Introduction

The NEB Grade XII Mathematics examination for 2083 (2026), Set ‘B’ (Sub Code 0081, New Course), was set for students whose registration numbers begin with 80, 81, or 82, carrying full marks of 75 across a 3-hour sitting. Unlike a subject built around memorized facts, Mathematics at this level tests whether students can apply formulas correctly under pressure, recognize which technique a question is really asking for, and carry a multi-step calculation through to a clean final answer without losing accuracy along the way — and this particular paper is a strong, well-rounded example of exactly that kind of testing.

What stands out immediately in this set is how many of its MCQs require an actual short calculation rather than pure recall. Working out the sine of a half-angle in a 3-4-5 triangle, determining the order of a differential equation from its integrating factor, and finding the time of flight of a projectile all demand that students apply a formula correctly within seconds, not just recognize it. The descriptive sections then build on this same expectation at greater length — an ambiguous-case triangle solution in Group B, a full simplex method optimization problem, and a related-rates calculus question involving water flowing into a conical container are all exactly the kind of multi-step, formula-heavy problems that reward calm, methodical working over rushed pattern-matching.

Algebra and calculus together account for well over half the paper’s marks, reflecting how central these two units are to the NEB Grade XII syllabus, but the paper doesn’t neglect anything — coordinate geometry (circles, parabolas, ellipses, and hyperbolas all appear), vectors, trigonometry, probability, statistics, linear programming, and even classical mechanics all make an appearance, giving students a genuinely comprehensive practice paper if they work through it in full.

In this article, we’ll break down the complete question pattern, map out exactly where the marks were concentrated unit by unit, work through the reasoning and solution for every MCQ, discuss the approach and key results for each descriptive question, and lay out a preparation strategy built specifically around what this paper demands. We’ve also answered the questions students most frequently ask about this exam.

Exam Overview

Full Question Paper

Question Pattern Analysis

This paper follows the standard three-group NEB structure used across Mathematics sets:

Group A — Multiple Choice Questions (11 × 1 = 11 marks) All 11 MCQs are compulsory, distributed separately 30 minutes after the exam begins, and span algebra, coordinate geometry, trigonometry, vectors, probability, calculus, and numerical methods. A genuinely high proportion of these require a short calculation to reach the answer rather than simple recall.

Group B — Short Answer Questions (8 × 5 = 40 marks) Questions 12 to 19 cover a mixed bag of quick conceptual statements (mathematical induction, De Moivre’s theorem, combination properties), word-and-complex-number problems, triangle solution and vector area calculations, conic section equations, correlation and regression statistics, a cluster of quick calculus facts, an L’Hospital’s rule limit combined with a differential equation, and finally a full linear programming problem with an alternative on classical mechanics.

Group C — Long Answer Questions (3 × 8 = 24 marks) Questions 20 to 22 test combinatorics and Cramer’s rule, a trigonometric identity proof alongside conic section and vector angle calculations, and a calculus-heavy question combining related rates, integration by partial fractions, and a differential equation solved through recognizing an exact differential.

A useful thread running through this paper is how often a seemingly small detail changes the entire approach to a question — recognizing that a 3-4-5 triangle is right-angled before computing a half-angle sine, noticing that xdy + ydx is simply the derivative of the product xy before attempting a differential equation, or realizing an SSA triangle setup has two valid solutions rather than one. This paper genuinely rewards careful reading over speed.

Topics Covered

Based on a full review of all 22 questions, this paper draws from the following core units of the NEB Grade XII Mathematics syllabus:

• Algebra — sums of powers of natural numbers, factorial properties, permutations and combinations, complex numbers and De Moivre’s theorem, mathematical induction, systems of linear equations and Cramer’s rule
• Trigonometry — solving triangles (including the ambiguous SSA case), half-angle formulas, trigonometric identities involving cotangent
• Coordinate Geometry / Conic Sections — circles, parabolas, ellipses, and hyperbolas, including tangents, foci, vertices, and directrices
• Vectors — projection of one vector onto another, cross product and area of a parallelogram, angle between two vectors
• Probability — conditional probability for dependent events
• Statistics — correlation coefficient and regression coefficients from raw bivariate data
• Calculus — limits (including L’Hospital’s rule), differentiation of hyperbolic and inverse hyperbolic functions, integration (standard forms and partial fractions), differential equations (variable separable, linear, and exact-differential recognition), related rates
• Linear Programming — the simplex method for constrained optimization
• Classical Mechanics — projectile motion, Newton’s second law, motion on an inclined plane
• Numerical Methods — interpreting row-reduced augmented matrices (Gauss elimination) to determine the nature of a solution set

