CBSE 11 Mathematics
Practice Paper 04
1. If x + iy = \(\frac{a + ib}{a - ib}\), prove that \(x^2 + y^2 = 1\).
2. If \(x - iy =\sqrt\frac{a - ib}{c - id}\) prove that \((x^2+y^2)^2=\frac{a^2 + b^2}{c^2 + d^2}\).
3. If \(z_1 = 2 – i, z_2 = 1 + i\), find \(|\frac{z_1 + z_2 + 1}{z_1 - z_2 + 1}|\).
4. Express each of the complex number given in the form a + ib.
(a) \((5i)(\frac{-1}{5} i)\)
(b) \(i^9+i^19\)
(c) \(i^-39\)
(d) \( (\sqrt3+\sqrt-2)\)\((2\sqrt3-i)\)
(e) \(\frac{5+\sqrt2 i}{1-\sqrt2 i}\)
5. Find the multiplicative inverse of each of the complex numbers
(a) \(4–3i\)
(b) \(\sqrt5 + 3i\)
(c) \(– i\)
6. Prove that \(Re(z_1z_2)=Rez_1Rez_2-Imz_1-Imz_2\).
7. If \((a+ib)^2=x+iy\) Prove that \((a^2+b^2)^2=x^2+y^2\).
8. Find the value of \(1+i^2 + i^4 + i^6 + i^8 + ---- + i^100\).
9. Solve for x and y, \(3x + (2x-y) i= 6 – 3i\).
10. Let \(z_1 = 2 – i, z_2 = -2+i\) Find \(Re \frac{(z_1z_2)}{z_1}\)
11. Find the conjugate and the modulus of each of the following
(a) \(3-\sqrt3 i\)
(b) \((2+2i)^2\)
(c) \(\frac{2+\sqrt3 i}{3-2i}\)
(d) \(\frac{4}{3-4i}\)
(e) \((3+\sqrt5 i)^2\)
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