Introduction

The HISSAN Central Examination continues to serve as one of the most reliable rehearsals for Grade XII students heading into their NEB board exams, and the 2082 (2026) Compulsory Mathematics paper (Set D1) is a good demonstration of how closely these papers track NEB’s own style and difficulty level. Worth 75 full marks across an 11-mark objective section and a 64-mark subjective section, this paper draws from a genuinely broad slice of the syllabus — permutations and combinations, complex numbers, coordinate geometry (parabolas, ellipses, hyperbolas, circles), vectors, statistics, differential and integral calculus, differential equations, systems of linear equations, and classical mechanics (projectiles, recoil, forces in equilibrium) all make an appearance in a single sitting.

What stands out in this paper is how many questions test whether a student can move fluidly between a formula and its geometric or physical meaning. Question 15, for instance, doesn’t just ask for the eccentricity of a conic — it asks students to complete the square on a slightly rearranged parabola equation first, then extract the focus, vertex, and directrix from the resulting standard form. Question 22’s related-rates problem about a draining or filling water tank is a classic calculus application question that requires connecting geometric similarity (the changing radius of the water’s surface) to a rate of volume change — exactly the kind of multi-concept integration that board examiners favor in the later stages of a Grade XII paper.

The mechanics content in this set — recoil velocity from conservation of momentum, and a projectile relation connecting horizontal range and maximum height — is also worth flagging, since these topics often get less classroom attention than calculus or algebra but appear reliably in Group B and C alternatives every year. Similarly, the statistics question on Karl Pearson’s correlation coefficient and a regression-based estimate is a near-guaranteed feature across HISSAN and NEB papers, so mastering the deviation-method calculation thoroughly pays dividends well beyond this one paper.

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

Full Question Paper

Quick Answer Key (Group A — Multiple Choice Questions)

Brief explanation of each MCQ:

1. The sum of the first n natural numbers is the classic arithmetic series result [latex]1+2+\cdots+n = \frac{n(n+1)}{2}[/latex].
2. latex^6 = \cos(90°)+i\sin(90°) = i[/latex], by De Moivre’s theorem.
3. By the law of sines, [latex]\sin^2A=\sin^2B+\sin^2C[/latex] translates directly to [latex]a^2=b^2+c^2[/latex], which is the Pythagorean relation confirming that angle A is the right angle, so A = 90°.
4. The circle latex^2+y^2=8[/latex] has center (2,0) and radius [latex]2\sqrt2[/latex]. For the line [latex]x+y=c[/latex] to be tangent, the perpendicular distance from the center to the line must equal the radius: [latex]\frac{|2-c|}{\sqrt2}=2\sqrt2[/latex], giving [latex]|2-c|=4[/latex], so c = 6 or c = -2. The positive value is 6.
5. The dot product of two vectors is defined as [latex]\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta[/latex], where θ is the angle between them.
6. [latex]P(A/B)=\frac{P(A\cap B)}{P(B)}=\frac{1/4}{5/8}=\frac{2}{5}[/latex].
7. Using the chain rule on [latex]\sinh^{-1}\frac{x}{2}[/latex]: [latex]\frac{d}{dx}\sinh^{-1}\frac{x}{2}=\frac{1/2}{\sqrt{1+(x/2)^2}}=\frac{1/2}{\sqrt{(4+x^2)/4}}=\frac{1}{\sqrt{4+x^2}}[/latex].
8. At the point (0,4) on the circle [latex]x^2+y^2=16[/latex], the tangent line is horizontal (parallel to the x-axis), since the radius to this point is vertical. A horizontal line has an inclination of 0.
9. See the note directly below the table — this option needs a closer look.
10. From [latex]\sin^{-1}\left(\frac{dy}{dx}\right)=x[/latex], we get [latex]\frac{dy}{dx}=\sin x[/latex]. Integrating both sides gives [latex]y=-\cos x+c[/latex], which rearranges to [latex]y+\cos x=c[/latex].
11. In the Gauss elimination method, the coefficient of the first variable in the pivot equation must be non-zero, since dividing by a zero coefficient would make the elimination process undefined.
    11-OR. Since O is the circumcenter, the central angles satisfy [latex]\angle BOC=2A[/latex], [latex]\angle COA=2B[/latex], [latex]\angle AOB=2C[/latex]. Applying Lami’s theorem to the three equilibrium forces along OA, OB, OC gives [latex]\frac{P}{\sin2A}=\frac{Q}{\sin2B}=\frac{R}{\sin2C}[/latex].

