10th CBSE MATHS SAMPLE PAPER 2025-26

Sample Question Paper (SET-1): Mathematics (CBSE Class X)

Maximum Marks: 80 | Time: 3 Hours | Total Questions: 38

General Instructions:

1. The question paper contains 38 questions, divided into five sections: MCQ, VSA, SA, LA, and Competency-Based.
2. All questions are compulsory. Internal choices are provided in some questions.
3. Use of calculators is not permitted.
4. Write answers clearly and legibly, ensuring proper labeling of question numbers.
5. Show all necessary calculations and steps for SA and LA questions.

---

Multiple Choice Questions (1 Mark Each, Total: 20 Marks)

1. The HCF of 72 and 126 is:
   a) 6
   b) 18
   c) 12
   d) 24

2. The roots of the quadratic equation ( x^2 - 8x + 15 = 0 ) are:
   a) 3, 5
   b) -3, -5
   c) 2, 6
   d) -2, -6

3. Assertion-Reason:
   Assertion (A): The sum of the angles of a polygon with ( n ) sides is ( (n-2) \cdot 180^\circ ).
   Reason (R): A polygon can be divided into ( (n-2) ) triangles.
   a) Both A and R are true, and R is the correct explanation of A.
   b) Both A and R are true, but R is not the correct explanation of A.
   c) A is true, but R is false.
   d) A is false, but R is true.

4. The value of ( \cos 30^\circ \cdot \sin 60^\circ ) is:
   a) ( \frac{\sqrt{3}}{4} )
   b) ( \frac{3}{4} )
   c) ( \frac{\sqrt{3}}{2} )
   d) ( \frac{1}{4} )

5. The probability of drawing a red card from a standard deck of 52 cards is:
   a) ( \frac{1}{2} )
   b) ( \frac{1}{4} )
   c) ( \frac{1}{3} )
   d) ( \frac{2}{3} )

6. The distance between the points (4, 3) and (-2, -1) is:
   a) ( \sqrt{52} ) units
   b) ( \sqrt{40} ) units
   c) ( \sqrt{20} ) units
   d) 10 units

7. The 12th term of the AP: 5, 8, 11, 14, … is:
   a) 38
   b) 41
   c) 44
   d) 35

8. If ( \tan \theta = \frac{8}{15} ), then ( \sin \theta ) is:
   a) ( \frac{8}{17} )
   b) ( \frac{15}{17} )
   c) ( \frac{8}{\sqrt{17}} )
   d) ( \frac{15}{\sqrt{17}} )

9. The median of the data: 6, 8, 10, 12, 14 is:
   a) 8
   b) 10
   c) 12
   d) 14

10. The radius of a circle with area 154 cm² is:
    a) 7 cm
    b) 14 cm
    c) 11 cm
    d) 22 cm

11. The sum of the first 15 terms of an AP with first term 2 and common difference 3 is:
    a) 360
    b) 330
    c) 315
    d) 345

12. The value of ( \sec 45^\circ ) is:
    a) ( \sqrt{2} )
    b) ( \frac{\sqrt{2}}{2} )
    c) 1
    d) 2

13. The slope of the line joining the points (1, 2) and (4, 8) is:
    a) 2
    b) 3
    c) 1
    d) -2

14. The probability of getting an even number when rolling a die is:
    a) ( \frac{1}{2} )
    b) ( \frac{1}{3} )
    c) ( \frac{2}{3} )
    d) ( \frac{1}{6} )

15. The area of a sector with radius 10 cm and sector angle ( 90^\circ ) is:
    a) 25π cm²
    b) 50π cm²
    c) 100π cm²
    d) 10π cm²

16. The coordinates of the point dividing the line segment joining (2, 3) and (8, 9) in the ratio 1:1 are:
    a) (5, 6)
    b) (4, 5)
    c) (6, 7)
    d) (3, 4)

17. If the discriminant of a quadratic equation is negative, the roots are:
    a) Real and equal
    b) Real and distinct
    c) Not real
    d) Positive

18. The 5th term of an AP with first term 10 and common difference -2 is:
    a) 2
    b) 4
    c) 6
    d) 8

19. The value of ( \sin^2 30^\circ + \cos^2 60^\circ ) is:
    a) 1
    b) ( \frac{1}{2} )
    c) ( \frac{3}{4} )
    d) ( \frac{1}{4} )

20. The circumference of a circle with diameter 28 cm is:
    a) 88 cm
    b) 44 cm
    c) 176 cm
    d) 56 cm

Very Short Answer Questions (2 Marks Each, Total: 10 Marks)

21. Find the value of ( k ) for which the quadratic equation ( x^2 + kx + 9 = 0 ) has equal roots.
22. If the angle of elevation of a balloon from a point is ( 60^\circ ) and the distance to the base is 30 m, find the height of the balloon.
23. Find the coordinates of the midpoint of the line segment joining (-3, 4) and (5, -2).
24. Calculate the mode of the data: 3, 4, 4, 5, 5, 5, 6, 7.
25. Find the ratio in which the point (2, 3) divides the line segment joining (0, 1) and (4, 5).

