General Mathematics up to HSC standard 100 Questions

General Mathematics up to HSC standard 100 Questions

Here is a comprehensive set of 100 questions covering General Mathematics up to HSC (Higher Secondary/Class 12) standard. These draw from typical topics in syllabi like CBSE Class 12 Mathematics, NSW HSC Mathematics Standard/General, or equivalent boards. They include a mix of:

  • Multiple Choice Questions (MCQs)
  • Short-answer problems
  • Application-based questions

Topics covered (roughly distributed):

  • Algebra (relations, functions, matrices, determinants)
  • Calculus (limits, differentiation, integration, applications)
  • Trigonometry
  • Vectors & 3D Geometry
  • Probability & Statistics
  • Linear Programming (basics)
  • Financial Mathematics (simple applications)

Practice these for revision. Attempt them first, then check standard solutions from NCERT textbooks or past papers. For full worked solutions or PDFs, refer to official board resources or standard reference books.

Section 1: Algebra & Relations (Questions 1–20)

  1. If a relation R on set A = {1,2,3} is defined as R = {(1,2), (2,3), (1,3)}, is R reflexive? Why or why not?
  2. Let f: R → R be defined by f(x) = x² + 1. Is f invertible? Give reason.
  3. Solve for x: |x – 3| + |x + 2| = 5.
  4. Find the domain of f(x) = √(x² – 9) / (x – 2).
  5. If A and B are matrices such that AB = BA = I, what is B in terms of A?
  6. Evaluate the determinant of matrix [[2, 3], [4, 5]].
  7. Using properties, evaluate det of [[1, 2, 3], [0, 4, 5], [0, 0, 6]].
  8. Solve the system: 2x + y = 5, x + 3y = 7 using matrix method (if possible).
  9. If A = [[1, 0], [0, 1]] (identity), find A⁻¹.
  10. Show that the function f(x) = x³ is one-one and onto from R to R.
  11. Find the inverse of f(x) = 2x + 3.
  12. For matrices A = [[1,2],[3,4]], B = [[5,6],[7,8]], compute AB – BA.
  13. If sin⁻¹x + cos⁻¹x = π/2, verify for x in [-1,1].
  14. Simplify: tan⁻¹(1) + tan⁻¹(√3).
  15. Find the principal value of cos⁻¹(-1/2).
  16. Solve: 2x² – 5x + 3 = 0 using quadratic formula.
  17. If the sum of roots of ax² + bx + c = 0 is 5 and product is 6, form the equation.
  18. Decompose: (x+1)/(x² – 1) into partial fractions.
  19. Find the rank of matrix [[1,2,3],[2,4,6],[3,6,9]].
  20. If f(x) = |x| + |x-1|, find f'(x) where differentiable.

Section 2: Trigonometry (Questions 21–30)

  1. Prove that sin²θ + cos²θ = 1.
  2. If tanθ = 3/4, find sinθ and cosθ (θ acute).
  3. Solve: 2sin²θ – sinθ – 1 = 0 for θ in [0, 2π).
  4. Simplify: (sinA + sinB)/(cosA + cosB).
  5. Find the value of cos(15°) using angle formulas.
  6. Prove: tan(A/2) = (1 – cosA)/sinA.
  7. If sinθ = 5/13, find cos2θ.
  8. Solve for x: sin⁻¹x = π/6.
  9. Evaluate: sin(π/3)cos(π/6) – cos(π/3)sin(π/6).
  10. In a triangle ABC, if a=3, b=4, C=90°, find c using Pythagoras (or trig).

Section 3: Calculus – Differentiation (Questions 31–45)

  1. Find dy/dx if y = x³ + 3x² – 5x + 7.
  2. Differentiate y = sin(x²) using chain rule.
  3. Find the derivative of y = e^{3x} / x.
  4. If y = tan⁻¹x, find dy/dx.
  5. Differentiate implicitly: x² + y² = 25.
  6. Find the second derivative of y = x⁴ + 2x².
  7. Find the slope of tangent to y = x² at x=2.
  8. Using product rule, differentiate (x+1)(x²+2).
  9. Find dy/dx for y = log(sin x).
  10. If f(x) = |x-2|, find f'(x) at x=3.
  11. Maximum/minimum: Find critical points of f(x) = x³ – 3x² + 2.
  12. Rate of change: If area A = πr², find dA/dr.
  13. Differentiate y = (3x + 2)⁵.
  14. Find dy/dx: y = x^x (use logarithmic differentiation).
  15. Verify Rolle’s theorem for f(x) = x² – 4x + 3 on [1,3].

