CM SHRI SCHOOL
Class X Mathematics (Standard) • Advanced Practice Worksheet
Subject: Mathematics (041) Max Marks: 100
Chapters: 5 Comprehensive Units Target: Board Exam Excellence (HOTS)
General Instructions: This worksheet comprises 50 curated problems split equally across 5 core chapters. It
blends conceptual Higher Order Thinking Skills (HOTS) questions with Standard Level CBSE Previous Years
Questions (PYQs) from 2020–2025. Show complete step-by-step methods for all answers.
Unit 1: Real Numbers
1. CBSE 2023 If two positive integers a and b are written as a = x3y2 and b = xy3, where x, y are prime
numbers, then find the value of HCF(a, b) × LCM(a, b).
2. HOTS Prove that √p + √q is an irrational number, where p and q are distinct prime numbers.
3. CBSE 2024 Explain clearly why the expression (7 × 11 × 13 × 15) + 15 is a composite number.
4. HOTS Show that the square of any positive integer cannot be of the form 5m + 2 or 5m + 3 for any integer
m.
5. CBSE 2020 Three bells toll together at intervals of 9, 12, and 15 minutes respectively. If they start tolling
simultaneously, after what minimum time will they next toll together?
6. HOTS Find the largest possible number that divides 2053 and 967 and leaves a remainder of 5 and 7
respectively.
7. CBSE 2025 Prove that √3 is an irrational number, and hence show that 5 - 2√3 is also an irrational number.
8. HOTS If the HCF of 408 and 1032 is expressible in the linear combination form 1032m - 408 × 5, determine
the value of m.
9. CBSE 2022 Find the smallest natural number by which √1200 should be multiplied so that the resulting
square root becomes a rational number.
10. HOTS Can two natural numbers have 16 as their HCF and 380 as their LCM? Give valid mathematical
reasons to support your answer.
CM Shri School • Mathematics Standard Page 1 of 5
, Unit 2: Linear Equations in Two Variables
11. CBSE 2024 Find the value(s) of k for which the pair of linear equations kx + 3y = k - 3 and 12x + ky = k has
infinitely many solutions.
12. HOTS Solve the following system of algebraic equations for variables x and y:
a/x - b/y = 0 and ab2/x + a2b/y = a2 + b2 (x ≠ 0, y ≠ 0)
13. CBSE 2023 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 original fraction.
14. HOTS Solve the given pair of simultaneous equations using an efficient method:
152x - 378y = -74
-378x + 152y = -604
15. CBSE 2020 Determine graphically the coordinates of the vertices of the triangle, the equations of whose
sides are given by 2y - x = 8, 5y - x = 14, and y - 2x = 1.
16. HOTS The total population of a village is 5000. If in a given year, the number of males increases by 5% and
that of females increases by 3%, the population becomes 5202. Find the initial number of males and females
in the village separately.
17. CBSE 2022 For what value of the parameter k will the following pair of linear equations yield no solution?
3x + y = 1 and (2k - 1)x + (k - 1)y = 2k + 1
18. HOTS A man travels 300 km to his home partly by train and partly by bus. He takes exactly 4 hours if he
travels 60 km by train and the remaining distance by bus. If he travels 100 km by train and the rest by bus, he
takes 10 minutes longer. Determine the individual speeds of the train and the bus.
19. CBSE 2023 Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, their ratio
becomes 4 : 5. Find the value of both numbers.
20. HOTS If the system of linear equations 2x + 3y = 7 and 2ax + (a + b)y = 28 possesses infinitely many
solutions, establish the specific algebraic relationship between a and b.
Unit 3: Arithmetic Progressions
21. CBSE 2023 If the sum of the first n terms of an AP is given by Sn = 3n2 + 5n, derive the formula for its n-th
term (an). Hence, compute its 15th term.
CM Shri School • Mathematics Standard Page 2 of 5
Class X Mathematics (Standard) • Advanced Practice Worksheet
Subject: Mathematics (041) Max Marks: 100
Chapters: 5 Comprehensive Units Target: Board Exam Excellence (HOTS)
General Instructions: This worksheet comprises 50 curated problems split equally across 5 core chapters. It
blends conceptual Higher Order Thinking Skills (HOTS) questions with Standard Level CBSE Previous Years
Questions (PYQs) from 2020–2025. Show complete step-by-step methods for all answers.
Unit 1: Real Numbers
1. CBSE 2023 If two positive integers a and b are written as a = x3y2 and b = xy3, where x, y are prime
numbers, then find the value of HCF(a, b) × LCM(a, b).
2. HOTS Prove that √p + √q is an irrational number, where p and q are distinct prime numbers.
3. CBSE 2024 Explain clearly why the expression (7 × 11 × 13 × 15) + 15 is a composite number.
4. HOTS Show that the square of any positive integer cannot be of the form 5m + 2 or 5m + 3 for any integer
m.
5. CBSE 2020 Three bells toll together at intervals of 9, 12, and 15 minutes respectively. If they start tolling
simultaneously, after what minimum time will they next toll together?
6. HOTS Find the largest possible number that divides 2053 and 967 and leaves a remainder of 5 and 7
respectively.
7. CBSE 2025 Prove that √3 is an irrational number, and hence show that 5 - 2√3 is also an irrational number.
8. HOTS If the HCF of 408 and 1032 is expressible in the linear combination form 1032m - 408 × 5, determine
the value of m.
9. CBSE 2022 Find the smallest natural number by which √1200 should be multiplied so that the resulting
square root becomes a rational number.
10. HOTS Can two natural numbers have 16 as their HCF and 380 as their LCM? Give valid mathematical
reasons to support your answer.
CM Shri School • Mathematics Standard Page 1 of 5
, Unit 2: Linear Equations in Two Variables
11. CBSE 2024 Find the value(s) of k for which the pair of linear equations kx + 3y = k - 3 and 12x + ky = k has
infinitely many solutions.
12. HOTS Solve the following system of algebraic equations for variables x and y:
a/x - b/y = 0 and ab2/x + a2b/y = a2 + b2 (x ≠ 0, y ≠ 0)
13. CBSE 2023 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 original fraction.
14. HOTS Solve the given pair of simultaneous equations using an efficient method:
152x - 378y = -74
-378x + 152y = -604
15. CBSE 2020 Determine graphically the coordinates of the vertices of the triangle, the equations of whose
sides are given by 2y - x = 8, 5y - x = 14, and y - 2x = 1.
16. HOTS The total population of a village is 5000. If in a given year, the number of males increases by 5% and
that of females increases by 3%, the population becomes 5202. Find the initial number of males and females
in the village separately.
17. CBSE 2022 For what value of the parameter k will the following pair of linear equations yield no solution?
3x + y = 1 and (2k - 1)x + (k - 1)y = 2k + 1
18. HOTS A man travels 300 km to his home partly by train and partly by bus. He takes exactly 4 hours if he
travels 60 km by train and the remaining distance by bus. If he travels 100 km by train and the rest by bus, he
takes 10 minutes longer. Determine the individual speeds of the train and the bus.
19. CBSE 2023 Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, their ratio
becomes 4 : 5. Find the value of both numbers.
20. HOTS If the system of linear equations 2x + 3y = 7 and 2ax + (a + b)y = 28 possesses infinitely many
solutions, establish the specific algebraic relationship between a and b.
Unit 3: Arithmetic Progressions
21. CBSE 2023 If the sum of the first n terms of an AP is given by Sn = 3n2 + 5n, derive the formula for its n-th
term (an). Hence, compute its 15th term.
CM Shri School • Mathematics Standard Page 2 of 5
