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what is the difference between the two ?

The prefix poly means many. So the word polynomial refers to one or more than one term in an expression. The relationship between these terms may be sums or differences. You call expression with a single term a monomial, an expression with two terms is a binomial, and expression with three terms is a trinomial. Here we are only discussing Binomial along with their exponents.

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)

The term independent of x in the expansion of 

(\frac{1}{60}-\frac{x^{8}}{81})\cdot (2x^{3}-\frac{3}{x^{2}})^{6} is equal to :

  • Option 1)

    -72

  • Option 2)

    36

  • Option 3)

    -36

  • Option 4)

    -108

 
The term independent of  in the expansion of  =>  =>  Option 1) Option 2) 36 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

If ^{20}C_{1}+(2^{2}) ^{20}C_{2}+(3^{2}) + ^{20}C_{3} + \cdots \cdots

+(20^{2}) ^{20}C_{20}=A(2^{\beta }), then the ordered pair (A, \beta ) is equal to :

 

  • Option 1)

    (420, 19)

  • Option 2)

    (420, 18)

     

  • Option 3)

    (380, 18)

     

  • Option 4)

    (380, 19)

     

  +  +   LHS :  Comparing LHS with RHS  Option 1)(420, 19)Option 2)(420, 18)  Option 3)(380, 18)  Option 4)(380, 19)  

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)

The coefficient of x^{18} in the product \left ( 1+x \right )\left ( 1-x \right )^{10}\left ( 1+x+x^{2} \right )^{9} is : 

  • Option 1)

    -84

  • Option 2)

    84

  • Option 3)

    126

  • Option 4)

    -126

So coefficient of       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

The smallest natural number n , such that the coefficient of x in

the expansion of (x^{2}+\frac{1}{x^{3}})^{n} is   ^{n}C_{23}  , is : 

  • Option 1)

    38

  • Option 2)

    58

  • Option 3)

    23

  • Option 4)

    35

 
In  the expansion of  General term is                                       For coefficient of x ,                                    So, we have     =>  =>   The minimum value of n be 38. So, option (1) is correct. Option 1) 38 Option 2) 58 Option 3) 23 Option 4) 35

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 the coefficients of x^{2} and x^{3} are both zero , in the expansion of the 

expression (1+ax+bx^{2})(1-3x)^{15} in powers of x , then the 

ordered pair ( a , b ) is equal to : 

  • Option 1)

    (28, 861)

  • Option 2)

    (-54 , 315 )

  • Option 3)

    ( 28, 315 )

  • Option 4)

    ( - 21 , 714 )

 
=      coefficient of  =>  => b - 45 a +945 = 0..........................................(1) coefficient of  => => -b + 21 a -273 = 0 ........................................(2) From (1) & (2) a = 28 & b = 315 correct option (3).   Option 1) (28, 861) Option 2) (-54 , 315 ) Option 3) ( 28, 315 ) Option 4) ( - 21 , 714 )

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 some three consecutive coefficients in the binomial expansion of (x+1)^{n} in powers of x are in the ratio 2:15:70, then the average of these three coefficients is :

  • Option 1)

    964

  • Option 2)

    232

  • Option 3)

    227

  • Option 4)

    625

 
                          putting value of  n in (i)                                                  Option 1) Option 2) Option 3) Option 4)
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