Unit-Wise Weightage

Grouping the 75 marks by broad mathematical unit gives a clear picture of where this paper concentrated its emphasis:

Calculus and algebra together make up well over half the paper, which matches the general emphasis NEB places on these two units throughout the Grade XII Mathematics syllabus. Coordinate geometry, trigonometry, and vectors form a solid secondary tier, while statistics and linear programming each contribute a single but substantial question. Probability and numerical methods appear only briefly here, though this shouldn’t be read as a signal to under-revise them, since their weighting can shift meaningfully between different sets.

Unit-Wise Discussion

Algebra — Series, Factorials, and Complex Numbers (Q1, Q2, Q12, Q13, Q20): This unit spans a wide range, from the closed-form sum of cubes and factorial divisibility in the MCQs, to quick conceptual recall of induction, binomial expansion, and De Moivre’s theorem in Q12, and genuinely substantial problems in Q13 (a vowels-together word arrangement combined with finding the cube roots of a complex number) and Q20 (a restricted committee-selection combinatorics problem, an induction proof, and a Cramer’s rule word problem). This breadth reflects how central algebra remains throughout the Grade XII syllabus.

Trigonometry (Q4, Q14a, Q21a): The MCQ on finding sin(A/2) in a 3-4-5 triangle rewards students who recognize the classic Pythagorean triple immediately, since the triangle being right-angled significantly simplifies the semi-perimeter-based half-angle formula. Q14a introduces the ambiguous SSA case in triangle solving, where two distinct valid triangles satisfy the given information. Q21a asks for a full identity proof connecting cotangent half-angles to the sides of a triangle — a good test of comfort working with the semi-perimeter and inradius-based formulas for tangent and cotangent of half-angles.

Coordinate Geometry and Conic Sections (Q3, Q15, Q21b): The circle MCQ tests the straightforward but easily-forgotten fact that a circle touching the x-axis has a radius equal to its center’s y-coordinate. Q15 covers both a parabola tangent in parametric form and constructing a hyperbola’s equation from its focus and eccentricity, while Q21b asks students to identify and describe an ellipse’s vertices, foci, and directrices from its general equation — testing whether students can correctly identify which axis is major simply by comparing the denominators after normalizing the equation.

Vectors (Q5, Q14b, Q21c): The projection MCQ tests a frequently confused formula — the projection of a onto b uses the magnitude of b in the denominator, not the magnitude of a. Q14b combines the cross product with a slightly unusual twist, asking students to verify numerically whether the resulting parallelogram area happens to be a whole number. Q21c is a standard angle-between-vectors calculation using the dot product formula.

Probability (Q6): This conditional probability MCQ for dependent events is a direct application of the definition P(A/B) = P(A∩B)/P(B), and it’s worth noting that this formula holds regardless of whether the events are independent or dependent — the “dependent events” phrasing in the question is really just context, not an indication that a different formula is needed.

Calculus (Q7, Q8, Q9, Q10, Q17, Q18, Q22): This is the most heavily weighted single unit in the paper, spanning differentiation of hyperbolic functions, evaluating a 0/0 limit through factoring, solving two different first-order differential equations (one variable-separable, one linear with an integrating factor), a rapid-fire set of calculus facts in Q17, an L’Hospital’s rule limit combined with another differential equation in Q18, and a substantial Group C question combining a related-rates cone problem, a partial-fractions integration, and a differential equation solved by recognizing an exact differential form. Students who are comfortable moving fluently between these different calculus sub-skills will find this the most rewarding unit to have prepared thoroughly.

Statistics (Q16): This question requires computing both the correlation coefficient and the regression coefficient of X on Y from a small raw dataset — a calculation-heavy question that rewards careful, organized tabulation of the required sums (ΣX, ΣY, ΣXY, ΣX², ΣY²) rather than attempting to compute everything in one pass.