Short Answer Questions (3 Marks Each, Total: 18 Marks)

26. Prove that ( \sqrt{11} ) is irrational.
27. Find the area of a triangle with vertices (1, 2), (4, 6), and (-2, 3).
28. Solve the pair of linear equations: ( 3x + 2y = 11 ) and ( x - y = 3 ).
29. A bag contains 3 red, 4 blue, and 5 green balls. Find the probability of drawing a blue ball.
30. Find the length of the arc of a circle with radius 21 cm and sector angle ( 60^\circ ).
31. If ( \sin \theta = \frac{3}{5} ), find the values of ( \cos \theta ) and ( \tan \theta ).

Long Answer Questions (5 Marks Each, Total: 20 Marks)

32. Derive the formula for the sum of the first ( n ) terms of a geometric progression.
33. Prove that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
34. OR Prove that the area of a parallelogram is the product of its base and height using coordinate geometry.
35. Solve the following system of equations graphically: ( 2x + y = 6 ) and ( x - 2y = -2 ). Verify the solution algebraically.

Competency-Based/Case Study-Based Questions (4 Marks Each, Total: 12 Marks)

36. A store sells two types of notebooks: plain and ruled. A plain notebook costs ₹20 and requires 1 sheet of paper, while a ruled notebook costs ₹30 and requires 2 sheets of paper. The store uses 50 sheets of paper and earns ₹650 in a day. How many plain and ruled notebooks were sold? (Solve using a system of equations and verify the solution.)
37. A ladder 10 m long is placed against a vertical wall. The foot of the ladder is 6 m away from the base of the wall.
    a) Find the height at which the ladder touches the wall. (2 marks)
    b) If the foot of the ladder is moved 2 m closer to the wall, find the new height. (2 marks)
38. A school organizes a charity event where students sell pens and pencils. A pen costs ₹10, and a pencil costs ₹5. A student sells 15 items and earns ₹110.
    a) How many pens and pencils were sold? (Solve using a system of equations.) (3 marks)
    b) If the student donates all earnings to charity, what value is reflected? (1 mark)

---

End of Question Paper

Sample Question Paper (SET-2): Mathematics (CBSE Class X)

Maximum Marks: 80 | Time: 3 Hours | Total Questions: 38

General Instructions:

1. The question paper contains 38 questions, divided into five sections: MCQ, VSA, SA, LA, and Competency-Based.
2. All questions are compulsory. Internal choices are provided in some questions.
3. Use of calculators is not permitted.
4. Write answers clearly and legibly, ensuring proper labeling of question numbers.
5. Show all necessary calculations and steps for SA and LA questions.

---

Multiple Choice Questions (1 Mark Each, Total: 20 Marks)

1. The LCM of 18 and 24 is:
   a) 36
   b) 72
   c) 48
   d) 12

2. The roots of the quadratic equation ( 2x^2 - 10x + 12 = 0 ) are:
   a) 2, 3
   b) -2, -3
   c) 3, 4
   d) -3, -4

3. Assertion-Reason:
   Assertion (A): The sum of the first ( n ) terms of an AP is ( \frac{n}{2} [2a + (n-1)d] ).
   Reason (R): The ( n )-th term of an AP is given by ( a + (n-1)d ).
   a) Both A and R are true, and R is the correct explanation of A.
   b) Both A and R are true, but R is not the correct explanation of A.
   c) A is true, but R is false.
   d) A is false, but R is true.

4. The value of ( \sin 45^\circ \cdot \cos 45^\circ ) is:
   a) ( \frac{1}{2} )
   b) ( \frac{\sqrt{2}}{2} )
   c) ( \frac{1}{\sqrt{2}} )
   d) 1

5. The probability of drawing a face card from a standard deck of 52 cards is:
   a) ( \frac{3}{13} )
   b) ( \frac{1}{4} )
   c) ( \frac{1}{13} )
   d) ( \frac{2}{13} )

6. The distance between the points (0, 5) and (5, 0) is:
   a) 5 units
   b) ( \sqrt{50} ) units
   c) ( \sqrt{25} ) units
   d) 10 units

7. The 7th term of the AP: 4, 9, 14, 19, … is:
   a) 34
   b) 29
   c) 39
   d) 24

8. If ( \cos \theta = \frac{4}{5} ), then ( \tan \theta ) is:
   a) ( \frac{3}{4} )
   b) ( \frac{4}{3} )
   c) ( \frac{5}{3} )
   d) ( \frac{3}{5} )

9. The mean of the data: 2, 4, 6, 8, 10 is:
   a) 6
   b) 5
   c) 7
   d) 4

10. The area of a circle with radius 10 cm is:
    a) 100π cm²
    b) 50π cm²
    c) 25π cm²
    d) 200π cm²

11. The sum of the first 10 terms of an AP with first term 1 and common difference 2 is:
    a) 100
    b) 110
    c) 90
    d) 120

12. The value of ( \cot 30^\circ ) is:
    a) ( \sqrt{3} )
    b) ( \frac{1}{\sqrt{3}} )
    c) 1
    d) 2

13. The slope of the line joining the points (-2, 3) and (4, -1) is:
    a) ( -\frac{2}{3} )
    b) ( \frac{2}{3} )
    c) ( -\frac{3}{2} )
    d) ( \frac{3}{2} )

14. The probability of getting a number less than 3 when rolling a die is:
    a) ( \frac{1}{3} )
    b) ( \frac{1}{2} )
    c) ( \frac{1}{6} )
    d) ( \frac{2}{3} )

