Subject Mathematics Medium ENGLISH
Faculty Renu Mam Status AVAILABLE
Category TOPIC BASED COURSE Lecture 67
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Polynomial functions, Roots (zeros), Solutions, Equation vs Identity, Questions,
Methods of finding roots (i) Factorisation

Methods of finding roots- (ii) Transformation method. (iii) Dharacharya Method (Perfect square),

Questions based on finding roots.

Nature of roots : in ax2 + bx + c = 0 (a≠0)
(1) When a, b, c, ∈ R
(2) When a, b, c, ∈ Q
(3) When a = 1, b, c, ∈ I and D is Perfect square of integer
(4) when a, b, c ∉ R
(5) when D1 + D2 ≥ 0 (in a1x2 + b1x+ c1 = 0 and a2x2 + b2x+ c2 = 0 where

D1 = b12 –4a1c1 and D2 = b2 –4a2c2)
(6) Intermediate Mean Value Theorem (IMVT)
Questions based on nature of roots.

Plotting of quadratic expression (Graph) when a > 0 or a < 0
in y = ax2 + bx + c (a≠0), Range of y = ax2 + bx + c when x ∈ R
Sign of a, b, c, D, Range in an interval x ∈[x1, x2],

Sign of quadratic expression, Range of
y =L/Q , y

Q/ Q

Range by substitution, condition of common roots-
(1) when 1 root common (2) when both the roots are common
Location of roots-
(1) When both the roots are greater than k (k∈R)
(2) When both the roots are less than k
(3) When 1 root < k and other root > k
(4) When both the roots lies in interval (k1, k2)
(5) When only 1 root lies in (k1, k2)

Questions based on location of Roots,
Pseudo-Quadratic equation, Questions based on it.


Arithmetic progression (AP), Standard terms, General term or last term (tn or 𝓁) of AP, Condition for 3 terms
in AP, Arithmetic mean (AM) of n numbers, Middle terms, Sum of First n terms (sn) of AP, Properties of AP,
n Arithmetic means inserted between 2 numbers, sum of n Arithmetic mean inserted between two numbers,
Properties of AP.

Questions based on Arithmetic progression and their properties.

Summation series based on AP, Geometric progression (GP), Standard terms, General term of GP, Sum of
first n terms of GP, Sum of ∞ terms of GP, supposition of terms in GP, n Geometric means between 2 positive
numbers, Properties of GP.

Questions based on GP and their properties.

Summation Series based on G.P., Harmonic Progression (HP), General term of Harmonic Progression,
Harmonic Means of n numbers, Questions based on Harmonic Progression.

Relation between AM, GM, HM, Solving inequalities based on AM ≥ GM ≥ HM.
Arithmetic Geometric Progression (AGP), General Term, Sum of first n terms of AGP,
Sum of ∞ numbers of terms in AGP, Summation series of AGP.

(a) Summation of series based on product of terms in GP but with non-AP; Summation of series, i.e.

             n                                                                             n      n      n        n 
     Sn = tr,  (Vn - Vn-1 )method, Evaluating the value of  1,  ∑r,  r2,  ∑r3,
             r=1                                                                         r=1    r=1   r=1    r=1

(b) Method of differences
(1) First difference in AP. (2) Second difference in AP
(3) First difference in GP. (4) Second difference in GP;
Questions Based on method of differences.

Miscellaneous Series

Miscellaneous Series


About factorial n (n!,⌊n ), Domain-Range and Properties of factorial n. About nCr, nPr, formulae based on n! ,
nCr and nPr, Binomial expansion (for n ∈ N), Pascal-Triangle, General term, mth term from ending, middle term
(for n odd, n even), Questions based on Binomial expansion and determining terms in Binomial expansion.

Questions based to determine middle term in Binomial expansion, Questions based to determine coefficient
of xr in Binomial expansion, Questions based to determine the term independent of x.

Questions based on determining coefficient in product of 2 Binomial expansions, Multinomial theorem.

Coefficient determining by concept of permutation and combination and by using multinomial theorem; total
number of terms in multinomial expansion; Number of terms free from fractional or irrational powers in
Binomial expansion.

Numerically-Greatest term in the expansion of (x + a)n (n ∈ N), Algebraically - Greatest and least term in the
expansion of (x + a)n (n ∈ N); Questions based on Ι + ƒ .

Questions based on Ι + ƒ , exponent of prime number (p) in ⌊n ; Questions based on divisibility and remainder,
Last digit by cyclicity, Last digit, Last two digits, Last 3 digits in a number.

Summation of series.

Questions based on Sigma, Summation of Binomial Coefficients taken two at a time, Summation when
upper index is variable.

Questions based on summation of Binomial coefficients taken two at a time when upper index is variable.

Double-Sigma, Binomial expansion for negative or fractional power, Some-important expansions,
Questions based on determining
Coefficient in negative or fractional power in Binomial expansion.


Introduction of factorial n ( ⌊n or n!) , nCr, nPr, Physical interpretation of n!, nCr, nPr.

Fundamental - Principles of counting
(i) Multiplication - Rule (ii) Addition- Rule
Basic Questions based on multiplication and addition-Rule; Sample-space.

Questions, Number Problems.

Number problems based on divisible by 3, 4, 5, 25,
Theorem-1: Selection and Permutation of r things out of n.
Theorem-2 : Permutation of n things in which some things are of same kind.

Questions considering the word “RAKESH MODI” Questions, problem of forming words of 7 letters taking 3
vowels and 4 consonants using letters of word : “DIFFERENTIATION”.
Problem of forming the words each consisting 3 consonants and 3 vowels by using letters of words

Rank (Position) of a word or numbers, sum of the numbers formed (No repetition or when repetition allowed),
Circular Permutation.

Number of selection of r consecutive things out of n distinct things, Geometrical Problems Number of total
lines, number of diagonals, number of triangles
(a) One side common with given polygon
(b) Two sides common with given polygon
(c) Three sides common with given polygon.
(d) None of the side common with given polygon.
Chess board problems - Number of total rectangles, Number of total squares.
Problem based on moving from left bottom corner to the right top corner in a chess board.

Groupings & distribution of n differents things into groups or bundles.

Selection of none, one or more things when given things are different or identical, Total number of divisors,
Total number of proper divisors, Sum of total divisors, Number of ways in which a number (N) can be resolved
as a product of two factors which are relatively prime or co-prime.

Multinomial theorem of permutation and combination, Beggar’s Method

Questions based on multinomial theorem, Dearrangement of n different things.

Miscellaneous questions

Miscellaneous questions

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