MP Board Class 12 Mathematics Solved Paper 2024 — Question-by-Question Analysis for 2027 Exam
📝 MP Board Class 12 Mathematics Solved Paper 2024 — Question-by-Question Analysis
The MP Board Class 12 Higher Mathematics exam 2024 (Set A, Code I0770) tested students on a comprehensive range of topics from the NCERT-based syllabus. This solved paper provides step-by-step solutions with marking schemes, common mistakes to avoid, and expert tips to help you score maximum marks in the 2027 board exam.
📋 Paper Overview
Exam: MP Board Class 12 Higher Mathematics 2024
Paper Code: I0770, Set A
Total Marks: 100 | Time: 3 Hours
Passing Marks: 33 | Level: Moderate
📑 Table of Contents
- 📊 Section A — Objective Questions (1 Mark Each)
- ✏️ Section B — Very Short Answer (2 Marks Each)
- 📐 Section C — Short Answer (3 Marks Each)
- 🧮 Section D — Long Answer (4 Marks Each)
- 🎯 Section E — Very Long Answer (6 Marks Each)
- 💡 Answer Key & Marking Scheme
- 🎓 Key Exam Tips for Mathematics
- ❓ Frequently Asked Questions
📊 Section A — Objective Questions (1 Mark Each)
This section contains 10 multiple-choice questions (Q.1–Q.10), each carrying 1 mark. Choose the correct option.
Q.1 If A = {1, 2, 3}, then number of equivalence relations containing (1, 2) is:
(A) 1 (B) 2 (C) 3 (D) 4
✅ Solution: (B) 2
Explanation: Since (1,2) must be in the equivalence relation, we need the smallest equivalence relation containing (1,2). Due to symmetry, (2,1) must be included. Due to transitivity, (1,1) and (2,2) must also be included. The possible equivalence relations are: R₁ = {(1,1), (2,2), (3,3), (1,2), (2,1)} and R₂ = {(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)}. Hence 2 such relations exist.
Q.2 sin⁻¹(sin(2π/3)) = ?
(A) 2π/3 (B) π/3 (C) -π/3 (D) π/2
✅ Solution: (B) π/3
Explanation: Principal value branch of sin⁻¹ is [-π/2, π/2]. Since 2π/3 ∉ [-π/2, π/2], we use sin(2π/3) = sin(π – π/3) = sin(π/3), and π/3 ∈ [-π/2, π/2]. So sin⁻¹(sin(2π/3)) = π/3.
Q.3 If A = [aᵢⱼ] is a 3×3 matrix such that aᵢⱼ = i + j, then A is:
(A) Symmetric (B) Skew-symmetric (C) Diagonal (D) Scalar
✅ Solution: (A) Symmetric
Explanation: aᵢⱼ = i + j, so aⱼᵢ = j + i = i + j = aᵢⱼ. Hence Aᵀ = A, making it a symmetric matrix. The elements would be: A₁₁=2, A₁₂=3, A₂₁=3…
Q.4 ∫₋₁¹ |x| dx = ?
(A) 0 (B) 1 (C) -1 (D) 2
✅ Solution: (B) 1
Explanation: |x| is an even function. ∫₋₁¹ |x| dx = 2∫₀¹ x dx = 2[x²/2]₀¹ = 2(½) = 1. Alternatively, the area under |x| from -1 to 1 forms two triangles of area ½ each, total = 1.
Q.5 The order of the differential equation (d²y/dx²)³ + (dy/dx)⁴ = x² is:
(A) 3 (B) 4 (C) 2 (D) 1
✅ Solution: (C) 2
Explanation: Order of a differential equation is the highest order derivative present. Here d²y/dx² is the highest derivative (second order), so the order is 2. The power/exponent does not affect the order.
💡 Quick Tip: In MP Board exams, the first 10 MCQs often test conceptual clarity — focus on definitions, properties, and standard formulas. Don’t rush; each mark counts!
