Geometry
1. Two triangles are similar when the ratio of sides is same and angles are same. If two triangles have same configuration i.e. same sides and same angles, then triangles are congruent.
2. Vertical opposite angles are equal.
3. Alternate angles are equal.
4. General Equation of the line is $ y = mx + c$ where m is the slope and c is the intercept.
5. Slope of any line passing through points $ (x_1, y_1) $ and $ (x_2, y_2) $ is $ m = \frac{y_2-y_1}{x_2-x_1} $. Slope is inclination of the line with positive x-axis. If $ \theta $ is the angle made by the line with positive x-axis then slope will be $ m = tan\theta $. For horizontal line, slope is 0 and for vertical line, slope is $ \infty $.
6. Equation of line passing through points $ (x_1, y_1) $ and $ (x_2, y_2) $ is $ (y-y_1) = \frac{y_2-y_1}{x_2-x_1} (x-x_1) $
7. If we are given two lines, then lines $y=m_1x+c_1$ and $y=m_2x+c_2$ are parallel only when these have the same slopes, i.e. $m_1 = m_2 $.
8. Two lines $y=m_1x+c_1$ and $y=m_2x+c_2$ are perpendicular when the product of the slopes of two lines is -1, i.e. $m_1m_2 = -1$.
9. Equation of the circle is of form, $ ax^2 + ay^2 + 2bx + 2cy + d = 0 $.
10. Equation of the parabola is either of the form $ (y-c)^2=4a(x-b) $, or $ (x-c)^2 = 4a(y-b) $ or $ y = ax^2 - bx $ or $ x = ay^2-by $.
11. Equation of ellipse is $ \frac{x^2}{a^2}+ \frac{y^2}{b^2} = 1 $.
12. Equation of hyperbola is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $.
Calculus
1. Function is defined as a operation in which element of the set first are related to elements of second set by some relation.
Domain of the function is the set all elements (values) which function can take. Range is set of values which function can give.
Let any function $ y = f(x) $. Let us consider that curve $ y=f(x) $ passes through $(x,y) $. If we need to find the tangent at this point, then slope of the curve will be given by $ m = \frac{dy}{dx}=\frac{d f(x)}{dx} = f'(x) $ where $ \frac{dy}{dx}$ is derivative of y with respect to x.
Derivative of some functions are given below:
1. $ y = constant \implies \frac{dy}{dx}=0$
2. $ y = x^n \implies \frac{dy}{dx} = nx^{n-1} $
3. $ y = sinx \implies \frac{dy}{dx}=cosx$
4. $ y = cosx \implies \frac{dy}{dx}= -sinx $
5. $ y = tanx \implies \frac{dy}{dx}= sec^2x $
6. $ y = lnx \implies \frac{dy}{dx}=\frac{1}{x} $
Some integral formulas:
1. $ \int x^n dx = \frac{x^{n+1}}{n+1} + C $
2. $ \int sinx dx = -cosx + C$
3. $ \int cosx dx = sinx + C $
4. $ \int tanx dx = log|sec x| + C$
Problem for Practice:
1. Find the equation of the line which is parallel to the given line $ y = 6x + 4 $ and passes through point (4, 6).
2. Find the equation of the curve for which every point of the curve is at same distance from the point (3,2).
3. Differentiate the following with respect to x:
(a) $ y = x^2 + 4x $
(b) $z = 5x^3+10$
(c) $y = sin4x + log|x|$
(d) $y = sin^2x$
(e)$ y = cos5x + tan2x + log|sin x| $
4. Integrate the following functions:
(a) $y = sinx$
(b)$y=5x^2 + 4x$
(c) $y = sin^2x$
(d) $ y = cos5x $
(e)$y = logx $
