CBSE Class 12 Mathematics Part 2 Exercise Solutions

How to Use This Page

This page provides comprehensive exercise solutions for CBSE Class 12 Mathematics Part 2, covering Chapters 7 through 13 of the NCERT textbook. Each section is designed to strengthen your conceptual understanding and problem-solving speed for the board examination.

Start by attempting every question on your own before consulting the solutions. When you check your work, focus on the method rather than just the final answer — board examiners award marks for correct steps, proper notation, and logical flow. If your approach differs from the solution shown, verify whether it is mathematically valid; many problems admit multiple correct methods.

Pay special attention to the Common Mistakes to Avoid and Exam Checklist sections at the end. These consolidate the most frequent errors students make and the essential items to verify before submitting your paper. Use the checklist as a final review tool in the days leading up to the exam.

For maximum benefit, solve the MCQs, fill-in-the-blanks, and short-answer questions first to build speed, then tackle the detailed long-answer problems. Time yourself: aim to complete each section within the marks-weighted time allocation (roughly 1.5 minutes per mark).

Key Concepts (Quick Recap)

Chapter 7: Integrals

• Indefinite integral: ∫ f(x) dx = F(x) + C, where F′(x) = f(x)

• Integration by parts (ILATE rule): ∫ u dv = uv − ∫ v du; choose u by priority — Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential
• Partial fractions: Decompose P(x)/Q(x) into simpler fractions based on factorisation of Q(x)
• Linear factor (x − a): A/(x − a)
• Repeated factor (x − a)²: A/(x − a) + B/(x − a)²
• Irreducible quadratic (x² + bx + c): (Ax + B)/(x² + bx + c)
• Definite integral properties:
• ∫[a to b] f(x) dx = ∫[a to b] f(a + b − x) dx
• ∫[0 to a] f(x) dx = ∫[0 to a] f(a − x) dx
• ∫[−a to a] f(x) dx = 2∫[0 to a] f(x) dx if f is even; = 0 if f is odd
• Fundamental Theorem: ∫[a to b] f(x) dx = F(b) − F(a)

Chapter 8: Applications of Integrals

• Area under a curve: Area = ∫[a to b] y dx = ∫[a to b] f(x) dx
• Area between two curves: Area = ∫[a to b] |f(x) − g(x)| dx
• Area with y-axis: Area = ∫[c to d] x dy = ∫[c to d] g(y) dy
• Circle: Area enclosed by x² + y² = a² is πa²
• Ellipse: Area enclosed by x²/a² + y²/b² = 1 is πab

Chapter 9: Differential Equations

• Order: Highest order derivative present
• Degree: Power of the highest order derivative (when polynomial in derivatives)
• Formation: Differentiate the given relation n times and eliminate constants
• Variable separable: dy/dx = f(x)g(y) → dy/g(y) = f(x) dx → integrate both sides
• Homogeneous DE: dy/dx = f(y/x); substitute y = vx, dy/dx = v + x dv/dx
• Linear DE: dy/dx + P(x) · y = Q(x); integrating factor IF = e^(∫P dx); Solution: y · IF = ∫(Q · IF) dx + C
• General solution: Contains as many arbitrary constants as the order
• Particular solution: Obtained by substituting given conditions into the general solution

Chapter 10: Vector Algebra

• Magnitude: |a⃗| = √(a₁² + a₂² + a₃²)
• Unit vector: â = a⃗/|a⃗|
• Direction cosines: l = a₁/|a⃗|, m = a₂/|a⃗|, n = a₃/|a⃗|; l² + m² + n² = 1
• Dot product: a⃗ · b⃗ = a₁b₁ + a₂b₂ + a₃b₃ = |a⃗||b⃗| cos θ
• Cross product: a⃗ × b⃗ = |î ĵ k̂; a₁ a₂ a₃; b₁ b₂ b₃|; |a⃗ × b⃗| = |a⃗||b⃗| sin θ
• Projection: Projection of a⃗ on b⃗ = (a⃗ · b⃗)/|b⃗|
• Perpendicularity: a⃗ ⊥ b⃗ ⟺ a⃗ · b⃗ = 0
• Parallelism: a⃗ ∥ b⃗ ⟺ a⃗ × b⃗ = 0⃗

