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Number Theory - Class 11 Commerce Applied Mathematics - Extra Questions

If Z1 and Z2 are two complex numbers and c>0, then prove that
|z1+z2|2(1+c)|z1|2+(1+c1)|z2|2



Let z1 and z2 be complex numbers such that z1z2 and |z1|=|z2|. If z1 has positive real part and z2 has negative imaginary part, then show that z1+z2z1z2 is purely imaginary.



Multiply 3752 by 37+52



Find the modulus of complex number 2+23i.



Compute :
(1+i)1



Locate the points representing the complex number for which log1/2|z1|+4|z1|2>1



compute.
[(cosθ+isinθ)(cosθisinθ)]=?



Prove the identity,|1+z1¯z2|2+|z1z2|2=(1+|z1|2)(1+|z2|2)



Solve (1+i1i)24



Prove the identity |1z1z2|2|z1z2|2=(1|z1|2)(1|z2|2)



The HCF of two or more prime numbers is always __________ 



If (a+ib)(c+id)=A+iB, then show that (a2+b2)(c2+d2)=A2+B2.



If (a+ib)(c+id)=A+iB, then show that (a2+b2)(c2+d2)=A2+B2



Define and Give one example of each :
a) Parallel lines
b) Intersecting lines
c) Perpendicular line
d) Prime Number
e) Twin Prime
f) Even Number



Find the values of the following:
(1+i3)3



Find the value of (1+3)2+(13)2.



Find the modules and the argument of the complex no Z=1+i



If a=1+3i2b=13i2 then show that a2=b and b2=a.



z=2+3i find |z|



Find the principal arguments of the following Complex Number
(i) 5+5i



If z1,z2C, then |z1z2|.



Let Z=x+iy and ω=1iZZi. If |ω|=1, show that Z is purely real.



Find the modulus of 1+i1i1i1+i



Solve: x2(322i)x2i=0



State whether the given statement is true or false ¯(z1)=(ˉz)1



Express 5+i22i in the form of x+iy.



If (1+i)22i=xiy, then find the value of x+y.



Find the real and imaginary parts of the complex number z=3i20i192i1



If (1+i1i)3(1i1+i)3=x+iy, then find (x,y),



For the complex number37+19.
Real part is 37 and imaginary is 19
Enter 1 if true or 0 for false



What are twin-primes ?



Express the following in the form of a = ib, a,bϵR i=1. State the values of a and b.
 (1+2i)(2+i)



If ABC are angles of a triangle such that x = cis A, y = cis B, z = cis C, then find value of xyz



Give three pairs of prime numbers whose difference is 2. [Remark: Two prime numbers whose difference is 2 are called twin primes].



If z \neq 0 is a complex number,then prove that Re(z) = 0 \Longrightarrow  Im(z^2)=0.



If (a + ib) (c + id) (e + if)(g +ih) = A + iB, then prove that (a^2 + b^2) (c^2 + d^2)(e^2 + f^2)(g^2 + h^2) = A^2 + B^2



If z = x + iy and w = \dfrac{(1 - iz)}{(z - i)} and \left|w\right| = 1, then prove that z is purely real.



If (x+iy)^3=u+iv, then prove that \displaystyle \frac{u}{x}+\frac{v}{y} =4(x^2 - y^2)



If z_1 and z_2 are complex numbers and u = \sqrt{z_1z_2}, then prove that 
\left|z_1\right| + \left|z_2\right| = \left|\displaystyle \frac{z_1 + z_2}{2}+u\right| + \left|\displaystyle \frac{z_1 + z_2}{2}-u\right|.



If  \left|z_1 + z_2\right| = \left|z_1\right| + \left|z_2\right|, then show that arg(z_1) = arg(z_2)



If \left|z_1 - z_2\right| = \left|z_1\right| + \left|z_2\right|, then prove that arg(z_1) - arg (z_2) = \pi . I



Find the minimum value of \left | 1+z \right |+\left | 1-z \right |.



2^{i}=e^{i(lnx)}.



