ML Aggarwal Class 10 Solutions ICSE Maths

Written by K. MenonPublished on 12 June 2026

ML Aggarwal Class 10 Solutions help ICSE students solve each Maths chapter with step-by-step working, reasons for each transformation, and final answers. This page provides chapter-wise and exercise-wise support for Understanding ICSE Mathematics Class 10, including Commercial Mathematics, Algebra, Matrices, Geometry, Mensuration, Trigonometry, Coordinate Geometry, Statistics, and Probability.

Use these solutions after you try the question from your printed textbook. Do not copy a final answer without checking the chapter name, exercise number, and edition, because exercise numbering may differ from one school edition to another.

ML Aggarwal Class 10 Solutions: Chapter-Wise ICSE Maths Index

The table below is a study index for ICSE Class 10 Maths solutions based on the usual topic flow in Understanding ICSE Mathematics Class 10 by M.L. Aggarwal. Treat the chapter order as a guide, not as a replacement for your own textbook contents page.

Chapter name	Major topics	Exercise links	Revision priority
Goods and Services Tax	Taxable value, GST rate, CGST, SGST, IGST, bill amount	GST worked solutions	First revision for commercial calculations
Banking	Recurring deposit, monthly instalment, interest, maturity value	Banking worked solutions	First revision for formula substitution
Shares and Dividends	Face value, market value, dividend, income, yield	Shares and dividends solutions	Formula revision and unit checking
Linear Inequations	Solving inequalities, number line, interval notation	Linear inequations solutions	First revision for sign rules
Quadratic Equations	Factorisation, formula method, nature of roots, word problems	Quadratic equations solutions	First revision for algebra accuracy
Ratio and Proportion	Direct proportion, inverse proportion, continued proportion	Ratio and proportion solutions	Useful for word problems
Remainder and Factor Theorems	Remainder theorem, factor theorem, polynomial division	Polynomial solutions	Practise signs and substitution
Arithmetic Progression	General term, sum of terms, inserted means	Progression solutions	Formula-based revision
Geometric Progression	Common ratio, general term, sum of finite terms	Progression solutions	Check powers carefully
Matrices	Order, equality, addition, subtraction, multiplication, equations	Matrices worked solutions	Practise row-by-column working
Similarity	Similar triangles, proportional sides, scale factor	Similarity proof solutions	Proof-writing practice
Circles	Cyclic quadrilaterals, tangents, intersecting chords, angle properties	Circle proof solutions	First revision for theorem use
Loci and Constructions	Locus conditions, ruler-compass construction steps	Loci and construction support	Practise diagrams neatly
Mensuration	Area, perimeter, surface area, volume, unit conversion	Mensuration solutions	First revision for formula use
Trigonometry	Ratios, identities, standard angles, simplification	Trigonometry solutions	First revision for identities
Heights and Distances	Angle of elevation, angle of depression, right triangle equations	Heights and distances solutions	Diagram-based practice
Coordinate Geometry	Distance, midpoint, section formula, slope, line, area of triangle	Coordinate geometry solutions	Substitution-based revision
Statistics	Mean, median, mode, ogive, grouped data tables	Statistics solutions	Table-format practice
Probability	Sample space, favourable outcomes, probability of an event	Probability solutions	Short-answer accuracy

How to use these solutions for daily homework and revision

First, solve the textbook question without looking at the answer. Then compare your working with the steps given here. If your final answer differs, check the line where the operation changes: sign change, formula substitution, square root, unit conversion, or theorem reason.

For revision, keep a notebook with three columns: question number, mistake made, corrected method. This helps you remember why a step is valid, not just what the answer looks like.

Chapter-wise table with exercise links and topic names

The index table gives exercise links by topic because chapter numbering can change between editions. If your book has a different exercise number, match the topic and the first question type before using a solution.

What each solution includes: question, steps, reasoning, and final answer

Every exercise solution should follow this format:

Restate the question before solving.
Write each step in order.
Give the reason for each important transformation, such as a formula, theorem, identity, or sign rule.
End with a clear final answer, written with the correct unit where required.

