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Option d

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#### 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

#### 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.

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#### 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.

#### 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

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#### Let be a solution of the differential equation, If then is equal to : Option: 1 Option: 2 Option: 3 Option: 4

Formation of Differential Equation and Solutions of a Differential Equation -

This is the general solution of the differential equation (2), which represents the family of the parabola (when a = 1) and one member of the family of parabola is given in Eq (1).

Also, Eq (1) is a particular solution of the differential equation (2).

The solution of the differential equation is a relation between the variables of the equation not containing the derivatives, but satisfying the given differential equation.

A general solution of a differential equation is a relation between the variables (not involving the derivatives) which contains the same number of the arbitrary constants as the order of the differential equation.

Particular solution of the differential equation obtained from the general solution by assigning particular values to the arbitrary constant in the general solution.

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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)