Mathematics
गणित · Class 10 · Code SEE
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chapters done
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The syllabus asks for
- Sets: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- n(A∪ B)=n(A)+n(B)-n(A∩ B); for disjoint sets n(A∩ B)=0.
- Only A: n(A)-n(A∩ B). Neither: n(U)-n(A∪ B).
- n(A∪ B∪ C)=n(A)+n(B)+n(C)-n(A∩ B)-n(B∩ C)-n(A∩ C)+n(A∩ B∩ C).
- Fill a three-set Venn diagram from the centre outwards; check that all regions add to n(U).
- Exactly two =Σ n(pair)-3 n(A∩ B∩ C); exactly one =n(A∪ B∪ C)-(exactly two)-n(A∩ B∩ C).
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The syllabus asks for
- Tax and Money Exchange: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- VAT =(r)/(100)× price; bill = price ×(1+(r)/(100)); price = bill ÷(1+(r)/(100)). Nepal's standard rate is 13%.
- Discount is allowed on the marked price first; VAT is charged on the selling price.
- Excise duty is charged on the factory price; VAT is then charged on factory price + excise.
- Final price with excise e% and VAT v%: factory ×(1+(e)/(100))(1+(v)/(100)).
- A seller pays VAT collected on sales minus VAT paid on purchases.
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The syllabus asks for
- Compound Interest: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- Simple interest I=(PTR)/(100); compound interest adds interest to the principal every period.
- A=Pleft(1+(R)/(100)right)^T and CI=A-P (compounded yearly).
- Half-yearly: A=Pleft(1+(R)/(200)right)^2T; quarterly: A=Pleft(1+(R)/(400)right)^4T.
- P=dfracA(1+R/100)^T; find R or T by matching A/P with a power.
- For 2 years, CI-SI=Pleft((R)/(100)right)^2.
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The syllabus asks for
- Population Growth and Depreciation: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- Growth: PT=P0left(1+(R)/(100)right)^T. Past population: divide by the factor.
- Net rate = birth - death + immigration - emigration (all in % per year).
- For different yearly rates multiply the factors; for a fixed number of migrants add or subtract at the correct time.
- Depreciation: V=Pleft(1-(R)/(100)right)^T; total depreciation =P-V.
- Depreciation each year is charged on the value at the start of that year, so the yearly loss shrinks.
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The syllabus asks for
- Plane Surfaces: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- Rectangle lb, 2(l+b); square a^2, 4a; triangle frac12bh; parallelogram bh; rhombus frac12d1d2; trapezium frac12(a+b)h.
- Heron: s=(a+b+c)/(2), Area =√s(s-a)(s-b)(s-c); equilateral frac√34a^2.
- Circle: C=2π r, A=π r^2; semicircle perimeter π r+2r, area frac12π r^2.
- Sector: arc =(θ)/(360)×2π r, area =(θ)/(360)×π r^2.
- Path or ring = larger area - smaller area; ring =π(R^2-r^2).
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The syllabus asks for
- Cylinder and Sphere: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- Cylinder: CSA =2π rh, TSA =2π r(r+h), V=π r^2h.
- Hollow cylinder: V=π h(R^2-r^2); outer CSA =2π Rh.
- Sphere: surface =4π r^2, V=frac43π r^3.
- Hemisphere: CSA =2π r^2, TSA =3π r^2, V=frac23π r^3.
- Hollow sphere: V=frac43π(R^3-r^3).
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The syllabus asks for
- Prism and Pyramid: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- Right prism: LSA =Ph, TSA =Ph+2A, V=Ah (P base perimeter, A base area).
- Cuboid: LSA =2h(l+b), TSA =2(lb+bh+hl), V=lbh, diagonal =√l^2+b^2+h^2.
- Cube: TSA =6a^2, V=a^3.
- Square pyramid: l^2=h^2+(frac a2)^2, LSA =2al, TSA =2al+a^2, V=frac13a^2h.
- A pyramid is one third of the prism with the same base and height.
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The syllabus asks for
- Highest Common Factor and Lowest Common Multiple: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- HCF = common factors at lowest powers; LCM = all factors at highest powers (include the numerical parts).
- For two numbers or expressions: HCFtimesLCM=A× B. Not valid for three.
- Factorisation toolkit: common factor, grouping, a^2-b^2, perfect squares, x^2+px+q, ax^2+bx+c, a^3± b^3.
- a^3+b^3=(a+b)(a^2-ab+b^2) and a^3-b^3=(a-b)(a^2+ab+b^2).
- Euclid's method: divide, then divide the divisor by the remainder, until the remainder is 0; the last divisor is the HCF.
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The syllabus asks for
- Radical and Surd: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- A surd √[n]a (a rational) is irrational; √16, √[3]27 are not surds; √2, √[3]5 are.
- √a×√b=√ab; frac√a√b=√frac ab; (√a)^2=a; but √a+b≠√a+√b.
- Add and subtract only like surds after simplifying: 5√2+3√2=8√2.
- Conjugate of a+√b is a-√b; product =a^2-b (rational).
- Rationalise frac1√a by √a; rationalise frac1√a±√b by √amp√b, giving denominator a-b.
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The syllabus asks for
- Indices: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- a^m× a^n=a^m+n, a^m÷ a^n=a^m-n, (a^m)^n=a^mn, (ab)^n=a^nb^n.
- a^0=1 and a^-n=(1)/(a^n); a negative index flips the base.
