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#### A random variable X has the following probability distribution:          $X: \: \: 1\; \; \: \: 2\; \;\: \: \: \: 3\; \; \: \: \: \: \: 4\: \: \: \; \; \: \: 5$ $P(X): \;K^{2}\; \; 2K\; \; K\; \; 2K\; \; 5K^{2}$ Then $P(X>2)$ is equal to:  Option: 1 Option: 2 Option: 3 Option: 4

7/12

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#### Let $a_{n}$ be the nth term of a G.P. of positive terms. If $\sum_{n=1}^{100}a_{2n+1}=200\: \: and\: \: \sum_{n=1}^{100}a_{2n}=100,\: \: then\: \sum_{n=1}^{200}a_n$ is equal to :    Option: 1 Option: 2 Option: 3  Option: 4

225

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#### If one end of a focal chord AB of the parabola  is at  then the equation of the tangent to it at B is : Option: 1 Option: 2 Option: 3 Option: 4

Length of the Latus rectum and parametric form -

Parametric Equation:

From the equation of the parabola, we can write

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Tangents of Parabola in Point Form -

Tangents of Parabola in  Point Form

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Correct Option 3

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#### If  then a value of  satisfying  is :    Option: 1   Option: 2   Option: 3   Option: 4

Option: 3

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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)

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#### 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 length of the minor axis (along y-axis) of an ellipse in the standard form is  If this ellipse touches the line,  then its eccentricity is :  Option: 1   Option: 2   Option: 3   Option: 4

What is Ellipse? -

Ellipse

Standard Equation of Ellipse:

The standard form of the equation of an ellipse with center (0, 0) and major axis on the x-axis is

1. a > b

2.  the length of the major axis is 2a

3.  the coordinates of the vertices are (±a, 0)

4.  the length of the minor axis is 2b

5.  the coordinates of the co-vertices are (0, ±b)

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Equation of Tangent of Ellipse in Parametric Form and Slope Form -

Slope Form:

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