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Option d

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

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ax2 – 2bx + 5 = 0 having equal roots or  and

Put  in the second equation

Correct Option 2

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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

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The root of the quadratic equation is given by the formula:

Where D is called the discriminant of the quadratic equation, given by  ,

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Let

Now the equation

Let

Only positive value possible so

x=0 is the only solution.

The least positive value of 'a' for which the equation, $2x^{2}+(a-10)x+\frac{33}{2}=2a$ 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.

$\\\mathrm{x_1 = \frac{-b + \sqrt{D}}{2a} } \;\mathrm{and \;\;x_2 = \frac{-b - \sqrt{D}}{2a} } \\\\\mathrm{Then,\;\; ax^2+bx +c =a(x-x_1)(x-x_2) }$

iii) if roots D = 0, then roots will be real and equal, then

$\\\mathrm{x_1=x_2 = \frac{-b}{2a} } \\\mathrm{Then, \;\; ax^2+bx +c =a(x-x_1)^2 }$

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${D \geqslant 0} \\\\ {(a-10)^{2}-8\left(\frac{33}{2}-2 a\right) \geq 0} \\\\ {a^{2}+100-20 a-132+16 a \geq 0}$

$\\ {a^{2}-4 a-32 \geqslant 0} \\\\ {a^{2}-8 a+4 a-32 \geq 0} \\\\ {(a+4)(a-8) \geq 0}$

$a \leq -4 \ \text{ or }\ a \geq \ 8$

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

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Let   be roots of the equation

so

So,

Correct Option 3

Let $\alpha$ and $\beta$ be two real roots of the equation $(k+1)\tan ^{2}x-\sqrt{2}\cdot\lambda \tan x=(1-k),$ where, $k(\neq-1 )$ and $\lambda$ are real numbers. If  then a value of $\lambda$ is : Option: 1   Option: 2   Option: 3   Option: 4

As we have learnt,

Sum of roots:

$\\\mathrm{\alpha + \beta =\frac{-b}{a}}$

Product of roots:

$\alpha \cdot \beta = \frac{c}{a}$

Trigonometric Ratio for Compound Angles (Part 2)

$\\\mathrm{\tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}}\\\\\mathrm{\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta}}$

Now,

$\\\tan \alpha + \tan{\beta } = \frac{\sqrt{2} \lambda}{1+k}\\\\\tan \alpha \times \tan{\beta } = \frac{k-1}{1+k}$

Since $\tan \alpha \& \tan \beta$ are the roots of the given equation

$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan{\beta }}{1-\tan{\alpha }\tan{\beta }}$ $= \frac{\frac{\sqrt{2}\lambda }{1+k}}{1- \frac{k-1}{k+1}} =\frac{\lambda}{\sqrt2}$

Now,

$\\ {\tan ^{2}(\alpha+\beta)=\frac{\lambda^{2}}{2}=50} \\ {\lambda=10}$

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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.

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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

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Given

Correct option (3)

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)

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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)

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Correct Option (3)

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Let $\alpha =\frac{-1+i\sqrt{3}}{2}.$ If $a=(1+\alpha )\sum_{k=0}^{100}\alpha ^{2k}$ and $b=\sum_{k=0}^{100}\alpha ^{3k},$ then a and b are the roots of the quadratic equation : Option: 1 Option: 2  Option: 3  Option: 4

Cube roots of unity

$\\\mathrm{\mathbf{\omega=\frac{-1+ i\sqrt{3}}{2}},\;\;\omega^2=\mathbf{\frac{-1- i\sqrt{3}}{2}}}$

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

$S_n=a( \frac{r^n-1}{r-1})$

Now,

$\alpha$ = $\omega$

b = 1 + $\omega$3 + $\omega$6 + ……$\omega$300= 101

a= (1+$\omega$) (1+$\omega$2+$\omega$4+$\omega$6.....+$\omega$200)

$a=(1+\omega)\frac{(\omega^{2(101)}-1)}{\omega^2-1}=\frac{1-\omega^2}{1-\omega^2}=1$

Now, equation with roots 1 and 101 is

x2 – (1+101)x + 101*1 = 0

x2 – 102x + 101 = 0

Correct Option (3)

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 :

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Correct option (3)