- a^1/n is the nth root of a and a^m/n=left(a^1/nright)^m: root first, then power.
- If a^x=a^y (a>0, a≠1) then x=y.
- To solve an exponential equation: common base, simplify, equate indices, check.
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The syllabus asks for
- Algebraic Fraction: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- Factorisation formulas: a^2-b^2=(a+b)(a-b); (a± b)^2; a^3± b^3=(a± b)(a^2mp ab+b^2); trinomials by sum and product or by splitting the middle term.
- Cancel only factors, never added terms.
- (p)/(q)×(r)/(s)=(pr)/(qs); (p)/(q)÷(r)/(s)=(p)/(q)×(s)/(r).
- To add or subtract: factorise the denominators, take the LCM, rewrite, combine numerators with brackets, simplify.
- Complex fractions: simplify the inside first; x^2-(1)/(x^2)=left(x+(1)/(x)right)left(x-(1)/(x)right).
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The syllabus asks for
- Equations: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- A pair of linear equations: solve by substitution, elimination or graph; check in both.
- Word problems: choose letters, form equations, solve, answer in words.
- Quadratic standard form ax^2+bx+c=0; zero product rule after factorising.
- Completing the square: add left((b)/(2a)right)^2 to both sides.
- Quadratic formula x=frac-b±√b^2-4ac2a; discriminant D=b^2-4ac: D>0 two roots, D=0 one root, D<0 no real root.
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The syllabus asks for
- Area of Triangles and Quadrilaterals: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- Parallelograms on the same base and between the same parallels have equal area.
- A triangle on the same base as a parallelogram and between the same parallels has half its area.
- Triangles on the same base (or equal bases) and between the same parallels are equal in area.
- A median divides a triangle into two triangles of equal area; triangles with the same height have areas in the ratio of their bases.
- The diagonals of a parallelogram divide it into four triangles of equal area.
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The syllabus asks for
- Construction: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- Constructions use only a ruler and compass (a protractor only for a given angle); keep all arcs.
- Parallel through a point: rhombus method with equal radii.
- Triangle = quadrilateral: join diagonal AC; through D draw DE∥ AC meeting BC produced at E; Δ ABE has the same area.
- Parallelogram = triangle: bisect the base; draw the given angle at the midpoint; parallels through the apex and through the end of the base.
- Rectangle = triangle: same, with 90^°. Square = rectangle: semicircle on a+b; the perpendicular at the junction is √ab.
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The syllabus asks for
- Circle: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- Angle at the centre =2× angle at the circumference on the same arc.
- Angles in the same segment are equal.
- Angle in a semicircle =90^°; the hypotenuse of a right triangle is a diameter of its circumcircle.
- Cyclic quadrilateral: opposite angles add to 180^°; exterior angle = interior opposite angle.
- Tangent ⊥ radius; tangent length =√OP^2-r^2; two tangents from a point are equal.
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The syllabus asks for
- Trigonometry: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- sinθ=(p)/(h), cosθ=(b)/(h), tanθ=(p)/(b)=(sinθ)/(cosθ); reciprocals csc,sec,cot. Memory aid SOH-CAH-TOA.
- Standard values for 30^°,45^°,60^° come from the triangles with sides 1,1,√2 and 1,√3,2.
- Complementary angles: sin(90^°-θ)=cosθ, cos(90^°-θ)=sinθ, tan(90^°-θ)=cotθ.
- Identities: sin^2θ+cos^2θ=1; 1+tan^2θ=sec^2θ; 1+cot^2θ=csc^2θ.
- For one given ratio, draw the triangle and find the third side with Pythagoras.
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The syllabus asks for
- Statistics: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- Range =L-S; coefficient (L-S)/(L+S).
- Q1 at (n+1)/(4)th item, Q3 at (3(n+1))/(4)th item (individual/discrete); (n)/(4), (3n)/(4) with the interpolation formula for continuous data.
- QD=(Q3-Q1)/(2); coefficient (Q3-Q1)/(Q3+Q1).
- MD=(Σ f|x-bar x|)/(n); coefficient (MD)/(bar x).
- σ=√(Σ f(x-bar x)^2)/(n)=√(Σ fx^2)/(n)-bar x^2; step deviation σ=h√(Σ fd'^2)/(n)-left((Σ fd')/(n)right)^2.
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The syllabus asks for
- Probability: concepts, formulas and worked problems
- Exercises of the SEE type
Key points
- P(E)=(n(E))/(n(S)) for equally likely outcomes; 0≤ P(E)≤1.
- Complement: P(E')=1-P(E). "At least one" =1-P(none).
- Addition rule: P(A∪ B)=P(A)+P(B)-P(A∩ B); for mutually exclusive events P(A∩ B)=0.
- Multiplication rule for independent events: P(A∩ B)=P(A)× P(B).
- Without replacement the second probability changes; use a tree diagram and multiply along each path.
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Objective questions
Compulsory Mathematics: Unit-wise Objective Question Bank
Subjective questions
Compulsory Mathematics: Unit-wise Subjective Question Bank
Model sets and predicted questions
Compulsory Mathematics: Model Question Sets and Predicted Questions
Exam guide
The SEE Examination: Pattern, Grid and Strategy
The paper
SEE, 3 hr, FM 75 written (+25 internal)
| Section | Questions | Marks each | Total |
|---|---|---|---|
| Compulsory Mathematics | — | — | — |