Introduction

The NEB Grade XII Mathematics board exam has a reputation among students for being one of the more demanding papers of the year, and the 2080 (2023) paper (Sub. Code 0081, New Course, Set I, for general stream) shows exactly why. Carrying 75 full marks across three groups, this particular paper draws from an unusually wide spread of the syllabus — group theory, quadratic equations, trigonometric equations, inverse trigonometric functions, conic sections, vectors, probability, differential calculus, integral calculus, systems of linear equations, mechanics, and even a brief touch of input-output economic analysis all appear within a single sitting.

What makes this paper worth working through carefully is the way it tests conceptual precision rather than just formula plugging. Take Question 1, which asks students to identify the identity property of a group — a topic that’s easy to gloss over but frequently trips students up when the associative, identity, and cancellation properties are presented side by side as near-identical-looking options. Or Question 5, which asks exactly when an ellipse becomes a circle — a question that seems simple until you’re actually asked to distinguish it precisely from the condition for zero eccentricity. These aren’t hard questions in isolation, but they demand that students have internalized definitions clearly enough to spot the one correct option among several plausible-looking distractors.

The subjective sections build on this same philosophy. Question 15 asks students to estimate price from a regression equation built on real supply-price data — a calculation that rewards careful, error-free arithmetic across seven data points. Question 19’s alternative option introduces an input-output economic model, asking students to solve a system of two equations for total industry output — a genuinely applied problem that blends linear algebra with basic economic reasoning. Meanwhile, questions like the simplex method problem (Q18) and the plane-through-intersection problem (Q21c) test whether students can execute a full multi-step procedure accurately from start to finish.

This article works through every question with brief, clear solutions, breaks the paper down chapter by chapter, estimates where the marks concentrate, and lays out a preparation strategy tailored to what this paper actually tests — so that instead of just checking your work against a key, you come away with a sharper sense of exactly how NEB expects Grade XII Mathematics to be studied.

Full Question Paper

Quick Answer Key (Group A — Multiple Choice Questions)

Brief explanation of each MCQ:

1. Group axioms define identity element e such that combining it with any element a (in either order) returns a unchanged: [latex]e \circ a = a \circ e = a[/latex]. The other options describe inverse, associative, and cancellation properties respectively, not the identity property.
2. Since coefficients are rational, irrational roots of a quadratic must occur in conjugate pairs. If one root is [latex]3-2\sqrt2[/latex], the other is [latex]3+2\sqrt2[/latex]. Sum of roots = 6, product = [latex]9-8=1[/latex], giving [latex]x^2-6x+1=0[/latex].
3. [latex]\cos\theta=0[/latex] holds at odd multiples of [latex]\pi/2[/latex], giving the general solution latex\frac{\pi}{2}[/latex].
4. [latex]\cos^{-1}\frac{1}{\sqrt2}=\frac{\pi}{4}[/latex], and [latex]\sin\frac{\pi}{4}=\frac{1}{\sqrt2}[/latex].
5. The ellipse [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=1[/latex] with foci at [latex](\pm c,0)[/latex] has [latex]c^2=a^2-b^2[/latex]. It becomes a circle when the foci coincide with the center, i.e., when [latex]c=0[/latex], which happens exactly when [latex]\sqrt{a^2-b^2}=0[/latex].
6. Using the vector identity connecting dot and cross products: latex^2 = a^2b^2-(\vec{a}\cdot\vec{b})^2[/latex], derived directly from [latex]\sin^2\theta = 1-\cos^2\theta[/latex].
7. [latex]P(A\cap B)=P(B)\times P(A/B)=0.7\times0.4=0.28[/latex].
8. Slope of tangent = [latex]-\frac{2}{5}[/latex], so slope of normal (negative reciprocal) = [latex]\frac{5}{2}[/latex].
9. Rewriting [latex]4x^2-9[/latex] as [latex]4\left(x^2-\frac{9}{4}\right)[/latex], the integral becomes [latex]\frac{1}{2}\int\frac{dx}{\sqrt{x^2-9/4}}=\frac{1}{2}\log\left|x+\sqrt{x^2-\frac{9}{4}}\right|+C[/latex].
10. A system of linear equations in exactly two variables must involve only x and y (no z), making [latex]x+y=5,;x-y=1[/latex] the only valid option.
11. For a projectile achieving maximum range (at 45° launch angle) on level ground, [latex]R_{max}=\frac{u^2}{g}[/latex]. Setting [latex]10=\frac{u^2}{10}[/latex] gives [latex]u^2=100[/latex], so [latex]u=10,ms^{-1}[/latex].
    11-OR. Revenue = price × quantity = [latex]y\times x = (20-4x)x = -4x^2+20x[/latex].