15. The area of a sector with radius 7 cm and sector angle ( 120^\circ ) is:
    a) ( \frac{49\pi}{3} ) cm²
    b) ( \frac{49\pi}{6} ) cm²
    c) ( 14\pi ) cm²
    d) ( 7\pi ) cm²

16. The coordinates of the point dividing the line segment joining (3, 4) and (7, 8) in the ratio 1:3 are:
    a) (4, 5)
    b) (5, 6)
    c) (6, 7)
    d) (4, 6)

17. If the discriminant of a quadratic equation is zero, the roots are:
    a) Real and distinct
    b) Real and equal
    c) Not real
    d) Negative

18. The 6th term of an AP with first term 7 and common difference -3 is:
    a) -8
    b) -5
    c) -2
    d) 1

19. The value of ( \sin^2 45^\circ + \cos^2 45^\circ ) is:
    a) 1
    b) ( \frac{1}{2} )
    c) ( \sqrt{2} )
    d) 2

20. The circumference of a circle with radius 14 cm is:
    a) 44 cm
    b) 88 cm
    c) 28 cm
    d) 56 cm

Very Short Answer Questions (2 Marks Each, Total: 10 Marks)

21. Find the value of ( k ) for which the quadratic equation ( 3x^2 - kx + 12 = 0 ) has real roots.
22. If the angle of elevation of a tree from a point 20 m away is ( 45^\circ ), find the height of the tree.
23. Find the coordinates of the centroid of a triangle with vertices (1, 1), (3, 5), and (5, 1).
24. Calculate the median of the data: 2, 4, 4, 5, 6, 7, 8.
25. Find the ratio in which the point (1, 2) divides the line segment joining (-1, 1) and (3, 4).

Short Answer Questions (3 Marks Each, Total: 18 Marks)

26. Prove that ( \sqrt{13} ) is irrational.
27. Find the area of a triangle with vertices (-1, 2), (3, 6), and (5, -2).
28. Solve the pair of linear equations: ( 4x + y = 14 ) and ( 2x - 3y = -2 ).
29. A box contains 6 red and 4 blue balls. Find the probability of drawing a red ball.
30. Find the area of a sector of a circle with radius 14 cm and sector angle ( 90^\circ ).
31. If ( \cos \theta = \frac{5}{13} ), find the values of ( \sin \theta ) and ( \cot \theta ).

Long Answer Questions (5 Marks Each, Total: 20 Marks)

32. Derive the formula for the area of a triangle using coordinate geometry.
33. Prove that the tangents drawn from an external point to a circle are equal in length.
34. OR Prove that the opposite sides of a quadrilateral circumscribing a circle are equal using coordinate geometry.
35. Solve the following system of equations graphically: ( x + 2y = 5 ) and ( 3x - y = 4 ). Verify the solution algebraically.

Competency-Based/Case Study-Based Questions (4 Marks Each, Total: 12 Marks)

36. A shop sells two types of calculators: basic and scientific. A basic calculator costs ₹100 and requires 1 battery, while a scientific calculator costs ₹200 and requires 2 batteries. The shop uses 15 batteries and earns ₹1900 in a day. How many basic and scientific calculators were sold? (Solve using a system of equations and verify the solution.)
37. A pole 12 m high casts a shadow of 8 m on the ground.
    a) Find the angle of elevation of the top of the pole from the end of the shadow. (2 marks)
    b) If the shadow length increases to 10 m, find the new angle of elevation. (2 marks)
38. A school organizes a fundraiser where students sell notebooks and pens. A notebook costs ₹25, and a pen costs ₹10. A student sells 20 items and earns ₹350.
    a) How many notebooks and pens were sold? (Solve using a system of equations.) (3 marks)
    b) What value is reflected by organizing such a fundraiser? (1 mark)

---

End of Question Paper

Sample Question Paper (SET-3): Mathematics (CBSE Class X)

Maximum Marks: 80 | Time: 3 Hours | Total Questions: 38

General Instructions:

1. The question paper contains 38 questions, divided into five sections: MCQ, VSA, SA, LA, and Competency-Based.
2. All questions are compulsory. Internal choices are provided in some questions.
3. Use of calculators is not permitted.
4. Write answers clearly and legibly, ensuring proper labeling of question numbers.
5. Show all necessary calculations and steps for SA and LA questions.

---

Multiple Choice Questions (1 Mark Each, Total: 20 Marks)

1. The HCF of 48 and 72 is:
   a) 12
   b) 24
   c) 16
   d) 8

2. The roots of the quadratic equation ( x^2 - 6x + 8 = 0 ) are:
   a) 2, 4
   b) -2, -4
   c) 3, 5
   d) -3, -5

3. Assertion-Reason:
   Assertion (A): The area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by ( \frac{1}{2} | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) | ).
   Reason (R): The area of a triangle can be found using the base and height formula.
   a) Both A and R are true, and R is the correct explanation of A.
   b) Both A and R are true, but R is not the correct explanation of A.
   c) A is true, but R is false.
   d) A is false, but R is true.

