Introduction

The NEB Grade XII board examination is the single most important assessment in a Nepali higher secondary student’s academic journey, and Mathematics remains one of the subjects where careful, methodical practice pays off more visibly than almost anywhere else. The 2082 (2025) Mathematics paper (Sub. Code 0081, New Course, Set R) carries 75 full marks across three groups, and it does an excellent job of sampling nearly every major branch of the Grade XII syllabus — algebra, trigonometry, coordinate geometry, vectors, probability, statistics, differential calculus, integral calculus, differential equations, linear programming, and mechanics all make an appearance.

What makes this paper particularly useful as a study resource is its balance. Group A’s eleven MCQs test whether students have internalized standard formulas correctly — the relationship between permutations and combinations, Cramer’s rule for solving simultaneous equations, the tangent condition for a line and circle, and the integrating factor of a linear differential equation are all classic formula-recall questions that reward precise memorization. Group B and Group C then shift toward genuine problem-solving: proving trigonometric identities in a triangle, working with vector projections, computing correlation and regression from raw tabulated data, applying the simplex method to a linear programming problem, and solving differential equations by variable separation.

This blend is exactly what NEB rewards in the actual board exam — a mix of quick, formula-based marks that students can pick up efficiently if their basics are solid, and longer, multi-step problems that test whether a student can actually work through calculus, coordinate geometry, or statistics methodically under time pressure. The statistics question (Q16), which asks students to calculate a correlation coefficient and then use regression to estimate a value, is a particularly good example of a question type that recurs almost every year in some form and is worth mastering thoroughly.

This article works through every MCQ and subjective question with brief solutions, breaks the paper down by chapter and topic, estimates where the marks concentrate, and offers focused preparation advice — so that this paper becomes a genuine study tool rather than just something to check answers against.

Quick Answer Key (Group A — Multiple Choice Questions)

Full Question Paper

Brief explanation of each MCQ:

1. Since [latex]P(n,r) = n!/(n-r)![/latex] counts arrangements and [latex]C(n,r) = n!/[r!(n-r)!][/latex] counts selections, multiplying a combination by the r! ways to arrange those r chosen items gives the number of arrangements: [latex]C(n,r) \times r! = P(n,r)[/latex].
2. By Cramer’s rule, y is found by replacing the y-column of the coefficient determinant with the constants column, then dividing by the original coefficient determinant — giving [latex]y = \frac{\begin{vmatrix}p_1 & r_1\p_2 & r_2\end{vmatrix}}{\begin{vmatrix}p_1 & q_1\p_2 & q_2\end{vmatrix}}[/latex].
3. Using the standard half-angle identity, [latex]\cos\frac{B}{2} = \sqrt{\frac{s(s-b)}{ac}}[/latex], so taking the reciprocal gives [latex]\sec\frac{B}{2} = \sqrt{\frac{ac}{s(s-b)}}[/latex].
4. For the line [latex]y=mx+c[/latex] to touch the circle [latex]x^2+y^2=a^2[/latex], the perpendicular distance from the origin to the line must equal the radius, giving [latex]\frac{|c|}{\sqrt{1+m^2}} = a[/latex], or [latex]c = \pm a\sqrt{1+m^2}[/latex].
5. Computing the determinant for [latex]\vec{a}\times\vec{b}[/latex] with [latex]\vec{a}=(0,0,1)[/latex] and [latex]\vec{b}=(0,1,0)[/latex] gives latex = (-1,0,0)[/latex].
6. [latex]P(B/A) = \frac{P(A\cap B)}{P(A)} = \frac{1/5}{1/3} = \frac{3}{5}[/latex].
7. For [latex]y=3x^2-x[/latex], [latex]\frac{dy}{dx}=6x-1[/latex]; at x=-1, slope of tangent = -7, so slope of normal = [latex]-\frac{1}{-7}=\frac{1}{7}[/latex].
8. Writing [latex]9x^2+1[/latex] as latex^2+1[/latex] and substituting u=3x (du=3dx) gives [latex]\int\frac{dx}{9x^2+1} = \frac{1}{3}\tan^{-1}(3x)+C[/latex].
9. A differential equation [latex]\frac{dy}{dx}=f(x,y)[/latex] is homogeneous only when the numerator and denominator of f(x,y) have the same degree; among the options, only [latex]\frac{x^2+y^2}{x^2-y^2}[/latex] has matching degree-2 numerator and denominator.
10. Rewriting [latex]\cos^2x\frac{dy}{dx}=1-y[/latex] as [latex]\frac{dy}{dx}+y\sec^2x=\sec^2x[/latex] (standard linear form), the integrating factor is [latex]e^{\int\sec^2x,dx}=e^{\tan x}[/latex].
11. Multiplying [latex]x-2y=1[/latex] by 3 gives [latex]3x-6y=3[/latex]; subtracting this from [latex]3x+4y=13[/latex] eliminates x and gives [latex]10y=10[/latex].
    11-OR. Maximum height [latex]=\frac{(u\sin\theta)^2}{2g} = \frac{(40\times0.5)^2}{2\times10} = \frac{400}{20}=20,m[/latex].

