1. Quadratic equation:

Roots of the quadratic equation $a x^{2} + b x + c = 0$ are $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Sum of roots, $x_{1} + x_{2} = - \frac{b}{a}$

Product of roots, $x_{1} x_{2} = \frac{c}{a}$

2. Binomial theorem:

${(1 + x)}^{n} = 1 + n x + \frac{n (n - 1)}{2} x^{2} + \frac{n (n - 1) (n - 2)}{6} x^{3} + \dots$

${(1 - x)}^{n} = 1 - n x + \frac{n (n - 1)}{2} x^{2} - \frac{n (n - 1) (n - 2)}{6} x^{3} + \dots$

If $x ≪ 1$ then ${(1 + x)}^{n} \approx 1 + n x$ & ${(1 - x)}^{n} \approx 1 - n x$

3. Logarithm:

$\log (m n) = \log (n) + \log (n)$

$l o g (\frac{m}{n}) = \log (m) - \log (n)$

$\log {(m)}^{n} = n \log (m)$

$\log_{e} (m) = 2.303 \log_{10} (m)$

$\log (2) = 0.3010$

$\log (3) = 0.4771$

4. Componendo and dividendo theorem:

If $\frac{p}{q} = \frac{a}{b}$ , then $\frac{p + q}{p - q} = \frac{a + b}{a - b}$ .

(i) Arithmetic progression (AP): $a, a + d, a + 2 d, a + 3 d, \dots a + (n - 1) d$ here $d =$ common difference

Sum of $n$ terms, $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

Note:

(i) $1 + 2 + 3 + 4 + 5 \dots + n = \frac{n (n + 1)}{2}$

(ii) $1^{2} + 2^{2} + 3^{2} + \dots + n^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

(ii) Geometrical progression (GP): $a, a r, {a r}^{2}, {a r}^{3}, \dots$ here, $r =$ common ratio

Sum of $n$ terms, $S_{n} = \frac{a (1 - r^{n})}{1 - r}$

Sum of $\infty$ terms, $S_{\infty} = \frac{a}{1 - r}$ , if $r < 1$ .