A note on Question 9: All four listed integrals — [latex]\int\frac{dx}{\sqrt{a^2+x^2}}[/latex], [latex]\int\frac{dx}{\sqrt{a^2-x^2}}[/latex], [latex]\int\frac{dx}{a^2-x^2}[/latex], and [latex]\int\frac{dx}{x^2-a^2}[/latex] — actually correspond to standard, well-known closed-form integrals taught in the NEB Grade XII syllabus (giving inverse hyperbolic/logarithmic, inverse sine, and logarithmic forms respectively). None of them is obviously “non-standard” in the usual sense, so this option likely depends on how your specific textbook’s “list of standard integrals” is worded — it’s worth checking your textbook’s exact list of six standard forms and confirming with your teacher which one is excluded from that particular list, rather than assuming a specific lettered answer here.

Brief Solutions to Group B & C Questions

Q12:
a. Circular permutations of n distinct objects, where clockwise and anticlockwise arrangements are considered the same: [latex]\frac{(n-1)!}{2}[/latex]
b. [latex]e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots[/latex]
c. Polar form of [latex]i[/latex]: since [latex]i=\cos90°+i\sin90°[/latex], this is its polar representation.
d. Principle of mathematical induction: a statement P(n) is true for all natural numbers n if P(1) is true, and P(k) being true implies P(k+1) is true.
e. The system [latex]ax+by+c=0[/latex] and [latex]dx+ey+f=0[/latex] cannot be solved (has no solution) when the lines are parallel but not identical, i.e., when [latex]\frac{a}{d}=\frac{b}{e}\neq\frac{c}{f}[/latex].

Q13:
a. PENTAGON has 8 letters (P,E,N,T,A,G,O,N) with N repeated twice, and 3 distinct vowels (E,A,O). There are 4 odd positions (1,3,5,7). Choosing and arranging the 3 vowels in 3 of these 4 odd positions: [latex]P(4,3)=4\times3\times2=24[/latex] ways. The remaining 5 letters (P,N,T,G,N, with N repeated) fill the remaining 5 positions: [latex]\frac{5!}{2!}=60[/latex] ways. Total arrangements = [latex]24\times60=1440[/latex].
b. The numerator of the nth term is the sum [latex]2+5+8+\cdots[/latex] to n terms (arithmetic, first term 2, common difference 3): [latex]\frac{n}{2}[2(2)+(n-1)(3)]=\frac{n(3n+1)}{2}[/latex]. So the nth term is [latex]\frac{3n^2+n}{2\cdot n!}[/latex]. Using the standard results [latex]\sum\frac{n}{n!}=e[/latex] and [latex]\sum\frac{n^2}{n!}=2e[/latex], the sum becomes [latex]\frac12\left[3(2e)+e\right]=\frac{7e}{2}[/latex], confirming the given identity.

Q14:
a. Proving [latex]\sin(A+B)=\sin A\cos B+\cos A\sin B[/latex] rigorously for all angles (not just acute ones in a right triangle) is most cleanly done using unit vectors placed at angles A and B from a reference axis; expressing these vectors in component form and using the vector cross product (whose magnitude directly involves the sine of the angle between two vectors) naturally produces the addition formula, which is why the geometric proof restricted to triangles doesn’t fully generalize without the vector framework.
b. Using the projection formula in a triangle, [latex]a=b\cos C+c\cos B[/latex] gives [latex]a-b\cos C=c\cos B[/latex], and [latex]c=a\cos B+b\cos A[/latex] gives [latex]c-b\cos A=a\cos B[/latex]. Dividing these: [latex]\frac{a-b\cos C}{c-b\cos A}=\frac{c\cos B}{a\cos B}=\frac{c}{a}[/latex], which equals [latex]\frac{\sin C}{\sin A}[/latex] by the law of sines.