Linear Programming (Q19) and Mechanics (Q19-Or, Q11-Or): The main Q19 question is a complete simplex-method (or equivalent corner-point) optimization problem with two constraints, while its alternative shifts entirely to classical mechanics, covering Newton’s second law and motion on a smooth inclined plane. Q11’s alternative similarly swaps a numerical methods question for a projectile motion time-of-flight calculation — a good reminder that NEB “Or” choices sometimes span entirely different units, not just different questions within the same one.

Numerical Methods (Q11): This MCQ tests whether students can correctly interpret a row-echelon form augmented matrix to determine the nature of a linear system’s solution set — recognizing that a row of all zeros (including in the constant column) signals a dependent system with infinitely many solutions, rather than no solution or a unique solution.

Difficulty Analysis

This paper sits at a moderate-to-high difficulty level typical of NEB Grade XII Mathematics, with its difficulty coming less from any single advanced concept and more from the sheer breadth of formulas and techniques a student must keep readily accessible. Group A cannot be treated as a quick warm-up, since several MCQs (the half-angle sine calculation, the differential equation solution, the integrating factor derivation) require genuine short calculations under time pressure. Group B is comfortably scorable for well-prepared students, though Q13’s cube-root-of-a-complex-number problem and Q14’s ambiguous-case triangle both demand careful, multi-step work rather than a single formula application. Group C is the most demanding section — Q19’s simplex method problem requires systematically checking every feasible corner point, and Q22’s cone-related-rates problem combined with its partial-fractions integration is genuinely one of the more calculation-intensive questions across the whole paper, rewarding students who can maintain accuracy across several algebraic steps in a row without shortcuts.

MCQ Explanations (Group A)

Q1. Closed-form expression for ∑n³ The correct answer is n²(n+1)²/4. This is the well-known identity that the sum of the first n cubes equals the square of the sum of the first n natural numbers, i.e., ∑n³ = [n(n+1)/2]² = n²(n+1)²/4.

Q2. HCF of n!, (n+1)!, and (n+2)! The correct answer is n!. Since (n+1)! = (n+1) × n! and (n+2)! = (n+2)(n+1) × n!, each factorial is simply n! multiplied by additional integers, meaning n! is a common factor of all three, and no larger common factor exists since n! itself doesn’t divide any larger common multiplier evenly across all three terms.

Q3. Equation of a circle touching the x-axis with center (3,4) The correct answer is x² + y² − 6x − 8y + 9 = 0. Since the circle touches the x-axis, its radius must equal the y-coordinate of its center, which is 4. Writing (x−3)² + (y−4)² = 4² and expanding gives x² + y² − 6x − 8y + 9 = 0.

Q4. Value of sin(A/2) in a 3-4-5 triangle The correct answer is 1/√10. With a = 3, b = 4, c = 5, the semi-perimeter s = 6, giving sin(A/2) = √[(s−b)(s−c)/(bc)] = √[(2)(1)/(4×5)] = √(1/10) = 1/√10.

Q5. Formula for the projection of vector a on vector b The correct answer is (a·b)/|b|. The projection of a onto b is found by dividing the dot product by the magnitude of the vector being projected onto — here, b — not the magnitude of a.

Q6. Conditional probability P(A/B) for dependent events The correct answer is 3/5. Using P(A/B) = P(A∩B)/P(B) = (1/5)/(1/3) = 3/5.

Q7. Derivative of log(cosh(x/a)) The correct answer is (1/a)tanh(x/a). Differentiating using the chain rule, the derivative of log(cosh(x/a)) is (1/cosh(x/a)) × sinh(x/a) × (1/a) = (1/a)tanh(x/a).

Q8. Value of the limit as x→1 of (x³−1)/(x²−1) The correct answer is 3/2. Factoring both numerator and denominator, (x−1)(x²+x+1)/[(x−1)(x+1)] simplifies to (x²+x+1)/(x+1), which evaluates to 3/2 at x = 1.

Q9. Solution of ydx − (x/2)dy = 0 The correct answer is x = c√y. Separating variables gives dx/x = dy/(2y), and integrating both sides leads to ln x = (1/2)ln y + C, which rearranges to x = c√y.