5. Trigonometry:

$2 π radian$ $= 360^{°} \Rightarrow 1 r a d = {57.3}^{°}$

$s i n θ = \frac{p e r p e n d i c u l a r}{h y p o t e n u s e}$

$c o s θ = \frac{b a s e}{h y p o t e n u s e}$

$t a n θ = \frac{p e r p e n d i c u l a r}{b a s e}$

$c o t θ = \frac{b a s e}{p e r p e n d i c u l a r}$

$s e c θ = \frac{h y p o t e n u s e}{b a s e}$

$c o s e c θ = \frac{h y p o t e n u s e}{p e r p e n d i c u l a r}$

$s i n θ = \frac{a}{\sqrt{a^{2} + b^{2}}}$

$c o s θ = \frac{b}{\sqrt{a^{2} + b^{2}}}$

$t a n θ = \frac{a}{b}$

$c o s e c θ = \frac{1}{s i n θ}$

$s e c θ = \frac{1}{c o s θ}$

$c o t θ = \frac{1}{t a n θ}$

$s i n (A \pm B) = s i n A c o s B \pm c o s A s i n B$

$c o s (A \pm B) = c o s A c o s B \mp s i n A s i n B$

$t a n (A \pm B) = \frac{t a n A \pm t a n B}{1 \mp t a n A t a n B}$

$t a n (2 A) = \frac{2 t a n A}{1 - t a n^{2} A}$

$s i n (90^{°} + θ) = c o s θ$	$s i n (180^{°} - θ) = s i n θ$	$s i n (- θ) = - s i n θ$	$s i n (90^{°} - θ) = c o s θ$
$c o s (90^{°} + θ) = - s i n θ$	$c o s (180^{°} - θ) = - c o s θ$	$c o s (- θ) = c o s θ$	$c o s (90^{°} - θ) = s i n θ$
$t a n (90^{°} + θ) = - c o t θ$	$t a n (180^{°} - θ) = - t a n θ$	$t a n (- θ) = - t a n θ$	$t a n (90^{°} - θ) = c o t θ$
$s i n (180^{°} + θ) = - s i n θ$	$s i n (270^{°} - θ) = - c o s θ$	$s i n (270^{°} + θ) = - c o s θ$	$s i n (360^{°} - θ) = - s i n θ$
$c o s (180^{°} + θ) = - c o s θ$	$c o s (270^{°} - θ) = - s i n θ$	$c o s (270^{°} + θ) = s i n θ$	$c o s (360^{°} - θ) = c o s θ$
$t a n (180^{°} + θ) = t a n θ$	$t a n (270^{°} - θ) = c o t θ$	$t a n (270^{°} + θ) = - c o t θ$	$t a n (360^{°} - θ) = - t a n θ$

$θ \to$	$0^{°}$ $(0)$	$30^{°}$ $(\frac{π}{6})$	$45^{°}$ $(\frac{π}{4})$	$60^{°}$ $(\frac{π}{3})$	$90^{°}$ $(\frac{π}{2})$	$120^{°}$ $(\frac{2 π}{3})$	$135^{°}$ $(\frac{3 π}{4})$	$150^{°}$ $(\frac{5 π}{6})$	$180^{°}$ $(π)$	$270^{°}$ $(\frac{3 π}{2})$	$360^{°}$ $(2 π)$
$s i n θ$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$	$- 1$	$0$
$c o s θ$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$	$- \frac{1}{2}$	$- \frac{1}{\sqrt{2}}$	$- \frac{\sqrt{3}}{2}$	$- 1$	$0$	$1$
$t a n θ$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	$\infty$	$- \sqrt{3}$	$- 1$	$- \frac{1}{\sqrt{3}}$	$0$	$\infty$	$0$

sine rule,

$\frac{s i n A}{a} = \frac{s i n B}{b} = \frac{s i n C}{c}$

cosine rule,

$c o s A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}, c o s B = \frac{c^{2} + a^{2} - b^{2}}{2 c a}, c o s C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$

For small angle $θ$ ,

$s i n θ \approx θ, c o s θ \approx 1, t a n θ \approx θ a n d s i n θ \approx t a n θ$

6. Differentiation:

$y = x^{n} \to \frac{d y}{d x} = n x^{n - 1}$

$y = l n x \to \frac{d y}{d x} = \frac{1}{x}$

$y = s i n x \to \frac{d y}{d x} = c o s x$

$y = c o s x \to \frac{d y}{d x} = - s i n x$

$y = e^{α x + β} \to \frac{d y}{d x} = α e^{α x + β}$

Product Rule, $y = u v \to \frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}$

$y = f (g (x)) \Rightarrow \frac{d y}{d x} = \frac{d f (g (x))}{d g (x)} \times \frac{d (g (x))}{d x}$

$y = k = constant$ $\Rightarrow \frac{d y}{d x} = 0$

Division Rule, $y = \frac{u}{v} \Rightarrow \frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$

7. Integration:

$\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C, n \neq - 1$

$\int \frac{1}{x} d x = l n x + C$

$\int s i n x d x = - c o s x + C$

$\int c o s x d x = s i n x + C$

$\int e^{a x + β} d x = \frac{1}{α} e^{a x + β} + C$

$\int {(α x + β)}^{n} d x = \frac{{(α x + β)}^{n + 1}}{α (n + 1)} + C$

8. Maxima & Minima of a function, $y = f (x)$ :

For maximum value $\frac{d y}{d x} = 0 & \frac{d^{2} y}{d x^{2}} = - v e$

For minimum value $\frac{d y}{d x} = 0 & \frac{d^{2} y}{d x^{2}} = + v e$

Average of a varying quantity or function, if $y = f (x)$ then $< y > = \bar{y} = \frac{\int_{x_{1}}^{x_{2}} y d x}{\int_{x_{1}}^{x_{2}} d x} = \frac{\int_{x_{1}}^{x_{2}} y d x}{(x_{2} - x_{1})}$