4. The value of ( \tan 45^\circ \cdot \cot 45^\circ ) is:
   a) 1
   b) ( \sqrt{2} )
   c) ( \frac{1}{\sqrt{2}} )
   d) 0

5. The probability of drawing a queen from a standard deck of 52 cards is:
   a) ( \frac{1}{13} )
   b) ( \frac{1}{26} )
   c) ( \frac{2}{13} )
   d) ( \frac{1}{4} )

6. The distance between the points (-3, 4) and (1, -2) is:
   a) ( \sqrt{52} ) units
   b) ( \sqrt{20} ) units
   c) ( \sqrt{40} ) units
   d) 10 units

7. The 9th term of the AP: 3, 7, 11, 15, … is:
   a) 35
   b) 39
   c) 31
   d) 43

8. If ( \sin \theta = \frac{12}{13} ), then ( \cos \theta ) is:
   a) ( \frac{5}{13} )
   b) ( \frac{12}{5} )
   c) ( \frac{13}{5} )
   d) ( \frac{5}{12} )

9. The mode of the data: 1, 2, 2, 3, 3, 3, 4 is:
   a) 2
   b) 3
   c) 4
   d) 1

10. The circumference of a circle with area 314 cm² is:
    a) 62.8 cm
    b) 31.4 cm
    c) 44 cm
    d) 88 cm

11. The sum of the first 12 terms of an AP with first term 5 and common difference 4 is:
    a) 324
    b) 294
    c) 354
    d) 264

12. The value of ( \sec 60^\circ ) is:
    a) 2
    b) ( \sqrt{2} )
    c) ( \frac{1}{\sqrt{2}} )
    d) 1

13. The slope of the line joining the points (2, 5) and (6, 11) is:
    a) ( \frac{3}{2} )
    b) ( \frac{2}{3} )
    c) ( \frac{3}{4} )
    d) ( \frac{4}{3} )

14. The probability of getting a multiple of 3 when rolling a die is:
    a) ( \frac{1}{3} )
    b) ( \frac{1}{2} )
    c) ( \frac{1}{6} )
    d) ( \frac{2}{3} )

15. The area of a sector with radius 12 cm and sector angle ( 60^\circ ) is:
    a) 24π cm²
    b) 12π cm²
    c) 36π cm²
    d) 48π cm²

16. The coordinates of the midpoint of the line segment joining (-2, 3) and (4, -1) are:
    a) (1, 1)
    b) (2, 2)
    c) (1, 2)
    d) (0, 1)

17. If the discriminant of a quadratic equation is positive, the roots are:
    a) Real and equal
    b) Real and distinct
    c) Not real
    d) Negative

18. The 8th term of an AP with first term 6 and common difference -2 is:
    a) -8
    b) -6
    c) -4
    d) -2

19. The value of ( \sin^2 60^\circ + \cos^2 30^\circ ) is:
    a) 1
    b) ( \frac{3}{2} )
    c) ( \frac{3}{4} )
    d) 2

20. The radius of a circle with circumference 132 cm is:
    a) 21 cm
    b) 42 cm
    c) 14 cm
    d) 28 cm

Very Short Answer Questions (2 Marks Each, Total: 10 Marks)

21. Find the value of ( k ) for which the quadratic equation ( 4x^2 + kx + 1 = 0 ) has equal roots.
22. If the angle of elevation of a tower from a point 50 m away is ( 30^\circ ), find the height of the tower.
23. Find the coordinates of the centroid of a triangle with vertices (2, 3), (4, 7), and (-2, 1).
24. Calculate the mean of the data: 3, 5, 7, 9, 11, 13.
25. Find the ratio in which the point (4, 5) divides the line segment joining (2, 3) and (6, 7).

Short Answer Questions (3 Marks Each, Total: 18 Marks)

26. Prove that ( \sqrt{17} ) is irrational.
27. Find the area of a triangle with vertices (0, 4), (4, 0), and (-2, 2).
28. Solve the pair of linear equations: ( 5x - 2y = 16 ) and ( 3x + y = 13 ).
29. A bag contains 5 white, 3 red, and 2 blue balls. Find the probability of drawing a white ball.
30. Find the length of the arc of a circle with radius 10 cm and sector angle ( 120^\circ ).
31. If ( \tan \theta = \frac{5}{12} ), find the values of ( \sin \theta ) and ( \cos \theta ).

Long Answer Questions (5 Marks Each, Total: 20 Marks)

32. Derive the formula for the distance between two points in coordinate geometry.
33. Prove that the opposite sides of a parallelogram are equal using coordinate geometry.
34. OR Prove that the diagonals of a rhombus are perpendicular to each other using coordinate geometry.
35. Solve the following system of equations graphically: ( 2x - y = 4 ) and ( x + 3y = 3 ). Verify the solution algebraically.

Competency-Based/Case Study-Based Questions (4 Marks Each, Total: 12 Marks)

36. A vendor sells two types of bags: small and large. A small bag costs ₹50 and requires 1 kg of material, while a large bag costs ₹80 and requires 2 kg of material. The vendor uses 20 kg of material and earns ₹1100 in a day. How many small and large bags were sold? (Solve using a system of equations and verify the solution.)
37. A flagpole 15 m high casts a shadow of 9 m on the ground.
    a) Find the angle of elevation of the top of the flagpole from the end of the shadow. (2 marks)
    b) If the shadow length decreases to 6 m, find the new angle of elevation. (2 marks)
38. A school club sells two types of tickets for a concert: student and adult. A student ticket costs ₹20, and an adult ticket costs ₹40. The club sells 25 tickets and earns ₹700.
    a) How many student and adult tickets were sold? (Solve using a system of equations.) (3 marks)
    b) What value is reflected by organizing such a concert? (1 mark)

---

End of Question Paper

Sample Question Paper (SET-5): Mathematics (CBSE Class X)

Maximum Marks: 80 | Time: 3 Hours | Total Questions: 38

General Instructions:

1. The question paper contains 38 questions, divided into five sections: MCQ, VSA, SA, LA, and Competency-Based.
2. All questions are compulsory. Internal choices are provided in some questions.
3. Use of calculators is not permitted.
4. Write answers clearly and legibly, ensuring proper labeling of question numbers.
5. Show all necessary calculations and steps for SA and LA questions.

