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Maths Class 11 Chapter 5 Part -1 Quadratic equations
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Then, f(x) =
a0 + a1x + a2x2 + … + anxn is called a real polynomial of real variable x with real coefficients
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Complex Polynomial: If a0, a1, a2, … , an be complex numbers and x is a varying complex
number, then f(x) = a0 + a1x + a2x2 + … + an – 1xn – 1 + anxn is called a complex polynomial or a
polynomial of complex variable with complex coefficients
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Degree of a Polynomial: A polynomial f(x) = a0 + a1x + a2x2 + a3x3 + … + anxn , real or
complex is a polynomial of degree n , if an ≠ 0
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Polynomial Equation: If f(x) is a polynomial, real or complex, then f(x) = 0 is called a
polynomial equation
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Quadratic Equation: A polynomial of second degree is called a quadratic polynomial
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A quadratic polynomial f(x) when equated to zero is called quadratic equation
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Roots of a Quadratic Equation: The values of variable x
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Important Points to be Remembered
An equation of degree n has n roots, real or imaginary
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An odd degree equation has at least one real root whose sign is opposite to that of its
last’ term (constant term), provided that the coefficient of highest degree term is
positive
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If an equation has only one change of sign it has one positive root
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Solution of Quadratic Equation
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Then, x = α and x = β will
satisfy the given equation
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Direct Formula: Quadratic equation ax2 + bx + c = 0 (a ≠ 0) has two roots, given by
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where D = Δ = b2 – 4ac is called discriminant of the equation
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Nature of Roots
Let quadratic equation be ax2 + bx + c = 0, whose discriminant is D
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(b) If D > 0, a = 1; b, c ∈ I and D is a perfect square
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(c) If D > and D is not a perfect square
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(iii) Conjugate Roots The irrational and complex roots of a quadratic equation always occur in
pairs
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(b) If one root be α + √β, then other root will be α – √β
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(i) If b = 0 => Roots are real/complex as (c < 0/c > 0) and equal in magnitude but of opposite
sign
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(iii) If b = C = 0 => Both roots are zero
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(v) If a > 0, c < 0, a < 0, c > 0} => Roots are of opposite sign
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(ix) If sign of b = sign of c ≠
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(x) If a + b + c = 0 => One root is 1 and second root is c/a
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Quadratic Equation: If roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0) are α and β, then
Sum of roost = S = α + β = -b/a = – coefficient of x / coefficient of x2 Product of roots = P = α
* β = c/a = constant term / coefficient of x2
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Then,
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Cubic Equation
If α, β and γ are the roots of cubic equation, then the equation is
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The quadratic function f(x) = ax2 + 2hxy + by2 + 2gx + 2fy + c is always resolvable into linear
factor, iff
abc + 2fgh – af2 – bg2 – ch2 = 0
Condition for Common Roots in a Quadratic Equation
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Both Roots are Common
The required condition is
a1 / a2 = b1 / b2 = c1 / c2
(i) To find the common root of two equations, make the coefficient of second degree term in
the two equations equal and subtract
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(ii) Two different quadratic equations with rational coefficient can not have single common
root which is complex or irrational as imaginary and surd roots always occur in pair
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f(b) < 0, then at least one or in general odd number of roots of the equation f(x) = 0 lies
between a and b
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f( b) > 0, then in general even number of roots of the equation f(x) = 0 lies between a
and b or no root exist f(a) = f(b), then there exists a point c between a and b such that f'(c) = 0,
a < c < b
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(iv) If one root is k times the other root of the quadratic equation ax2 + bx + c = 0 ,then
(k + 1)2 / k = b2 / ac
Quadratic Expression
An expression of the form ax2 + bx + c, where a, b, c ∈ R and a ≠ 0 is called a quadratic
expression in x
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Graph of a Quadratic Expression
We have
y = ax2 + bx + c = f(x)
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Let y + D/4a = Y and x + D / 2a = X
Y = a * X2 => X2 = Y / a
(i) The graph of the curve y = f(x) is parabolic
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(iii) If a > 0, then the parabola opens upward
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(i) For D > 0, parabola cuts X-axis in two real and distinct points
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(iii) For D < O,parabola does not cut X-axis (i
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, imaginary value of x)
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Maximum and Minimum Values of Quadratic Expression
(i) If a > 0, quadratic expression has least value at x = b / 2a
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But their is no greatest
value
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This greatest value is given
by 4ac – b2 / 4a = – D/4a