9. Formulae for determination of area:

Area of a square $=$ ${(s i d e)}^{2}$

Area of rectangle $=$ length $\times$ breadth

Area of a triangle $= \frac{1}{2} \times$ base $\times$ height

Area of a trapezoid $= \frac{1}{2} \times$ (distance between parallel sides) $\times$ (sum of parallel sides)

Area enclosed by a circle $= π r^{2}$ , ( $r =$ radius)

Surface area of a sphere $= 4 π r^{2}$ , ( $r =$ radius)

Area of the curved surface of cylinder $= 2 π r l$ , ( $r =$ radius and $l =$ length)

Area of the whole surface of cylinder $= 2 π r (r + l)$ , ( $l =$ length)

Area of ellipse $= π a b$ , ( $a & b$ are semi-major and semi-minor axis respectively)

Surface area of a cube $= 6 {(s i d e)}^{2}$

Total surface area of a cone $= π r^{2} + π r l$ , where $π r l = π r \sqrt{r^{2} + h^{2}}$ .

10. Formulae for determination of volume:

Volume of a rectangular slab $= l e n g t h \times b r e a d t h \times h e i g h t = a b t$

Volume of a cube $= {(s i d e)}^{3}$

Volume of a sphere $= \frac{4}{3} π r^{3}$ , ( $r =$ radius)

Volume of a cylinder $= π r^{2} l$ , ( $r =$ radius and $l =$ length)

Volume of a cone $= \frac{1}{3} π r^{2} h$ , ( $r =$ radius and $h =$ height)

11. Key points:

To convert an angle from degree to radian, we have to multiply it by $\frac{π}{180^{°}}$ and to convert an angle from radian to degree, we have to multiply it by $\frac{180^{°}}{π}$ .

With the help of differentiation, if $y$ is given, we can find $\frac{d y}{d x}$ and by the help of integration, if $\frac{d y}{d x}$ is given, we can find $y$ .

The maximum and minimum values of the function, $y = A c o s θ + B s i n θ$ are $\sqrt{A^{2} + B^{2}}$ and $- \sqrt{A^{2} + B^{2}}$ respectively.

12. Important Identities:

${(a + b)}^{2} = a^{2} + b^{2} + 2 a b$

${(a - b)}^{2} = a^{2} + b^{2} - 2 a b$

$(a + b) (a - b) = a^{2} - b^{2}$

${(a + b)}^{3} = a^{3} + b^{3} + 3 a b (a + b)$

${(a - b)}^{3} = a^{3} - b^{3} - 3 a b (a - b)$

13. Vectors:

A physical quantity that requires magnitude and a particular direction when it is expressed.

(i) Triangle law of vector addition:

$\vec{R} = \vec{A} + \vec{B}$

$R = \sqrt{A^{2} + B^{2} + 2 A B c o s θ}$

$t a n α = \frac{B s i n θ}{A + B c o s θ}$

If $A = B$ then $R = 2 A c o s \frac{θ}{2} & α = \frac{θ}{2}$

$R_{m a x} = A + B$ for $θ = 0^{°}$

$R_{m i n} = A - B$ for $θ = 180^{°}$

(ii) Parallelogram Law of Addition of Two Vectors: If two vectors are represented by two adjacent sides of a parallelogram which are directed away from their common point then their sum (i.e. resultant vector) is given by the diagonal of the parallelogram passing away through that common point.

$\vec{A B} + \vec{A D} = \vec{A C} = \vec{R}$

or $\vec{A} + \vec{B} = \vec{R} \Rightarrow R = \sqrt{A^{2} + B^{2} + 2 A B c o s θ}$

$t a n α = \frac{B s i n θ}{A + B c o s θ}$ and $t a n β = \frac{A s i n θ}{B + A c o s θ}$

14. Vector subtraction:

$\vec{R} = \vec{A} - \vec{B} \Rightarrow \vec{R} = \vec{A} + (- \vec{B})$

$R = \sqrt{A^{2} + B^{2} - 2 A B c o s θ}$ , $t a n α = \frac{B s i n θ}{A - B c o s θ}$

If $A = B$ then, $R = 2 A s i n (\frac{θ}{2})$

15. Addition of More than Two Vectors (Law of Polygon):

If some vectors are represented by sides of a polygon in the same order, then their resultant vector is represented by the closing side of a polygon in the opposite order.

