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#### If $x=\sum_{n=0}^{\infty }(-1)^{n}\tan ^{2n}\theta \: \: and\: \: y=\sum_{n=0}^{\infty }\cos ^{2n}\theta ,$ for $0<\theta < \frac{\pi }{4},$ then Option: 1 Option: 2 Option: 3 Option: 4

$x=\sum_{n=0}^{\infty}(-1)^{n} \tan ^{2 n} \theta=1-\tan^2\theta+\tan^4\theta..........$

$y=\sum_{n=0}^{\infty} \cos ^{2 n} \theta=1+\cos^2\theta+\cos^4\theta......$

Use $\text S_{\infty}=\frac{1}{1-r}$

${x=\frac{1}{1+\tan ^{2} \alpha}=\cos ^{2} \theta} \\ {y=\frac{1}{1-\cos ^{2} \theta}=\frac{1}{\sin ^{2} \theta}}$

$\Rightarrow (1-x)= \sin ^{2} \theta$

$\Rightarrow y(1-x)=1$

Correct Option (3)

#### If  be a complex number satisfying  then  cannot be :  Option: 1   Option: 2   Option: 3   Option: 4

Complex number -

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form as a + bi where a is the real part and b is the imaginary part. For example, 5 + 2i is a complex number. So, too, is 3 + 4i√3.

We write the complex number by C or z = a + ib, a and b are real number (a, b ∈ R).

• a is real part of the complex number and denoted by Re(z),

• b is the imaginary part of the complex number and denoted by Im(z),

E.g :    z = 2 + 3i is a complex number.

With Re(z) = 2 and Im(z) = 3

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Area of triangle, circle (formula) -

Equation of Circle:

The equation of the circle whose center is at the point   and have radius r is given by

If the center is origin then, , hence equation reduces to |z| = r

Interior of the circle is represented by

The exterior is represented by

Here z can be represented as x + iy and is represented by

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z = x + iy

|x| + |y| = 4

Minimum value of |z| =

Maximum value of |z| = 4

So |z| can't be

Correct Option (1)

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#### The following system of linear equations   has  Option: 1 infinitely many solutions,  satisfying  Option: 2 infinitely many solutions,  satisfying  Option: 3 no solution Option: 4 only the trivial solution.

System of Homogeneous linear equations -

If ? ≠ 0, then x= 0, y = 0, z = 0 is the only solution of the above system. This solution is also known as a trivial solution.

If ? = 0, at least one of x, y and z are non-zero. This solution is called a non-trivial solution.

Explanation: using equation (ii) and (iii), we have

This is the condition for a system have Non-trivial solution.

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so infinite non-trivial solution exist

now equation (1) + 3 equation (3)

10x - 20z = 0

x = 2z

Correct Option 2

#### Let   If  then :    Option: 1 Option: 2 Option: 3 Option: 4

Elementary row operations -

Elementary row operations

Row transformation: Following three types of operation (Transformation) on the rows of a given matrix are known as elementary row operation (transformation).

i) Interchange of ith row with jth row, this operation is denoted by

ii) The multiplication of ith row by a constant k (k≠0) is denoted by

iii) The addition of ith row to the elements of jth row multiplied by constant k (k≠0) is denoted by

In the same way, three-column operations can also be defined too.

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Correct Option (3)

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#### Let  denote the greatest integer  and  Then the function,  is discontinuous, when x is equal to:  Option: 1   Option: 2   Option: 3   Option: 4

Correct Option (1)

#### 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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#### In the expansion of $\left ( \frac{x}{cos\theta }+\frac{1}{x\sin \theta } \right )^{16}$, if    is the least value of the term independent of $x$ when $\frac{\pi }{8}\leq \theta \leq \frac{\pi }{4}$ and    is the least value of the term independent of $x$ when $\frac{\pi }{16}\leq \theta \leq \frac{\pi }{8}$, then the ratio   is equal to :  Option: 1 Option: 2 Option: 3 Option: 4

General Term of Binomial Expansion$\left(T_{r+1}\right)^{\mathrm{th}} \text { term is called as general term in }(x+y)^{n}\;\text{and general term is given by}$

$\mathrm{T}_{\mathrm{r}+1}=^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \mathrm{\;x}^{\mathrm{n}-\mathrm{r}} \cdot \mathrm{y}^{\mathrm{r}}$

Term independent of x