Brief Solutions to Group B & C Questions

Q12:
a. i. [latex]\arg(z_1z_2)=\theta_1+\theta_2[/latex]
ii. Modulus of [latex]z_1z_2 = r_1r_2[/latex]
iii. Polar form: [latex]z_1z_2 = r_1r_2\left[\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)\right][/latex]
b. latex^{50}=(1+x)^{100}[/latex]. Total number of terms = 101 (since latex^n[/latex] has n+1 terms). Middle term = 51st term = [latex]T_{51}=\binom{100}{50}x^{50}[/latex].

Q13:
a. “OBJECTIVE” has 9 letters with E repeated twice. Arrangements = [latex]\frac{9!}{2!}=181440[/latex].
b. Each term follows the pattern [latex]\frac{1+2+\cdots+n}{(n+1)!}=\frac{n(n+1)/2}{(n+1)!}=\frac{n}{2\cdot n!}[/latex]. Summing from n=1 to ∞: [latex]\sum\frac{n}{2\cdot n!}=\frac{1}{2}\sum\frac{1}{(n-1)!}=\frac{e}{2}[/latex]. Multiplying by 2 gives exactly e, confirming the identity.

Q14:
a. [latex]2\tan^{-1}\frac{1}{2}=\tan^{-1}\left(\frac{2\times\frac12}{1-\frac14}\right)=\tan^{-1}\frac{4}{3}[/latex]. Then [latex]\tan^{-1}\frac43+\tan^{-1}\frac17=\tan^{-1}\left(\frac{\frac43+\frac17}{1-\frac43\cdot\frac17}\right)=\tan^{-1}\left(\frac{31/21}{17/21}\right)=\tan^{-1}\frac{31}{17}[/latex].
b. With [latex]\vec{a}=2\vec{i}+\vec{k}[/latex] and [latex]\vec{b}=\vec{i}+\vec{j}+\vec{k}[/latex]: [latex]\vec{a}\times\vec{b}=\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\2&0&1\1&1&1\end{vmatrix}=-\vec{i}-\vec{j}+2\vec{k}[/latex].

Q15: Using x̄ = 89, ȳ = 15 (means of supply and price), and computing deviations: [latex]\Sigma dx,dy=87[/latex], [latex]\Sigma dx^2=238[/latex]. Regression coefficient of y on x: [latex]b_{yx}=\frac{87}{238}\approx0.366[/latex]. Regression line: [latex]y-15=0.366(x-89)[/latex]. At x=110: [latex]y=15+0.366(21)\approx15+7.68=22.68[/latex]. Estimated price ≈ Rs. 22.68 per kg when supply is 110 kg.

Q16:
a. [latex]\frac{d}{dx}\sinh^{-1}x=\frac{1}{\sqrt{1+x^2}}[/latex]; [latex]\frac{d}{dx}\tanh x=\text{sech}^2x[/latex]
b. The integral of f(x) represents the antiderivative F(x) — a function whose derivative equals f(x) — plus an arbitrary constant of integration.
c. For [latex]\left[1+\left(\frac{dy}{dx}\right)^2\right]^3=\left(\frac{d^3y}{dx^3}\right)^2[/latex]: the order is 3 (highest derivative present is the third derivative), and the degree is 2 (the power to which that highest-order derivative is raised once the equation is free of radicals/fractions in derivatives).