About Understanding ICSE Mathematics Class 10 by ML Aggarwal

Book, subject, board, and class covered

Understanding ICSE Mathematics Class 10 by M.L. Aggarwal is a Mathematics textbook used by many ICSE schools for Class 10 practice. The solutions on this page are written for ICSE Class 10 Maths topics such as Commercial Mathematics, Algebra, Geometry, Mensuration, Trigonometry, Coordinate Geometry, Statistics, and Probability.

This page is not a board publication and does not claim approval from CISCE or any publisher. It is a teacher-written solution resource to help students understand the working behind answers.

How this differs from other ICSE Maths books

Different ICSE Maths books may teach the same syllabus topics in a different order. The examples, exercise numbering, and wording of questions may also differ. That is why you should not mix an answer from one book with an exercise from another book unless the question is exactly the same.

Students looking for nearby levels can also use ml aggarwal class 7 solutions, ml aggarwal class 8 solutions, and ml aggarwal class 11 solutions for class-wise practice.

Why students should match the chapter name and exercise before copying an answer

Before using any ML Aggarwal Class 10 maths solution, check four things: chapter name, exercise number, question number, and the exact numbers in the question. A small difference in rate of GST, number of months, radius, height, or angle can change the final answer.

ICSE vs CBSE warning: Use this page only for ICSE Class 10 ML Aggarwal Maths. CBSE-style books may include different chapter order, exercise numbering, examples, and exam-writing expectations. If your book mentions CBSE or follows a different syllabus pattern, do not use these ICSE solutions as your answer key.

Commercial Mathematics Solutions

Commercial Mathematics questions are usually formula-based, but marks are lost when students skip the taxable value, month count, face value, or market value. Write the formula first, substitute values second, and give the final answer with rupees or percentage.

Commercial Mathematics formula table

Topic	Formula	Use
GST amount	$\text{GST} = \frac{\text{Taxable value} \times \text{GST rate}}{100}$	Find tax on goods or services
Invoice value	$\text{Invoice value} = \text{Taxable value} + \text{GST}$	Find final bill amount
CGST and SGST	$\text{CGST} = \text{SGST} = \frac{\text{GST}}{2}$	Used for intra-state transactions when GST is split equally
Recurring deposit interest	$I = \frac{P \times n(n+1) \times r}{2 \times 12 \times 100}$	Find interest when $P$ is monthly instalment, $n$ is months, and $r$ is annual rate
Recurring deposit maturity value	$M = Pn + I$	Find amount received at maturity
Dividend per share	$\text{Dividend per share} = \frac{\text{Face value} \times \text{Dividend rate}}{100}$	Find income from one share
Total dividend income	$\text{Income} = \text{Number of shares} \times \text{Dividend per share}$	Find annual dividend income
Yield percentage	$\text{Yield} = \frac{\text{Income}}{\text{Investment}} \times 100$	Compare return with market price paid

Goods and Services Tax: step-by-step GST calculations

Question: A product has a taxable value of $\text{₹}12,000$ . GST is charged at $18\%$ . Find the GST and the invoice value.

Step 1: Write the formula for GST.

\text{GST} = \frac{\text{Taxable value} \times \text{GST rate}}{100}

Step 2: Substitute the taxable value and GST rate. GST is calculated on the taxable value, not on the final invoice value.

\text{GST} = \frac{12000 \times 18}{100} = 2160

Step 3: Add GST to the taxable value to get the invoice value.

\text{Invoice value} = 12000 + 2160 = 14160

Final answer: GST $= \text{₹}2,160$ , and invoice value $= \text{₹}14,160$ .

Common mistakes in GST

Calculating GST on the amount after adding tax instead of the taxable value.
Mixing CGST, SGST, and IGST. In many school problems, intra-state GST is split as $\text{CGST} = \text{SGST}$ .
Forgetting to add GST to the taxable value when the question asks for the invoice value.
Writing only the tax amount when the question asks for the total bill.

