What is the quadratic formula in CBSE Class 10 Maths?

For a quadratic equation ax^2 + bx + c = 0 (a not equal to 0), the roots are x = (-b plus or minus the square root of (b^2 - 4ac)) / 2a. The discriminant is D = b^2 - 4ac: D > 0 gives two distinct real roots, D = 0 gives two equal real roots, and D < 0 gives no real roots.

What is the formula for the nth term and sum of an AP?

The nth term of an AP is a_n = a + (n - 1)d. The sum of the first n terms is S_n = (n/2)[2a + (n - 1)d], or S_n = (n/2)(a + l), where a is the first term, d the common difference and l the last term.

What are the mean formulas for grouped data in Class 10 Statistics?

Direct method: mean = (sum of f_i x_i) / (sum of f_i). Assumed-mean method: mean = a + (sum of f_i d_i)/(sum of f_i). Step-deviation method: mean = a + h x (sum of f_i u_i)/(sum of f_i), where u_i = (x_i - a)/h.

What is the relationship between HCF and LCM of two numbers?

For two positive integers, HCF x LCM = product of the two numbers. This follows from the Fundamental Theorem of Arithmetic, which says every composite number can be factorised as a unique product of primes.

What are the surface area and volume formulas for a sphere and cone?

Sphere: surface area = 4*pi*r^2, volume = (4/3)*pi*r^3. Cone: curved surface area = pi*r*l, total surface area = pi*r*(l + r), volume = (1/3)*pi*r^2*h, where slant height l = square root of (r^2 + h^2).

NCERT Class 10 Maths Formulas — Complete CBSE Formula Sheet (All Chapters)

Q: What is the formula for the nth term and sum of an AP?

The nth term of an AP is a_n = a + (n - 1)d. The sum of the first n terms is S_n = (n/2)[2a + (n - 1)d], or S_n = (n/2)(a + l), where a is the first term, d the common difference and l the last term.

Q: What are the mean formulas for grouped data in Class 10 Statistics?

Direct method: mean = (sum of f_i x_i) / (sum of f_i). Assumed-mean method: mean = a + (sum of f_i d_i)/(sum of f_i). Step-deviation method: mean = a + h x (sum of f_i u_i)/(sum of f_i), where u_i = (x_i - a)/h.

Q: What is the relationship between HCF and LCM of two numbers?

For two positive integers, HCF x LCM = product of the two numbers. This follows from the Fundamental Theorem of Arithmetic, which says every composite number can be factorised as a unique product of primes.

Q: What are the surface area and volume formulas for a sphere and cone?

Sphere: surface area = 4*pi*r^2, volume = (4/3)*pi*r^3. Cone: curved surface area = pi*r*l, total surface area = pi*r*(l + r), volume = (1/3)*pi*r^2*h, where slant height l = square root of (r^2 + h^2).

Ch 1Real Numbers

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a unique product of primes (apart from the order of the factors).

HCF and LCM relation

$\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$

Euclid's division lemma

$a = bq + r,\quad 0 \le r \lt b$

Full solutions: Real Numbers →

Ch 2Polynomials

Zeroes of a quadratic polynomial

For $ax^2+bx+c$: sum of zeroes $\alpha+\beta = -\dfrac{b}{a}$, product $\alpha\beta = \dfrac{c}{a}$.

Cubic polynomial

For $ax^3+bx^2+cx+d$: $\alpha+\beta+\gamma = -\dfrac{b}{a}$, $\alpha\beta+\beta\gamma+\gamma\alpha = \dfrac{c}{a}$, $\alpha\beta\gamma = -\dfrac{d}{a}$.

Full solutions: Polynomials →

Ch 3Pair of Linear Equations in Two Variables

Conditions for solutions

For $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$:

Unique solution (intersecting)

$\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$

Infinitely many (coincident)

$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$

No solution (parallel)

$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$

Full solutions: Linear Equations →

Ch 4Quadratic Equations

Quadratic formula

$x = \dfrac{-b \pm \sqrt{\,b^2 - 4ac\,}}{2a}$ for $ax^2+bx+c=0$

Discriminant

$D = b^2 - 4ac$. $D\gt0$: distinct real roots; $D=0$: equal real roots; $D\lt0$: no real roots.

