- Mathematical Logic
- Trigonometric Functions
- Pair of Straight Lines (Misc. Ex. 4)
- Three Dimensional Geometry (Misc. Ex. 6)
- Linear Programming
A median of a triangle is the line joining the midpoint of a side and the opposite vertex.
An altitude is the perpendicular line drawn from a vertex on the opposite side of the triangle.
A parallelopiped is a 3-D object, each of whose faces is a parallelogram.
A rectangular parallelopiped is the one, whose faces are rectangular.
A cube is a rectangular parallelopiped, whose edges are of equal length.
A tetrahedron is a 3-D object, whose all faces are triangular. It can be shown that a parallelopiped can be decomposed into 6 tetrahedra.
For a visual proof : 1 Parallelopipded = 6 Tetrahedra
Let there be 2 such objects, whose coterminus edges are identical. Then, the volume of the tetrahedron is th of that of the parallelopiped. The expression can also be obtained using the following theorem :
The section formula is one of the important results in the geometry. We will obtain the section formula for internal as well as external division, using vectors.
In this blog, we will prove fundamental theorems in vector algebra. These theorems are in accordance with the +2 curriculum.
The value which satisfies an equation is known as a solution. For example, is an equation. On simplifying,
So, the solution is .
The equations involving trigonometric functions are known as trigonometric equations. The unknown values, which are to be found, represent the measure of an angle. For example, is satisfied by or .
The solution, which lies between and , is known as the principal solution.
We saw that the measure of full circle is radians or . Hence, after adding to above angle, we will still get a solution. So,
Thus, there are infinitely many solutions to above equations, apart from . These solutions are known as general solutions. Consider another example:
Note: We will have to use allied angles formulas and the formula sheet to solve problems of this kind, where we are asked to find the general solutions.
The Cartesian coordinates and the polar coordinates are inter-convertible. See the figure below:
A triangle has 3 sides, say, and 3 angles, . By solving a triangle, we mean, to find values of all lengths of sides and all angles of a triangle. See the figure below.
The side opposite to is denoted by and so on.
There are certain rules to be used to solve these problems. These are:
is the radius of the circumcircle.
is the semi-perimeter.
The area of triangle is given by,
It is also given by
. This is known as Heron’s formula.
If a function is defined from set A to set B (which is one-one and onto), we can define an inverse function, from set B to set A. So, if , then and .
If , then . If , then .
Note that is different from . is , which is .
On the other hand, is the angle, whose sine is .
Corresponding to each of six trigonometric functions, we have 6 inverse trigonometric functions, i.e. .
Recall : Principal value is the value of angle, which lies between and .
Consider 2 functions, and . Let be a value at which these functions are defined.
I) If and , then the limit takes the form .
II) If , then takes the form
III) If and , then the limit takes the form
IV) If , and , then the limit takes the form .
V) If , then the limit takes the form .
VI) If and and , then the limit takes the form
VII) If and and , then the limit takes the form .
These are known as the indeterminate forms. The limits are evaluated either by L’Hosptial’s rule or by substituting an equivalent infinitesimal.
The rule can be proved using Taylor’s theorem. It says, if and are at or and , then
This rule is sometimes applied on th derivatives, if all derivatives of lesser orders are .
This is used for evaluation of form. One of the functions can be replaced by another, if they both converge to at a point and the limit of their ratio at that point is . For example,
One can try substituting a value of closer to (but not equal to) the actual limit. Evaluate the function using the calculator. The answer will be closer to the actual limit. We’ve actually used the concept of limit here.
Explanation: Consider the limit
This is of the form . Let’s put in the function .
We get as .
On substituting , a value closer to , we get .
On substituting , a value closer to and , we get . Clearly, the limit is .