Brief Solutions to Group B & C Questions

Q12:
a. [latex]\log_e(1-x) = -\left(x+\frac{x^2}{2}+\frac{x^3}{3}+\cdots\right)[/latex] for |x|<1
b. Total permutations of n distinct elements = [latex]n![/latex]
c. De Moivre’s theorem: latex^n = \cos(n\theta)+i\sin(n\theta)[/latex] for any integer n
d. Sum of cubes of first n natural numbers = [latex]\left[\frac{n(n+1)}{2}\right]^2[/latex]
e. Augmented matrix for [latex]3x+2y=1,;4x+y=3[/latex]: [latex]\begin{bmatrix}3 & 2 & | & 1\4 & 1 & | & 3\end{bmatrix}[/latex]

Q13:
a. [latex]C(6,3)=20[/latex]. Since [latex]C(a,2)\times20=120[/latex], [latex]C(a,2)=6[/latex], so [latex]\frac{a(a-1)}{2}=6[/latex], giving [latex]a=4[/latex].
b. Since the two square roots are negatives of each other, the original complex number is [latex]z=(\sqrt3+i)^2 = 3+2\sqrt3i-1=2+2\sqrt3i[/latex]. Modulus [latex]=\sqrt{4+12}=4[/latex]; argument [latex]=\tan^{-1}(\sqrt3)=60°[/latex]. Polar form: [latex]z=4(\cos60°+i\sin60°)[/latex].

Q14:
a. Using the sine rule and half-angle substitutions, expanding [latex]p\sin^2\frac{R}{2}+r\sin^2\frac{P}{2}=\frac{q}{2}[/latex] with [latex]\sin^2\frac{\theta}{2}=\frac{1-\cos\theta}{2}[/latex] and the projection formula leads to [latex]p+r=2q\cdot\text{(ratio term)}[/latex] simplifying to [latex]2q=p+r[/latex], which is precisely the condition for p, q, r to be in arithmetic progression.
b. Projection of b on a [latex]=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|}[/latex]. [latex]\vec{a}\cdot\vec{b}=12+6+8=26[/latex], [latex]|\vec{a}|=\sqrt{16+9+4}=\sqrt{29}[/latex]. Projection [latex]=\frac{26}{\sqrt{29}}[/latex].

Q15:
a. Rewriting [latex]3x^2-4y^2-6x=0[/latex] as [latex]3(x-1)^2-4y^2=3[/latex], then dividing by 3 gives [latex]\frac{(x-1)^2}{1}-\frac{y^2}{3/4}=1[/latex] — a hyperbola with [latex]a^2=1,,b^2=\frac34[/latex]. Eccentricity [latex]e=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{1+\frac34}=\frac{\sqrt7}{2}[/latex].
b. With major axis = 4× minor axis, [latex]a=2b[/latex]. Substituting the point (4,2) into [latex]\frac{x^2}{4b^2}+\frac{y^2}{b^2}=1[/latex] gives [latex]\frac{16}{4b^2}+\frac{4}{b^2}=1[/latex], so [latex]b^2=8,,a^2=32[/latex]. Eccentricity [latex]e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{1-\frac14}=\frac{\sqrt3}{2}[/latex].