Banking: recurring deposit, interest, and maturity value problems

Question: A person deposits $\text{₹}500$ every month in a recurring deposit account for $2$ years. If the rate of interest is $6\%$ per annum, find the interest and maturity value.

Step 1: Identify the values. Monthly instalment $P = 500$ , number of months $n = 24$ , rate $r = 6$ .

Step 2: Use the recurring deposit interest formula.

I = \frac{P \times n(n+1) \times r}{2 \times 12 \times 100}

Step 3: Substitute the values. The factor $n(n+1)$ is used because each monthly deposit earns interest for a different length of time.

I = \frac{500 \times 24 \times 25 \times 6}{2 \times 12 \times 100}

Step 4: Simplify.

I = \frac{1800000}{2400} = 750

Step 5: Find the maturity value.

M = Pn + I = 500 \times 24 + 750 = 12000 + 750 = 12750

Final answer: Interest $= \text{₹}750$ , and maturity value $= \text{₹}12,750$ .

Common mistakes in banking

Using years instead of months in the recurring deposit formula. If the deposit is for $2$ years, then $n = 24$ .
Forgetting the denominator $2 \times 12 \times 100$ .
Giving interest as the final maturity value.
Multiplying $P$ by the number of years instead of the number of months.

Shares and dividends: formula-based worked examples

Question: A man buys $100$ shares of face value $\text{₹}10$ at a market value of $\text{₹}25$ each. If the company declares $20\%$ dividend, find his annual income and yield percentage.

Step 1: Dividend is calculated on face value, not market value.

\text{Dividend per share} = \frac{10 \times 20}{100} = 2

Step 2: Find total annual income.

\text{Income} = 100 \times 2 = 200

Step 3: Find total investment using market value.

\text{Investment} = 100 \times 25 = 2500

Step 4: Find yield percentage.

\text{Yield} = \frac{200}{2500} \times 100 = 8\%

Final answer: Annual income $= \text{₹}200$ , and yield $= 8\%$ .

Common mistakes in commercial maths calculations

Using market value instead of face value while calculating dividend.
Leaving out rupee units in money answers.
Rounding too early in percentage questions.
Copying the formula but substituting values in the wrong places.

Algebra Solutions

Algebra solutions must show each operation clearly. In ICSE-style answers, the examiner should be able to see why a sign changed, why a root was rejected, or why an interval was chosen.

Linear inequations: number line representation and interval answers

Question: Solve $3x - 5 \leq 10$ for $x \in \mathbb{R}$ , and write the answer in interval form.

Step 1: Add $5$ to both sides. This keeps the inequality sign unchanged because the same number is added to both sides.

3x - 5 + 5 \leq 10 + 5

3x \leq 15

Step 2: Divide both sides by $3$ . The inequality sign does not reverse because $3$ is positive.

x \leq 5

Step 3: Write the interval form.

x \in (-\infty, 5]

Final answer: $x \leq 5$ , or $x \in (-\infty, 5]$ .

Common mistakes in linear inequations

Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
Using an open circle on the number line for $\leq$ or $\geq$ . These require a closed point.
Writing $(-\infty, 5)$ instead of $(-\infty, 5]$ when $x = 5$ is included.
Writing only the number line and not the algebraic solution.

Quadratic equations: factorisation, formula method, and word problems

Question: Solve $x^2 - 5x + 6 = 0$ .

Step 1: Factorise the quadratic expression. We need two numbers whose product is $6$ and sum is $-5$ .

x^2 - 5x + 6 = x^2 - 2x - 3x + 6

Step 2: Factor by grouping.

x(x-2) - 3(x-2) = 0

(x-2)(x-3) = 0

Step 3: Use the zero-product rule.

x - 2 = 0 \quad \text{or} \quad x - 3 = 0

x = 2 \quad \text{or} \quad x = 3

Final answer: $x = 2$ or $x = 3$ .

Question: Solve $2x^2 + 3x - 2 = 0$ by the formula method.

