CBSE Class 10 Mathematics Syllabus

Describe the method of completing the square to solve ax² + bx + c = 0.

Make the coefficient of x² equal to 1, add and subtract the square of half the coefficient of x to form a perfect square (x + b/2a)², then take the square root and solve for x.

What is the discriminant of a quadratic equation ax² + bx + c = 0?

The discriminant is D = b² − 4ac.

How does the discriminant determine the nature of the roots of a quadratic equation?

If b²−4ac > 0: two distinct real roots; if b²−4ac = 0: two equal (coincident) real roots; if b²−4ac < 0: no real roots.

What is an Arithmetic Progression (AP)?

A list of numbers in which each term is obtained by adding a fixed number (the common difference d) to the preceding term.

For an AP with first term a and common difference d, how is the common difference found from consecutive terms?

d = a_(k+1) − a_k, i.e. any term minus its preceding term (it is constant throughout the AP).

What is the formula for the nth term (a_n) of an AP with first term a and common difference d?

a_n = a + (n − 1)d.

What is the formula for the sum of the first n terms of an AP, given a and d?

S_n = n/2 [2a + (n − 1)d].

What is the formula for the sum of the first n terms of an AP when the first term a and last term l are known?

S_n = n/2 (a + l).

When are two figures said to be similar?

Two figures are similar if they have the same shape (not necessarily the same size); their corresponding angles are equal and their corresponding sides are in the same ratio (proportion).

State the Basic Proportionality Theorem (Thales' Theorem).

If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then the other two sides are divided in the same ratio.

State the AAA (or AA) similarity criterion for triangles.

If in two triangles the corresponding angles are equal (it suffices that two pairs of angles are equal), then the triangles are similar.

State the SSS similarity criterion for triangles.

If in two triangles the corresponding sides are in the same ratio (proportional), then their corresponding angles are equal and the triangles are similar.

State the SAS similarity criterion for triangles.

If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar.

State the theorem relating the areas of two similar triangles to their sides.

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

State the Distance Formula for the distance between points (x₁, y₁) and (x₂, y₂).

Distance = √[(x₂ − x₁)² + (y₂ − y₁)²].

What is the distance of a point (x, y) from the origin?

√(x² + y²).

State the Section Formula for the point dividing the line segment joining (x₁,y₁) and (x₂,y₂) internally in the ratio m₁:m₂.

( (m₁x₂ + m₂x₁)/(m₁+m₂) , (m₁y₂ + m₂y₁)/(m₁+m₂) ).

What is the formula for the midpoint of the line segment joining (x₁,y₁) and (x₂,y₂)?

( (x₁+x₂)/2 , (y₁+y₂)/2 ).

State the formula for the area of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃).

Area = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|.

What does it mean if the area of a triangle formed by three points is zero?

The three points are collinear (they lie on the same straight line).

Define the six trigonometric ratios of an acute angle A in a right triangle.

sin A = opposite/hypotenuse; cos A = adjacent/hypotenuse; tan A = opposite/adjacent; cosec A = 1/sin A; sec A = 1/cos A; cot A = 1/tan A.

Give the values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°.

sin: 0, 1/2, 1/√2, √3/2, 1 | cos: 1, √3/2, 1/√2, 1/2, 0 | tan: 0, 1/√3, 1, √3, not defined.

State the relationship for trigonometric ratios of complementary angles.

sin(90°−A) = cos A, cos(90°−A) = sin A, tan(90°−A) = cot A, cot(90°−A) = tan A, sec(90°−A) = cosec A, cosec(90°−A) = sec A.

State the three fundamental Pythagorean trigonometric identities.

sin²A + cos²A = 1; 1 + tan²A = sec²A; 1 + cot²A = cosec²A.

In heights and distances, what are the angle of elevation and the angle of depression?

Angle of elevation is the angle between the horizontal and the line of sight when the object is above the observer; angle of depression is that angle when the object is below the observer.

Define a tangent to a circle.

A tangent to a circle is a line that intersects (touches) the circle at exactly one point, called the point of contact.

State the theorem about a tangent and the radius at the point of contact.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

How many tangents can be drawn to a circle from a point, depending on the point's position?

From a point inside the circle: no tangent; from a point on the circle: exactly one tangent; from a point outside the circle: exactly two tangents.

State the theorem about the lengths of tangents drawn from an external point to a circle.

The lengths of the two tangents drawn from an external point to a circle are equal.

Outline the construction for dividing a line segment AB in the ratio m:n.

Draw a ray AX making an acute angle with AB, mark off (m+n) equal arcs on AX, join the (m+n)th point to B, then draw a line parallel to it through the mth point to meet AB at the dividing point.

What is the circumference (perimeter) and area of a circle of radius r?

Circumference = 2πr; Area = πr².

