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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)

Option d

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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: It means term containing x0,

Now,

\\\mathrm{T}_{\mathrm{r}+1}=^{16} \mathrm{C}_{\mathrm{r}}\left(\frac{\mathrm{x}}{\cos \theta}\right)^{16-\mathrm{r}}\left(\frac{1}{\mathrm{x} \sin \theta}\right)^{\mathrm{r}}\\\text{for r = 8 term is free from 'x' }\\\begin{aligned} &\mathrm{T}_{9}=^{16} \mathrm{C}_{8} \frac{1}{\sin ^{8} \theta \cos ^{8} \theta}\\ &\mathrm{T}_{9}=^{16} \mathrm{C}_{8} \frac{2^{8}}{(\sin 2 \theta)^{8}}\\ &\text { in } \theta \in\left[\frac{\pi}{8}, \frac{\pi}{4}\right], L_{1}=^{16} \mathrm{C}_{8} 2^{8} \end{aligned}

$\\\because \text{Min value of L}_1\;\text{at }\theta=\pi/4\\\text { in } \theta \in\left[\frac{\pi}{16}, \frac{\pi}{8}\right], L_{2}=16 \mathrm{C}_{8} \frac{2^{8}}{\left(\frac{1}{\sqrt{2}}\right)^{8}}=^{16} \mathrm{C}_{8} \cdot 2^{8} \cdot 2^{4}\\\\\because \text{Min value of L}_2\;\text{at }\theta=\pi/8\\\frac{L_{2}}{L_{1}}=\frac{16 \mathrm{C}_{8} \cdot 2^{8} 2^{4}}{^{16} \mathrm{C}_{8} \cdot 2^{8}}=16$

Correct option 1

#### Let  and  be differentiable functions on  such that  is the identity function. If for some  then  is equal to :    Option: 1 Option: 2 Option: 3 Option: 4

Rules of Differentiation (Chain Rule) -

Rules of Differentiation (Chain Rule)

Chain Rule or Derivation of Composite Function:

The chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Let f and g be functions. For all x in the domain of g for which g is differentiable at x and f is differentiable at g(x), the derivative of the composite function

h(x) = (f?g)(x) = f (g(x)) Is given by

h′(x) = f’(g(x))?g’(x)

Composites of Three or More Functions

For all values of x for which the function is differentiable, if k(x) = h(f(g(x)))Then,

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Correct Option 4