
Where You Are Now: You're attempting LEVEL 3-4 questions! This means you're aiming for 90%+ score! 🎯
📊 TRACK YOUR PROGRESS
Create Your Score Sheet:
| Chapter | Attempted | Correct | Wrong | To Retry | Date |
|---|---|---|---|---|---|
| Ch 1 (20Q) | __/20 | __/20 | __/20 | Q__ | //__ |
| Ch 2 (20Q) | __/20 | __/20 | __/20 | Q__ | //__ |
| Ch 3 (20Q) | __/20 | __/20 | __/20 | Q__ | //__ |
| Ch 4 (20Q) | __/20 | __/20 | __/20 | Q__ | //__ |
| Ch 5 (20Q) | __/20 | __/20 | __/20 | Q__ | //__ |
| TOTAL | __/100 | __/100 | __/100 | -- | -- |
Target Score: 80/100 or above = Board Exam Ready! 🏆
💬 COMMUNITY CHALLENGE
Share Your Score!
- Comment below with your score
- Which chapter was toughest?
- Which questions stumped you?
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Top Performers: We'll feature students scoring 90+/100 in our next post!
Study Groups: Connect with fellow students in comments!
🎓 FINAL MOTIVATION
These 100 questions represent the TOUGHEST challenges in Class 10 Maths (Chapters 1-5).
If you can solve 80+ of these:
- ✨ Your concepts are crystal clear
- ✨ You're ready for ANY board exam question
- ✨ 90%+ score is guaranteed
- ✨ You've mastered problem-solving
Remember:
- 🌟 Every tough question you solve = +1 confidence level
- 🌟 Every concept you master = +5 marks secured
- 🌟 Every hour you practice = Closer to your dream score
You've got this! 💪
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✅ YOUR ACTION PLAN
This Week:
- Attempt Chapter 1 questions (20Q)
- Mark difficult questions
- Study concepts from our notes
- Retry marked questions
Next Week:
- Attempt Chapters 2-3 questions (40Q)
- Improve time per question
- Track your accuracy
- Identify weak areas
Following Week:
- Complete Chapters 4-5 questions (40Q)
- Solve all 100 questions
- Calculate final score
- Celebrate if 80+! 🎉
🏆 CHALLENGE ACCEPTED?
This question bank is for SERIOUS students who:
- Want to score 90%+ in boards
- Are willing to work hard
- Don't give up easily
- Aim for excellence
Are you one of them?
Then start solving NOW! ⏰
Last Updated: November 2024 Question Count: 100 Hard Questions Chapters: 1-5 (Real Numbers to AP) Difficulty: 🔥 HARD Level Only Source: PYQs + SPQs + Board Pattern
Created with 💚 by NCERT Nation Team
