Std. 10
Mathematics

GST is an indirect tax on goods and services. In ICSE problems the applicable rate is supplied; calculate the taxable value first and then the tax.

GST = taxable value × rate/100 Amount paid = taxable value + GST For an intra-state supply, the total GST is divided into CGST and SGST.

A fixed amount P is deposited every month for n months. I = P·n(n+1)/24 × r/100 and MV = Pn + I, where r is annual interest rate.

Dividend is calculated on face value, while return is based on actual investment. Income = shares × FV × dividend rate/100 Return = income/investment ×100.

Market value above face value is a premium; below face value is a discount. Brokerage and fractional shares are excluded from the prescribed problems.

Solve inequalities algebraically and express the solution in set notation and on a number line. When multiplying or dividing by a negative number, reverse the inequality sign.

A quadratic has the form ax²+bx+c=0, a≠0. It may be solved by factorisation or the quadratic formula.

x = (−b ± √(b²−4ac))/(2a) The discriminant is D=b²−4ac.

D>0 gives two distinct real roots; D=0 gives two equal real roots; D<0 gives no real roots.

Translate word problems into an equation, solve it, and reject any mathematical root that does not satisfy the original conditions.

A ratio compares quantities in the same units. A proportion states equality of ratios, for example a/b=c/d. Cross multiplication gives ad=bc.

If y varies directly as x, y=kx; the ratio y/x is constant.

If y varies inversely as x, xy=k; increasing one quantity decreases the other proportionally.

When f(x) is divided by (x−a), the remainder is f(a).

If f(a)=0, then (x−a) is a factor of f(x). This is especially useful for cubic factorisation.

A matrix is an arrangement of numbers in rows and columns. Its order is written rows × columns.

Equal matrices have the same order and equal corresponding entries. Addition and subtraction require equal orders.

For AB to exist, the number of columns of A must equal the number of rows of B. Each entry is found by multiplying a row of A by a column of B and adding.

For a 2×2 matrix, |a b; c d| = ad−bc.

A 2×2 matrix has an inverse when its determinant is non-zero. The prescribed inverse formula is A⁻¹ = 1/(ad−bc) [[d,−b],[−c,a]].

An arithmetic progression is a sequence in which the difference between consecutive terms is constant. That difference is the common difference d.

aₙ = a+(n−1)d where a is the first term.

Sₙ = n/2[2a+(n−1)d] and, when the last term l is known, Sₙ = n/2(a+l).

Identify the first term, common difference and required term/number of terms before substituting. Check that n is appropriate for the situation.

If P divides A(x₁,y₁), B(x₂,y₂) internally in ratio m:n, P=((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)).

The midpoint is ((x₁+x₂)/2,(y₁+y₂)/2).

For three vertices, the centroid is ((x₁+x₂+x₃)/3,(y₁+y₂+y₃)/3).

m=(y₂−y₁)/(x₂−x₁). Parallel lines have equal slopes; perpendicular lines have product of slopes −1 when both slopes are defined.

In x-axis: (x,y)→(x,−y); in y-axis: (x,y)→(−x,y); in origin: (x,y)→(−x,−y).

Similar figures have equal corresponding angles and proportional corresponding sides. For triangles use AA, SAS or SSS similarity criteria as applicable.

A line parallel to one side of a triangle divides the other two sides proportionally. The converse is also useful for proving parallelism.

For similar triangles, Area₁/Area₂=(corresponding side₁/corresponding side₂)².

In a right triangle, hypotenuse²=base²+perpendicular². The converse can establish whether a triangle is right-angled.

Name corresponding sides and angles carefully. Do not jump from a diagram to a result; state the theorem or criterion that justifies each major step.

The angle subtended by an arc at the centre is twice the angle subtended by the same arc at the circumference.

Angles standing on the same chord in the same segment are equal.

Opposite angles of a cyclic quadrilateral are supplementary: A+C=180°. The exterior angle equals the opposite interior angle.

The tangent at a point is perpendicular to the radius through that point. Tangents drawn from an external point are equal.

If two chords intersect inside a circle, PA·PB=PC·PD. Use the correct segments before applying the relation.

Use compass and straightedge accurately. Construction is a geometric process, not a measurement exercise.

A line segment can be divided internally in a given ratio using an auxiliary ray and parallel-line construction.

Tangents at a point and tangents from an external point are constructed using perpendiculars, equal radii and the standard circle construction.

Perpendicular and angle bisectors are constructed using equal-radius arcs. Leave essential construction arcs visible.

Use a sharp pencil, accurate arcs and clear labels. Do not erase the construction evidence needed to justify the result.

CSA=2πrh TSA=2πr(h+r) V=πr²h

l=√(r²+h²) CSA=πrl TSA=πr(l+r) V=⅓πr²h

Surface area=4πr² Volume=4/3πr³

TSA=3πr² Volume=2/3πr³

Split a solid into familiar shapes. Add or subtract only the surfaces and volumes that actually form the required quantity. Keep all dimensions in consistent units.

sinθ=opposite/hypotenuse cosθ=adjacent/hypotenuse tanθ=opposite/adjacent

cosecθ=1/sinθ secθ=1/cosθ cotθ=1/tanθ

sin²θ+cos²θ=1 1+tan²θ=sec²θ 1+cot²θ=cosec²θ

sin(90°−θ)=cosθ cos(90°−θ)=sinθ tan(90°−θ)=cotθ

Draw the right triangle first. Mark the angle of elevation or depression, identify opposite/adjacent/hypotenuse and then choose the ratio.

For a frequency distribution, Mean=Σfx/Σf. The assumed-mean method uses Mean=A+Σfd/Σf.

Mean=A+h(Σfu/Σf) where u=(x−A)/h.

For grouped continuous data, Median=l+[(N/2−cf)/f]h. Identify the median class using cumulative frequency first.

For grouped data, Mode=l+[(f₁−f₀)/(2f₁−f₀−f₂)]h. The modal class has the highest frequency.

A cumulative-frequency curve represents accumulated frequencies. The median can be estimated graphically using the prescribed ogive method.

Probability measures how likely an event is. For equally likely outcomes, divide favourable outcomes by total outcomes.

P(E)=favourable outcomes/total outcomes and 0≤P(E)≤1.

P(not E)=1−P(E). This is useful when the complementary event is easier to count.

If A and B cannot occur together, P(A∪B)=P(A)+P(B).

For independent events, P(A∩B)=P(A)P(B). Always identify the event structure before using a multiplication rule.