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Maths derivatives files, Summaries of Mathematics

Mathematics

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Typology: Summaries

2025/2026

Uploaded on 01/18/2026

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Class 12 CBSE Mathematics – Solved Questions
Derivatives, Matrices & Inverse Trigonometric
Functions
Chapter 1: Derivatives (Solved)
Q1. Find dy/dx of y = x² sin x.
Solution: Using product rule: dy/dx = 2x sin x + x² cos x.
Q2. Differentiate y = ln(x² + 1).
Solution: dy/dx = (2x)/(x² + 1).
Q3. Find dy/dx if y = e³■ cos x.
Solution: dy/dx = e³■(3cos x − sin x).
Q4. Differentiate y = tan■¹(2x).
Solution: dy/dx = 2/(1 + 4x²).
Q5. Find dy/dx of y = (x² + 1)/(x − 1).
Solution: Using quotient rule: dy/dx = (x² − 2x − 1)/(x − 1)².
Q6. Differentiate y = sin■¹x.
Solution: dy/dx = 1/√(1 − x²).
Q7. Find dy/dx if y = x■.
Solution: dy/dx = x■(1 + ln x).
Q8. Differentiate y = √(x² + 4x + 5).
Solution: dy/dx = (x + 2)/√(x² + 4x + 5).
Q9. Differentiate y = ln(sin x).
Solution: dy/dx = cot x.
Q10. Differentiate y = (sin x + cos x)².
Solution: dy/dx = 2(sin x + cos x)(cos x − sin x).
Chapter 2: Matrices (Solved)
Q1. Find the order of matrix A = [[2,3,1],[4,5,6]].
Solution: Order = 2 × 3.
Q2. Find transpose of A = [[1,2],[3,4]].
Solution: A■ = [[1,3],[2,4]].
Q3. If A = [[x,2],[3,4]] is symmetric, find x.
Solution: x = 3.
Q4. Find AB where A=[[1,0],[2,3]], B=[[2,1],[1,0]].
Solution: AB = [[2,1],[7,2]].
Q5. Find determinant of [[2,1],[5,3]].
Solution: |A| = (2×3 − 5×1) = 1.
Q6. Find |A| if A=[[1,2],[3,4]].
Solution: |A| = −2.
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Class 12 CBSE Mathematics – Solved Questions

Derivatives, Matrices & Inverse Trigonometric

Functions

Chapter 1: Derivatives (Solved)

Q1. Find dy/dx of y = x² sin x. Solution: Using product rule: dy/dx = 2x sin x + x² cos x. Q2. Differentiate y = ln(x² + 1). Solution: dy/dx = (2x)/(x² + 1). Q3. Find dy/dx if y = e³n cos x. Solution: dy/dx = e³n(3cos x − sin x). Q4. Differentiate y = tann¹(2x). Solution: dy/dx = 2/(1 + 4x²). Q5. Find dy/dx of y = (x² + 1)/(x − 1). Solution: Using quotient rule: dy/dx = (x² − 2x − 1)/(x − 1)². Q6. Differentiate y = sinn¹x. Solution: dy/dx = 1/√(1 − x²). Q7. Find dy/dx if y = xn. Solution: dy/dx = xn(1 + ln x). Q8. Differentiate y = √(x² + 4x + 5). Solution: dy/dx = (x + 2)/√(x² + 4x + 5). Q9. Differentiate y = ln(sin x). Solution: dy/dx = cot x. Q10. Differentiate y = (sin x + cos x)². Solution: dy/dx = 2(sin x + cos x)(cos x − sin x).

Chapter 2: Matrices (Solved)

Q1. Find the order of matrix A = [[2,3,1],[4,5,6]]. Solution: Order = 2 × 3. Q2. Find transpose of A = [[1,2],[3,4]]. Solution: An = [[1,3],[2,4]]. Q3. If A = [[x,2],[3,4]] is symmetric, find x. Solution: x = 3. Q4. Find AB where A=[[1,0],[2,3]], B=[[2,1],[1,0]]. Solution: AB = [[2,1],[7,2]]. Q5. Find determinant of [[2,1],[5,3]]. Solution: |A| = (2×3 − 5×1) = 1. Q6. Find |A| if A=[[1,2],[3,4]]. Solution: |A| = −2.

Q7. Verify (AB)n = BnAn. Solution: LHS = RHS, hence verified. Q8. Find inverse of [[2,1],[5,3]]. Solution: An¹ = [[3,−1],[−5,2]]. Q9. Check invertibility of [[1,2],[2,4]]. Solution: Determinant = 0, not invertible. Q10. Solve AX=B for A=[[1,1],[1,−1]] and B=[[4],[2]]. Solution: X = [[3],[1]].

Chapter 3: Inverse Trigonometric Functions (Solved)

Q1. Find sinn¹(1/2). Solution: sinn¹(1/2) = π/6. Q2. Evaluate tann¹(1). Solution: tann¹(1) = π/4. Q3. Find cosn¹(−1/2). Solution: cosn¹(−1/2) = 2π/3. Q4. Simplify tann¹x + cotn¹x. Solution: Result = π/2. Q5. Find d/dx of sinn¹x. Solution: 1/√(1 − x²). Q6. Evaluate sinn¹(sin 5π/6). Solution: Answer = π/6. Q7. Find domain of cosn¹x. Solution: − 1 ≤ x ≤ 1. Q8. Evaluate tann¹(√3). Solution: Answer = π/3. Q9. Simplify cosn¹x + sinn¹x. Solution: Result = π/2. Q10. Find tann¹(1/x), x>0. Solution: Result = π/2 − tann¹x.