Find quick revision notes of chapters Quadratic Equation and Sequence and Series. This quick revision notes is very important for competitive examination as it will help in revising complete unit in very less time. In UPSEE, WBJEE and other engineering entrance examinations some questions are asked directly on formula basis. This revision notes will help you in revising formulae.
Theory of Equations
- The general form of a quadratic equation in x is ax^{2} + bx + c = 0
- The roots of the quadratic equation are
- Let a and b be the roots of a quadratic equation ax^{2} + bx + c = 0 with a ≠ 0, then
UPSEE 2017 Solved Sample Paper Set-1
- Quadratic equation can be formed if the sum of roots and product of roots is given
x^{2} - (sum of roots)x + (product of roots) = 0
- For a quadratic equation ax^{2} + bx + c = 0, where a ≠ 0, and its discriminant is
D = b^{2} – 4ac ,
(1) if D > 0, the equation has two distinct real roots.
(2) if D = 0, the equation has one double real root.
(3) if D < 0, the equation has no real root, it has two distinct unreal roots.
- The graph of a quadratic function is a parabola. If a > 0, the parabola opens upward;
if a < 0, the parabola opens downward.
- Let x_{1} and x_{2} be the roots of ax^{2} + bx + c = 0 and D be the discriminant of the equation
JEE Main Mathematics Solved Sample Paper Set-VII
Sequence and Series
- A sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it of the type {1, 2, 3...k}.
- A sequence containing finite number of terms is called a finite sequence otherwise it is called infinite sequence
- Let a_{1}, a_{2}, a_{3},…,a_{n}, be a given sequence.
Then, the expression a_{1 }+ a_{2} + a_{3} +,… + a_{n} + ... is called the series associated with the given sequence .
- A sequence a_{1}, a_{2}, a_{3},…,a_{n}, is called arithmetic sequence or arithmetic progression if
a_{n + 1} = a_{n} + d, n ∈ N
where, a_{1} = first term, d = common difference
- If a fixed number is added to (subtracted from) each term of a given AP, then the resulting sequence is also an AP and it has the same common difference as that of the given AP.
- If each term of an AP is multiplied by a fixed constant (or divided by a non-zero fixed constant), then the resulting sequence is also an AP.
- The nth term (general term) of the A.P. is denoted by a_{n}_{ }.
Mathematically, a_{n}_{ }= a + (n – 1) d
- Let a, a + d, a + 2d, …, a + (n – 1)d be a given AP.
then, l = a + (n – 1) d
where, S_{n}= sum to n terms of A.P, n = the number of terms, d = common difference
a = the first term, l = the last term
- If a number A is inserted between two numbers a and b so that a, A, b is in A.P., then the number A is called the arithmetic mean (A.M.) of the numbers a and b.
JEE Main Physics Solved Sample Paper Set-VII
- Let A_{1}, A_{2}, A_{3}, …, A_{n} be n numbers between a and b such that a, A_{1}, A_{2}, A_{3}, …, A_{n}, b is an A.P.
- A sequence a_{1}, a_{2}, a_{3},…,a_{n}, is called geometric progression, if each term is non-zero and
- a, ar, ar^{2}, ar^{3} ..... are in GP
where, a = first term, r = common ratio
- The nth term of a G.P. is given by,a_{n} = ar^{n}^{-1}
- Let the first term of a G.P. be a and the common ratio be r.
- Let G_{1}, G_{2},…, G_{n} be n numbers between positive numbers a and b such that a, G_{1}, G_{2},…, G_{n}, b is a G.P. then,
- Let A and G be A.M. and G.M. of two given positive real numbers a and b, respectively. Then
A ≥ G
- Sum of first n natural numbers is given as
JEE Main Chemistry Solved Sample Paper Set-VII