---

Multiple Choice Questions (1 Mark Each, Total: 20 Marks)

1. The LCM of 20 and 36 is:
   a) 180
   b) 72
   c) 60
   d) 90

2. The roots of the quadratic equation ( x^2 - 9x + 20 = 0 ) are:
   a) 4, 5
   b) -4, -5
   c) 3, 6
   d) -3, -6

3. Assertion-Reason:
   Assertion (A): The sum of the opposite angles of a cyclic quadrilateral is ( 180^\circ ).
   Reason (R): A cyclic quadrilateral can be inscribed in a circle.
   a) Both A and R are true, and R is the correct explanation of A.
   b) Both A and R are true, but R is not the correct explanation of A.
   c) A is true, but R is false.
   d) A is false, but R is true.

4. The value of ( \sin 30^\circ + \cos 60^\circ ) is:
   a) 1
   b) ( \frac{1}{2} )
   c) ( \frac{\sqrt{3}}{2} )
   d) 2

5. The probability of drawing a king from a standard deck of 52 cards is:
   a) ( \frac{1}{13} )
   b) ( \frac{1}{26} )
   c) ( \frac{2}{13} )
   d) ( \frac{1}{4} )

6. The distance between the points (2, -3) and (-4, 5) is:
   a) 10 units
   b) ( \sqrt{52} ) units
   c) ( \sqrt{100} ) units
   d) ( \sqrt{40} ) units

7. The 10th term of the AP: 1, 4, 7, 10, … is:
   a) 28
   b) 31
   c) 34
   d) 37

8. If ( \cos \theta = \frac{3}{5} ), then ( \tan \theta ) is:
   a) ( \frac{4}{3} )
   b) ( \frac{3}{4} )
   c) ( \frac{5}{4} )
   d) ( \frac{4}{5} )

9. The mean of the data: 5, 10, 15, 20, 25 is:
   a) 15
   b) 12
   c) 18
   d) 10

10. The area of a circle with circumference 44 cm is:
    a) 154 cm²
    b) 88 cm²
    c) 308 cm²
    d) 616 cm²

11. The sum of the first 8 terms of an AP with first term 3 and common difference 5 is:
    a) 184
    b) 164
    c) 144
    d) 124

12. The value of ( \cot 60^\circ ) is:
    a) ( \sqrt{3} )
    b) ( \frac{1}{\sqrt{3}} )
    c) 1
    d) 2

13. The slope of the line joining the points (3, 4) and (7, 10) is:
    a) ( \frac{3}{2} )
    b) ( \frac{2}{3} )
    c) ( \frac{3}{4} )
    d) ( \frac{4}{3} )

14. The probability of getting a number greater than 2 when rolling a die is:
    a) ( \frac{2}{3} )
    b) ( \frac{1}{3} )
    c) ( \frac{1}{2} )
    d) ( \frac{1}{6} )

15. The area of a sector with radius 8 cm and sector angle ( 45^\circ ) is:
    a) 4π cm²
    b) 8π cm²
    c) 16π cm²
    d) 32π cm²

16. The coordinates of the midpoint of the line segment joining (1, -2) and (5, 4) are:
    a) (3, 1)
    b) (2, 1)
    c) (3, 2)
    d) (2, 2)

17. If the discriminant of a quadratic equation is negative, the roots are:
    a) Real and equal
    b) Real and distinct
    c) Not real
    d) Positive

18. The 7th term of an AP with first term 8 and common difference -4 is:
    a) -16
    b) -12
    c) -8
    d) -4

19. The value of ( \sin^2 45^\circ + \cos^2 45^\circ ) is:
    a) 1
    b) ( \frac{1}{2} )
    c) ( \sqrt{2} )
    d) 2

20. The radius of a circle with area 616 cm² is:
    a) 14 cm
    b) 28 cm
    c) 7 cm
    d) 21 cm

Very Short Answer Questions (2 Marks Each, Total: 10 Marks)

21. Find the value of ( k ) for which the quadratic equation ( 2x^2 - kx + 5 = 0 ) has equal roots.
22. If the angle of elevation of a cloud from a point 40 m away is ( 60^\circ ), find the height of the cloud.
23. Find the coordinates of the centroid of a triangle with vertices (-1, 2), (3, 6), and (5, -2).
24. Calculate the median of the data: 4, 6, 6, 7, 8, 9, 10.
25. Find the ratio in which the point (3, 4) divides the line segment joining (1, 2) and (5, 6).

