PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions

Find the conjugate of the following complex number:

\[\frac{(3 - i )^2}{2 + i}\]

Find the conjugate of the following complex number:

\[\frac{(1 + i)(2 + i)}{3 + i}\]

Find the conjugate of the following complex number:

\[\frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)}\]

Find the modulus of \[\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}\].

Find the modulus and argument of the following complex number and hence express in the polar form:

1 + i

Find the modulus and argument of the following complex number and hence express in the polar form:

\[\sqrt{3} + i\]

Find the modulus and argument of the following complex number and hence express in the polar form:

1 − i

Find the modulus and argument of the following complex number and hence express in the polar form:

\[\frac{1 - i}{1 + i}\]

Find the modulus and argument of the following complex number and hence express in the polar form:

\[\frac{1}{1 + i}\]

Find the modulus and argument of the following complex number and hence express in the polar form:

\[\frac{1 + 2i}{1 - 3i}\]

Find the modulus and argument of the following complex number and hence express in the polar form:

 sin 120° - i cos 120° 

Find the modulus and argument of the following complex number and hence express in the polar form:

 \[\frac{- 16}{1 + i\sqrt{3}}\]

If z₁, z₂ and z₃, z₄ are two pairs of conjugate complex numbers, prove that \[\arg\left( \frac{z_1}{z_4} \right) + \arg\left( \frac{z_2}{z_3} \right) = 0\].

Write the conjugate of \[\frac{2 - i}{\left( 1 - 2i \right)^2}\] .

If (1+i)(1 + 2i)(1+3i)..... (1+ ni) = a+ib,then 2 ×5 ×10 ×...... ×(1+n²) is equal to.

If (1 + i) (1 + 2i) (1 + 3i) .... (1 + ni) = a + ib, then 2.5.10.17.......(1+n²)=

If \[\frac{( a^2 + 1 )^2}{2a - i} = x + iy, \text { then } x^2 + y^2\] is equal to

Two finite sets have m and n elements. The number of elements in the power set of first set is 48 more than the total number of elements in power set of the second set. Then, the values of m and n are: 

In a class of 175 students the following data shows the number of students opting one or more subjects. Mathematics 100; Physics 70; Chemistry 40; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. How many students have offered Mathematics alone? 

Suppose \[A_1 , A_2 , . . . , A_{30}\] are thirty sets each having 5 elements and \[B_1 , B_2 , . . . , B_n\] are n sets each with 3 elements. Let \[\cup^{30}_{i = 1} A_i = \cup^n_{j = 1} B_j = S\] and each element of S belong to exactly 10 of the \[A_i 's\]and exactly 9 of the\[B_j 's\] then n is equal to