Q10. Integrating factor for dy/dx − y/x = x² The correct answer is 1/x. Comparing with the standard linear form dy/dx + P(x)y = Q(x), here P(x) = −1/x, so the integrating factor is e^∫(−1/x)dx = e^(−ln x) = 1/x.

Q11. Nature of the solution from a row-reduced augmented matrix The correct answer is Infinitely many solutions. Since the last row reduces entirely to zeros, including the constant term (0 = 0), this represents a dependent equation that provides no new information, leaving one variable free and resulting in infinitely many solutions.

Q11-Or. Time of flight of a projectile The correct answer is 40/g. Using the standard formula for time of flight, T = 2u sinθ/g = 2(40)(sin30°)/g = 2(40)(0.5)/g = 40/g.

Group B and C Answer Discussion

Q12 (Quick conceptual statements): The principle of mathematical induction states that if a statement holds for the base case and its truth for any value k guarantees its truth for k+1, then it holds for all natural numbers from the base case onward. The general term of (x+a)ⁿ is given by T_(r+1) = ⁿCᵣ x^(n−r) aʳ. A system of linear equations becomes inconsistent when the rank of the coefficient matrix differs from the rank of the augmented matrix, meaning no solution satisfies all equations simultaneously — geometrically, this corresponds to parallel lines or planes that never meet. De Moivre’s theorem for a positive integral power n states that (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). A key property of combinations is the symmetry relation ⁿCᵣ = ⁿC₍ₙ₋ᵣ₎.

Q13 (Word arrangement and cube roots): Treating the three vowels in “DAUGHTER” (A, U, E) as a single block alongside the five consonants gives 6 units to arrange in 6! ways, multiplied by 3! ways to arrange the vowels within their block, giving 6! × 3! = 720 × 6 = 4320 total words. For the cube roots of −1/2 + (√3/2)i, converting to polar form gives modulus 1 and argument 120°, and applying De Moivre’s theorem for cube roots yields three roots at angles 40°, 160°, and 280°, each with modulus 1 — specifically cos40°+isin40°, cos160°+isin160°, and cos280°+isin280°.

Q14 (Ambiguous triangle case and vector cross product): With a = 2, b = √6, and A = 45°, applying the sine rule gives sinB = √3/2, which has two valid solutions in this SSA setup: B = 60° (giving C = 75° and c = √3+1) or B = 120° (giving C = 15° and c = √3−1) — both triangles are genuinely valid given the information provided. For the vectors a = 2i−3j+4k and b = −i+5j+2k, computing the cross product gives a×b = −26i −8j +7k, so the parallelogram’s area is |a×b| = √(26²+8²+7²) = √789, which is not a whole number (√789 ≈ 28.09 square units).

Q15 (Parabola tangent and hyperbola equation): The tangent to y² = 4ax at the parametric point (at₁², 2at₁) is given by t₁y = x + at₁². For the hyperbola with focus at (−7,0) and eccentricity 7/4, since ae = 7 and e = 7/4, solving gives a = 4, and then b² = a²(e²−1) = 16 × (33/16) = 33, giving the equation x²/16 − y²/33 = 1.

Q16 (Correlation and regression): Organizing the height (X) and weight (Y) data into the required sums — ΣX = 441, ΣY = 447, ΣXY = 28180, ΣX² = 27811, ΣY² = 28571 — and applying the standard correlation coefficient formula gives r ≈ 0.69, indicating a moderately strong positive correlation between height and weight in this sample. The regression coefficient of X on Y, calculated as [nΣXY − ΣXΣY]/[nΣY² − (ΣY)²], comes out to approximately 0.71.

Q17 (Quick calculus facts): For a particle moving along S = f(t), velocity is v = ds/dt = f'(t). The derivative of sinh⁻¹x with respect to x is 1/√(x²+1). The integral of cosh x dx is sinh x + C. The integral of dx/(x²−a²) is (1/2a)ln|(x−a)/(x+a)| + C. For the differential equation d²y/dx² + y(dy/dx) = 5x (as corrected in the exam’s own printed note), the order is 2, since the highest derivative present is the second derivative.

