Mathematics Learning Path
High School (11–12) Mathematics
Explore structured mathematics courses, solved problems, simulations and Python activities.
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Sample Spaces and Events
A bag holds a red, a green and a blue ball, and you take out two. How many results are possible? The answer depends on whether you put the first ball back, and on whether the order matters. Learn to write the list of all outcomes for coins, dice, cards and bags, to name the kinds of event, and to combine events with union, intersection and complement. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Probability by Counting
What is the chance that a hand of five cards holds exactly two aces? You could list every hand, but there are more than two and a half million of them. Count them with combinations instead. Learn the classical definition and its rules, the addition theorem for two and three events, odds and the trick of the complement for at least one. Then use the counting tools of the permutations and combinations chapter on cards, committees and arrangements. It covers the second part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Dice Sums, Matching and Geometric Probability
Why do two dice show a total of seven more often than a total of twelve? Why do two people in a class of twenty-three often share a birthday? Why is a hat check so often wrong for everyone? Count the ways, and the surprises explain themselves. Learn the number of ways to get a dice total, the birthday and matching problems, probability on a line segment, and how a short simulation checks an answer. It covers the third part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Coordinates of a Point in Space
Where is the light in the corner of your room? Three perpendicular lines through one corner give every point three coordinates. Learn the three axes, the three coordinate planes and the eight octants. Then find the foot of a perpendicular and the image of a point in a plane or an axis. Last, measure the distance of a point from the origin, from an axis and from a plane. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching ideas of A-Level and IB.
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Distance and Section Formulae in Space
A cable runs from the top of one mast to the top of another. How long is it, and where does a clamp sit if it divides the cable in the ratio 2 to 3? Two formulae answer both questions for any two points in space. Use them for the midpoint, for dividing a segment inside and outside, and for the centroid of a triangle and of a tetrahedron. Then test collinear points, and tell a parallelogram, a rectangle, a rhombus or a right triangle from its corners. It covers the Indian syllabus (NCERT, JEE Main) and the matching ideas of A-Level and IB.
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Loci, Spheres and Cuboid Problems
A beacon is placed so that it is as far from one tower as from another. Where can it be? A balloon is always 5 metres from a pole top. What shape does it trace? These are loci in space. Meet the sphere as the set of points at a fixed distance and the plane halfway between two points. Then study other loci from the distance formula, and the cube and the cuboid with coordinates. It covers the Indian syllabus at exam depth (NCERT, JEE Main) and the matching ideas of A-Level and IB.
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The Circle: Equations, Lines and Tangents
Where is the centre of a round pond when you know three points on its shore? Start from the distance of a point from a fixed centre and write the equation of a circle in its standard and general forms. Then place a point and a line against a circle, find the length of a chord, and draw the tangent and the normal. You end with the tangents from a point outside, the length of a tangent and the chord of contact. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topic of A-Level Pure Maths.
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Systems of Circles: Radical Axis, Common Tangents and Families
Two round ponds overlap, and a fence is to run along the line through the two meeting points. Where does that line lie? Look at two circles together: how they touch, cut or miss, how many common tangents they have, and what their radical axis is. Then meet orthogonal circles, the circles through the meeting points of two curves, and the locus questions that the entrance exams love. It covers the second part of the Indian syllabus unit (NCERT, JEE Main) and goes beyond the circle work of A-Level Pure Maths.
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The Parabola
Throw a ball and it traces a parabola. Hold a torch at the focus of a mirror and the beam goes out straight. This course defines the parabola as the set of points as far from a fixed point as from a fixed line. You get the four standard equations, the latus rectum, the parametric form, the tangent and the normal in three forms, the focal chord and the reflection property. It covers the parabola in the Indian syllabus unit (NCERT, JEE Main, BITSAT) and in A-Level Further Maths.
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The Ellipse and the Hyperbola
A planet circles the Sun on an ellipse, and a sonic boom spreads along a hyperbola. Both curves come from the same focus and directrix idea as the parabola, with the ratio e a little different. You write the standard equations, find the foci, the latus rectum and the eccentricity, and use the auxiliary circle, the parametric form, tangents, normals and the asymptotes. You end by telling a conic from its general equation. It covers the second half of the conics unit in the Indian syllabus (NCERT, JEE Main) and in A-Level Further Maths.
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Limits and Standard Limits
How close can you get to a place you may not stand on? A limit is the value a function heads for as x comes close to a point, even when the function has a hole at that point. You read limits from a table and a graph, from the left and the right, and you see when a limit does not exist. Then you find them by factorising, rationalising and dividing by the highest power, and you learn the standard limits of sine, powers, exponentials and logarithms. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topic of A-Level, IB and AP.
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Derivatives from First Principles to the Product Rule
A speedometer shows how fast the distance changes at one instant. A tangent shows how steep a curve is at one point. Both are the same limit. Draw a secant line, move its second point closer and closer, and the secant becomes the tangent. The slope is the derivative. You find it from the definition for powers, sine, cosine and tangent, then speed up with the sum, product and quotient rules, and use the derivative as a rate. It covers the second part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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