CBSE CLASS 10 MATHEMATICS -SAMPLE QUESTION PAPER

 

SECTION B

Questions 21 to 26 carry 2 marks each

  1. Find the HCF of 867 and 255 using Euclid's division algorithm.
  2. OR Express 140 as a product of its prime factors.
  3. If α and β are the zeros of the polynomial 2x² + 7x + 5, find the value of α + β + αβ.
  4. Solve for x: 1/(x-1) + 2/(x-2) = 3/(x-3), x ≠ 1, 2, 3
  5. OR Find the roots of the equation x² - 3x - 10 = 0 by factorization method.
  6. In an AP, if a = 5, d = 3 and aₙ = 50, find n and Sₙ.
  7. Prove that: (sin θ + cos θ)² + (sin θ - cos θ)² = 2
  8. Find the area of a triangle whose vertices are (1, -1), (-4, 6) and (-3, -5).

    SECTION C

    Questions 27 to 34 carry 3 marks each

    1. Prove that √3 is an irrational number.
    2. OR Find the LCM and HCF of 12, 15 and 21 by prime factorization method.
    3. 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 original speed of the train.
    4. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.
    5. OR How many terms of the AP: 9, 17, 25, ... must be taken to give a sum of 636?
    6. In △ABC, AD is the median. Prove that AB² + AC² = 2(AD² + BD²).
    7. A pole 6 m high casts a shadow 2√3 m long on the ground. Find the sun's elevation.
    8. Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, -3). Also find the value of x.
    9. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
    10. Two circles touch each other externally at point P. AB is a common tangent to the circles touching them at A and B. Find the value of ∠APB.                                                                            ANSWER KEY

      SECTION B (2 marks each)

      21. Using Euclid's division algorithm:

      • 867 = 255 × 3 + 102
      • 255 = 102 × 2 + 51
      • 102 = 51 × 2 + 0
      • HCF = 51

      22 (OR). 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7

      22. For 2x² + 7x + 5:

      • α + β = -7/2
      • αβ = 5/2
      • α + β + αβ = -7/2 + 5/2 = -1

      23. Solving: 1/(x-1) + 2/(x-2) = 3/(x-3)

      • Cross multiply and simplify
      • x² - 9x + 14 = 0
      • (x - 7)(x - 2) = 0
      • x = 7 (x = 2 is rejected)

      24 (OR). x² - 3x - 10 = 0

      • (x - 5)(x + 2) = 0
      • x = 5 or x = -2

      24. Given: a = 5, d = 3, aₙ = 50

      • 5 + (n-1)3 = 50
      • n = 16
      • Sₙ = n/2(a + aₙ) = 16/2(5 + 50) = 440

      25. LHS = sin²θ + cos²θ + 2sinθcosθ + sin²θ + cos²θ - 2sinθcosθ

      • = 1 + 1 = 2 = RHS (Proved)

      26. Area = 1/2|1(6+5) + (-4)(-5+1) + (-3)(-1-6)|

      • = 1/2|11 + 16 + 21| = 24 sq. units

      SECTION C (3 marks each)

      27. Let √3 = a/b where a, b are coprime integers

      • 3 = a²/b², so a² = 3b²
      • Therefore 3 divides a², so 3 divides a
      • Let a = 3c, then 9c² = 3b², so 3c² = b²
      • Therefore 3 divides b
      • This contradicts that a and b are coprime
      • Hence √3 is irrational (Proved)

      28 (OR).

      • 12 = 2² × 3
      • 15 = 3 × 5
      • 21 = 3 × 7
      • HCF = 3
      • LCM = 2² × 3 × 5 × 7 = 420

      28. Let original speed = x km/h

      • 360/x - 360/(x+5) = 1
      • 360(x + 5) - 360x = x(x + 5)
      • x² + 5x - 1800 = 0
      • (x - 40)(x + 45) = 0
      • x = 40 km/h

      29. Let first term = a, common difference = d

      • a₄ + a₈ = 24 → 2a + 10d = 24 → a + 5d = 12
      • a₆ + a₁₀ = 44 → 2a + 14d = 44 → a + 7d = 22
      • Solving: d = 5, a = -13
      • First three terms: -13, -8, -3

      30 (OR). AP: 9, 17, 25, ...

      • a = 9, d = 8
      • Sₙ = n/2[2a + (n-1)d] = 636
      • n/2[18 + 8n - 8] = 636
      • 4n² + 5n - 636 = 0
      • n = 12

      30. Using Apollonius theorem (statement and proof required)

      31. tan θ = 6/(2√3) = √3

      • θ = 60°

      32. Using section formula:

      • 2 = [5k + (-3)]/(k + 1)
      • 2k + 2 = 5k - 3
      • k = 5/3
      • Ratio = 5:3
      • x = [12(3) + 4(5)]/8 = 7

      33. (Theorem proof with diagram required)

      34. ∠APB = 90° (Tangent property theorem)

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