✏️ Section B — Very Short Answer Questions (2 Marks Each)
Attempt any 5 out of 7 questions. Each carries 2 marks.
Q.11 Find the principal value of tan⁻¹(√3) − sec⁻¹(−2).
✅ Solution:
tan⁻¹(√3) = π/3 (principal value branch: (-π/2, π/2))
sec⁻¹(−2) = cos⁻¹(−½) = π − cos⁻¹(½) = π − π/3 = 2π/3 (principal value branch: [0,π] − {π/2})
Answer: π/3 − 2π/3 = −π/3
Q.12 Evaluate: ∫₀^{π/2} sin²x dx.
✅ Solution:
∫₀^{π/2} sin²x dx = ∫₀^{π/2} (1−cos2x)/2 dx
= ½ [x − (sin2x)/2]₀^{π/2}
= ½ [(π/2 − 0) − (0 − 0)]
= ½ × π/2 = π/4
⚡ Common Mistake Alert!
Many students forget to apply the identity sin²x = (1−cos2x)/2 and try to integrate directly. Remember: Use standard trigonometric identities whenever you see squares of sine or cosine in integrals!
📐 Section C — Short Answer Questions (3 Marks Each)
Attempt any 5 out of 8 questions. Show all working steps for partial credit.
Q.18 Find the value of k so that the function f is continuous at x = 0:
f(x) = { (sin kx)/x, if x ≠ 0; 2, if x = 0 }
✅ Solution:
For continuity at x = 0: lim_{x→0} f(x) = f(0)
lim_{x→0} (sin kx)/x = k × lim_{x→0} (sin kx)/(kx) = k × 1 = k
f(0) = 2
∴ k = 2
Q.20 Solve the differential equation: dy/dx + y = e⁻ˣ
✅ Solution:
This is a linear differential equation of the form dy/dx + Py = Q
Here P = 1, Q = e⁻ˣ
Integrating factor (IF) = e^{∫P dx} = e^{∫1 dx} = eˣ
Solution: y·eˣ = ∫e⁻ˣ·eˣ dx = ∫1 dx = x + C
∴ y = xe⁻ˣ + Ce⁻ˣ
🧮 Section D — Long Answer Questions (4 Marks Each)
Attempt any 3 out of 5 questions. Step-by-step working is essential.
Q.23 Find the inverse of the matrix A = [[1, −1, 2], [0, 2, −3], [3, −2, 4]] using elementary row transformations.
✅ Solution (Outline):
Step 1: Write A = IA: [A|I] = [1 −1 2 | 1 0 0; 0 2 −3 | 0 1 0; 3 −2 4 | 0 0 1]
Step 2: R₃ → R₃ − 3R₁: [1 −1 2 | 1 0 0; 0 2 −3 | 0 1 0; 0 1 −2 | −3 0 1]
Step 3: R₂ → ½R₂: [1 −1 2 | 1 0 0; 0 1 −3/2 | 0 ½ 0; 0 1 −2 | −3 0 1]
Step 4: R₃ → R₃ − R₂: [1 −1 2 | 1 0 0; 0 1 −3/2 | 0 ½ 0; 0 0 −½ | −3 −½ 1]
Step 5: R₃ → −2R₃; then back-substitute to get A⁻¹ = [ [2, 0, −1], [−9, 2, 3], [−6, 1, 2] ]
🎯 Section E — Very Long Answer Questions (6 Marks Each)
Attempt any 2 out of 4 questions. Detailed derivations expected.
Q.28 Evaluate: ∫₀^{π/2} log(sin x) dx.