Q16:
a. Using deviations from the means (X̄=58, Ȳ=80): [latex]\Sigma dxdy=595,,\Sigma dx^2=446,,\Sigma dy^2=850[/latex]. Correlation coefficient [latex]r=\frac{595}{\sqrt{446\times850}}\approx0.97[/latex] — indicating a very strong positive relationship between supply and price.
b. Regression equation of X on Y: [latex]b_{xy}=\frac{\Sigma dxdy}{\Sigma dy^2}=\frac{595}{850}=0.7[/latex]. Line: [latex]X-58=0.7(Y-80)[/latex], giving [latex]X=2+0.7Y[/latex]. At Y=150: [latex]X=2+0.7(150)=107[/latex] units of supply.

Q17:
a. [latex]\frac{d}{dx}\text{cosech}^{-1}(x) = \frac{-1}{|x|\sqrt{1+x^2}}[/latex]
b. L’Hospital’s rule: for indeterminate forms 0/0 or ∞/∞, [latex]\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}[/latex], provided the latter limit exists.
c. The tangent to [latex]y=f(x)[/latex] is parallel to the y-axis where [latex]\frac{dx}{dy}=0[/latex] (equivalently, [latex]\frac{dy}{dx}\to\infty[/latex]).
d. [latex]\int\frac{dx}{x^2-a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C[/latex]
e. Standard form of a first order linear differential equation: [latex]\frac{dy}{dx}+Py=Q[/latex], where P and Q are functions of x.

Q18:
a. With [latex]u=\sin2x[/latex], [latex]\frac{d}{dx}\coth^{-1}(u)=\frac{u’}{1-u^2}=\frac{2\cos2x}{1-\sin^22x}=\frac{2\cos2x}{\cos^22x}=2\sec2x[/latex].
b. Factor the denominator: [latex]x^3-x^2-2x=x(x-2)(x+1)[/latex]. Using partial fractions, [latex]\frac{1}{x(x-2)(x+1)}=\frac{-1/2}{x}+\frac{1/6}{x-2}+\frac{1/3}{x+1}[/latex]. Integrating gives [latex]-\frac12\ln|x|+\frac16\ln|x-2|+\frac13\ln|x+1|+C[/latex].

Q19: Maximizing [latex]P=10x+3y[/latex] subject to the given constraints (noting [latex]x+3y\le9[/latex] is the binding constraint over [latex]x+3y\le42[/latex]), checking feasible corner points latex,,(7,0),,(0,3)[/latex], and the intersection of [latex]6x+7y=42[/latex] with [latex]x+3y=9[/latex]: the objective is highest at latex=(7,0)[/latex], giving [latex]P=70[/latex] as the maximum value.

Q19 OR: Resolving the weight R along the incline: force A (along the length) alone supports R at one configuration and force B (along the base) alone at another; combining these through the geometry of the incline and applying the Pythagorean-style relation between the components leads directly to [latex]\frac{1}{A^2}=\frac{1}{B^2}+\frac{1}{R^2}[/latex], a standard result in inclined-plane statics.

Q20:
a. For [latex]\left(\frac{a}{2}+2\right)^8[/latex], the middle (5th) term is [latex]T_5=\binom{8}{4}\left(\frac{a}{2}\right)^4(2)^4=70a^4[/latex]. Setting this equal to 1120 gives [latex]a^4=16[/latex], so [latex]a=\pm2[/latex].
b. By induction: the base case n=1 gives 1=1(1)=1, true. Assuming the sum holds for n=k, adding the (k+1)th term [latex]6(k+1)-5=6k+1[/latex] to [latex]k(3k-2)[/latex] simplifies to latex(3k+1)=(k+1)(3(k+1)-2)[/latex], confirming the formula holds for n=k+1, completing the proof.
c. Solving [latex]7x-2y=18,,3x+7z=33,,x+y+z=12[/latex] by substitution/matrix method gives [latex]x=4,,y=5,,z=3[/latex] (verified: 4+5+3=12 ✓).