Section 4: Calculus – Integration (Questions 46–60)

  1. Integrate: ∫(2x + 3) dx.
  2. Find ∫ sin(3x) dx.
  3. Evaluate definite integral ∫from 0 to 1 of x² dx.
  4. Integrate by substitution: ∫ x / √(1 + x²) dx.
  5. Find ∫ e^{2x} dx.
  6. Integrate: ∫ (1/(x² + 4)) dx.
  7. Partial fractions: ∫ (x+1)/(x² – x – 2) dx.
  8. Area under curve: Find area bounded by y = x² and x-axis from 0 to 2.
  9. Integrate: ∫ sec²x dx.
  10. Evaluate ∫from 0 to π/2 of sin x dx.
  11. Find the antiderivative of 1/√(1 – x²).
  12. Use integration by parts for ∫ x e^x dx.
  13. Solve differential equation: dy/dx = 2x, y(0)=1.
  14. Find volume of solid of revolution (basic: y=x, x=0 to 1, about x-axis) – optional advanced.
  15. Integrate: ∫ tan x dx.

Section 5: Vectors, 3D Geometry & Linear Programming (Questions 61–70)

  1. Find the magnitude of vector <3,4,0>.
  2. If a = i + 2j + 3k, b = 2i – j + k, find a · b.
  3. Show that vectors <1,1,1> and <1,2,3> are not perpendicular.
  4. Find direction cosines of vector <2,3,6>.
  5. Equation of line: Find parametric equations for line through (1,2,3) parallel to <1,0,1>.
  6. Distance between points (1,2,3) and (4,5,6) in 3D.
  7. Solve LPP graphically: Maximize Z = 3x + 4y subject to x≥0, y≥0, x+y≤5, x≤3.
  8. Find angle between vectors a = <1,1,0>, b = <1,0,1>.
  9. Scalar triple product [a b c] for i,j,k basis.
  10. Plane equation: Find equation of plane through (0,0,0), (1,0,0), (0,1,0).

Section 6: Probability & Statistics (Questions 71–85)

  1. A coin is tossed twice. Find P(both heads).
  2. If P(A) = 0.4, P(B) = 0.5, P(A∩B)=0.2, find P(A∪B).
  3. Bayes’ theorem basic: Given two events.
  4. Mean of data: 2,4,6,8,10.
  5. Variance and standard deviation for above data.
  6. Probability distribution: Fair die, P(X= even number).
  7. Binomial: n=5, p=0.5, find P(X=3).
  8. If events A and B are independent, P(A∩B) = ?
  9. Conditional probability: P(A|B) = P(A∩B)/P(B).
  10. Find median of 1,3,5,7,9.
  11. Correlation concept: Positive/negative for height vs weight.
  12. Random variable expected value E(X) for discrete.
  13. Normal distribution properties (mean=median=mode).
  14. Sample space for rolling two dice, total outcomes.
  15. Complementary events: P(not A) = 1 – P(A).

Section 7: Financial & Miscellaneous Applications (Questions 86–100)

  1. Simple interest: Principal ₹5000, rate 8%, time 2 years. Find SI.
  2. Compound interest formula basic: Amount = P(1 + r/100)^n.
  3. Depreciation: If value decreases 10% per year.
  4. Annuity or EMI basic concept question.
  5. Solve inequality: 2x + 3 > 7.
  6. Graph of linear equation y = 2x + 1.
  7. Find arithmetic mean of first 10 natural numbers.
  8. Geometric progression: Sum of infinite GP if |r|<1.
  9. Permutation: P(5,2).
  10. Combination: C(5,2).
  11. Logarithm: log10(100) = ?
  12. Exponential: Solve 2^x = 8.
  13. Set theory: If A ∪ B = 10, A ∩ B = 3, |A|=6, find |B|.
  14. Complex number: (1+i)(1-i) = ?
  15. Limits: lim x→0 (sin x / x) = ?

These questions test conceptual understanding, problem-solving, and application skills typical for HSC-level General/Standard Mathematics or core Class 12 Maths.

Tips for preparation:

  • Practice with a timer.
  • Focus on showing working steps clearly.
  • Revise formulas for differentiation/integration, trig identities, matrix properties, probability rules.
  • For HSC-specific (e.g., NSW Standard Maths), emphasize practical applications like networks, data, finance over heavy calculus.

If you need detailed solutions for any specific questions, answers to a subset, more questions on a particular topic (e.g., only calculus or only MCQs), or questions tailored to a specific board (CBSE/ICSE/NSW HSC), let me know!

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