Step 1: Compare with $ax^2 + bx + c = 0$ . Here, $a = 2$ , $b = 3$ , and $c = -2$ .

Step 2: Use the quadratic formula.

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Step 3: Substitute the values.

x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)}

x = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm 5}{4}

Step 4: Find the two roots.

x = \frac{-3 + 5}{4} = \frac{1}{2}

x = \frac{-3 - 5}{4} = -2

Final answer: $x = \frac{1}{2}$ or $x = -2$ .

Common mistakes in quadratic equations

Writing $b^2 - 4ac$ incorrectly when $c$ is negative.
Dropping the $\pm$ sign in the quadratic formula.
Factorising $x^2 - 5x + 6$ as $(x+2)(x+3)$ , which gives the wrong middle term.
Keeping an impossible root in a word problem, such as a negative length or negative time.

Ratio and proportion: direct, inverse, and continued proportion questions

Question: Find the third proportional to $4$ and $12$ .

Step 1: Let the third proportional be $x$ . Then $4:12 = 12:x$ .

Step 2: Convert the proportion into an equation.

\frac{4}{12} = \frac{12}{x}

Step 3: Cross-multiply.

4x = 144

Step 4: Solve for $x$ .

x = 36

Final answer: The third proportional is $36$ .

Arithmetic and geometric progression solutions where applicable

For arithmetic progression, write $a$ , $d$ , and $n$ before using $a_n = a + (n-1)d$ . For geometric progression, write $a$ , $r$ , and $n$ before using $a_n = ar^{n-1}$ . Most errors happen when students copy the wrong common difference or common ratio.

Common mistakes in signs, roots, and simplification

Changing $-5x$ to $+5x$ without performing the same operation on both sides.
Taking square root on one side but not using both possible roots when required.
Not checking whether an algebraic answer satisfies the original word problem.

Matrices Solutions

Matrices questions should show order first. If the order is not suitable, the operation is not defined. For multiplication, write row-by-column products instead of jumping directly to the answer.

Matrix order, equality, addition, and subtraction

If two matrices are equal, corresponding elements are equal. If two matrices are added or subtracted, they must have the same order.

Matrix multiplication with row-column working

Question: If $A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ and $B = \begin{bmatrix}2 & 0\\1 & 5\end{bmatrix}$ , find $AB$ .

Step 1: Check the order. Both matrices are of order $2 \times 2$ , so $AB$ is defined and will also be of order $2 \times 2$ .

Step 2: Multiply the first row of $A$ by the first column of $B$ .

(1)(2) + (2)(1) = 4

Step 3: Multiply the first row of $A$ by the second column of $B$ .

(1)(0) + (2)(5) = 10

Step 4: Multiply the second row of $A$ by the first column of $B$ .

(3)(2) + (4)(1) = 10

Step 5: Multiply the second row of $A$ by the second column of $B$ .

(3)(0) + (4)(5) = 20

Final answer: $AB = \begin{bmatrix}4 & 10\\10 & 20\end{bmatrix}$ .

Solving simultaneous equations using matrices where required

When simultaneous equations are solved using matrices, first write them in the form $AX = B$ . Then use $X = A^{-1}B$ , provided $A^{-1}$ exists. If the determinant is $0$ , the inverse matrix does not exist.

Common mistakes in matrix order and multiplication

Assuming $AB = BA$ . Matrix multiplication is not generally commutative.
Multiplying corresponding entries instead of using row-by-column multiplication.
Trying to add matrices of different orders.
Writing the answer without checking the order of the product.

Geometry Solutions

Geometry solutions need reasons. A correct diagram helps, but the proof must name the theorem or property used. Write statements and reasons in parallel when the question asks you to prove something.

Similarity problems with proportional sides

For similar triangles, first prove equality of angles or proportionality of sides. Then write the corresponding sides in the same order. If $\triangle ABC \sim \triangle PQR$ , then $AB$ corresponds to $PQ$ , $BC$ corresponds to $QR$ , and $AC$ corresponds to $PR$ .