$\vec{R} = \vec{A} + \vec{B} + \vec{C} + \vec{D}$

Rectangular component of a $3 D$ vector, $\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$

Angle made with $x$ -axis, $c o s α = \frac{A_{x}}{A} = \frac{A_{x}}{\sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}} = l$

Angle made with $y$ -axis, $c o s β = \frac{A_{y}}{A} = \frac{A_{y}}{\sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}} = m$

Angle made with $z$ -axis, $c o s γ = \frac{A_{z}}{A} = \frac{A_{z}}{\sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}} = n$

Here, $l, m a n d n$ are called direction cosines.

$l^{2} + m^{2} + n^{2} = c o s^{2} α + c o s^{2} β + c o s^{2} γ = \frac{A_{x}^{2} + A_{y}^{2} + A_{2}^{2}}{{(\sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}})}^{2}} = 1$

or ${s i n}^{2} α + {s i n}^{2} β + {s i n}^{2} γ = 2$

The general vector in $x$ - $y$ plane

$\vec{r} = x \hat{i} + y \hat{j} = r (c o s θ \hat{i} + s i n θ \hat{j})$

16. Scalar product (Dot Product):

$\vec{A} \cdot \vec{B} = A B c o s θ \Rightarrow$ Angle between two vectors $θ = c o s^{- 1} (\frac{\vec{A} \cdot \vec{B}}{A B})$

If $\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k} & \vec{B} = B_{x} \hat{i} + B_{y} \hat{j} + B_{z} \hat{k}$ then

$\vec{A} \cdot \vec{B} = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$ and the angle between $\vec{A} & \vec{B}$ is given by

$c o s θ = \frac{\vec{A} \cdot \vec{B}}{A B} = \frac{A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}}{\sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}} \sqrt{B_{x}^{2} + B_{y}^{2} + B_{z}^{2}}}$

$\hat{i} . \hat{i} = 1, \hat{j} . \hat{j} = 1, \hat{k} . \hat{k} = 1, \hat{i} . \hat{j} = 0, \hat{i} . \hat{k} = 0, \hat{j} . \hat{k} = 0$

Component of vector $\vec{b}$ along vector $\vec{a}$ , ${\vec{b}}_{| |} = (\vec{b} \cdot \hat{a}) \hat{a}$

Component of vector $\vec{b}$ perpendicular to $\vec{a}$ , ${\vec{b}}_{⊥} = \vec{b} - {\vec{b}}_{‖} = \vec{b} - (\vec{b} \cdot \hat{a}) \hat{a}$

17. Cross Product (Vector product):

$\vec{A} \times \vec{B} = A B s i n θ \hat{n}$ where $\hat{n}$ is a vector perpendicular to $\vec{A} & \vec{B}$ or their plane and its direction given by the right-hand thumb rule.

(i) $\vec{A} \times \vec{B} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ A_{x} & A_{y} & A_{z} \\ B_{x} & B_{y} & B_{z} \end{matrix}| = \hat{i} (A_{y} B_{z} - A_{z} B_{y}) - \hat{j} (A_{x} B_{z} - B_{x} A_{z}) + \hat{k} (A_{x} B_{y} - B_{x} A_{y})$

(ii) $\vec{A} \times \vec{B} = - \vec{B} \times \vec{A}$

(iii) $(\vec{A} \times \vec{B}) \cdot \vec{A} = (\vec{A} \times \vec{B}) \cdot \vec{B} = 0$

(iv) $\hat{i} \times \hat{i} = \vec{0}, \hat{j} \times \hat{j} = \vec{0}, \hat{k} \times \hat{k} = \vec{0}$

