Q-1) Find the value of S, where
Q-2) If \(z_1,z_2\) be 2 complex numbers in the argand plane the n
\(|z_1+\sqrt{z_1^2-z_2^2}|+|z_1-\sqrt{z_1^2-z-2^2}|\) =
a) \(z_1\) b)\(\)|z_2| c)\(|z_1+z_2|\) d)None of these
Q-3) If \(S(n)=i^n+i^{2n}\), where \(n \in N\), find the number of distinct elements in \(S(n)\).
Q-4) The roots of the cubic equation \((z+\alpha \beta)^3=\alpha ^3\) \((\alpha \ne 0)\) represent the vertices of a triangle whose sides are of length
a)\(\frac{1}{\sqrt{3}}|\alpha \beta|\) b)\(\sqrt{3}|\alpha|\) c)\(\sqrt{3}|\beta|\) d)\(\frac{1}{\sqrt{3}}|\alpha|\)
Q-5) \((1+i)(1+2i)(1+3i)…(1+ni)=\alpha + i\beta\), then find the value of \(2.5.10..(1+n^2)\) in terms of \(\alpha, \beta\).
Q-6) If \(|z_1|=|z_2|=…|z_n|=1\), then show that \(|z_1+z_2+…z_n|=\left| \frac{1}{z_1}+\frac{1}{z_2}+…\frac{1}{z_n} \right|\)
Q-7) If \(z_1,z_2,z_3\) are 3 complex numbers and \(a,b,c \in R^+\), such that
then \(\frac{a^2}{z_2-z_3}+\frac{b^2}{z_3-z_1} + \frac{c^2}{z_1-z_2} = \)
a) 0 b) abc c) 3abc d) a+b+c
Q-8) If \(z_1=a+ib\) and \(z_2=c+id\) be 2 complex numbers such that \(|z_1|=|z_2|=1\) and \(\text{Re}(z_1\overline{z_2})=0\), then the pair of complex numbers \(\omega_1=a+ic\) and \(\omega_2=b+id\) satisfies
a) \(|\omega_1|=1\) b) \(|\omega_2|=1\) c) \(\text{Re}(\omega_1 \overline{\omega_2})=0\) d) \(\omega_1 \overline{\omega_2}=0\)
Q-9)Let \(z_1,z_2\) be 2 distinct complex numbers with \(|z_1|=|z_2|\). Also \(\text{Re}(z_1)>0\) & \(\text{Img}(z_2)<0\), then \(\frac{z_1+z_2}{z_1-z_2}\) may be
a) 0 b) real & positive c) real & negative d) purely imaginary
Q-10) If \(\alpha , \beta\) be the roots of the equation \(x^2-2x+4=0\), then find the value of \(\alpha^6+\beta^6\).
Q-11) If \(z=\frac{q+ir}{1+p}\), then for \(\frac{p+iq}{1+r}=\frac{1+iz}{1-iz}\) to be true we must have
a) \(p^2+q^2+r^2=1\) b) \(p^2+q^2+r^2=2\) c) \(p^2+q^2+r^2=3\) d)None of these
Q-12) If \(x=a+b\), \(y=a\omega+b\omega^2\), \(z=a\omega^2+b\omega\), then \(x^3+y^3+z^3\) equals
a) \(a^3+b^3\) b) \((a+b)^3\) c) \(3(a^3+b^3)\) d) None of these
Q-13) If \(1,\omega, \omega^2…\omega^{n-1}\) be the nth roots of unity and \(z_1,z_2\) be any non-zero complex numbers, then find the value of
Q-14) If \(X\) be the set of all complex numbers \(z\) such that \(|z|=1\). Define a relation \(R\) on \(X\) by \(z_1Rz_2=|\arg(z_1)-\arg(z_2)|=\frac{2\pi}{3}\), then \(R\) is
a)Reflexive b)Symmetric c)Transitive d)Anti-Symmetric
(You can read about basic relation terminology over here)
Q-15) If \(\alpha\) is a complex constant such that \(\alpha z^2+z+\overline{\alpha}=0\) has a real root then
a) \(\alpha + \overline{\alpha}=1\) b) \(\alpha + \overline{\alpha}=0\) c) \(\alpha + \overline{\alpha}=-1\) d) The absolute value of the real root is 1
<<In case there is any confusion over any example (solved or not), problem or concept, please feel free to create a thread in the appropriate forum. Our memebrs/experts will try their best to help solve the problem for you>>