Compute the modulus of complex number z=\sqrt5-2i



If \displaystyle a+ib=\dfrac { { \left( x+i \right)  }^{ 2 } }{ { 2x }^{ 2 }+1 } , prove that \displaystyle { a }^{ 2 }+{ b }^{ 2 }=\dfrac { { \left( { x }^{ 2 }+1 \right)  }^{ 2 } }{ { \left( 2x^2+1 \right)  }^{ 2 } } .



Express \cfrac{2+i}{(1+i)(1-2i)} in the form of a+ib. Find its modulus and argument.



Find the real and imaginary parts of the complex number \dfrac {a + ib}{a - ib}



Find the number of integral solution of (1-i)^x=2^x.



If z and \alpha complex numbers such that |z| = |\alpha| = r, r > 0 and \omega = \dfrac {z - \overline {\alpha}}{r^{2} + z\overline {\alpha}}, find Re(\omega).



If the ratio \left (\dfrac {1 - z}{1 + z}\right ) is purely imaginary, then find value of |z|.



If a=\cfrac { 1+i }{ \sqrt { 2 }  } , find the value of { a }^{ 6 }+{ a }^{ 4 }+{ a }^{ 2 }+1



If \left| { z }_{ 1 } \right| =2,\left| { z }_{ 2 } \right| =3,\left| { z }_{ 3 } \right| =4 and \left| 2{ z }_{ 1 }+3{ z }_{ 2 }+4{ z }_{ 3 } \right| =10, then absolute value 8{ z }_{ 2 }{ z }_{ 3 }+27{ z }_{ 3 }{ z }_{ 1 }+64{ z }_{ 1 }{ z }_{ 2 } must be equal to-



Find the points of the plane which satisfy the following equations.
\displaystyle \left | \, \frac{z \, - \, 2}{z \, + \, 3} \, \right | \, = \, 1.



Find the points of the plane which satisfy the following equations.
\displaystyle \left | \, \frac{z \, + \, i}{z \, - \, 3i} \, \right | \, = \, 1.



Simplify the following :
\dfrac{(1 \, - \, i)^3}{1 \, - \, i^3}



Express the following  in the form A + iB :
\dfrac{1}{1 \, - \, \cos \theta \, + \, 2i \, \sin \theta}



Put the following in the form A + iB :
\dfrac{(a \, + \, ib)^2}{(a \, - \, ib)} \, - \, \dfrac{(a \, - \, ib)^2}{(a \, + \, ib)}



If Z_r \, = \, \left ( \cos\dfrac{r\pi }{10} \, + \, i \, \sin \dfrac{r\pi }{10}\right ). Then,  find the value of Z_1 \cdot Z_2 \cdot Z_3 \cdot Z_4



If i{ z }^{ 3 }+{ z }^{ 2 }-z+i=0, then show that \left| z \right| =1.



The\quad complex\quad number\quad \cfrac { { 2 }^{ n } }{ { (1+i) }^{ 2n } } +\cfrac { { (1+i) }^{ 2n } }{ { 2 }^{ n } } ,\quad n\quad \in Z,\quad is\quad equal\quad to\quad 



Find all the values of z which satisfy the equation
exp\left| \dfrac { { \left| z \right|  }^{ 2 }-\left| z \right| +4 }{ { \left| z \right|  }^{ 2 }+1 } \log2 \right| ={ \log }_{ \sqrt { 2 }  }\left| 3\sqrt { 15 } +11i \right|



Find the modulus and argument of the complex number:
\dfrac{1}{1+i}



For any two complex number z_1 and z_2 prove that Re(z_1z_2)= Rez_1Rez_2-I mz_1Imz_1



let z_1 and z_2 be two complex number such that |1-{z_1}z_2|^2-|z_1-z_2|^2=k\left(1-|z_1|^2\right)\left(1-|z_2|^2\right)  find the value of k



If \left( {a + ib} \right) = \frac{{1 + i}}{{1 - i}} , then prove that \left( {{a^2} + {b^2}} \right) = 1



How solve and what is its meaning
z_{1}=2+3i
z_{2}=3+4i
\left|z_{1}+z_{2}\right|=?



Let \dfrac{1}{a+ib}=c+id for non-zero a,b, then find c,d.



Find the modulus and the arguments of each of the complex numbers is z=-\sqrt{3}+i



Find real values of \theta for which\left( {\dfrac{{4 + 3i\;\sin \theta }}{{1 - 2i\;\sin \theta }}} \right) is purely real.  