Circles: angle properties, cyclic quadrilaterals, tangents, and proofs

Question: In a circle, $ABCD$ is a cyclic quadrilateral. If $\angle A = 70^\circ$ , find $\angle C$ .

Step 1: Use the theorem: opposite angles of a cyclic quadrilateral are supplementary.

\angle A + \angle C = 180^\circ

Step 2: Substitute $\angle A = 70^\circ$ .

70^\circ + \angle C = 180^\circ

Step 3: Solve for $\angle C$ .

\angle C = 180^\circ - 70^\circ = 110^\circ

Final answer: $\angle C = 110^\circ$ .

Common mistakes in circles

Using the cyclic quadrilateral theorem without first stating that the quadrilateral is cyclic.
Confusing angle in the same segment with angle at the centre.
Using tangent-radius perpendicularity without identifying the point of contact.
Writing only numerical steps in a proof question and omitting reasons.

Loci and constructions where included in the exercise

For loci, write the condition first: equal distance from two fixed points, fixed distance from a point, or equal distance from two intersecting lines. For construction, use a sharp pencil, mark arcs clearly, and keep the final answer visible. A construction answer is incomplete if the required point, line, or angle is not labelled.

How to write geometry proof answers in ICSE style

What to write	Why it matters
Given statement	Shows what information is allowed to be used
To prove	Shows the target of the proof
Construction, if needed	Explains any extra line or point drawn
Statement with reason	Shows the theorem used at each step
Conclusion	Connects the last step to what was asked

Exam-writing note: In theorem-based geometry, do not write a chain of angles without reasons. A line such as $\angle ABC = \angle ADC$ should be followed by a reason, for example, angles in the same segment are equal.

Mensuration Solutions

Mensuration questions are calculation-heavy. Write the formula before substitution, convert units before calculation, and keep $\pi$ as instructed by the question. If the question does not specify a value of $\pi$ , use the value your teacher expects for that exercise.

Formula table for quick revision

Figure or solid	Formula	Meaning
Rectangle	$\text{Area} = l \times b$	Area of rectangular region
Circle	$\text{Area} = \pi r^2$	Area enclosed by a circle
Circle circumference	$C = 2\pi r$	Distance around a circle
Cylinder curved surface area	$2\pi rh$	Area of curved part only
Cylinder total surface area	$2\pi r(h+r)$	Curved part plus two circular ends
Cylinder volume	$\pi r^2h$	Space occupied by the cylinder
Cone curved surface area	$\pi rl$	Uses slant height $l$
Cone total surface area	$\pi r(l+r)$	Curved surface plus base
Cone volume	$\frac{1}{3}\pi r^2h$	One-third of corresponding cylinder volume
Sphere surface area	$4\pi r^2$	Total surface area of a sphere
Sphere volume	$\frac{4}{3}\pi r^3$	Volume of a sphere
Hemisphere total surface area	$3\pi r^2$	Curved surface plus circular base

Area and perimeter problems

For area and perimeter questions, check whether the answer must be in $\text{cm}^2$ , $\text{m}^2$ , $\text{cm}$ , or $\text{m}$ . Perimeter is a length; area is a square unit.

Surface area and volume of solids

Question: Find the volume of a cylinder of radius $7\ \text{cm}$ and height $10\ \text{cm}$ . Take $\pi = \frac{22}{7}$ .

Step 1: Write the formula for the volume of a cylinder.

V = \pi r^2h

Step 2: Substitute $r = 7$ and $h = 10$ . Radius and height are already in centimetres, so no unit conversion is needed.

V = \frac{22}{7} \times 7^2 \times 10

Step 3: Simplify.

V = \frac{22}{7} \times 49 \times 10 = 1540

Final answer: Volume $= 1540\ \text{cm}^3$ .

Conversion of units before substitution

If one measurement is in metres and another is in centimetres, convert before substituting in the formula. Do not calculate first and convert later unless you are certain the power of the unit is handled correctly. For example, $1\ \text{m} = 100\ \text{cm}$ , but $1\ \text{m}^2 = 10000\ \text{cm}^2$ .