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(ii) a < 0 and D < 0, so f(x) < 0 for all x ∈ R i
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, f(x) is negative for all real values of x
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(iv) a < 0 and D = 0, so f(x) ≤ 0 for all x ∈ R i
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, f(x) is negative for all real values of x except
at vertex, where f(x) = 0
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(vi) a < 0 and D > 0
Let f(x) = 0 have two real roots α and β (α < β)
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,
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For this we impose conditions on a, b and c
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(i) Both the roots are positive i
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, they lie in (0,∞), if and only if roots are real, the sum of the
roots as well as the product of the roots is positive
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, α + β = -b/a < 0 and αβ = c/a > 0 with b2 – 4ac ≥ 0
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(ii) Both the roots are greater than a given number k, iFf the following conditions are satisfied
D ≥ 0, -b/2a > k and f(k) > 0
(iii) Both
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In particular, the roots of the equation
will be of opposite sign, iff 0 lies between the roots
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a1 < a2 < a3 < … < an – 1 < an
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If kn is even, we put plus sign
on the left of anand if kn is odd, then we put minus sign on the left of an In the next interval we
put a sign according to the following rule
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if kn – 1 is an odd
number and the polynomial f(x) has same sign if kn – 1 is an even number
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Thus, we consider all the intervals
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Descarte’s Rule of Signs
The maximum number of positive real roots of a polynomial equation f(x) = 0 is the number of
changes of sign from positive to negative and negative to positive in f(x)
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Rational Algebraic In equations
(i) Values of Rational Expression P(x)/Q(x) for Real Values of x, where P(x) and Q(x) are
Quadratic Expressions To find the values attained by rational expression of the form a1x2 +
b1x + c1 / a2x2 + b2x + c2
for real values of x
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(b) Obtain a quadratic equation in x by simplifying the expression,
(c) Obtain the discriminant of the quadratic equation
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The values of y so obtained determines
the set of values attained by the given rational expression
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To solve these in equations we use the sign method as
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(a) Obtain P(x) and Q(x)
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(c) Make the coefficient of x positive in all factors
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(e) Plot the critical points on the number line
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(f) In the right most region the expression P(x) / Q(x) bears positive sign and in other region the
expression bears positive and negative signs depending on the exponents of the factors
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Also, a
polynomial function is everywhere continuous and differentiable, then there exist θ ∈ (α, β)
such that f'(θ) = 0
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Equation and In equation Containing Absolute Value
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g(x) ≥ 0
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g(x) ≤ 0
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In equation Containing Absolute Value
(i) |x| < a ⇒ – a < x < a (a > 0)
(ii) |x| ≤ a ⇒ – a ≤ x ≤ a
(iii) |x| > a ⇒ x < – a or x > a
(iv) |x| ≥ a ⇒ x le; – a or x ≥ a
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(vi) |x – y| ≥ ||x| – |y||
Inequalities
Let a and b be real numbers
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Important Points to be Remembered
(i) If a > b and b > c, then a > c
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, an – 1 > an, then a1 > an
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Arithmetico-Geometric and Harmonic Mean Inequality
(i) If a, b > 0 and a ≠ b, then
(ii) if ai > 0, where i = 1,2,3,…,n, then
(iii) If a1, a2,…, an are n positive real numbers and m1, m2,…,mn are n positive rational
numbers, then
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, Weighted AM > Weighted GM
(iv) If a1, a2,…, an are n positive distinct real numbers, then
(a)
(b)
(c) If a1, a2,…, an and b1, b2,…, bn are rational numbers and M is a rational number, then
(d)
(v) If a1, a2, a3,…, an are distinct positive real numbers and p, ,q, r are natural numbers, then
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Tchebychef’s Inequality
Let a1, a2,…, an and b1, b2,…, bn are real numbers, such that
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(i) If a1 ≤ a2 ≤ a3 ≤… ≤ an and b1 ≤ b2 ≤ b3 ≤… ≤ bn, then
n(a1b1 + a2b2 + a3b3 + …+ anbn) ≥ (a1 + a2 + …+ an) (b1 + b2 + …+ bn)
(ii) If If a1 ≥ a2 ≥ a3 ≥… ≥ an and b1 ≥ b2 ≥ b3 ≥… ≥ bn, then
n(a1b1 + a2b2 + a3b3 + …+ anbn) ≤ (a1 + a2 + …+ an) (b1 + b2 + …+ bn)
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Logarithm Inequality
(i) (a) When y > 1 and logy x > z ⇒ x > yz
(b) When y > 1 and logy x < z ⇒ 0 < x < yz
(ii) (a) When 0 < y < 1 and logy x > z ⇒ 0 < x < yz
(b) hen 0 < y < 1 and logy x < z ⇒ x > yz
Application of Inequalities to Find the Greatest and Least Values
(i) If xl,x2,…,xn are n positive variables such that xl + x2 +…+ xn = c (constant), then the
product xl * x2 *…
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(ii) If xl,x2,…,xn are positive variables such that xl,x2,…,xn = c (constant), then the sum xl +
x2 +…
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(iii) If xl,x2,…,xn are variables and ml,m2,…,mn are positive real number such that xl + x2 +…
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+ xn / ml + m2 +…
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