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Let \alpha and \beta   be the roots of equation

x^{2}-6x-2=0.\; if a_{n}=\alpha ^{n}-\beta ^{n},\; for\: n\geq 1,\; then\; the \: value\; of\; \frac{a_{10}-2a_{8}}{2a_{9}}

is equal to:

  • Option 1)

    6

  • Option 2)

    -6

  • Option 3)

    3

  • Option 4)

    -3

3

All the pairs ( x, y ) that satisfy the inequality 

2^{\sqrt{sin^{2}x-2sinx+5}\cdot \frac{1}{4^{sin^{2}y}}}\leq 1  also satisfy the equation : 

  • Option 1)

    2|sinx|=3siny

  • Option 2)

    2 sinx=siny

  • Option 3)

    sinx=2siny

  • Option 4)

    sinx=|siny|

 
=>  &  So, correct option is (4). Option 1) Option 2) Option 3) Option 4)

If \alpha ,\beta \: \: and\: \: \gamma are three consecutive terms of a non-constant

G.P. such that the equations \alpha x^{2}+2\beta x+\gamma =0  and 

x^{2}+ x-1=0 have a common root , then \alpha (\beta +\gamma ) is equal to :

  • Option 1)

    0

  • Option 2)

    \alpha \beta

  • Option 3)

    \alpha \gamma

  • Option 4)

    \beta \gamma

 
  are in G.P. =>  For equation,                  Hence, roots are equal & equals to  Since, given equation have common roots , hence   must be root of                     Option 1) 0 Option 2) Option 3) Option 4)

Let z\epsilon C  with Im(z)=10  and  it satisfies 

\frac{2z-n}{2z+n}=2i-1  for some natural number n . Then : 

  • Option 1)

     n = 20 and Re(z) = -10

  • Option 2)

     n = 40 and Re(z) = 10

  • Option 3)

     n = 40 and Re(z) = -10

  • Option 4)

     n = 20 and Re(z) = 10

 
 ,  ,  Option 1)  n = 20 and Re(z) = -10 Option 2)  n = 40 and Re(z) = 10 Option 3)  n = 40 and Re(z) = -10 Option 4)  n = 20 and Re(z) = 10

A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to :

 

  • Option 1)

    28

  • Option 2)

    27

  • Option 3)

    25

  • Option 4)

    24

 
atleast one boy & one girl :  ( 1B & 2G) + ( 2B & 1G)   As, n cannot be -ve so, n = 25 Option 1) 28 Option 2) 27 Option 3) 25 Option 4) 24

If B=\begin{bmatrix} 5 &2\alpha &1 \\ 0 &2 &1 \\ \alpha &3 &-1 \end{bmatrix} is the inverse of a 3\times 3 matrix A, then the sum of all values of \alpha for which det \left ( A \right )+1=0, is : 

 

 

 

 

  • Option 1)

    0

  • Option 2)

    1

  • Option 3)

    2

  • Option 4)

    -1

 
                                                                                                              Option 1) Option 2) Option 3) Option 4)

The equation \left | z-i \right |=\left | z-1 \right |,i=\sqrt{-1}, represents : 

 

  • Option 1)

    a circle of radius \frac{1}{2}.

     

  • Option 2)

    a circle of radius 1.

  • Option 3)

    the line through the origin with slope 1.

  • Option 4)

    the line through the origin with slope -1.

The line through the origin with slope 1Option 1)a circle of radius .  Option 2)a circle of radius Option 3)the line through the origin with slope Option 4)the line through the origin with slope 

if \alpha \: and\: \beta are roots of the equation 375x^{2}-25x-2=0, then \lim_{n\rightarrow \infty }\sum_{r=1}^{n}\alpha ^{r}+\lim_{n\rightarrow \infty }\sum_{r=1}^{n}\beta ^{r} is equal to : 

  • Option 1)

    \frac{1}{12}

  • Option 2)

    \frac{7}{116}

  • Option 3)

    \frac{21}{346}

  • Option 4)

    \frac{29}{358}

 
                                                                                    Option 1) Option 2) Option 3) Option 4)

Let a,b and c be in G.P. with common ratio r , where a\neq 0 and

0<r\leq \frac{1}{2}. If  3a , 7b and 15c are the first three terms of an A.P., 

then the 4th term of this A.P. is :

  • Option 1)

    \frac{2}{3}a

  • Option 2)

    5 a

  • Option 3)

    \frac{7}{3}a

  • Option 4)

    a

 
Since a,b,c are in G.P. with common ratio r then  b = ar ,  Also 3a, 7b and 15c are in A.P. =>  =>  =>  =>  =>  =>  =>  So, terms are                         or                         =>    or    So, 4th term     or    So, option (4) is correct.   Option 1) Option 2) 5 a Option 3) Option 4) a

The sum of the real roots of the equation 

\begin{vmatrix} x & -6 &-1 \\ 2 &-3x &x-3 \\ -3& 2x &x+2 \end{vmatrix}=0, is equal to : 

  • Option 1)

    6

  • Option 2)

    0

  • Option 3)

    1

  • Option 4)

    -4

=>  =>  Root of equation (-3,1,2) So, Sum of real root of equation = -3+1+2=0 So, option (2) is correct.Option 1)6Option 2)0Option 3)1Option 4)-4

If z and w are two complex numbers such that |zw|=1 

and arg(z)-arg(w)=\frac{\pi}{2},  then : 

  • Option 1)