Trigonometry Solutions

Trigonometry questions usually require one of three methods: use a standard angle value, simplify an identity, or form an equation from a right triangle. Draw a diagram for heights and distances before writing the trigonometric ratio.

Trigonometry formula table

Concept	Formula or value	Use
Basic ratios	$\sin\theta = \frac{\text{perpendicular}}{\text{hypotenuse}}$ , $\cos\theta = \frac{\text{base}}{\text{hypotenuse}}$ , $\tan\theta = \frac{\text{perpendicular}}{\text{base}}$	Right triangle questions
Reciprocal ratios	$\cosec\theta = \frac{1}{\sin\theta}$ , $\sec\theta = \frac{1}{\cos\theta}$ , $\cot\theta = \frac{1}{\tan\theta}$	Convert reciprocal functions
Pythagorean identity	$\sin^2\theta + \cos^2\theta = 1$	Simplification and proof
Tangent identity	$1 + \tan^2\theta = \sec^2\theta$	Convert between $\tan\theta$ and $\sec\theta$
Cotangent identity	$1 + \cot^2\theta = \cosec^2\theta$	Convert between $\cot\theta$ and $\cosec\theta$
Standard values	$\sin 30^\circ = \frac{1}{2}$ , $\cos 60^\circ = \frac{1}{2}$ , $\tan 45^\circ = 1$	Direct evaluation

Trigonometric ratios and standard angles

Memorise the standard values for $0^\circ$ , $30^\circ$ , $45^\circ$ , $60^\circ$ , and $90^\circ$ . Do not use decimal approximations unless the question asks for an approximate answer.

Heights and distances word problems

Question: From a point on the ground, the angle of elevation of the top of a tower is $30^\circ$ . If the point is $20\ \text{m}$ from the foot of the tower, find the height of the tower.

Step 1: Draw a right triangle. The height of the tower is the perpendicular, and the distance from the tower is the base.

Step 2: Use $\tan\theta$ , because the question gives base and asks for perpendicular.

\tan 30^\circ = \frac{\text{height}}{20}

Step 3: Substitute $\tan 30^\circ = \frac{1}{\sqrt{3}}$ .

\frac{1}{\sqrt{3}} = \frac{h}{20}

Step 4: Solve for $h$ .

h = \frac{20}{\sqrt{3}} = \frac{20\sqrt{3}}{3}

Final answer: Height of the tower $= \frac{20\sqrt{3}}{3}\ \text{m}$ .

Trigonometric identities with step-by-step simplification

Question: Prove that $\frac{1 - \cos^2\theta}{\sin^2\theta} = 1$ .

Step 1: Start with the left-hand side.

\frac{1 - \cos^2\theta}{\sin^2\theta}

Step 2: Use the identity $1 - \cos^2\theta = \sin^2\theta$ .

\frac{1 - \cos^2\theta}{\sin^2\theta} = \frac{\sin^2\theta}{\sin^2\theta}

Step 3: Simplify the fraction.

\frac{\sin^2\theta}{\sin^2\theta} = 1

Hence proved.

Common mistakes in trigonometry

Choosing $\sin\theta$ when the given sides require $\tan\theta$ or $\cos\theta$ .
Using $\tan 30^\circ = \sqrt{3}$ instead of $\tan 30^\circ = \frac{1}{\sqrt{3}}$ .
Changing both sides of an identity at the same time and losing the proof structure.
Forgetting to rationalise where the teacher expects a rationalised denominator.

Coordinate Geometry Solutions

Coordinate geometry answers should be written by substitution. State the formula, substitute the coordinates in the same order, simplify carefully, and then write the final answer.

Coordinate geometry formula table

Concept	Formula	Use
Distance formula	$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$	Distance between two points
Midpoint formula	$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$	Midpoint of a line segment
Section formula	$\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)$	Point dividing a segment internally in the ratio $m:n$
Slope	$m = \frac{y_2-y_1}{x_2-x_1}$	Inclination of a line
Equation of a line	$y - y_1 = m(x - x_1)$	Line through one point with slope $m$
Area of triangle	$\frac{1}{2}\left\|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right\|$	Area from three coordinates

Distance formula and section formula problems

Question: Find the distance between $A(2,3)$ and $B(8,11)$ .