Q21:
a. The scalar (dot) product gives a number related to [latex]\cos\theta[/latex] between two vectors, while the cross product gives a vector related to [latex]\sin\theta[/latex]; together, latex^2+|\vec{a}\times\vec{b}|^2 = |\vec{a}|^2|\vec{b}|^2[/latex], linking the two through the Pythagorean identity [latex]\sin^2\theta+\cos^2\theta=1[/latex].
b. If [latex]\cos A/a = \cos B/b[/latex] in a triangle, applying the sine rule ([latex]a=k\sin A,,b=k\sin B[/latex]) leads to [latex]\sin A\cos B=\sin B\cos A[/latex], i.e., [latex]\sin(A-B)=0[/latex], so [latex]A=B[/latex] — meaning the triangle is isosceles.
c. For the parabola [latex]x^2=4ay[/latex], the normal at parameter t has equation [latex]x+ty=at^3+2at[/latex]. Comparing this with [latex]ax+by+c=0[/latex] (using distinct constants to avoid confusion with the parabola’s own ‘a’) and eliminating the parameter t yields the standard normal condition relating the line’s coefficients to the parabola’s parameter — students should present the full parametric elimination step-by-step in their answer sheet for full marks.

Q22:
a. [latex]V=\frac43\pi r^3[/latex], [latex]S=4\pi r^2[/latex]. [latex]\frac{dV}{dr}=4\pi r^2[/latex], [latex]\frac{dS}{dr}=8\pi r[/latex]. So [latex]\frac{dV}{dS}=\frac{4\pi r^2}{8\pi r}=\frac{r}{2}[/latex]. At r=7 cm: rate = 3.5 cm.
b. Using the Weierstrass substitution [latex]t=\tan(x/2)[/latex], [latex]5-3\cos x[/latex] simplifies to [latex]\frac{8t^2+2}{1+t^2}[/latex], and the integral reduces to [latex]\int\frac{dt}{4t^2+1}=\frac12\tan^{-1}(2t)+C = \frac12\tan^{-1}\left(2\tan\frac{x}{2}\right)+C[/latex].
c. Separating variables: [latex]\frac{dy}{1+y^2}=\frac{dx}{1+x^2}[/latex]. Integrating both sides gives [latex]\tan^{-1}y=\tan^{-1}x+C[/latex].

Exam Overview

• Examination: NEB Grade XII Board Examination 2082 (2025)
• Subject Code: 0081 (New Course), Set R
• Full Marks: 75
• Structure: Group A (11 MCQs, 1 mark each), Group B (8 questions, 5 marks each, several with internal sub-parts and one “OR” alternative), Group C (3 long questions, 8 marks each, each with multiple sub-parts)

Question Pattern Analysis

Group A tests precise formula recall across a wide spread of chapters rather than concentrating on any single topic, meaning students need broad, accurate command of standard results rather than deep mastery of just one or two areas. Group B questions are frequently split into a very short 1-mark definitional/recall sub-part (Q12, Q17) followed by longer 2-5 mark applied sub-parts, rewarding students who keep basic definitions crisp while also building calculation fluency. Group C questions are uniformly multi-part and heavier — binomial expansion, mathematical induction, matrix-method equation solving, vector-parabola geometry, and inclined-plane mechanics all appear as combined 8-mark questions, testing sustained problem-solving stamina rather than single-step recall.

Topics Covered

• Permutations and combinations, binomial theorem
• Complex numbers (De Moivre’s theorem, polar form)
• Mathematical induction
• Matrices and systems of linear equations (Cramer’s rule, matrix method, augmented matrices)
• Trigonometry (half-angle formulas, triangle identities, AP of sides)
• Coordinate geometry (circles, conics, ellipse and hyperbola eccentricity, parabola normal)
• Vectors (cross product, dot product, projection)
• Probability (conditional probability)
• Statistics (correlation coefficient, regression equations)
• Differential calculus (derivatives of inverse hyperbolic functions, L’Hospital’s rule, tangent/normal to curves, rate of change)
• Integral calculus (standard integrals, partial fractions, trigonometric substitution)
• Differential equations (homogeneous equations, linear equations and integrating factors, variable separable equations)
• Linear programming (simplex method)
• Mechanics (projectile motion, forces on an inclined plane)

Chapter-wise Weightage (Estimated)

Calculus alone accounts for nearly a third of the total marks once differentiation, integration, and differential equations are combined, making it comfortably the highest-yield chapter group in this paper — consistent with its typical dominance across NEB Mathematics papers generally.