Q17:
a. Separating variables: [latex]y(y+1),dy=\left(x+\frac1x\right)dx[/latex]. Integrating: [latex]\frac{y^3}{3}+\frac{y^2}{2}=\frac{x^2}{2}+\ln|x|+C[/latex].
b. Using partial fractions, [latex]\frac{8}{(2x-1)(2x+1)}=\frac{4}{2x-1}-\frac{4}{2x+1}[/latex]. Integrating: [latex]2\ln|2x-1|-2\ln|2x+1|+C = 2\ln\left|\frac{2x-1}{2x+1}\right|+C[/latex].

Q18: Maximizing [latex]P=3x+5y[/latex] subject to [latex]x+2y\le40,;2x+y\le50[/latex]. Corner points: (0,0), (25,0), (0,20), and the intersection of the two constraint lines at (20,10). Evaluating P at each: 0, 75, 100, and [latex]3(20)+5(10)=110[/latex]. Maximum P = 110 at (x,y) = (20,10).

Q19: This is a standard statics proof. Placing the rod MN along a coordinate axis with forces S₁ and S₂ acting at M and N respectively, the resultant of the original like parallel forces acts at a point dividing MN in the ratio [latex]S_2:S_1. When equal and opposite forces P are added to S₁ and S₂, the new resultant’s magnitude stays [latex]S_1+S_2[/latex] but its line of action shifts; working through the moment balance about the original resultant’s point of action shows the displacement equals [latex]\frac{Pk}{S_1+S_2}[/latex], as required.

Q19 OR: Using the input-output table, technical coefficients are [latex]a_{11}=700/2000=0.35[/latex], [latex]a_{21}=500/2000=0.25[/latex], [latex]a_{12}=600/1600=0.375[/latex], [latex]a_{22}=400/1600=0.25[/latex]. New final demands: [latex]0.9\times300=270[/latex] for A, [latex]1.05\times250=262.5[/latex] for B. Setting up [latex]X_1=0.35X_1+0.375X_2+270[/latex] and [latex]X_2=0.25X_1+0.25X_2+262.5[/latex] and solving simultaneously gives approximately X₁ ≈ 764.3 and X₂ ≈ 604.8 as the new total outputs.

Q20:
a. Writing the equations as [latex]x-2y=0,;3x+2y=8[/latex]: [latex]D=\begin{vmatrix}1&-2\3&2\end{vmatrix}=8[/latex], [latex]D_x=\begin{vmatrix}0&-2\8&2\end{vmatrix}=16[/latex], [latex]D_y=\begin{vmatrix}1&0\3&8\end{vmatrix}=8[/latex]. So [latex]x=16/8=2[/latex], [latex]y=8/8=1[/latex].
b. Ways to draw 2 socks of the same color: [latex]\binom{8}{2}+\binom{6}{2}=28+15=43[/latex].
c. Base case n=1: [latex]2^3-1=7[/latex], divisible by 7. Assuming [latex]2^{3k}-1=7m[/latex] holds, then [latex]2^{3(k+1)}-1=8\cdot2^{3k}-1=8(2^{3k}-1)+7=8(7m)+7=7(8m+1)[/latex], which is divisible by 7 — completing the induction.

Q21:
a. Rewriting [latex]9y^2-4x^2=36[/latex] as [latex]\frac{y^2}{4}-\frac{x^2}{9}=1[/latex] (a hyperbola with [latex]a^2=4,,b^2=9[/latex]), the latus rectum [latex]=\frac{2b^2}{a}=\frac{18}{2}=9[/latex].
b. Direction ratios of AB: latex=(-1,-1,0)[/latex]. Magnitude = [latex]\sqrt2[/latex]. Direction cosines: [latex]\left(-\frac{1}{\sqrt2},,-\frac{1}{\sqrt2},,0\right)[/latex].
c. Family of planes through the intersection: latex+\lambda(2x-3y+4z-7)=0[/latex]. Substituting (1,-1,1) gives [latex]10+2\lambda=0[/latex], so [latex]\lambda=-5[/latex]. The required plane simplifies to [latex]9x-13y+17z-39=0[/latex].

