Free Class 12 Solutions and Answers - All Boards, Books and Exams 2021

Let \(A=\left\{\theta \in\left(-\frac{\pi}{2}, \pi\right): \frac{3+2 i \sin \theta}{1-2 i \sin \theta} \text { is purely imaginary }\right\}\).Then, the sum of the elements in \(A\) is（）

A. \(\frac {3\pi }{4}\)

B. \(\frac {5\pi }{6}\)

C. \(\pi\)

D. \(\frac {2\pi }{3}\)

Let \(A=\left\{\theta \in\left(-\frac{\pi}{2}, \pi\right): \frac{3+2 i \sin \theta}{1-2 i \sin \theta} \text { is purely imaginary }\right\}\).Then, the sum of the elements in \(A\) is（）

A. \(\frac {3\pi }{4}\)

B. \(\frac {5\pi }{6}\)

C. \(\pi\)

D. \(\frac {2\pi }{3}\)

Math|Class 12

Correct12

A value of \(θ\) for which \(\frac {2+3i\sin \theta }{1-2i\sin \theta }\) is purely imaginary, is（）

A. \(\frac {\pi }{3}\)

B. \(\frac {\pi }{6}\)

C. \(\sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)\)

D. \(\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)

A value of \(θ\) for which \(\frac {2+3i\sin \theta }{1-2i\sin \theta }\) is purely imaginary, is（）

A. \(\frac {\pi }{3}\)

B. \(\frac {\pi }{6}\)

C. \(\sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)\)

D. \(\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)

Math|Class 12

Correct39

If \(\begin{vmatrix} 6i&-3i&1\\ 4&3i&-1\\ 20&3&i\end{vmatrix} =x+iy\), then（）

A. \(x=3,y=1\)

B. \(x=1,y=1\)

C. \(x=0,y=3\)

D. \(x=0,y=0\)

If \(\begin{vmatrix} 6i&-3i&1\\ 4&3i&-1\\ 20&3&i\end{vmatrix} =x+iy\), then（）

Correct6

The value of sum \(\sum\limits _{n=1}^{13}(i^{n}+i^{n+1}),\) where \(i=\sqrt {-1}\) , equals（）

A. \(i\)

B. \(i-1\)

C. \(-i\)

D. 0

The value of sum \(\sum\limits _{n=1}^{13}(i^{n}+i^{n+1}),\) where \(i=\sqrt {-1}\) , equals（）

Correct4

The smallest positive integer \(n\) for which \((\frac {1+i}{1-i})^{n}=1\), is（）

A. \(8\)

B. \(16\)

C. \(12\)

D. None of these

The smallest positive integer \(n\) for which \((\frac {1+i}{1-i})^{n}=1\), is（）

Correct13

Let \(a, b, x\) and \(y\) be real numbers such that \(a-b=1\) and \(y \neq 0 .\) If the complex number \(z=x+i y\) satisfies \(\operatorname{Im}\left(\frac{a z+b}{z+1}\right)=y,\) then which of the following is(are) possible value(s) of \(x ?\)（）

A. \(1-\sqrt{1+y^{2}}\)

B. \(-1-\sqrt{1-y^{2}}\)

C. \(1+\sqrt{1+y^{2}}\)

D. \(-1+\sqrt{1-y^{2}}\)

Let \(a, b, x\) and \(y\) be real numbers such that \(a-b=1\) and \(y \neq 0 .\) If the complex number \(z=x+i y\) satisfies \(\operatorname{Im}\left(\frac{a z+b}{z+1}\right)=y,\) then which of the following is(are) possible value(s) of \(x ?\)（）

A. \(1-\sqrt{1+y^{2}}\)

B. \(-1-\sqrt{1-y^{2}}\)

C. \(1+\sqrt{1+y^{2}}\)

D. \(-1+\sqrt{1-y^{2}}\)

Math|Class 12

Correct34

The equation \(|z-i|=|z-1|,i=\sqrt {-1},\) , represents （）

A. a circle of radius \(\frac {1}{2}\)

B. the line passing through the origin with slope 1

C. a circle of radius 1

D. the line passing through the origin with slope \(-1\)

The equation \(|z-i|=|z-1|,i=\sqrt {-1},\) , represents （）

A. a circle of radius \(\frac {1}{2}\)

B. the line passing through the origin with slope 1

C. a circle of radius 1

D. the line passing through the origin with slope \(-1\)

Math|Class 12

Correct11

If \(a\gt 0\) and \(z=\frac {(1+i)^{2}}{a-i}\), has magnitude \(\sqrt {\frac {2}{5}}\) then\(\overline {z}\) is equal to（）

A. \(\frac{1}{5}-\frac{3}{5} i\)

B. \(-\frac{1}{5}-\frac{3}{5} i\)

C. \(-\frac{1}{5}+\frac{3}{5} i\)

D. \(-\frac{3}{5}-\frac{1}{5} i\)

If \(a\gt 0\) and \(z=\frac {(1+i)^{2}}{a-i}\), has magnitude \(\sqrt {\frac {2}{5}}\) then\(\overline {z}\) is equal to（）