Short Answer Questions (3 Marks Each, Total: 18 Marks)

26. Prove that ( \sqrt{19} ) is irrational.
27. Find the area of a triangle with vertices (2, 1), (4, 5), and (-2, 3).
28. Solve the pair of linear equations: ( 2x + 3y = 8 ) and ( x - 2y = -3 ).
29. A box contains 4 red, 3 blue, and 5 green balls. Find the probability of drawing a green ball.
30. Find the length of the arc of a circle with radius 12 cm and sector angle ( 90^\circ ).
31. If ( \sin \theta = \frac{8}{17} ), find the values of ( \cos \theta ) and ( \tan \theta ).

Long Answer Questions (5 Marks Each, Total: 20 Marks)

32. Derive the formula for the area of a sector of a circle.
33. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
34. OR Prove that the diagonals of a rectangle are equal in length using coordinate geometry.
35. Solve the following system of equations graphically: ( 3x + y = 7 ) and ( x - 2y = 1 ). Verify the solution algebraically.

Competency-Based/Case Study-Based Questions (4 Marks Each, Total: 12 Marks)

36. A shop sells two types of pens: gel and ballpoint. A gel pen costs ₹15 and requires 1 unit of ink, while a ballpoint pen costs ₹10 and requires 2 units of ink. The shop uses 30 units of ink and earns ₹300 in a day. How many gel and ballpoint pens were sold? (Solve using a system of equations and verify the solution.)
37. A ladder 13 m long is placed against a vertical wall. The foot of the ladder is 5 m away from the base of the wall.
    a) Find the height at which the ladder touches the wall. (2 marks)
    b) If the foot of the ladder is moved 2 m closer to the wall, find the new height. (2 marks)
38. A school organizes a sale of notebooks and pencils. A notebook costs ₹30, and a pencil costs ₹5. A student sells 20 items and earns ₹400.
    a) How many notebooks and pencils were sold? (Solve using a system of equations.) (3 marks)
    b) What value is reflected by participating in such a sale? (1 mark)

---

End of Question Paper

Sample Question Paper (SET-5): Mathematics (CBSE Class X)

Maximum Marks: 80 | Time: 3 Hours | Total Questions: 38

General Instructions:

1. The question paper contains 38 questions, divided into five sections: MCQ, VSA, SA, LA, and Competency-Based.
2. All questions are compulsory. Internal choices are provided in some questions.
3. Use of calculators is not permitted.
4. Write answers clearly and legibly, ensuring proper labeling of question numbers.
5. Show all necessary calculations and steps for SA and LA questions.

---

Multiple Choice Questions (1 Mark Each, Total: 20 Marks)

1. The LCM of 20 and 36 is:
   a) 180
   b) 72
   c) 60
   d) 90

2. The roots of the quadratic equation ( x^2 - 9x + 20 = 0 ) are:
   a) 4, 5
   b) -4, -5
   c) 3, 6
   d) -3, -6

3. Assertion-Reason:
   Assertion (A): The sum of the opposite angles of a cyclic quadrilateral is ( 180^\circ ).
   Reason (R): A cyclic quadrilateral can be inscribed in a circle.
   a) Both A and R are true, and R is the correct explanation of A.
   b) Both A and R are true, but R is not the correct explanation of A.
   c) A is true, but R is false.
   d) A is false, but R is true.

4. The value of ( \sin 30^\circ + \cos 60^\circ ) is:
   a) 1
   b) ( \frac{1}{2} )
   c) ( \frac{\sqrt{3}}{2} )
   d) 2

5. The probability of drawing a king from a standard deck of 52 cards is:
   a) ( \frac{1}{13} )
   b) ( \frac{1}{26} )
   c) ( \frac{2}{13} )
   d) ( \frac{1}{4} )

6. The distance between the points (2, -3) and (-4, 5) is:
   a) 10 units
   b) ( \sqrt{52} ) units
   c) ( \sqrt{100} ) units
   d) ( \sqrt{40} ) units

7. The 10th term of the AP: 1, 4, 7, 10, … is:
   a) 28
   b) 31
   c) 34
   d) 37

8. If ( \cos \theta = \frac{3}{5} ), then ( \tan \theta ) is:
   a) ( \frac{4}{3} )
   b) ( \frac{3}{4} )
   c) ( \frac{5}{4} )
   d) ( \frac{4}{5} )

9. The mean of the data: 5, 10, 15, 20, 25 is:
   a) 15
   b) 12
   c) 18
   d) 10

10. The area of a circle with circumference 44 cm is:
    a) 154 cm²
    b) 88 cm²
    c) 308 cm²
    d) 616 cm²

11. The sum of the first 8 terms of an AP with first term 3 and common difference 5 is:
    a) 184
    b) 164
    c) 144
    d) 124

12. The value of ( \cot 60^\circ ) is:
    a) ( \sqrt{3} )
    b) ( \frac{1}{\sqrt{3}} )
    c) 1
    d) 2

13. The slope of the line joining the points (3, 4) and (7, 10) is:
    a) ( \frac{3}{2} )
    b) ( \frac{2}{3} )
    c) ( \frac{3}{4} )
    d) ( \frac{4}{3} )

14. The probability of getting a number greater than 2 when rolling a die is:
    a) ( \frac{2}{3} )
    b) ( \frac{1}{3} )
    c) ( \frac{1}{2} )
    d) ( \frac{1}{6} )

15. The area of a sector with radius 8 cm and sector angle ( 45^\circ ) is:
    a) 4π cm²
    b) 8π cm²
    c) 16π cm²
    d) 32π cm²

