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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)

If a_1,a_2,a_3,................. are in A.P. such that a_1+a_7+a_{16}=40,

then the sum of the first 15 terms of this A.P. is :

  • Option 1)

    200

  • Option 2)

    280

  • Option 3)

    120

  • Option 4)

    150

Given ,                 ..................(1) We have to find out , ..............(2) Substituting the value of (1) in (2), Option 1)200Option 2)280Option 3)120Option 4)150

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 Sn denote the sum of the first terms of an A.P.. If S_{4}=16 and S_{6}=-48 then S_{10} is equal to : 

 

 

  • Option 1)

    -380

  • Option 2)

    -320

  • Option 3)

    -260

  • Option 4)

    -410

 
       So (2) - (1)                                    Option 1) Option 2) Option 3) Option 4)

Let a_1,a_2,a_3,......... be an A.P. with a_6=2 . Then the 

common difference of this A.P., which maximises the product 

a_1a_4a_5 , is :

  • Option 1)

    \frac{3}{2}

  • Option 2)

  • Option 3)

    \frac{6}{5}

  • Option 4)

    \frac{2}{3}

 
Assuming the first term of A.P. is a and difference is d. Then, Let  =>  So,  will be maximum at  So, option (2) is correct. 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)

The sum   \mathrm{1+\frac{1^3+2^3}{1+2}+\frac{1^3+2^3+3^3}{1+2+3}+..........+\frac{1^3+2^3+3^3+......+15^3}{1+2+3+.......+15}\:-\:\frac{1}{2}\left(1+2+3+.......+15\right)} 

is equal to :

  • Option 1)

    620

  • Option 2)

    1240

  • Option 3)

    1860

  • Option 4)

    660

 
  So, option (1) is correct.   Option 1) 620 Option 2) 1240 Option 3) 1860 Option 4) 660

The sum

\frac{3\times 1^{3}}{1^{2}}+\frac{5\times (1^{3}+2^{3})}{1^{2}+2^{2}}+\frac{7\times (1^{3}+2^{3}+3^{3})}{1^{2}+2^{2}+3^{2}}+.......... 

upto 10th term , is :

  • Option 1)

    680

  • Option 2)

    600

  • Option 3)

    660

  • Option 4)

    620

Given, general term will be                                                               So, correct option is (3).Option 1)680Option 2)600Option 3)660Option 4)620

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)

If a_{1},a_{2},a_{3},..........a_{n} are in A.P. and a_{1}+a_{4}+a_{7}..........+a_{16}=114,

then a_{1}+a_{6}+a_{11}+a_{16} is equal to : 

  • Option 1)

    98

  • Option 2)

    76

  • Option 3)

    38

  • Option 4)

    64

 
correct option is (2). Option 1) 98 Option 2) 76 Option 3) 38 Option 4) 64

The sum of the series  1+2\times3+3\times 5+4\times7+.......upto\:\:\:11^{th} term is :

  • Option 1)

    915

  • Option 2)

    946

  • Option 3)

    945

  • Option 4)

    916

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