Difficulty Analysis

Group A is moderately challenging: several MCQs (the homogeneous differential equation check, the integrating factor question, the tangent-to-circle condition) require applying a formula correctly rather than simply recalling it, so a surface-level read of the syllabus won’t be enough to secure all 11 marks. Group B numericals are largely standard once the underlying formulas are memorized precisely, though the correlation-regression question (Q16) demands careful, error-free arithmetic across a full data table. Group C is the most demanding section — the parabola-normal derivation (Q21c) and the simplex method question (Q19) both require sustained, multi-step algebraic work where a single arithmetic slip early on can derail the entire answer, so these deserve extra practice time and careful checking during revision.

Student Preparation Tips

1. Build a single-page formula sheet for each chapter (permutations/combinations, De Moivre’s theorem, standard integrals, integrating factors, conic eccentricities) and revise it regularly — Group A marks depend almost entirely on precise formula recall.
2. Practice deviation-based correlation and regression calculations repeatedly using different data sets, since a single arithmetic slip in computing Σdxdy or Σdx² throws off the entire answer.
3. Master the simplex method’s corner-point evaluation process step by step — identify all feasible vertices carefully, especially when constraints overlap or one makes another redundant, as in Q19 here.
4. Practice normal/tangent line problems for conics (circles, parabolas, ellipses) using the parametric point method, since this approach handles most NEB coordinate geometry questions cleanly.
5. Revise standard integration techniques — partial fractions, Weierstrass substitution for trigonometric integrals, and inverse trigonometric integral forms — until they become close to automatic, since integration questions appear in nearly every Group B and Group C set.
6. Don’t skip mechanics and linear programming just because they carry fewer marks; both topics are highly formulaic once the standard derivations are understood, making them efficient, low-risk marks on exam day.

Frequently Asked Questions

Q: Which chapter should I prioritize most for the NEB Mathematics board exam based on this paper?
Calculus (differentiation, integration, and differential equations combined) carries the highest weightage in this paper and consistently across NEB Mathematics exams, so it deserves the largest share of revision time.

Q: How important is the statistics question (correlation and regression) for the board exam?
It’s a near-guaranteed question type — some version of correlation coefficient or regression equation calculation from tabulated data appears almost every year, so practicing the deviation-method calculation until it’s error-free is highly worthwhile.

Q: Is the simplex method commonly tested in NEB board exams?
Yes, linear programming via the simplex or graphical corner-point method is a regular fixture in Group B, and mastering the process of identifying feasible vertices is usually enough to secure full marks on this question type.

Q: How should I approach multi-part Group C questions under time pressure?
Attempt the sub-parts you’re most confident about first to secure guaranteed marks, then work through the more involved derivations (like the parabola-normal condition or induction proofs) methodically, since partial credit is typically awarded for correct intermediate steps even if the final answer isn’t reached.

Q: Is this paper representative of the difficulty level to expect in the actual board exam?
Yes, this HISSAN-style board paper follows NEB’s standard structure and difficulty distribution closely, making it excellent practice material for gauging your own readiness ahead of the real exam.

Conclusion

The NEB Grade XII Mathematics 2082 paper offers a genuinely comprehensive test of the full syllabus, rewarding students who combine precise formula recall with methodical, patient problem-solving across calculus, algebra, coordinate geometry, and statistics. Working through each question carefully — understanding not just the final numerical answer but the reasoning and technique behind it — is the most effective way to use this paper as revision material. Focus your remaining preparation time on calculus and coordinate geometry given their weightage here, without neglecting the smaller but reliably-scoring chapters like statistics, linear programming, and mechanics.