Q22:
a. L’Hospital’s rule resolves indeterminate forms like 0/0 or ∞/∞ by differentiating numerator and denominator separately. Example: [latex]\lim_{x\to0}\frac{\sin x}{x}[/latex] gives 0/0 directly, but applying the rule gives [latex]\lim_{x\to0}\frac{\cos x}{1}=1[/latex].
b. Yes — the integral [latex]\int\frac{dx}{a+b\cos x}[/latex] takes two different forms depending on whether [latex]a>b[/latex] or [latex]a<b[/latex] (an arctangent form versus a logarithmic form), since the sign of [latex]a^2-b^2[/latex] under the Weierstrass substitution changes the nature of the resulting integral. For a=3, b=2 (where a>b), the result is [latex]\frac{2}{\sqrt5}\tan^{-1}\left(\sqrt{\frac15}\tan\frac{x}{2}\right)+C[/latex].
c. Rewriting [latex]x\frac{dy}{dx}+y-x\log x=0[/latex] as [latex]\frac{dy}{dx}+\frac{y}{x}=\log x[/latex] (linear form), the integrating factor is [latex]e^{\int\frac{dx}{x}}=x[/latex]. Solving gives [latex]xy=\frac{x^2}{2}\log x-\frac{x^2}{4}+C[/latex], or [latex]y=\frac{x}{2}\log x-\frac{x}{4}+\frac{C}{x}[/latex].

Exam Overview

• Examination: NEB Grade XII Board Examination 2080 (2023)
• Subject Code: 0081 (New Course), Set I, for General Stream
• Full Marks: 75
• Time: 3 hours
• Structure: Group A (11 MCQs, 1 mark each), Group B (8 questions, 5 marks each, split into multiple short sub-parts, with one “OR” alternative), Group C (3 long questions, 8 marks each, each with multiple sub-parts)

Question Pattern Analysis

Group A in this paper leans heavily on precise definitions and conceptual distinctions — group axioms, conic section properties, and vector identities are tested in a way that requires exact recall rather than approximate familiarity. Group B questions consistently break into several small sub-parts worth 1-2 marks each, which rewards students who keep a broad set of quick facts (derivatives of hyperbolic functions, order/degree of differential equations, expansion term counts) sharp and ready to recall instantly. Group C questions are noticeably applied and multi-step — determinant-based equation solving, mathematical induction, conic section properties, 3D geometry (direction cosines, planes through intersection), and integral calculus with parameter-dependent results all appear, testing sustained accuracy across longer derivations.

Topics Covered

• Group theory (identity, associative, and cancellation properties)
• Quadratic equations (conjugate irrational roots)
• Trigonometric equations (general solutions)
• Inverse trigonometric functions
• Conic sections (ellipse becoming a circle, latus rectum of hyperbola)
• Vectors (cross product identities, direction cosines)
• Probability (conditional probability)
• Complex numbers (polar form, argument, modulus of products)
• Binomial theorem (number of terms, middle term)
• Permutations and combinations (arrangement of letters, combinations of same-colored objects)
• Series and exponential function (e series proof)
• Inverse trigonometric identities (tan⁻¹ addition formula)
• Statistics (regression, price estimation from supply data)
• Differential calculus (hyperbolic function derivatives, order/degree of differential equations, L’Hospital’s rule)
• Integral calculus (partial fractions, trigonometric integrals with parameters)
• Differential equations (variable separable, linear first-order)
• Systems of linear equations (determinant method)
• Linear programming (simplex method)
• Mechanics (parallel forces, projectile motion)
• 3D geometry (planes through line of intersection)
• Applied mathematics (input-output economic analysis)

Chapter-wise Weightage (Estimated)

Algebra and calculus together dominate this paper’s weightage, together accounting for well over half the total marks — reflecting how central these two areas remain to NEB’s Grade XII Mathematics curriculum year after year.