Q15:
a. Completing the square on [latex]y^2-6y+12x-45=0[/latex]: latex^2=-12x+54=-12(x-4.5)[/latex]. This is a parabola of the form latex^2=-4p(x-h)[/latex] with [latex]4p=12[/latex] (so p=3), vertex (h,k) = (4.5, 3), focus = (h-p, k) = (1.5, 3), and directrix: x = 7.5.
b. With vertex at (2,0) giving a=2, and focus at (-5,0) giving c=5 (vertex and focus on opposite sides of the center at the origin, which is standard for a hyperbola with two symmetric vertices and foci), [latex]b^2=c^2-a^2=25-4=21[/latex]. Standard equation: [latex]\frac{x^2}{4}-\frac{y^2}{21}=1[/latex].

Q16:
a. With [latex]\bar{X}=18.5[/latex] and [latex]\bar{Y}\approx12.167[/latex], computing deviations gives [latex]\Sigma dx,dy\approx68.5[/latex], [latex]\Sigma dx^2=119.5[/latex], [latex]\Sigma dy^2\approx42.83[/latex]. Karl Pearson’s correlation coefficient: [latex]r=\frac{68.5}{\sqrt{119.5\times42.83}}\approx0.96[/latex] — indicating a very strong positive correlation between X and Y.
b. Regression line of y on x: [latex]y-\bar{y}=b_{yx}(x-\bar{x})[/latex] → [latex]y-36=0.2(x-28)[/latex] → [latex]y=30.4+0.2x[/latex]. At X=40: [latex]y=30.4+0.2(40)=38.4[/latex].

Q17:
a. Differentiating [latex]x^2+y^2=25[/latex] implicitly: [latex]2x+2y\frac{dy}{dx}=0[/latex], so [latex]\frac{dy}{dx}=-\frac{x}{y}[/latex]. At (4,-3): slope [latex]=-\frac{4}{-3}=\frac43[/latex].
b. Velocity = [latex]f'(x)=4x-5[/latex]. At x=1: [latex]4(1)-5=-1[/latex].
c. [latex]\int\frac{dx}{x^2-a^2}=\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C[/latex]
d. [latex]\int\cosh(x),dx=\sinh(x)+C[/latex]
e. For [latex]y=x\frac{dy}{dx}+\sqrt{1+\left(\frac{dy}{dx}\right)^2}[/latex]: the order is 1 (only a first derivative appears). Isolating the radical and squaring both sides to clear it produces a polynomial equation where [latex]\left(\frac{dy}{dx}\right)^2[/latex] appears, giving a degree of 2.

Q18:
a. Expanding [latex]\cos x = 1-\frac{x^2}{2}+\frac{x^4}{24}-\cdots[/latex], [latex]2\cos x-2+x^2 = \frac{x^4}{12}-\cdots[/latex]. Dividing by [latex]x^4[/latex] and taking the limit as [latex]x\to0[/latex] gives [latex]\frac{1}{12}[/latex].
b. Writing [latex]\cosh x=\frac{e^x+e^{-x}}{2}[/latex] and substituting [latex]t=e^x[/latex] converts the integral to [latex]\int\frac{dt}{2t^2+3t+2}[/latex]. Completing the square, [latex]2t^2+3t+2=2\left(t+\frac34\right)^2+\frac78[/latex], giving the result [latex]\frac{2}{\sqrt7}\tan^{-1}\left(\frac{4e^x+3}{\sqrt7}\right)+C[/latex].