(v) $\hat{i} \times \hat{j} = \hat{k}; \hat{j} \times \hat{k} = \hat{i}$

(vi) $\hat{k} \times \hat{i} = \hat{j}, \hat{j} \times \hat{j} = - \hat{k}$

(vii) $\hat{k} \times \hat{j} = - \hat{i}, \hat{i} \times \hat{k} = - \hat{j}$

18. Differentiation:

(i) $\frac{d}{d t} (\vec{A} \cdot \vec{B}) = \frac{d \vec{A}}{d t} \cdot \vec{B} + \vec{A} \cdot \frac{d \vec{B}}{d t}$

(ii) $\frac{d}{d t} (\vec{A} \times \vec{B}) = \frac{d \vec{A}}{d t} \times \vec{B} + \vec{A} \times \frac{d \vec{B}}{d t}$

When a particle moved from $(x_{1}, y_{1}, z_{1})$ to $(x_{2}, y_{2}, z_{2})$ then its displacement vector

$\vec{r} = {\vec{r}}_{2} - {\vec{r}}_{1} = (x_{2} \hat{i} + y_{2} \hat{j} + z_{2} \hat{k}) - (x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k})$

Magnitude, $r = |\vec{r}| = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}$

19. Lami's theorem:

$\frac{s i n A}{a} = \frac{s i n B}{b} = \frac{s i n C}{c}$

$\frac{F_{1}}{s i n θ_{1}} = \frac{F_{2}}{s i n θ_{2}} = \frac{F_{3}}{s i n θ_{3}}$

20. Area of triangle:

Area $= \frac{| \vec{A} \times \vec{B} |}{2} = \frac{1}{2} A B s i n θ$

21. Area of parallelogram:

Area $= |\vec{A} \times \vec{B}| = A B s i n θ$

For Parallel vectors $\vec{A} \times \vec{B} = \vec{0}$

For perpendicular vectors $\vec{A} \cdot \vec{B} = 0$

For coplanar vectors $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$

22. Examples of dot products:

Work, $W = \vec{F} \cdot \vec{d} = F d c o s θ$ where $F \to$ force, $d \to$ displacement

Power, $P = \vec{F} \cdot \vec{v} = F v c o s θ$ where $F \to$ force, $v \to$ velocity

Electric flux, $ϕ_{E} = \vec{E} \cdot \vec{A} = E A c o s θ$ where $E \to$ electric field, $A \to$ Area

Magnetic flux, $ϕ_{B} = \vec{B} \cdot \vec{A} = B A c o s θ$ where $B \to$ magnetic field, $A \to$ Area

Potential energy of dipole in a uniform field, $U = - \vec{p} \cdot \vec{E}$ where $p \to$ dipole moment, $E \to$ Electric field

23. Examples of cross products:

Torque, $\vec{τ} = \vec{r} \times \vec{F}$ where $r \to$ position vector, $F \to$ force

Angular momentum, $\vec{J} = \vec{r} \times \vec{p}$ where $r \to$ position vector, $p \to$ linear momentum

Linear velocity, $\vec{v} = \vec{ω} \times \vec{r}$ where $r \to$ position vector, $ω \to$ angular velocity

Torque on dipole placed in electric field, $\vec{τ} = \vec{p} \times \vec{E}$ where $p \to$ dipole moment, $E \to$ electric field

24. KEY POINTS :

(i) Tensor: A quantity that has different values in different directions is called tensor. Ex. Moment of Inertia

In fact, tensors are merely a generalisation of scalars and vectors; a scalar is a zero rank tensor, and a vector is a first-rank tensor.

(ii) Electric current is not a vector as it does not obey the law of vector addition.

(iii) A unit vector has no unit.

(iv) To a vector only a vector of the same type can be added, and the resultant is a vector of the same type.

(v) A scalar or a vector can never be divided by a vector.

The angle, between A→ and the resultant of 2 A→+3 B→ and 4 A→-3 B→ is

Important Questions on Mathematical Tools in Physics

Learn and Prepare for any exam you want !

The angle, between $\vec{A}$ and the resultant of $2 \vec{A} + 3 \vec{B}$ and $4 \vec{A} - 3 \vec{B}$ is