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√3e

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

-

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

-

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.

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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

-

Let   be roots of the equation

so

So,

Correct Option 3

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#### The most abundant elements by mass in the body of a healthy human adult are : Oxygen (61.4%); Carbon (22.9%), Hydrogen (10.0%); and Nitrogen (2.6%). The weight (in kg) which a 75 kg person would gain if all $\dpi{100} ^{1}H$ atoms are replaced by $\dpi{100} ^{2}H$ atoms is :   Option: 1 7.5 Option: 2 10 Option: 3 15 Option: 4 37.5

Given that

Mass of the person = 75 kg

Mass of 1H1 present in person = 10% of 75 kg = 7.5 kg

Since Mass of 1H2 is double the Mass of 1H1

So, Mass of 1H2 will be in person = 2 X 7.5 kg =15 kg

Thus, increase in weight = 15 - 7.5 = 7.5 kg

Therefore, Option (1) is correct

#### 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.

-

Correct Option (3)

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#### The correct order of the atomic radii of C, Cs, Al and S is : Option: 1 Option: 2 Option: 3 Option: 4

In a period from left to right the effective nuclear charge increases because the next electron fills in the same shell. So the atomic size decrease.

- wherein

The attraction between the outer electrons and the nucleus increases as the atomic radius decreases in a period.

- wherein

Size of atom and ion in a group -

In a group moving from top to the bottom the number of shell increases.So the atomic size increases.

- wherein

As we know that

From Left to right in a period size decreases and when going down the group size increases

Therefore, Option(2) is correct

#### The IUPAC symbol for the element with atomic number 119 would be: Option: 1 uue Option: 2une Option: 3 unh Option: 4 uun

Nomenclature of elements with atomic number >100 -

The name is derived directly from the atomic number of the element using the following numerical roots:

0 = nil

1 = un

2 = bi

3 = tri

5 = pent

6 = hex

7 = sept

8 = oct

9 = enn

Eg:

 Atomic number Name Symbol 101 Mendelevium (Unnilunium) Md (Unu) 102 Nobelium (Unnilbium) No (Unb)

-

uue

1  1  9

Un Un ennium

Therefore, Option(1) is correct.