Step 1: Use the distance formula.

AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Step 2: Substitute $x_1 = 2$ , $y_1 = 3$ , $x_2 = 8$ , and $y_2 = 11$ .

AB = \sqrt{(8-2)^2 + (11-3)^2}

Step 3: Simplify.

AB = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10

Final answer: $AB = 10$ units.

Equation of a line and slope-based questions

Question: Find the equation of the line passing through $(1,2)$ with slope $3$ .

Step 1: Use the point-slope form.

y - y_1 = m(x - x_1)

Step 2: Substitute $m = 3$ , $x_1 = 1$ , and $y_1 = 2$ .

y - 2 = 3(x - 1)

Step 3: Expand and simplify.

y - 2 = 3x - 3

y = 3x - 1

Final answer: The equation of the line is $y = 3x - 1$ .

Midpoint and area of triangle problems

For midpoint problems, do not subtract coordinates. Add the two $x$ -coordinates and divide by $2$ ; add the two $y$ -coordinates and divide by $2$ . For triangle area, use the modulus sign because area cannot be negative.

Step-by-step coordinate substitution format

Exam-writing note: In coordinate geometry, write the labelled coordinates before substitution. This prevents mistakes when the points are named $A$ , $B$ , and $C$ but the formula uses $(x_1,y_1)$ , $(x_2,y_2)$ , and $(x_3,y_3)$ .

Statistics and Probability Solutions

Statistics questions often become long because of tables. Keep columns aligned and write totals clearly. Probability questions are shorter, but the sample space must be understood before writing the fraction.

Statistics and probability formula table

Topic	Formula	Use
Mean for ungrouped data	$\bar{x} = \frac{\sum x}{n}$	Average of raw observations
Mean for grouped data	$\bar{x} = \frac{\sum f_ix_i}{\sum f_i}$	Average using frequency and class mark
Median class formula	$\text{Median} = l + \frac{\frac{N}{2} - c_f}{f} \times h$	Median of grouped data
Mode formula	$\text{Mode} = l + \frac{f_1-f_0}{2f_1-f_0-f_2} \times h$	Mode of grouped data
Probability	$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of equally likely outcomes}}$	Chance of an event

Mean, median, mode, and grouped data calculations

Question: Find the mean of the following distribution.

Marks	Frequency
$10$	$2$
$20$	$3$
$30$	$5$

Step 1: Make a calculation table with $fx$ .

$x$	$f$	$fx$
$10$	$2$	$20$
$20$	$3$	$60$
$30$	$5$	$150$
Total	$10$	$230$

Step 2: Use the mean formula.

\bar{x} = \frac{\sum fx}{\sum f}

Step 3: Substitute and simplify.

\bar{x} = \frac{230}{10} = 23

Final answer: Mean $= 23$ .

Ogive and graphical interpretation questions

For ogive questions, convert the grouped frequency table into cumulative frequency first. Plot class boundaries on the horizontal axis and cumulative frequency on the vertical axis. The median is read from $\frac{N}{2}$ on the cumulative frequency axis.

Probability problems with sample space explanation

Question: A fair die is thrown once. Find the probability of getting an even number.

Step 1: Write the sample space.

S = \{1,2,3,4,5,6\}

Step 2: Write the favourable outcomes. Even numbers are $2$ , $4$ , and $6$ .

E = \{2,4,6\}

Step 3: Use the probability formula.

P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}} = \frac{3}{6} = \frac{1}{2}

Final answer: Probability of getting an even number $= \frac{1}{2}$ .

Table format for long statistical calculations

For long statistics solutions, use a table with headings such as class interval, frequency, class mark, cumulative frequency, and product. Do not write scattered calculations below the question; it becomes hard to check errors.