Chapter 11: Three-dimensional Geometry

• Direction cosines (l, m, n): cosines of angles with positive x, y, z axes; l² + m² + n² = 1
• Direction ratios: [a, b, c] where l = a/√(a²+b²+c²), etc.
• Equation of line (through point, parallel to direction): r⃗ = a⃗ + λb⃗ or (x−x₁)/a = (y−y₁)/b = (z−z₁)/c
• Equation of line through two points: (x−x₁)/(x₂−x₁) = (y−y₁)/(y₂−x₁) = (z−z₁)/(z₂−z₁)
• Shortest distance between skew lines: d = |[(a⃗₂−a⃗₁) · (b⃗₁×b⃗₂)] / |b⃗₁×b⃗₂||
• Angle between two lines: cos θ = |l₁l₂ + m₁m₂ + n₁n₂| = |b⃗₁ · b⃗₂|/(|b⃗₁||b⃗₂|)
• Coplanarity condition: [(a⃗₂−a⃗₁) b⃗₁ b⃗₂] = 0

Chapter 12: Linear Programming

• Objective function: Z = ax + by (to maximise or minimise)
• Constraints: Linear inequalities defining the feasible region
• Feasible region: Intersection of all constraint half-planes
• Corner Point Method: Evaluate Z at each vertex of the feasible region; optimum occurs at a corner
• Iso-profit/iso-cost lines: Parallel lines representing constant Z; slide to find optimum
• Unbounded region: May have no optimal value; verify by checking neighbouring points

Chapter 13: Probability

• Conditional probability: P(A|B) = P(A ∩ B)/P(B), P(B) ≠ 0
• Multiplication theorem: P(A ∩ B) = P(A) · P(B|A) = P(B) · P(A|B)
• Bayes' theorem: P(Eᵢ|A) = [P(Eᵢ) · P(A|Eᵢ)] / [Σ P(Eⱼ) · P(A|Eⱼ)]
• Independence: A and B independent ⟺ P(A ∩ B) = P(A) · P(B)
• Random variable: A real-valued function on the sample space
• Probability distribution: P(X = xᵢ) = pᵢ, where Σpᵢ = 1
• Mean: E(X) = μ = Σ xᵢpᵢ
• Variance: Var(X) = Σ xᵢ²pᵢ − (Σ xᵢpᵢ)² = E(X²) − [E(X)]²
• Binomial distribution: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, where q = 1 − p; Mean = np; Variance = npq
• Total probability: P(A) = Σ P(Eᵢ) · P(A|Eᵢ)

I. Choose the Best Answer (MCQs)

1. ∫ x · eˣ dx equals:

(A) eˣ(x + 1) + C   (B) eˣ(x − 1) + C   (C) eˣ(x/2) + C   (D) xeˣ + C

2. The value of ∫[0 to π/2] sin³x dx is:

(A) 2/3   (B) 1/3   (C) π/3   (D) 0

3. The order and degree of [d²y/dx²]³ + [dy/dx]² + y = 0 are respectively:

(A) 2, 3   (B) 3, 2   (C) 2, 2   (D) 1, 3

4. If a⃗ = 2î − 3ĵ + k̂ and b⃗ = î + ĵ − 2k̂, then a⃗ · b⃗ equals:

(A) 7   (B) −7   (C) −3   (D) 3

5. The direction cosines of the z-axis are:

(A) (1, 0, 0)   (B) (0, 1, 0)   (C) (0, 0, 1)   (D) (1, 1, 1)

6. The integrating factor of the differential equation dy/dx + y/x = x² is:

(A) x   (B) eˣ   (C) 1/x   (D) e^(1/x)