Full solutions: Quadratic Equations →

Ch 5Arithmetic Progressions

nth term

$a_n = a + (n-1)d$

Sum of n terms

$S_n = \dfrac{n}{2}\big[2a + (n-1)d\big] = \dfrac{n}{2}(a + l)$

Full solutions: Arithmetic Progressions →

Ch 6Triangles

Basic Proportionality (Thales)

A line parallel to one side of a triangle divides the other two sides in the same ratio: $\dfrac{AD}{DB}=\dfrac{AE}{EC}$.

Similarity criteria

AAA (AA), SSS and SAS.

Ratio of areas of similar triangles

$\dfrac{\text{ar}(\triangle ABC)}{\text{ar}(\triangle PQR)} = \left(\dfrac{AB}{PQ}\right)^2$

Full solutions: Triangles →

Ch 7Coordinate Geometry

Distance formula

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

Section formula

$\left(\dfrac{m x_2 + n x_1}{m+n},\ \dfrac{m y_2 + n y_1}{m+n}\right)$

Midpoint

$\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$

Full solutions: Coordinate Geometry →

Ch 8Introduction to Trigonometry

Basic ratios

$\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$

Pythagorean identities

$\sin^2\theta + \cos^2\theta = 1$; $1 + \tan^2\theta = \sec^2\theta$; $1 + \cot^2\theta = \mathrm{cosec}^2\theta$

Complementary angles

$\sin(90^\circ-\theta) = \cos\theta$, $\tan(90^\circ-\theta) = \cot\theta$, $\sec(90^\circ-\theta) = \mathrm{cosec}\,\theta$

Full solutions: Introduction to Trigonometry →

Ch 9Some Applications of Trigonometry

Heights and distances

The angle of elevation is measured upward from the horizontal; the angle of depression is measured downward from the horizontal. Use $\tan\theta = \dfrac{\text{height}}{\text{distance}}$ with right triangles.

Full solutions: Applications of Trigonometry →

Ch 10Circles

Tangent and radius

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

Tangents from an external point

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

Full solutions: Circles →

Ch 11Areas Related to Circles

Circle

Area $= \pi r^2$; Circumference $= 2\pi r$.

Area of a sector (angle θ)

$\dfrac{\theta}{360^\circ} \times \pi r^2$

Length of an arc (angle θ)

$\dfrac{\theta}{360^\circ} \times 2\pi r$

Full solutions: Areas Related to Circles →

Ch 12Surface Areas and Volumes

Cylinder

CSA $= 2\pi rh$; TSA $= 2\pi r(h+r)$; Volume $= \pi r^2 h$.

Cone

CSA $= \pi r l$; TSA $= \pi r(l+r)$; Volume $= \dfrac{1}{3}\pi r^2 h$; $l = \sqrt{r^2+h^2}$.

Sphere

Surface area $= 4\pi r^2$; Volume $= \dfrac{4}{3}\pi r^3$.

Hemisphere

CSA $= 2\pi r^2$; TSA $= 3\pi r^2$; Volume $= \dfrac{2}{3}\pi r^3$.

Full solutions: Surface Areas and Volumes →

Ch 13Statistics

Mean — direct method

$\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}$

Mean — assumed-mean method

$\bar{x} = a + \dfrac{\sum f_i d_i}{\sum f_i}$, where $d_i = x_i - a$

Mean — step-deviation method

$\bar{x} = a + \left(\dfrac{\sum f_i u_i}{\sum f_i}\right) \times h$, where $u_i = \dfrac{x_i - a}{h}$

Mode (grouped data)

$\text{Mode} = l + \left(\dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$

Median (grouped data)

$\text{Median} = l + \left(\dfrac{\tfrac{n}{2} - cf}{f}\right) \times h$

Full solutions: Statistics →

Ch 14Probability

Probability of an event

$P(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$

Range

$0 \le P(E) \le 1$

Complement

$P(\overline{E}) = 1 - P(E)$

Full solutions: Probability →

Frequently asked questions

What is the quadratic formula?: $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$; discriminant $D = b^2-4ac$.
What is the nth term and sum of an AP?: $a_n = a+(n-1)d$; $S_n = \dfrac{n}{2}[2a+(n-1)d] = \dfrac{n}{2}(a+l)$.
What is the step-deviation formula for mean?: $\bar{x} = a + \left(\dfrac{\sum f_i u_i}{\sum f_i}\right)h$, where $u_i = \dfrac{x_i-a}{h}$.
What is the relation between HCF and LCM?: $\text{HCF} \times \text{LCM} = $ product of the two numbers.

These formulas follow the latest CBSE / NCERT Class 10 Maths syllabus. For worked examples and step-by-step answers, open each chapter's full NCERT solutions linked above.