Q18 (L’Hospital’s rule and differential equation): Applying L’Hospital’s rule twice to the limit of (x−sinx)/x² as x→0 (since it’s initially in 0/0 form, and remains so after the first differentiation) eventually gives sinx/2, which evaluates to 0 as x→0. For the equation ∛(dy/dx + y/x) = x, cubing both sides gives the linear differential equation dy/dx + y/x = x³; using the integrating factor x (derived the same way as in Q10, but with a positive coefficient here), the solution works out to y = x⁴/5 + C/x.

Q19 (Simplex method / Or: Newton’s second law and inclined plane): For the linear programming problem maximizing Z = 8x + 36y subject to 2x+6y ≤ 30 and x+6y ≤ 18 (with x,y ≥ 0), evaluating Z at each corner point of the feasible region — (0,0), (15,0), (12,1), and (0,3) — shows the maximum value Z = 132 occurs at (x,y) = (12,1). In the alternative, a 10 kg mass starting from rest and reaching 20 m/s in 5 seconds has an acceleration of 4 m/s² (from v = u + at), giving a force of F = ma = 40 N; separately, a 5 kg mass sliding down a smooth 30° incline experiences an acceleration of g sin30° = g/2 (about 4.9 m/s² using g = 9.8 m/s²), with a normal reaction force of mg cos30° ≈ 42.4 N.

Q20 (Combinatorics, induction, and Cramer’s rule): For a 5-member committee from 6 boys and 4 girls with at most 3 girls allowed, subtracting the disallowed case (exactly 4 girls and 1 boy, which gives C(4,4)×C(6,1) = 6 ways) from the total unrestricted selections (C(10,5) = 252) gives 252 − 6 = 246 valid ways. The induction proof for 1+3+5+…+(2n−1) = n² follows the standard pattern: verifying the base case n=1, then showing that adding (2k+1) to both sides of the assumed true statement for n=k produces exactly (k+1)² on the right side, completing the inductive step. For the pen-and-book pricing problem, setting up 2x+3y=1100 and 4x+y=700 and applying Cramer’s rule gives a pen cost of Rs. 100 and a book cost of Rs. 300.

Q21 (Triangle identity, ellipse, and vector angle): The identity (c+a−b)[cot(C/2)+cot(A/2)] = 2b·cot(B/2) can be proven by expressing each half-angle cotangent in terms of the semi-perimeter s and inradius r (cot(A/2) = (s−a)/r, and similarly for B and C), which after simplification reduces both sides to the same expression, 2b(s−b)/r, confirming the identity. For the conic 9x²+4y²=36, dividing through by 36 gives x²/4 + y²/9 = 1, an ellipse with its major axis along the y-axis (since 9 > 4); this gives vertices at (0,±3), foci at (0,±√5), and directrices at y = ±9/√5. For the angle between a = i+2j−k and b = i−j+k, the dot product formula gives cosθ = −2/(√6×√3) = −√2/3, so θ ≈ 118.1°.

Q22 (Related rates, integration, and differential equation): For the inverted conical container with height 24 cm and top radius 16 cm, similar triangles give the water’s radius-to-height ratio as 2/3 at any depth, so the volume as a function of depth h is V = (4π/27)h³; differentiating with respect to time and substituting dV/dt = 8 cm³/s and h = 12 cm gives a rate of depth change of dh/dt = 1/(8π) cm/s. The integral of (y²+1)/[(y²+49)(y²+4)] dy is solved using partial fractions in terms of y², giving the result (16/105)tan⁻¹(y/7) − (1/30)tan⁻¹(y/2) + C. Finally, recognizing that xdy + ydx is exactly the differential of the product xy, the equation xdy + ydx = √(x²+a²)dx integrates directly to xy = (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C, using the standard integral formula for √(x²+a²).

