If be a complex number satisfying
then
cannot be :
Option: 1
Option: 2
Option: 3
Option: 4
Option d
View Full Answer(3)
Let be such that the equation,
has a repeated root
, which is also a root of the equation,
. If
is the other root of this equation, then
is equal to:
Option: 1
Option: 2
Option: 3
Option: 4
Nature of Roots -
Let the quadratic equation is ax2 + bx + c = 0
D is the discriminant of the equation.
iii) if roots D = 0, then roots will be real and equal, then
-
ax2 – 2bx + 5 = 0 having equal roots or and
Put in the second equation
Correct Option 2
View Full Answer(1)The number of real roots of the equation, is :
Option: 1
Option: 2 4
Option: 3 1
Option: 4 2
Transcendental function -
Transcendental functions: the functions which are not algebraic are called transcendental functions. Exponential, logarithmic, trigonometric and inverse trigonometric functions are transcendental functions.
Exponential Function: function f(x) such that is known as an exponential function.
Logarithmic function: function f(x) such that is called logarithmic function
If a > 1 If a < 1
Properties of Logarithmic Function
-
Quadratic Equation -
The root of the quadratic equation is given by the formula:
Where D is called the discriminant of the quadratic equation, given by ,
-
Let
Now the equation
Let
Only positive value possible so
x=0 is the only solution.
View Full Answer(1)The least positive value of 'a' for which the equation, has real roots is
Option: 1 8
Option: 2 6
Option: 3 4
Option: 42
Nature of Roots -
Let the quadratic equation is ax2 + bx + c = 0
D is the discriminant of the equation.
ii) If D > 0, then roots will be real and distinct.
iii) if roots D = 0, then roots will be real and equal, then
-
least positive value is 8.
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If the equation, has conjugate complex roots and they satisfy
, then:
Option: 1
Option: 2
Option: 3
Option: 4
Nature of Roots -
Let the quadratic equation is ax2 + bx + c = 0
D is the discriminant of the equation.
i) if D < 0, then root are in the form of complex number,
If a,b,c ∈ R (real number) then roots will be conjugate of each other, means if p + iq is one of
the roots then other root will be p - iq
-
Let be roots of the equation
so
So,
Correct Option 3
View Full Answer(1)Let and
be two real roots of the equation
where,
and
are real numbers. If
then a value of
is :
Option: 1
Option: 2
Option: 3
Option: 4
As we have learnt,
Sum of roots:
Product of roots:
Trigonometric Ratio for Compound Angles (Part 2)
Now,
Since are the roots of the given equation
Now,
View Full Answer(1)
Let be a root of the equation
and the matrix
Then the matrix
is equal to :
Option: 1
Option: 2
Option: 3
Option: 4
Cube roots of unity -
z is a complex number
Let z3 = 1
⇒ z3 - 1 = 0
⇒ (z - 1)(z2 + z + 1) = 0
⇒ z - 1 = 0 or z2 + z + 1 = 0
If the second root is represented by ?, then the third root will be represented by ?2.
Properties of cube roots:
i) 1 + ? + ?2 = 0 and ?3 = 1
ii) to find ?n , first we write ? in multiple of 3 with remainder belonging to 0,1,2 like n=3q + r
Where r is from 0,1,2. Now ?n = ?3q + r = (?3)q·?r = ?r.
-
Multiplication of two matrices -
Matrix multiplication:
Two matrices A and B are conformable for the product AB if the number of columns in A and the number of rows in B is equal. Otherwise, these two matrices will be non-conformable for matrix multiplication. So on that basis,
i) AB is defined only if col(A) = row(B)
ii) BA is defined only if col(B) = row(A)
If
For examples
-
Given
Correct option (3)
View Full Answer(1)If Re where
then the point
lies on a :
Option: 1 circle whose centre is at
Option: 2 straight line whose slope is
Option: 3 circle whose diameter is
Option: 4 straight line whose slope is
Conjugate of complex numbers and their properties -
The complex conjugate of a complex number a + ib (a, b are real numbers and b ≠ 0) is a − ib.
It is denoted as .
i.e. if z = a + ib, then its conjugate is = a - ib.
Conjugate of complex numbers is obtained by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.
Note:
When a complex number is added to its complex conjugate, the result is a real number. i.e. z = a + ib, = a - ib
Then the sum, z + = a + ib + a - ib = 2a (which is real)
When a complex number is multiplied by its complex conjugate, the result is a real number i.e. z = a + ib, = a - ib
Then the product, z? = (a + ib)?(a - ib) = a2 - (ib)2
= a2 + b2 (which is real)
-
Circle(Definition) -
General Form:
The equation of a circle with centre at (h,k) and radius r is
This is known as the general equation of the circle.
Compare eq (i) and eq (ii)
h = -g, k = -h and c=h2+k2-r2
Coordinates of the centre (-g,-f)
Radius =g2+f2-c
-
Correct Option (3)
View Full Answer(1)Let If
and
then a and b are the roots of the quadratic equation :
Option: 1
Option: 2
Option: 3
Option: 4
Cube roots of unity
Sum of n-term of a GP
Let Sn be the sum of n terms of the G.P. with the first term ‘a’ and common ratio ‘r’. Then
Now,
=
,
b = 1 + 3 +
6 + ……
300= 101
a= (1+) (1+
2+
4+
6.....+
200)
Now, equation with roots 1 and 101 is
x2 – (1+101)x + 101*1 = 0
x2 – 102x + 101 = 0
Correct Option (3)
View Full Answer(1)Let and
be the roots of the equation
If
then which one of the following statements is not true ?
Option: 1
Option: 2
Option: 3
Option: 4
Polynomial Equation of Higher Degree, Remainder theorem -
Is known as the polynomial equation of degree n which have n and only n roots.
For example, suppose n = 3 and ax3 + bx2 +cx + d = 0 is polynomial equation with a ≠ 0 and ?, ? and ? are the roots of the equation then :
-
Correct option (3)
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If be a complex number satisfying
Let be such that the equation,
The number of real roots of the equation, is :
The least positive value of 'a' for which the equation, has real roots is
If the equation, has conjugate complex roots and they satisfy
Let and be two real roots of the equation