16. The coordinates of the midpoint of the line segment joining (1, -2) and (5, 4) are:
    a) (3, 1)
    b) (2, 1)
    c) (3, 2)
    d) (2, 2)

17. If the discriminant of a quadratic equation is negative, the roots are:
    a) Real and equal
    b) Real and distinct
    c) Not real
    d) Positive

18. The 7th term of an AP with first term 8 and common difference -4 is:
    a) -16
    b) -12
    c) -8
    d) -4

19. The value of ( \sin^2 45^\circ + \cos^2 45^\circ ) is:
    a) 1
    b) ( \frac{1}{2} )
    c) ( \sqrt{2} )
    d) 2

20. The radius of a circle with area 616 cm² is:
    a) 14 cm
    b) 28 cm
    c) 7 cm
    d) 21 cm

Very Short Answer Questions (2 Marks Each, Total: 10 Marks)

21. Find the value of ( k ) for which the quadratic equation ( 2x^2 - kx + 5 = 0 ) has equal roots.
22. If the angle of elevation of a cloud from a point 40 m away is ( 60^\circ ), find the height of the cloud.
23. Find the coordinates of the centroid of a triangle with vertices (-1, 2), (3, 6), and (5, -2).
24. Calculate the median of the data: 4, 6, 6, 7, 8, 9, 10.
25. Find the ratio in which the point (3, 4) divides the line segment joining (1, 2) and (5, 6).

Short Answer Questions (3 Marks Each, Total: 18 Marks)

26. Prove that ( \sqrt{19} ) is irrational.
27. Find the area of a triangle with vertices (2, 1), (4, 5), and (-2, 3).
28. Solve the pair of linear equations: ( 2x + 3y = 8 ) and ( x - 2y = -3 ).
29. A box contains 4 red, 3 blue, and 5 green balls. Find the probability of drawing a green ball.
30. Find the length of the arc of a circle with radius 12 cm and sector angle ( 90^\circ ).
31. If ( \sin \theta = \frac{8}{17} ), find the values of ( \cos \theta ) and ( \tan \theta ).

Long Answer Questions (5 Marks Each, Total: 20 Marks)

32. Derive the formula for the area of a sector of a circle.
33. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
34. OR Prove that the diagonals of a rectangle are equal in length using coordinate geometry.
35. Solve the following system of equations graphically: ( 3x + y = 7 ) and ( x - 2y = 1 ). Verify the solution algebraically.

Competency-Based/Case Study-Based Questions (4 Marks Each, Total: 12 Marks)

36. A shop sells two types of pens: gel and ballpoint. A gel pen costs ₹15 and requires 1 unit of ink, while a ballpoint pen costs ₹10 and requires 2 units of ink. The shop uses 30 units of ink and earns ₹300 in a day. How many gel and ballpoint pens were sold? (Solve using a system of equations and verify the solution.)
37. A ladder 13 m long is placed against a vertical wall. The foot of the ladder is 5 m away from the base of the wall.
    a) Find the height at which the ladder touches the wall. (2 marks)
    b) If the foot of the ladder is moved 2 m closer to the wall, find the new height. (2 marks)
38. A school organizes a sale of notebooks and pencils. A notebook costs ₹30, and a pencil costs ₹5. A student sells 20 items and earns ₹400.
    a) How many notebooks and pencils were sold? (Solve using a system of equations.) (3 marks)
    b) What value is reflected by participating in such a sale? (1 mark)

---

End of Question Paper

Sample Question Paper (SET-6): Mathematics (CBSE Class X)

Maximum Marks: 80 | Time: 3 Hours | Total Questions: 38

General Instructions:

1. The question paper contains 38 questions, divided into five sections: MCQ, VSA, SA, LA, and Competency-Based.
2. All questions are compulsory. Internal choices are provided in some questions.
3. Use of calculators is not permitted.
4. Write answers clearly and legibly, ensuring proper labeling of question numbers.
5. Show all necessary calculations and steps for SA and LA questions.

---

Multiple Choice Questions (1 Mark Each, Total: 20 Marks)

1. The HCF of 54 and 90 is:
   a) 18
   b) 9
   c) 12
   d) 6

2. The roots of the quadratic equation ( x^2 - 10x + 21 = 0 ) are:
   a) 3, 7
   b) -3, -7
   c) 2, 8
   d) -2, -8

3. Assertion-Reason:
   Assertion (A): The diagonals of a parallelogram bisect each other.
   Reason (R): A parallelogram has opposite sides equal and parallel.
   a) Both A and R are true, and R is the correct explanation of A.
   b) Both A and R are true, but R is not the correct explanation of A.
   c) A is true, but R is false.
   d) A is false, but R is true.

