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TGBIE · 1st Year · MPC

Mathematics IA — Important Questions

High-frequency SAQ and LAQ questions from TS Inter 1st Year Mathematics IA — Functions, Matrices, Vectors, Trigonometry, and Properties of Triangles. Based on TGBIE exam patterns 2025-26.

Ch. 1 Functions — Important Questions

  1. 1Let f: A → B and g: B → C be bijections. Prove that gof: A → C is also a bijection.
  2. 2If f(x) = 2x+1 and g(x) = x²−1, find (fog)(x) and (gof)(x). Are they equal?
  3. 3Show that the function f: ℝ → ℝ defined by f(x) = 3x − 7 is a bijection and find its inverse.
  4. 4If A = {1, 2, 3} and B = {a, b, c}, define a bijective function from A to B. How many bijections are possible?
  5. 5Let f: ℝ → ℝ be defined by f(x) = (3x+5)/2. Find f⁻¹ and verify that fof⁻¹ = I.

Ch. 2 Mathematical Induction — Important Questions

  1. 1Prove by mathematical induction: 1² + 2² + 3² + ... + n² = n(n+1)(2n+1)/6.
  2. 2Prove by induction: 1³ + 2³ + 3³ + ... + n³ = [n(n+1)/2]².
  3. 3Show using induction that 3.2² + 3².2³ + 3³.2⁴ + ... + 3ⁿ.2ⁿ⁺¹ = (6/5)(6ⁿ − 1).
  4. 4Prove by induction that 2.3 + 3.4 + 4.5 + ... + n(n+1)(n+2)/3 terms = n(n+1)(n+2)/3.
  5. 5Using mathematical induction, prove that 4ⁿ + 15n − 1 is divisible by 9 for all positive integers n.

Ch. 3 Matrices — Important Questions (LAQ)

  1. 1For matrix A = [[1, 0, 0],[2, 3, 4],[5, −6, x]], find x if A is a singular matrix. Hence find A⁻¹ if it exists.
  2. 2Solve the system: x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 using Cramer's rule.
  3. 3Using row operations, find the rank of the matrix: [[1,2,3],[2,3,4],[3,4,5]].
  4. 4If A = [[1,2],[3,4]], verify A·adj(A) = |A|·I.
  5. 5Solve using Gauss-Jordan method: 2x − y + 3z = 9, x + y + z = 6, x − y + z = 2.

Ch. 4 & 5 Vectors — Important Questions

  1. 1If a⃗ = 2i + 4j − 5k, b⃗ = i + 2j + 3k, find a⃗ + b⃗, a⃗ − b⃗, |a⃗ + b⃗| and the unit vector along a⃗ + b⃗.
  2. 2Find the angle between a⃗ = 2i − j + k and b⃗ = i − 3j + 5k.
  3. 3Find the area of the parallelogram with adjacent sides a⃗ = i − 2j + 3k and b⃗ = 2i + j − k.
  4. 4Find the volume of the parallelepiped with coterminous edges a⃗ = i − j, b⃗ = 2i + j − k, c⃗ = 3i + j + k.
  5. 5Show that the vectors a⃗ = i − 2j + 3k, b⃗ = −2i + 3j − 4k, c⃗ = i − 3j + 5k are coplanar.
  6. 6Find the unit vector perpendicular to both a⃗ = 2i + j + k and b⃗ = i − j + 2k.

Ch. 6–9 Trigonometry — Important Questions

  1. 1Prove that sin²(π/10) + cos²(π/5) = 3/4 without using tables.
  2. 2If A + B = 45°, prove that (1 + tanA)(1 + tanB) = 2. Hence find tan 22.5°.
  3. 3Prove that cos⁴(π/8) + cos⁴(3π/8) + cos⁴(5π/8) + cos⁴(7π/8) = 3/2.
  4. 4If tanθ = 1/7 and sinφ = 1/√10, show that θ + 2φ = π/4 (0 < θ, φ < π/2).
  5. 5Solve: 2cos²θ − √3·sinθ + 1 = 0 for θ ∈ [0, 2π].
  6. 6Find the general solution of: sinθ + sin2θ + sin3θ = 0.
  7. 7Prove that tan⁻¹(1/2) + tan⁻¹(2/11) = tan⁻¹(3/4).

Ch. 10 Properties of Triangles — Important Questions (LAQ)

  1. 1In ΔABC, if a = 13, b = 14, c = 15, find: (i) cos A (ii) sin A (iii) area of triangle.
  2. 2If cosA/a = cosB/b, show that the triangle is either isosceles or right-angled.
  3. 3Prove that a·sin(B−C) + b·sin(C−A) + c·sin(A−B) = 0.
  4. 4In ΔABC, prove that (a−b)² cos²(C/2) + (a+b)² sin²(C/2) = c².
  5. 5Show that the circumradius R of ΔABC = abc / (4Δ) where Δ is the area.
  6. 6In ΔABC with s = 56, r = 3, r₁ = 21, find a. Also find R.

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