✅ Solution:
Let I = ∫₀^{π/2} log(sin x) dx …(1)
Using the property ∫₀^a f(x)dx = ∫₀^a f(a−x)dx:
I = ∫₀^{π/2} log(sin(π/2 − x)) dx = ∫₀^{π/2} log(cos x) dx …(2)
Adding (1) and (2): 2I = ∫₀^{π/2} [log(sin x) + log(cos x)] dx = ∫₀^{π/2} log(sin x·cos x) dx
= ∫₀^{π/2} log((sin 2x)/2) dx = ∫₀^{π/2} log(sin 2x) dx − ∫₀^{π/2} log 2 dx
Put 2x = t, dx = dt/2, limits: x=0→t=0, x=π/2→t=π:
= ½∫₀^π log(sin t) dt − (π/2)log 2
Using ∫₀^π log(sin t) dt = 2∫₀^{π/2} log(sin t) dt = 2I:
= I − (π/2)log 2
Hence 2I = I − (π/2)log 2 ⇒ I = −(π/2)log 2
💡 Answer Key & Marking Scheme
*Theory/internal assessment carries the remaining marks to total 100. Students must attempt exactly 25 questions out of 34.
Quick MCQ Answer Key
🎓 Key Exam Tips for Mathematics
- 📖 Master NCERT First: 90% of MP Board questions are directly from NCERT textbook examples and exercises. Solve every NCERT example at least twice.
- ⏱️ Time Management: Allocate 30 min for Section A, 45 min for B+C, 60 min for D+E, and 45 min for revision. Never spend more than 10 min on one question.
- 📝 Show All Steps: MP Board awards partial marks for correct steps even if the final answer is wrong. Never skip the middle working.
- 🔢 Formula Sheet: Create a one-page formula summary for integration, differentiation, trigonometry, and vectors. Revise it daily.
- ⚡ Common Pitfalls: Watch out for sign errors in integration, forgetting the constant C, and misapplying inverse trigonometric principal values.
- 🎯 Choice Questions: In sections with “any 5 out of 7”, quickly scan all questions first and pick the ones you’re most confident about. Don’t attempt questions you’re unsure of.
📘 Chapter-wise Weightage Analysis (2024 Paper)
Top scoring chapters: Calculus (Differentiation + Integration) ~35 marks, Algebra (Matrices, Determinants) ~20 marks, Vectors & 3D Geometry ~15 marks, Probability ~10 marks, Linear Programming ~6 marks, Relations & Functions ~8 marks, Differential Equations ~6 marks.
❓ Frequently Asked Questions
Q: Is solving previous year papers enough to score 90+ in MP Board Maths?
Yes, combined with thorough NCERT study. Solving the last 5 years’ papers familiarizes you with the exact question pattern, difficulty level, and marking scheme. Aim to solve at least 10 previous year papers under timed conditions.
Q: How many questions should I attempt in the optional sections?
Always attempt only the minimum required (e.g., 5 out of 7 in Section B). Don’t waste time attempting extra questions — focus on getting every attempted question fully correct.
Q: What if I get stuck on a 6-mark question?
Write whatever you know — relevant formulas, partial steps, correct setup. MP Board awards partial marks generously. Never leave a question completely blank.
Q: Is the 2024 paper easier than 2023?
The 2024 paper was slightly more conceptual. While 2023 had more direct formula-based questions, 2024 introduced more application-based problems, especially in Calculus and Vectors. This trend is expected to continue for 2027.
Q: Can I use a calculator in the exam?
No, calculators are NOT permitted in MP Board Mathematics exams. All calculations must be done manually. Practice mental arithmetic and approximation techniques.
Q: Where can I download the original 2024 MP Board Maths paper PDF?
You can download the official MP Board 2024 Higher Mathematics question paper (Set A, Code I0770) from mpboardonline.com or the official MPBSE website at mpbse.nic.in.
🎯 Final Words
Mathematics in MP Board is all about practice and presentation. Solve at least 2 questions from each chapter daily in the months leading up to the exam. Write neatly, show every step, and mark your final answer with a box. With consistent effort, scoring 90+ in Maths is absolutely achievable. Good luck!
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