4. The value of ( \cos 60^\circ \cdot \sin 30^\circ ) is:
   a) ( \frac{1}{4} )
   b) ( \frac{\sqrt{3}}{2} )
   c) ( \frac{1}{2} )
   d) ( \frac{\sqrt{3}}{4} )

5. The probability of drawing an ace from a standard deck of 52 cards is:
   a) ( \frac{1}{13} )
   b) ( \frac{1}{26} )
   c) ( \frac{2}{13} )
   d) ( \frac{1}{4} )

6. The distance between the points (-2, 1) and (4, -3) is:
   a) ( \sqrt{52} ) units
   b) ( \sqrt{20} ) units
   c) ( \sqrt{40} ) units
   d) 8 units

7. The 11th term of the AP: 2, 6, 10, 14, … is:
   a) 42
   b) 46
   c) 38
   d) 50

8. If ( \tan \theta = \frac{15}{8} ), then ( \cos \theta ) is:
   a) ( \frac{8}{17} )
   b) ( \frac{15}{17} )
   c) ( \frac{17}{8} )
   d) ( \frac{8}{15} )

9. The median of the data: 3, 5, 7, 9, 11, 13 is:
   a) 7
   b) 8
   c) 9
   d) 6

10. The circumference of a circle with area 201 cm² is approximately:
    a) 50.3 cm
    b) 25.1 cm
    c) 44 cm
    d) 88 cm

11. The sum of the first 9 terms of an AP with first term 4 and common difference 3 is:
    a) 144
    b) 162
    c) 180
    d) 126

12. The value of ( \sin 45^\circ ) is:
    a) ( \frac{\sqrt{2}}{2} )
    b) ( \sqrt{2} )
    c) ( \frac{1}{\sqrt{2}} )
    d) 1

13. The slope of the line joining the points (1, 3) and (5, 9) is:
    a) ( \frac{3}{2} )
    b) ( \frac{2}{3} )
    c) ( \frac{4}{3} )
    d) ( \frac{3}{4} )

14. The probability of getting a number divisible by 2 when rolling a die is:
    a) ( \frac{1}{2} )
    b) ( \frac{1}{3} )
    c) ( \frac{2}{3} )
    d) ( \frac{1}{6} )

15. The area of a sector with radius 6 cm and sector angle ( 60^\circ ) is:
    a) 6π cm²
    b) 12π cm²
    c) 18π cm²
    d) 9π cm²

16. The coordinates of the point dividing the line segment joining (2, 5) and (8, 11) in the ratio 1:2 are:
    a) (4, 7)
    b) (5, 8)
    c) (6, 9)
    d) (3, 6)

17. If the discriminant of a quadratic equation is zero, the roots are:
    a) Real and distinct
    b) Real and equal
    c) Not real
    d) Negative

18. The 5th term of an AP with first term 9 and common difference -3 is:
    a) -3
    b) 0
    c) 3
    d) -6

19. The value of ( \cos^2 30^\circ + \sin^2 60^\circ ) is:
    a) 1
    b) ( \frac{3}{2} )
    c) ( \frac{3}{4} )
    d) 2

20. The radius of a circle with circumference 176 cm is:
    a) 28 cm
    b) 14 cm
    c) 56 cm
    d) 42 cm

Very Short Answer Questions (2 Marks Each, Total: 10 Marks)

21. Find the value of ( k ) for which the quadratic equation ( 5x^2 - kx + 7 = 0 ) has equal roots.
22. If the angle of elevation of a kite from a point 30 m away is ( 45^\circ ), find the height of the kite.
23. Find the coordinates of the centroid of a triangle with vertices (0, 2), (4, 6), and (2, 0).
24. Calculate the mode of the data: 2, 3, 3, 4, 4, 4, 5, 6.
25. Find the ratio in which the point (2, 3) divides the line segment joining (0, 1) and (4, 5).

Short Answer Questions (3 Marks Each, Total: 18 Marks)

26. Prove that ( \sqrt{23} ) is irrational.
27. Find the area of a triangle with vertices (-3, 2), (1, 5), and (3, -1).
28. Solve the pair of linear equations: ( 3x - y = 7 ) and ( 2x + 3y = 1 ).
29. A bag contains 2 red, 5 blue, and 3 green balls. Find the probability of drawing a blue ball.
30. Find the area of a sector of a circle with radius 10 cm and sector angle ( 120^\circ ).
31. If ( \cos \theta = \frac{7}{25} ), find the values of ( \sin \theta ) and ( \tan \theta ).

Long Answer Questions (5 Marks Each, Total: 20 Marks)

32. Derive the formula for the sum of the first ( n ) terms of an arithmetic progression.
33. Prove that the angle in a semicircle is a right angle.
34. OR Prove that the diagonals of a square are equal and bisect each other at right angles using coordinate geometry.
35. Solve the following system of equations graphically: ( x + y = 6 ) and ( 2x - y = 3 ). Verify the solution algebraically.

Competency-Based/Case Study-Based Questions (4 Marks Each, Total: 12 Marks)

36. A vendor sells two types of notebooks: spiral and regular. A spiral notebook costs ₹40 and requires 2 sheets of paper, while a regular notebook costs ₹25 and requires 1 sheet of paper. The vendor uses 25 sheets of paper and earns ₹475 in a day. How many spiral and regular notebooks were sold? (Solve using a system of equations and verify the solution.)
37. A pole 10 m high casts a shadow of 6 m on the ground.
    a) Find the angle of elevation of the top of the pole from the end of the shadow. (2 marks)
    b) If the shadow length increases to 8 m, find the new angle of elevation. (2 marks)
38. A school organizes a sale of pens and erasers. A pen costs ₹12, and an eraser costs ₹3. A student sells 15 items and earns ₹135.
    a) How many pens and erasers were sold? (Solve using a system of equations.) (3 marks)
    b) What value is reflected by participating in such a sale? (1 mark)

---

End of Question Paper