What are the formulas for the length of an arc and the area of a sector of angle θ (in degrees) in a circle of radius r?

Arc length = (θ/360) × 2πr; Area of sector = (θ/360) × πr².

What is the area of a minor segment of a circle (radius r, sector angle θ)?

Area of segment = Area of sector − Area of corresponding triangle = (θ/360)πr² − ½ r² sin θ.

What is the curved (lateral) surface area and total surface area of a right circular cylinder of radius r and height h?

Curved surface area = 2πrh; Total surface area = 2πr(r + h).

What is the curved surface area and total surface area of a cone with radius r, height h and slant height l (l = √(r²+h²))?

Curved surface area = πrl; Total surface area = πr(l + r).

What are the surface area formulas for a sphere of radius r and a hemisphere of radius r?

Sphere surface area = 4πr²; Hemisphere curved surface area = 2πr²; Hemisphere total surface area = 3πr².

State the volume formulas for a cylinder, cone and sphere.

Cylinder = πr²h; Cone = (1/3)πr²h; Sphere = (4/3)πr³.

What principle is used when a solid is converted (recast/melted) from one shape into another?

The volume remains constant (volume is conserved), so the volume of the original solid equals the volume of the new solid.

What is a frustum of a cone?

A frustum is the solid obtained when a cone is cut by a plane parallel to its base, removing the smaller cone on top; it has two circular ends of different radii.

State the volume of a frustum of a cone with radii r₁ and r₂ and height h.

Volume = (1/3)πh(r₁² + r₂² + r₁r₂).

State the curved surface area and total surface area of a frustum with radii r₁, r₂ and slant height l.

Curved surface area = π(r₁ + r₂)l; Total surface area = π(r₁ + r₂)l + πr₁² + πr₂², where l = √[h² + (r₁ − r₂)²].

State the direct method formula for the mean of grouped data.

Mean x̄ = Σf_i x_i / Σf_i, where x_i are the class marks and f_i the corresponding frequencies.

State the assumed mean (deviation) method formula for the mean of grouped data.

Mean x̄ = a + (Σf_i d_i / Σf_i), where a is the assumed mean and d_i = x_i − a.

State the step-deviation method formula for the mean of grouped data.

Mean x̄ = a + h × (Σf_i u_i / Σf_i), where u_i = (x_i − a)/h and h is the class size.

State the formula for the mode of grouped data.

Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where l is the lower limit of the modal class, f₁ its frequency, f₀ and f₂ the frequencies of the preceding and succeeding classes, and h the class size.

State the formula for the median of grouped data.

Median = l + [(n/2 − cf)/f] × h, where l is the lower limit of the median class, n = Σf_i, cf the cumulative frequency before the median class, f the frequency of the median class, and h the class size.

State the empirical relationship between mean, median and mode.

3 Median = Mode + 2 Mean (equivalently, Mode = 3 Median − 2 Mean).

What are the two types of ogives used to graphically represent a cumulative frequency distribution?

The 'less than' ogive and the 'more than' ogive (both are cumulative frequency curves).

How can the median be obtained graphically from ogives?

It is the x-coordinate of the point of intersection of the 'less than' and 'more than' ogives (or the x-value corresponding to cumulative frequency n/2 on a single ogive).

State the classical (theoretical) definition of the probability of an event E.

P(E) = (Number of outcomes favourable to E) / (Total number of equally likely outcomes), assuming all outcomes are equally likely.

What is the range of the probability of any event E?

0 ≤ P(E) ≤ 1.

What are a sure (certain) event and an impossible event, and what are their probabilities?

A sure event is certain to happen with probability 1; an impossible event cannot happen and has probability 0.

What are complementary events, and what is the relation between their probabilities?

E and 'not E' are complementary events; P(E) + P(not E) = 1, so P(not E) = 1 − P(E).

What is an elementary event in probability?

An event having only one (single) outcome of the experiment; the sum of the probabilities of all elementary events of an experiment is 1.

How does the SSS similarity criterion differ from the SSS congruence criterion?

SSS similarity requires corresponding sides to be in the same ratio (proportional), giving same shape; SSS congruence requires corresponding sides to be exactly equal, giving same shape and size.

Compare consistent and inconsistent pairs of linear equations.

A consistent pair has at least one solution (intersecting lines give one solution; coincident lines give infinitely many); an inconsistent pair has no solution (parallel lines).

Outline the steps to construct a tangent to a circle from an external point P.

Join P to centre O, find midpoint M of OP, draw a circle with centre M and radius MO; this circle cuts the given circle at two points; lines from P to these points are the required tangents.

How do you find the area of a combination of plane figures (e.g., a shape made of a rectangle and semicircles)?

Decompose the figure into known basic shapes, calculate each area separately, then add or subtract these areas as appropriate to get the total area.