Prove that {\left( {1 + i} \right)^4}{\left( {1 + \dfrac{1}{i}} \right)^4} = 16



\dfrac{4+3i}{(2+3i)(4-3i)}.



For any two complex numbers z_1,z_2 and any two real numbers a and b,
|az_1-bz_2|^2+|bz_1+az_2|^2=.....  .



Among the complex numbers Z satisfying the condition |z+3-\sqrt{3}i| = \sqrt{3}, find the number having the least positive argument.



Show that:
\left(\dfrac{1+i}{\sqrt{2}}\right)^{8}+\left(\dfrac{1-i}{\sqrt{2}}\right)^{8}=2



Find the value of { x }^{ 3 }+2{ x }^{ 2 }-3x+21 if x=1+2i



Let z_{1}=2-i,z_{2}=-2+i. Find Re\left(\dfrac {z_{1} z_{2}}{ \vec {z_{1}}}\right)



Find the expanded form when z is divided by z-0.5



Show that if \left| \dfrac { z - 3 i } { z + 3 i } \right| = 1 , then z is a real number.



If |z-1|+|z-3| \le 8, then find the range of |z-4|.



 Express the following in the form of a = ib, a,b\epsilonR i = \sqrt{-1}. State the values of a and b.
 (1+i)(1-i)^{-1}



Show that (-1 + \sqrt 3 i)^3 is a real number.



Express (5-3i)^{3} in the form a+ib.



Simplify: \left( -\sqrt { 3 } +\sqrt { -2 }  \right) \left( 2\sqrt { 3 } -i \right) =(a + ib)\left( -\sqrt { 3 } +\sqrt { -2 }  \right) . Find value of a and b.



Simplify and express the result in the form of a+ib 
(\sqrt{3}+i)^6



Find the possible missing twins for the following number so that they become twin primes.
89.



Express the following number as the sum of twin primes.
36.



Find the possible missing twins for the following number so that they become twin primes.
101.



Find the possible missing twins for the following number so that they become twin primes.
29.



Evaluate (\sqrt {-36}\times \sqrt{-25}).



Define twin prime numbers.



Given the output of the following,
when num1 = 4, num2 = 3, num3 = 2
1. num1 + = num2 + num3
    print (num1)
2. num1 = num1 * (num2 + num3)
    print (numl)
3. num1 = num2* (num2 + num 3)
    print(num1)
4. num1 = '5' + '5'
    print (numl)



Prove that if \displaystyle x \, + \,\frac{1}{x} \, = \, 2 \, cos \, \alpha, \, then \, x^n \, + \, \frac{1}{x^n} \, = \, 2 \, cos \, n\alpha.



Given: { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 }=A;{ z }_{ 1 }+{ z }_{ 2 }w+{ z }_{ 3 }{ w }^{ 2 }=B;{ z }_{ 1 }+{ z }_{ 2 }{ w }^{ 2 }+{ z }_{ 3 }{ w }^{  }=C where w is cube rott of unity
Prove: { \left| A \right|  }^{ 2 }+{ \left| B \right|  }^{ 2 }+{ \left| C \right|  }^{ 2 }=3\left( { \left| { z }_{ 1 } \right|  }^{ 2 }+{ \left| { z }_{ 2 } \right|  }^{ 2 }+{ \left| { z }_{ 3 } \right|  }^{ 2 } \right)



Perform the indicated operations.
\displaystyle (- \, 1 \, + \, \sqrt{3i})^3.



Solve \left(\dfrac {1}{1-4i}-\dfrac {2}{1+i}\right)\left(\dfrac {3-4i}{5+i}\right)



Solve \dfrac {1}{1+i}



Express the following number given in the form a+ib.
(5i)\left(-\dfrac{3}{5}i\right).



If |z + 1| = z + 2 (1 + i), then find z.



Define twin primes.Determine twin primes between 50 and 1000



What are twin-primes? Write all pairs of twin-primes between 50 and 100.



Express the following number as the sum of twin primes.
120.



Express the following number as the sum of twin primes.
84.



Class 11 Commerce Applied Mathematics Extra Questions