Difficulty Analysis

Group A is conceptually demanding despite its short time allocation — several questions (the group identity property, the ellipse-to-circle condition, the vector cross-product square identity) require precise definitional clarity rather than surface familiarity, so a quick skim of the syllabus won’t guarantee full marks here. Group B numericals are generally approachable with regular practice, though the regression-based price estimation (Q15) demands careful, error-free arithmetic across seven data points. Group C is where the paper is most rigorous: the plane-through-intersection problem (Q21c) and the parallel forces displacement proof (Q19) both require sustained multi-step derivations, and the input-output economics alternative (Q19-OR) introduces an applied context that many students find unfamiliar territory compared to standard algebra or calculus problems — all worth extra practice time.

Student Preparation Tips

1. Nail down group theory definitions precisely — identity, associativity, inverse, and cancellation properties look deceptively similar in symbolic form, so practice distinguishing them explicitly rather than relying on pattern recognition alone.
2. Practice regression and correlation calculations on varied data sets until the deviation-method arithmetic becomes fast and reliable, since this question type appears reliably across NEB papers.
3. Work through input-output economic analysis problems specifically — this topic is often under-practiced compared to core algebra and calculus, yet it appears as a recurring alternative question in Group B/C.
4. Master the full simplex method procedure, including identifying all feasible corner points correctly, since a single missed vertex can lead to an incorrect maximum value.
5. Build fluency in 3D geometry techniques — direction cosines, the family-of-planes method for finding a plane through a line of intersection, and similar standard techniques — as these appear consistently in Group C.
6. Revise integral calculus techniques that depend on parameter conditions, like the trigonometric integral in Q22(b) here, where the correct method depends on comparing the relative sizes of constants (a vs b) before choosing an approach.

Frequently Asked Questions

Q: Which topics carry the most weight in this NEB Mathematics paper?
Calculus and algebra together make up more than half the total marks, so prioritizing differentiation, integration, differential equations, complex numbers, and systems of equations is the most efficient use of revision time.

Q: Is the input-output economic analysis question common in NEB Mathematics exams?
It appears periodically as an alternative (“OR”) option rather than every year, but since many students skip this topic, practicing it can be a good way to gain an edge over less-prepared peers.

Q: How should I approach questions with multiple short sub-parts, like Q12 or Q16?
Treat each sub-part as an independent quick-recall question — write down the standard formula or definition immediately rather than trying to derive it from scratch, since these sub-parts are designed to reward fast, accurate recall.

Q: Is the simplex method or the graphical corner-point method preferred for linear programming questions?
Both are acceptable for two-variable problems like this one, and the graphical/corner-point method is often faster and less error-prone for board exam conditions, though the question explicitly asks for the simplex method here, so following the requested method is important for full marks.

Q: How difficult is this paper compared to a typical NEB Mathematics board exam?
It’s representative of standard NEB difficulty, though the conceptual precision required in Group A and the applied economics alternative in Group B/C push it slightly above average in terms of testing genuine understanding over memorization.

Conclusion

The NEB Grade XII Mathematics 2080 paper tests a genuinely wide cross-section of the syllabus, rewarding students who combine precise conceptual understanding with steady, accurate multi-step problem-solving. Working through each question here — understanding not just the final answer but why each step follows logically from the last — will meaningfully sharpen your readiness for the board exam. Focus your remaining preparation on calculus and algebra given their weightage, while making sure 3D geometry, statistics, and applied topics like input-output analysis don’t get left behind in the final stretch of revision.