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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 \frac{7}{12}
Option: 2 \frac{23}{36}
Option: 3 \frac{1}{36}
Option: 4 \frac{1}{6}
 

7/12

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Posted by

Kamlesh kumar

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: 1 300
Option: 2 175
Option: 3 225
Option: 4 150
 

225

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Posted by

Anakha

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If one end of a focal chord AB of the parabola y^{2}=8x is at A\left ( \frac{1}{2},-2 \right ), then the equation of the tangent to it at B is :
Option: 1 x+2y+8=0
Option: 2 2x-y-24=0
Option: 3 x-2y+8=0
Option: 4 2x+y-24=0
 

 

 

Length of the Latus rectum and parametric form -

Parametric Equation:

From the equation of the parabola, we can write 
\\\frac{y}{2a}=\frac{2x}{y}=t\text{ here, t is a parametewr}\\\\\text{Then, }x=at^2 \text{ and }y=2at\text{ are called the parametric equations }\\\text{and the point } (at^2,2at)\text{ lies on the parabola.}

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Tangents of Parabola in Point Form -

Tangents of Parabola in  Point Form

\\ {\text { Equation of the tangent to the parabola } \mathrm{y}^{2}=4 \mathrm{ax} \text { at the point } \mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right) \text { is }} \\ {\mathrm{y} \mathrm{y}_{1}=2 \mathrm{a}\left(\mathrm{x}+\mathrm{x}_{1}\right)}

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y^{2}=8 x \text { then } A\left(2 t_{1}^{2}, 4 t_1\right) \\ {\text { given } A\left(\frac{1}{2},-2\right) \Rightarrow t_{1}=-1 / 2}

t_1\cdot t_2=-1

\text { then } t_{2}=2 \Rightarrow B(8,8) \\ \text { Equation of tangent } 8 y=4(x+8) \\ {2 y=x+8}

Correct Option 3

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Posted by

avinash.dongre

If \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{xy}{x^{2}+y^{2}};y(1)=1; then a value of x satisfying y(x)=e is :   
Option: 1 \sqrt{3}\: e
 
Option: 2 \frac{1}{2}\sqrt{3}\: e
 
Option: 3 \sqrt{2}\: e
 
Option: 4 \frac{e}{\sqrt{2}}
 
 

Option: 3

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Posted by

Ayesha Sabeela

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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 y(1+x)=1
Option: 2 x(1-y)=1
Option: 3 y(1-x)=1
Option: 4 x(1+y)=1
 

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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Posted by

avinash.dongre

If z be a complex number satisfying \left | Re\left ( z \right ) \right |+\left | Im(z) \right |=4, then \left | z \right | cannot be : 
Option: 1 \sqrt{7}
 
Option: 2 \sqrt{\frac{17}{2}}
 
Option: 3 \sqrt{10}
 
Option: 4 \sqrt{8}
 
 

 

 

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 z_0  and have radius r is given by

|z-z_0| = r   

If the center is origin then, z_0=0, hence equation reduces to |z| = r

Interior of the circle is represented by |z-z_0| < r  

The exterior is represented by |z-z_0| > r

Here z can be represented as x + iy and z_0 is represented by  x_0 + iy_0

 

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z = x + iy

|x| + |y| = 4

Minimum value of |z| = 2\sqrt2

Maximum value of |z| = 4

z\in[\sqrt8,\sqrt{16}]

So |z| can't be \sqrt7

Correct Option (1)

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Posted by

avinash.dongre

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The length of the minor axis (along y-axis) of an ellipse in the standard form is \frac{4}{\sqrt{3}}. If this ellipse touches the line, x+6y=8; then its eccentricity is : 
Option: 1 \frac{1}{2}\sqrt{\frac{5}{3}}
 
Option: 2 \frac{1}{2}\sqrt{\frac{11}{3}}
 
Option: 3 \sqrt{\frac{5}{6}}
 
Option: 4 \frac{1}{3}\sqrt{\frac{11}{3}}
 
 

 

 

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

\mathbf{\frac{\mathbf{x}^{2}}{\mathbf{a}^{2}}+\frac{\mathbf{y}^{2}}{\mathbf{b}^{2}}=1} \quad \text { where }, \mathrm{b}^{2}=\mathrm{a}^{2}\left(1-\mathrm{e}^{2}\right)_{(\mathrm{a}>\mathrm{b})}

 

  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:

\\ {\text { The equation of tangent of slope m to the ellipse, } \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \text { are }} \\ {y=m x \pm \sqrt{a^{2} m^{2}+b^{2}} \text { and coordinate of point of contact is }} \\ {\left(\mp \frac{a^{2} m}{\sqrt{a^{2} m^{2}+b^{2}}}, \pm \frac{b^{2}}{\sqrt{a^{2} m^{2}+b^{2}}}\right)}

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\\\begin{array}{l}{2 \mathrm{b}=\frac{4}{\sqrt{3}} \quad \Rightarrow \quad \mathrm{b}=\frac{2}{\sqrt{3}}} \\ {\text { Equation of tangent } \equiv \mathrm{y}=\mathrm{mx} \pm \sqrt{\mathrm{a}^{2} \mathrm{m}^{2}+\mathrm{b}^{2}}}\end{array}\\\text { comparing with } \equiv y=\frac{-x}{6}+\frac{4}{3}\\