  • Option 2)

    z\bar w=\frac{-1+i}{\sqrt2}

  • Option 3)

    \bar z w=-i

  • Option 4)

    z\bar w=\frac{1-i}{\sqrt2}

 and  Let    =>                                                                  Option 1)Option 2)Option 3)Option 4)

If \alpha and \beta are the roots of the quadratic equation,

x^{2}+xsin\theta -2sin\theta =0,\theta \epsilon (0,\frac{\pi}{2}), then 

\frac{\alpha ^{12}+\beta ^{12}}{(\alpha ^{-12}+\beta ^{-12})\cdot (\alpha -\beta )^{24}} is equal to :

  • Option 1)

    \frac{2^{12}}{(sin\theta-4)^{12}}

  • Option 2)

    \frac{2^{12}}{(sin\theta+8)^{12}}

  • Option 3)

    \frac{2^{12}}{(sin\theta-8)^{6}}

  • Option 4)

    \frac{2^{6}}{(sin\theta+8)^{12}}

 
 and  are the roots of the equation  Now,                                                                                                                                                                   correct option (2) Option 1) Option 2) Option 3) Option 4)

If a>0 and z=\frac{(1+i)^{2}}{a-i} , has magnitude \sqrt{\frac{2}{5}} ,

then \bar{z} is equal to :

  • Option 1)

    -\frac{1}{5}-\frac{3}{5}i

  • Option 2)

    -\frac{3}{5}-\frac{1}{5}i

  • Option 3)

    \frac{1}{5}-\frac{3}{5}i

  • Option 4)

    -\frac{1}{5}+\frac{3}{5}i

 
     given that  Option (1) is correct. Option 1) Option 2) Option 3) Option 4)

Let z \:\varepsilon \:C be such that \left | z \right |<1. If \omega =\frac{5+3z}{5(1-z)} ,

then :

  • Option 1)

    5\:Re (\omega )>4

  • Option 2)

    5\:Im (\omega )>5

  • Option 3)

    5\:Re (\omega )>1

  • Option 4)

    5\:Im (\omega )<1

     Option 1)Option 2)Option 3)Option 4)

If m is chosen in the quadratic equation \left ( m^{2}+1 \right )x^{2}-3x+\left ( m^{2}+1 \right )^{2}=0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is  :

  • Option 1)

    10\sqrt{5}

  • Option 2)

    8\sqrt{3}

  • Option 3)

    8\sqrt{5}

  • Option 4)

    4\sqrt{3}

 
  for sum of root to be greatest     should be minimum  now equation         Option 1) Option 2) Option 3) Option 4)

Let p,q\; \epsilon\; \textbf{R}. If 2-\sqrt{3} is a root of the quadratic equation,x^{2}+px+q=0, then :

  • Option 1)

     p^{2}-4q+12=0          

  • Option 2)

     q^{2}-4p-16=0

  • Option 3)

    q^{2}+4p+14=0

  • Option 4)

    p^{2}-4q-12=0

 
Given that one root is  then another root will by  Sum of roots =  Product of roots=  Question is wrong   Option 1)             Option 2)   Option 3) Option 4)

All the points in the set 

S=\left \{ \frac{\alpha +i}{\alpha -i}:\alpha\; \epsilon\; \mathbf{R} \right \}\left ( i=\sqrt{-1} \right )  lie on a :

  • Option 1)

    straight line whose slope is 1.

  • Option 2)

    circle whose radius is 1.

  • Option 3)

    circle whose radius is \sqrt{2} .

  • Option 4)

    straight line whose slop is -1

 
Given ,     Option 1) straight line whose slope is . Option 2) circle whose radius is  Option 3) circle whose radius is  . Option 4) straight line whose slop is 

Let \alpha and \beta be the roots of the equation x^{2}+x+1=0. Then for y\neq 0 in R,\begin{vmatrix} y+1 & \alpha & \beta \\ \alpha & y+\beta & 1\\ \beta & 1& y+\alpha \end{vmatrix} is equal to :

  • Option 1)

    y\left ( y^{2}-1 \right )

  • Option 2)

    y\left ( y^{2}-3 \right )

  • Option 3)

    y^{3}

  • Option 4)

    y^{3}-1

 are roots of      Option 1)Option 2)Option 3)Option 4)

If three distinct numbers a,b,c  are in G.P. and the equations ax^{2}+2bx+c=0 and dx^{2}+2ex+f=0 have a common root, then which one of the following statements is correct ? 
 

  • Option 1)

    \frac{d}{a},\frac{e}{b},\frac{f}{c}  are in A.P.

  • Option 2)

    d,e,f are in A.P.

  • Option 3)

    d,e,f are in G.P.

     

  • Option 4)

    \frac{d}{a},\frac{e}{b},\frac{f}{c} are in G.P. 

 
Given  are in G.P. roots are =  root of this eq =  or     divide by ac Option 1)   are in A.P. Option 2)  are in A.P. Option 3)  are in G.P.   Option 4)  are in G.P. 

The number of integral values of m for which the equation \left ( 1+m^{2} \right )x^{2}-2\left ( 1+3m \right )x+\left ( 1+8m \right )=0 has no real root is : 


 

  • Option 1)

    1

  • Option 2)

    2

  • Option 3)

    infinitely many

  • Option 4)

    3

 
Given equation is  given eq. has no real root Infinite value of m satisfies this.  Option 1) Option 2) Option 3) infinitely many Option 4)
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