7. In a binomial distribution with n = 6 and p = 1/2, the mean is:

(A) 3   (B) 6   (C) 2   (D) 1

8. The area bounded by the curve y = x² and the line y = 4 is:

(A) 16/3   (B) 32/3   (C) 8/3   (D) 4

9. If P(A) = 1/2, P(B) = 1/3 and P(A ∩ B) = 1/6, then A and B are:

(A) Independent   (B) Mutually exclusive   (C) Dependent   (D) Complementary

10. The maximum value of Z = 3x + 2y subject to x + y ≤ 4, x ≥ 0, y ≥ 0 is:

(A) 8   (B) 12   (C) 6   (D) 10

II. Fill in the Blanks

1. ∫ dx/√(1 − x²) = sin⁻¹x + C

2. The degree of the differential equation [d²y/dx² + (dy/dx)³]^(2/3) = 0 is 2 (after raising both sides to power 3 to make it polynomial in derivatives, the equation becomes (d²y/dx²)² + higher terms = 0)

3. If a⃗ × b⃗ = 0⃗ and a⃗ ≠ 0⃗, b⃗ ≠ 0⃗, then the angle between a⃗ and b⃗ is 0° or 180° (vectors are parallel)

4. The value of ∫[−1 to 1] x³ dx is 0 (since x³ is an odd function, the integral over a symmetric interval is zero)

8. The variance of a binomial distribution with parameters n and p is npq where q = 1 − p

9. For a linear programming problem, the optimal value of the objective function occurs at a corner point (vertex) of the feasible region

10. The cross product a⃗ × b⃗ is maximum when the angle between a⃗ and b⃗ is 90° (π/2)

III. True or False

1. True. ∫[a to b] f(x) dx = ∫[a to b] f(t) dt — the variable of integration is a dummy variable.

2. False. The degree of a differential equation is always defined. → Correction: The degree is defined only when the differential equation is polynomial in its derivatives. For example, dy/dx + sin(dy/dx) = 0 has no defined degree.

3. True. Two vectors are perpendicular if and only if their dot product is zero.

4. False. The cross product of two vectors is commutative. → Correction: The cross product is anti-commutative: a⃗ × b⃗ = −(b⃗ × a⃗).

6. True. If P(A ∪ B) = P(A) + P(B), then A and B are mutually exclusive.

7. False. A linear programming problem always has a unique optimal solution. → Correction: An LPP can have a unique optimal solution, infinitely many optimal solutions, or no feasible solution at all.

8. True. For an odd function f(x), ∫[−a to a] f(x) dx = 0.

IV. Match the Following

V. Answer Briefly

VI. Answer in Detail

Common Mistakes to Avoid

1. Forgetting the constant of integration (+C) in indefinite integrals. Every indefinite integral must include "+ C". Board examiners deduct marks for this omission.

2. Wrong ILATE choice in integration by parts. Remember the priority order: Inverse trig > Logarithmic > Algebraic > Trigonometric > Exponential. Choosing the wrong function as u leads to a more complicated integral instead of a simpler one.

3. Mixing up order and degree of differential equations. Order = highest derivative present; Degree = power of the highest order derivative (only when the DE is polynomial in derivatives). The degree of dy/dx + sin(dy/dx) = 0 is NOT defined.

4. Using wrong limits in area problems. Always find the correct points of intersection before setting up the definite integral. Sketching the curves helps identify which function is on top.

5. Confusing dot product and cross product properties. Dot product gives a scalar; cross product gives a vector. Dot product is zero for perpendicular vectors; cross product is zero for parallel vectors. Do not interchange these conditions.

Exam Checklist

• [ ] Indefinite integrals: Every answer ends with "+ C" — no exceptions.
• [ ] Integration by parts: Applied ILATE rule correctly; verified by differentiating the result.
• [ ] Definite integrals: Substituted both limits (upper and lower) properly; checked for even/odd function shortcuts.
• [ ] Area problems: Drew a sketch, found points of intersection, identified the top function, and used correct limits.
• [ ] Differential equations: Checked that the DE is in standard form before finding the integrating factor; included the constant of integration before multiplying by IF.
• [ ] Vector/cross products: Wrote the determinant correctly (î, ĵ, k̂ in the first row); verified that a⃗ × b⃗ = −(b⃗ × a⃗).
• [ ] 3D geometry: Used direction cosines (not direction ratios) in angle formulae; verified l² + m² + n² = 1.
• [ ] Probability: Checked that all probabilities in a distribution sum to 1; used Bayes' theorem only when prior and conditional probabilities are both given.