Student Preparation Tips

1. Master the standard series and factorial identities cold. Sums like ∑n, ∑n², and ∑n³ appear across many NEB Mathematics papers in various disguises, and factorial-based HCF/divisibility reasoning is a recurring MCQ style worth practicing until it’s instant.
2. Practice recognizing the ambiguous SSA case in triangle problems. Whenever you’re given two sides and a non-included angle, always check whether more than one valid triangle satisfies the given information, as Q14 demonstrates clearly.
3. Build a personal reference sheet of conic section formulas. Circles, parabolas, ellipses, and hyperbolas each have their own vertex, focus, and directrix formulas, and this paper alone tested three different conic types — keeping them organized by type prevents formula mix-ups under time pressure.
4. Get comfortable computing correlation and regression coefficients from raw data quickly. Organizing the five key sums (ΣX, ΣY, ΣXY, ΣX², ΣY²) in a clear table before plugging into formulas saves significant time and reduces arithmetic errors.
5. Practice both variable-separable and linear differential equations until the choice of method is automatic. This paper tested both types, and recognizing which form you’re looking at immediately (rather than trying one method and switching) saves valuable exam time.
6. Learn to recognize exact differentials like d(xy) = xdy + ydx on sight. Q22’s differential equation becomes almost trivial once this pattern is spotted, but very difficult if approached as a standard separable or linear equation instead.
7. Practice simplex/corner-point method problems by checking every feasible vertex systematically. Skipping a vertex or miscalculating one intersection point is the most common way marks are lost in linear programming questions.
8. Revise both linear programming and classical mechanics, since “Or” questions can span entirely different units. As this paper shows with Q19 (linear programming versus mechanics) and Q11 (numerical methods versus projectile motion), an “Or” choice sometimes offers a completely different topic rather than just a different angle on the same one.

Frequently Asked Questions

Q1. What is the full marks and time duration for the NEB Grade XII Mathematics 2083 Set B paper? The paper carries full marks of 75 and is to be completed within 3 hours, with the MCQ section (Group A) distributed 30 minutes after the exam begins.

Q2. How many groups are there in this Mathematics paper, and how are marks distributed? There are three groups: Group A (11 MCQs, 1 mark each = 11 marks), Group B (8 short-answer questions, 5 marks each = 40 marks), and Group C (3 long-answer questions, 8 marks each = 24 marks).

Q3. Which unit carries the highest weightage in this Mathematics paper? Calculus — covering limits, differentiation, integration, differential equations, and related rates — carries the highest weightage at roughly 29%, closely followed by algebra at around 27%.

Q4. Are “Or” choice questions common in NEB Mathematics, and how should students use them? Yes, this paper includes “Or” choices in Q11 and Q19, and notably, both offer an alternative from a completely different unit rather than just a different version of the same topic. Students should quickly assess which unit they’re more confident in before committing to one option.

Q5. What kind of calculus questions should students expect based on this paper? Expect a genuine mix — quick-fire conceptual facts (derivatives and integrals of standard functions), full differential equation solutions using both variable-separable and linear methods, L’Hospital’s rule limits, and applied related-rates problems involving geometric shapes like cones.

Q6. How important is coordinate geometry for the NEB Mathematics board exam? Quite important — this paper alone tests circles, parabolas, ellipses, and hyperbolas, showing that all major conic sections are fair game in a single sitting, so revision should cover each conic type’s full formula set (equation, vertex, focus, directrix, tangent) rather than favoring just one or two.

Q7. Is this question paper suitable for practice by students preparing for a different set (A, C, etc.)? Yes. While registration numbers determine which exact set a student receives, the syllabus, unit weightage, and overall difficulty level are consistent across sets, so this Set B paper remains valuable practice material regardless of which set a student is actually assigned.

Conclusion

The NEB Grade XII Mathematics 2083 Set B paper reflects a well-balanced test of the full Grade XII syllabus, with calculus and algebra together accounting for over half the total marks, while coordinate geometry, trigonometry, vectors, statistics, linear programming, probability, and mechanics all still make meaningful appearances. What distinguishes this particular paper is how consistently it rewards careful reading and recognition of the “right approach” over quick pattern-matching — from spotting the ambiguous SSA triangle case, to recognizing an exact differential hiding inside what looks like a standard differential equation. Students preparing for their own examination should use the MCQ reasoning and descriptive answer discussion above as a template for practicing this style of question, since building genuine fluency across every unit — rather than mastering only a few favorite topics — is clearly what this kind of comprehensive paper is designed to reward.