Q18 OR:
a. By conservation of momentum (initial momentum = 0), [latex]m_{bullet}v_{bullet}=m_{rifle}v_{recoil}[/latex]: [latex]0.015,kg\times100,kmh^{-1}=3,kg\times v_{recoil}[/latex], giving [latex]v_{recoil}=0.5,kmh^{-1}[/latex].
b. Using [latex]R=\frac{u^2\sin2\theta}{g}[/latex] and [latex]h=\frac{u^2\sin^2\theta}{2g}[/latex], computing [latex]\frac{R^2}{16h}[/latex] simplifies (after substituting [latex]\sin2\theta=2\sin\theta\cos\theta[/latex]) to [latex]\frac{u^2\cos^2\theta}{2g}[/latex]. Adding h gives [latex]h+\frac{R^2}{16h}=\frac{u^2}{2g}(\sin^2\theta+\cos^2\theta)=\frac{u^2}{2g}[/latex], so [latex]\sqrt{2g\left(h+\frac{R^2}{16h}\right)}=u[/latex], as required.

Q20:
a. In [latex]\left(x-\frac{1}{2x}\right)^{20}[/latex], the general term is [latex]T_{r+1}=\binom{20}{r}x^{20-r}\left(\frac{-1}{2x}\right)^r[/latex]. Setting the power of x to zero: [latex]20-2r=0\Rightarrow r=10[/latex]. Term independent of x: [latex]\binom{20}{10}\cdot\frac{1}{2^{10}}=\frac{184756}{1024}=\frac{46189}{256}[/latex].
b. The terms are products [latex]3n\times4n[/latex] for n=1,2,3,…(giving 3×4, 6×8, 9×12, 12×16,…). nth term = [latex]12n^2[/latex]. Sum to n terms: [latex]12\sum n^2 = 12\cdot\frac{n(n+1)(2n+1)}{6}=2n(n+1)(2n+1)[/latex].
c. Rewriting the system as [latex]x-2y-z=-8,;2x+y-z=3,;3x+4y+z=22[/latex], computing determinants gives [latex]D=10[/latex], [latex]D_x=20[/latex], [latex]D_y=30[/latex], [latex]D_z=40[/latex]. So x=2, y=3, z=4 (verified against all three original equations).

Q21:
a. Law of sines: [latex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R[/latex]. Writing [latex]a=k\sin A,,b=k\sin B[/latex], the ratio [latex]\frac{a-b}{a+b}=\frac{\sin A-\sin B}{\sin A+\sin B}[/latex]. Applying sum-to-product identities, this simplifies to [latex]\cot\left(\frac{A+B}{2}\right)\tan\left(\frac{A-B}{2}\right)[/latex]. Since [latex]A+B=180°-C[/latex], [latex]\cot\left(\frac{A+B}{2}\right)=\tan\left(\frac{C}{2}\right)[/latex], giving [latex]\frac{a-b}{a+b}=\tan\frac{C}{2}\tan\left(\frac{A-B}{2}\right)[/latex], which rearranges to the required identity.
b. Completing the square on [latex]x^2+5y^2+3x-10y+\frac{71}{4}=0[/latex] gives [latex]\left(x+\frac32\right)^2+5(y-1)^2=\frac{29}{4}-\frac{71}{4}[/latex], which comes out negative with the constant as printed — a strong signal of a possible transcription issue in the original paper (the constant term is likely meant to be smaller, so that the right side is positive as required for a genuine ellipse). The method itself is standard: after completing the square correctly, divide through to reach the form [latex]\frac{(x+3/2)^2}{A}+\frac{(y-1)^2}{B}=1[/latex], then read off the eccentricity as [latex]e=\sqrt{1-B/A}[/latex] (for A>B) along with the center-shifted vertices and foci. It’s worth double-checking the exact constant against your official answer key before finalizing this solution.
c. For [latex]\vec{a}=(1,-2,4)[/latex] and [latex]\vec{b}=(2,7,m)[/latex] to be orthogonal, [latex]\vec{a}\cdot\vec{b}=0[/latex]: [latex]1(2)+(-2)(7)+4(m)=0[/latex] → [latex]2-14+4m=0[/latex] → m = 3.