ML Aggarwal Class 10 PDF and Book Access: What Is Safe and Legal

Why students should avoid unauthorized PDF downloads

Searches for ML Aggarwal Class 10 PDF, ML Aggarwal Class 10 ICSE book PDF, or ML Aggarwal Class 10 Maths book PDF often lead to unauthorized copies. Avoid downloading or sharing pirated textbook PDFs. They may violate copyright, may be incomplete, and may not match your school edition.

How to use the printed textbook with online solutions

This article provides solutions and explanations, not unauthorized textbook PDF downloads. Keep your printed textbook open, read the question from the book, and use the worked steps here to check the method and final answer.

How to identify the correct ICSE edition before solving

Check the cover, class, board, subject, edition year if printed, and contents page. Then match the chapter title and exercise number. If your exercise begins with different questions, use the topic heading instead of the question number alone.

What to do if your exercise numbers differ from this page

If exercise numbers differ, do not assume the answer is wrong. First compare the question wording and values. If the values match, the working method should still help; if the values differ, follow the same steps with your own numbers.

ICSE vs CBSE: Make Sure You Are Using the Right ML Aggarwal Solutions

How ICSE Class 10 Maths solutions differ from CBSE-style solutions

ICSE Maths solutions often require detailed working, theorem reasons in geometry, and structured calculation steps. CBSE-style books may have different sequencing, question types, and marking expectations. Use the solution set that matches your board and textbook.

Why syllabus, chapter order, and exercise numbering may differ

Even when a topic such as quadratic equations or trigonometry appears in both boards, the exercise arrangement can differ. This is why the safest method is to match the exact question, not only the chapter name.

When not to use this ICSE solution set

Do not use this ICSE solution set if your textbook does not say ICSE Class 10 Mathematics, if the chapter list is different from your school syllabus, or if your teacher has assigned a different book. Use your school-prescribed book as the first reference.

Exam-Writing Notes for ML Aggarwal Class 10 Maths

How to present theorem-based geometry proofs

Write geometry answers in a statement-reason format. If you use a theorem, name it clearly. For example, if two angles are equal because they stand on the same chord, write the reason as angles in the same segment are equal.

How to present calculation-heavy answers

For GST, banking, mensuration, coordinate geometry, and statistics, show the formula before substitution. Keep units with the final answer. Do not round intermediate values unless the question instructs you to round.

Exam-weightage note

The exact distribution of marks can change by examination year and school paper pattern. Instead of memorising fixed marks from old papers, revise by skill: formula substitution, algebraic manipulation, theorem proof, graph reading, and word-problem translation.

Frequently Asked Questions

Are ML Aggarwal Class 10 Solutions enough for ICSE Maths revision?

They are useful for checking methods and correcting mistakes, but they should not replace solving the textbook questions yourself. Use the solutions after attempting each exercise.

Can I use these solutions if my exercise numbers are different?

Yes, but only after matching the chapter name, topic, and exact question values. Exercise numbering can differ by edition, so the question text is more reliable than the number alone.

Does this page provide the ML Aggarwal Class 10 Maths book PDF?

No. This page provides worked solutions and study support. It does not provide unauthorized textbook PDF downloads.

How should I write final answers in Maths?

Write the final answer clearly and include the correct unit wherever needed. For example, money answers should include rupees, area answers should use square units, and volume answers should use cubic units.

Why are reasons important in geometry solutions?

Geometry is not only about getting the angle value. ICSE-style proof answers need the theorem or property used at each step, such as cyclic quadrilateral property, tangent-radius property, or angles in the same segment.

Should I memorise every formula table?

You should know the main formulas and practise substituting values correctly. Formula memory helps, but marks are usually lost when students skip steps, use wrong units, or copy values incorrectly.

ICSE Class 10 Syllabus Papers Books Notes 2026 Plan ICSE Board: Free PDF Study Materials for Classes 6–10 ICSE Board: Free PDF Study Materials for Classes 6–10 ICSE Class 10 Syllabus Papers Books Notes 2026 Plan