CBSE Class 10 Board Exam Mathematics (Standard) Syllabus

Name the methods used to find the roots of a quadratic equation.

Factorisation (splitting the middle term), completing the square, and the quadratic formula.

Define an Arithmetic Progression (AP) and its common difference.

An AP is a sequence in which each term is obtained by adding a fixed number to the preceding term. The fixed number is the common difference d = aₙ₊₁ − aₙ.

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

aₙ = a + (n − 1)d.

What are the two formulas for the sum of the first n terms of an AP?

Sₙ = n/2 [2a + (n − 1)d], or equivalently Sₙ = n/2 (a + l), where l is the last (nth) term.

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

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

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

Distance from 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.

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

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

Midpoint = ( (x₁ + x₂)/2, (y₁ + y₂)/2 ).

Using distances of plotted points, how do you verify that four points form a square versus a rhombus?

For both, all four sides are equal. For a square, the two diagonals are also equal; for a rhombus, the diagonals are unequal. Compute side lengths and diagonals with the distance formula to distinguish them.

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 it divides those two sides in the same ratio.

State the criteria for similarity of two triangles.

AAA (or AA): corresponding angles equal. SSS: corresponding sides proportional. SAS: one pair of equal included angles with the two including sides proportional.

What is the relationship between the areas of two similar triangles and their corresponding sides?

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides (or medians, altitudes, angle bisectors).

State the Pythagoras Theorem and its converse.

Pythagoras Theorem: In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Converse: If the square of one side equals the sum of the squares of the other two sides, the angle opposite the first side is a right angle.

How many tangents can be drawn to a circle from a point inside, on, and outside the circle?

From a point inside: 0 tangents. From a point on the circle: exactly 1 tangent. From a point outside: exactly 2 tangents.

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

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

State the theorem about the lengths of two 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.

How do you divide a line segment in a given ratio m : n by construction?

Draw a ray from one endpoint making an acute angle; mark m + n equal arcs along it; join the last point to the other endpoint of the segment; draw a parallel line through the mth point. It meets the segment at the dividing point.

How do you construct a tangent to a circle from an external point P?

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

Define the six trigonometric ratios sin, cos, tan, cosec, sec, cot for an acute angle in a right triangle.

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

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, undefined.

What are the reciprocal relationships among the trigonometric ratios?

cosec θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ. Also tan θ = sin θ/cos θ and cot θ = cos θ/sin θ.

State the three Pythagorean trigonometric identities.

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

What are the trigonometric ratios of complementary angles (θ and 90° − θ)?

sin(90° − θ) = cos θ; cos(90° − θ) = sin θ; tan(90° − θ) = cot θ; cot(90° − θ) = tan θ; sec(90° − θ) = cosec θ; cosec(90° − θ) = sec θ.

Define the angle of elevation and the angle of depression.

Angle of elevation: the angle formed by the line of sight with the horizontal when the object viewed is above the horizontal level. Angle of depression: the angle formed by the line of sight with the horizontal when the object is below the horizontal level.

Give the formulas for the area and circumference of a circle of radius r.

Area = πr²; Circumference = 2πr.

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

Length of arc = (θ/360) × 2πr; Area of sector = (θ/360) × πr².

What is the formula for the area of a segment of a circle?

Area of segment = Area of corresponding sector − Area of the triangle formed by the two radii and the chord = (θ/360)πr² − area of triangle.

Give the formulas for the curved surface area, total surface area, and volume of a cylinder of radius r and height h.

CSA = 2πrh; TSA = 2πr(r + h); Volume = πr²h.

Give the formulas for the curved surface area, total surface area, and volume of a cone with radius r, height h, and slant height l (l = √(r² + h²)).

CSA = πrl; TSA = πr(l + r); Volume = (1/3)πr²h.

Give the surface area and volume formulas for a sphere (radius r) and a hemisphere (radius r).

Sphere: Surface area = 4πr², Volume = (4/3)πr³. Hemisphere: CSA = 2πr², TSA = 3πr², Volume = (2/3)πr³.

State the three formulas (methods) for finding the mean of grouped data.

Direct method: x̄ = Σfᵢxᵢ / Σfᵢ. Assumed mean method: x̄ = a + Σfᵢdᵢ / Σfᵢ (dᵢ = xᵢ − a). Step-deviation method: x̄ = a + (Σfᵢuᵢ / Σfᵢ) × h, where uᵢ = (xᵢ − a)/h.

State the formula for the mode of grouped data.

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

State the formula for the median of grouped data and the empirical relationship between mean, median, and mode.

Median = l + [(n/2 − cf) / f] × h, where l = lower limit of median class, n = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class size. Empirical relation: 3 Median = Mode + 2 Mean.

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

P(E) = (Number of outcomes favourable to E) / (Total number of equally likely outcomes), where 0 ≤ P(E) ≤ 1.

What is the relationship between the probability of an event E and the probability of its complement (not E)?

P(E) + P(not E) = 1, so P(not E) = 1 − P(E). 'Not E' is the complement of E.

What are the probabilities of a sure (certain) event and an impossible event?

Probability of a sure event = 1; probability of an impossible event = 0.