Q22:
a. The cone has radius:height ratio 2:4 = 1:2, so at water depth y, the water surface radius is [latex]\frac{y}{2}[/latex]. Volume [latex]V=\frac13\pi\left(\frac{y}{2}\right)^2y=\frac{\pi y^3}{12}[/latex]. Differentiating: [latex]\frac{dV}{dt}=\frac{\pi y^2}{4}\frac{dy}{dt}[/latex]. At y=3 with [latex]\frac{dV}{dt}=2,m^3/min[/latex]: [latex]2=\frac{9\pi}{4}\frac{dy}{dt}[/latex], giving [latex]\frac{dy}{dt}=\frac{8}{9\pi}\approx0.283,m/min[/latex].
b. Completing the square: [latex]2ax-x^2=a^2-(x-a)^2[/latex]. So [latex]\int\frac{dx}{\sqrt{2ax-x^2}}=\sin^{-1}\left(\frac{x-a}{a}\right)+C[/latex].
c. A standard example of a homogeneous first-order, first-degree differential equation is [latex]\frac{dy}{dx}=\frac{x^2+y^2}{2xy}[/latex]. Substituting y=vx and simplifying leads to [latex]\frac{2v}{1-v^2}dv=\frac{dx}{x}[/latex]; integrating both sides and substituting back gives the general solution [latex]x^2-y^2=Cx[/latex].

Exam Overview

• Examination: HISSAN Central Examination 2082 (2026)
• Class: XII
• Subject: Compulsory Mathematics (Code 0081 D1)
• Full Marks: 75
• Time: 3 hours
• Structure: Group A (11 MCQs, 1 mark each, with an “OR” alternative on Q11), Group B (8 questions, 5 marks each, split into short sub-parts), Group C (3 long questions, 8 marks each, split into sub-parts)

Question Pattern Analysis

Group A blends quick formula-recall questions (sum of natural numbers, dot product definition) with questions requiring an intermediate derivation step (the tangent-to-circle condition, the sin² triangle relation, De Moivre’s theorem application), so students need both memorized formulas and the ability to apply them under light time pressure. Group B consistently splits into several 1-2 mark sub-parts covering definitions, standard results, and short calculations side by side with longer 2-3 mark applied problems — a structure that rewards students who keep foundational facts (series expansions, standard integrals, principle of induction) instantly accessible. Group C questions are the most demanding, combining binomial expansion, series summation, Cramer’s rule, trigonometric identity proofs, conic section analysis, vector orthogonality, and a calculus-based related-rates problem — testing sustained accuracy across genuinely varied mathematical territory within each single question.

Topics Covered

• Sequences and series (sum of natural numbers, arithmetic-geometric hybrid series, series involving factorials and e)
• Complex numbers (De Moivre’s theorem)
• Trigonometry (law of sines, Napier’s analogy, triangle side-angle relations)
• Coordinate geometry (circles, parabolas, ellipses, hyperbolas — foci, vertices, directrix, eccentricity)
• Vectors (dot product, orthogonality)
• Probability (conditional probability)
• Permutations and combinations (circular permutations, word arrangement with positional constraints)
• Mathematical induction (principle statement)
• Systems of linear equations (Gauss elimination, Cramer’s rule)
• Statistics (Karl Pearson’s correlation coefficient, regression line and estimation)
• Differential calculus (derivatives of inverse hyperbolic functions, slope of tangent/normal, velocity from position function, L’Hospital’s rule, order and degree of differential equations, related rates)
• Integral calculus (standard integral forms, integration involving hyperbolic functions, binomial expansion term extraction)
• Differential equations (variable separable, homogeneous first-order first-degree equations)
• Mechanics (conservation of momentum/recoil velocity, projectile motion, forces in equilibrium at the circumcenter)

Chapter-wise Weightage (Estimated)

Calculus and algebra together make up over half the paper’s marks, confirming their consistent position as the highest-yield areas across HISSAN and NEB Mathematics papers, with coordinate geometry and vectors forming a solid third pillar of the mark distribution.

Difficulty Analysis

Group A demands more than surface recall — questions like the sin² triangle relation, the tangent-to-circle condition, and the equilibrium-forces alternative all require applying a geometric or trigonometric identity correctly rather than simply remembering a formula in isolation. Group B numericals (correlation coefficient, regression estimate, Cramer’s rule-adjacent identities) are approachable with steady practice, though the correlation calculation demands careful, error-free arithmetic across six data pairs. Group C is the most rigorous section — the related-rates water tank problem (Q22a) requires connecting geometric similarity to calculus cleanly, and the ellipse eccentricity question (Q21b) is worth extra scrutiny since the numbers as commonly transcribed can produce an inconsistent (negative) result, meaning careful, patient completion-of-the-square work is essential rather than rushing to a formula.

Student Preparation Tips

1. Practice completing the square on conic section equations repeatedly across circles, parabolas, ellipses, and hyperbolas until identifying the vertex, focus, and directrix from an unstandardized equation becomes fast and reliable.
2. Build a clean reference sheet of standard integral and derivative forms, especially those involving inverse hyperbolic functions and hyperbolic functions themselves (cosh, sinh), since these appear more often in Compulsory Mathematics papers than students expect.
3. Master related-rates problems methodically — always start by expressing volume (or the relevant quantity) purely in terms of the one variable that’s changing, using geometric similarity where needed, before differentiating.
4. Drill Karl Pearson’s correlation coefficient and regression line calculations using the deviation method on multiple data sets, since arithmetic slips are the most common source of lost marks here, not conceptual misunderstanding.
5. Revisit the classical mechanics topics — conservation of momentum, projectile range and height relations, and forces in equilibrium — since these are often deprioritized in favor of calculus and algebra but appear as guaranteed alternative questions in Group B/C.
6. Practice deriving trigonometric identities like Napier’s analogy from the law of sines, rather than memorizing the final result alone, since exam questions often ask you to prove the identity from scratch.

Frequently Asked Questions

Q: Which topics should I prioritize most heavily based on this paper’s weightage?
Calculus (differentiation, integration, differential equations) and algebra (series, complex numbers, systems of equations) together account for over half the marks, making them the highest-priority areas for revision.

Q: How often does a Karl Pearson’s correlation coefficient question appear in HISSAN and NEB Mathematics papers?
Very consistently — some version of correlation coefficient calculation or regression-based estimation shows up almost every year, making it one of the most reliably practiced topics for guaranteed marks.

Q: Is the related-rates calculus problem (like the water tank question) a common NEB exam pattern?
Yes, related-rates problems involving cones, spheres, or similar geometric shapes appear periodically in Group C, and mastering the general method (expressing volume in one variable using similar triangles, then differentiating) prepares you for most variations.

Q: Should I be concerned if my calculated answer for a conic section question doesn’t match a clean number?
It’s worth double-checking your algebra first, but exam papers occasionally contain transcription or printing inconsistencies (as flagged for Q21b in this analysis) — if your method is sound and the numbers still don’t resolve cleanly, cross-check the exact figures against your official answer key or teacher before assuming an error in your own work.

Q: How does this HISSAN paper compare in difficulty to a typical NEB board Mathematics paper?
It’s closely aligned with NEB’s own structure and difficulty distribution, making it excellent, representative practice material for gauging genuine board exam readiness.

Conclusion

The HISSAN 2082 Compulsory Mathematics paper offers a thorough test of algebra, calculus, coordinate geometry, statistics, and mechanics working together, rewarding students who can move fluidly between formula recall and multi-step application. Working through this paper carefully — understanding the reasoning behind each solution rather than just matching a final numerical answer — will meaningfully strengthen board exam readiness. Use the chapter-wise weightage and preparation tips above to direct your remaining study time toward calculus and algebra first, while keeping steady practice going on coordinate geometry, statistics, and the often-overlooked mechanics topics.keeping steady practice going on coordinate geometry, statistics, and the often-overlooked mechanics topics.