A. \(\frac{1}{5}-\frac{3}{5} i\)

B. \(-\frac{1}{5}-\frac{3}{5} i\)

C. \(-\frac{1}{5}+\frac{3}{5} i\)

D. \(-\frac{3}{5}-\frac{1}{5} i\)

Math|Class 12

Correct12

Let \(z_{1}\) and \(z_{2}\) be two complex numbers satisfying \(|z_{1}|=9\) and \(|z_{2}-3-4i|=4\) Then, the minimum value of \(|z_{1}-z_{2}|\) is（）

A. 1

B. 2

C. \(\sqrt2\)

D. 0

Let \(z_{1}\) and \(z_{2}\) be two complex numbers satisfying \(|z_{1}|=9\) and \(|z_{2}-3-4i|=4\) Then, the minimum value of \(|z_{1}-z_{2}|\) is（）

Correct32

If \(\frac {z-\alpha }{z+\alpha }(\alpha \in R)\) is a purely imaginary number and \([z]=2,\) then a value of a is （）

A. \(\sqrt {2}\)

B. \(\frac {1}{2}\)

C. 1

D. 2

If \(\frac {z-\alpha }{z+\alpha }(\alpha \in R)\) is a purely imaginary number and \([z]=2,\) then a value of a is （）

Correct47

Let \(z\) be a complex number such that \(|z|+z=3+i\) (where \(i=\sqrt {-1})\)
Then, \(|z|\) is equal to（）

A. \(\frac {\sqrt {34}}{3}\)

B. \(\frac {5}{3}\)

C. \(\frac {\sqrt {41}}{4}\)

D. \(\frac {5}{4}\)

Let \(z\) be a complex number such that \(|z|+z=3+i\) (where \(i=\sqrt {-1})\)
Then, \(|z|\) is equal to（）

A. \(\frac {\sqrt {34}}{3}\)

B. \(\frac {5}{3}\)

C. \(\frac {\sqrt {41}}{4}\)

D. \(\frac {5}{4}\)

Math|Class 12

Correct29

A complex number \(z\) is said to be unimodular, if \(|z|\neq 1\). If \(z_{1}\) and \(z_{2}\) are complex numbers such that \(\frac {z_{1}-2z_{2}}{2-z_{1}\bar z_{2}}\) is unimodular and \(z_{2}\) is not unimodular. Then, the point \(z_{1}\) lies on a（）

A. straight line parallel to \(X\)-axis

B. straight line parallel to \(Y\)-axis

C. circle of radius \(2\)

D. circle of radius \(\sqrt {2}\)

A complex number \(z\) is said to be unimodular, if \(|z|\neq 1\). If \(z_{1}\) and \(z_{2}\) are complex numbers such that \(\frac {z_{1}-2z_{2}}{2-z_{1}\bar z_{2}}\) is unimodular and \(z_{2}\) is not unimodular. Then, the point \(z_{1}\) lies on a（）

A. straight line parallel to \(X\)-axis

B. straight line parallel to \(Y\)-axis

C. circle of radius \(2\)

D. circle of radius \(\sqrt {2}\)

Math|Class 12

Correct9

If \(z\) is a complex number such that \(|z|\geq 2,\) then the minimum value of \(|z+\frac {1}{2}|\) （）

A. is equal to\(\frac{5}{2}\)

B. lies in the interval \((1,2)\)

C. is strictly greater than\(\frac{5}{2}\)

D. is strictly greater than \(\frac{3}{2}\) but less than \(\frac{5}{2}\)

If \(z\) is a complex number such that \(|z|\geq 2,\) then the minimum value of \(|z+\frac {1}{2}|\) （）

A. is equal to\(\frac{5}{2}\)

B. lies in the interval \((1,2)\)

C. is strictly greater than\(\frac{5}{2}\)

D. is strictly greater than \(\frac{3}{2}\) but less than \(\frac{5}{2}\)

Math|Class 12

Correct24

Let complex numbers a and \(1/\overline {\alpha }\) lies on circles \((x-x_{0})^{2}+(y-y_{0})^{2}=r^{2}\) and \((x-x_{0})^{2}+(y-y_{0})^{2}=4r^{2}\)
respectively.
If \(z_{0}=x_{0}+iy_{0}\) ) satisfies the equation \(2|z_{0}|^{2}=r^{2}+2\) , then \(|\alpha |\) is equal to（）

A. \(\frac {1}{\sqrt {2}}\)

B. \(\frac {1}{2}\)

C. \(\frac {1}{\sqrt {7}}\)

D. \(\frac {1}{3}\)

Let complex numbers a and \(1/\overline {\alpha }\) lies on circles \((x-x_{0})^{2}+(y-y_{0})^{2}=r^{2}\) and \((x-x_{0})^{2}+(y-y_{0})^{2}=4r^{2}\)
respectively.
If \(z_{0}=x_{0}+iy_{0}\) ) satisfies the equation \(2|z_{0}|^{2}=r^{2}+2\) , then \(|\alpha |\) is equal to（）

A. \(\frac {1}{\sqrt {2}}\)

B. \(\frac {1}{2}\)

C. \(\frac {1}{\sqrt {7}}\)

D. \(\frac {1}{3}\)

Math|Class 12

Correct40

Let \(z\) be a complex number such that the imaginary part of \(z\) is non-zero and \(a=z^{2}+z+1\) is real. Then, \(a\) cannot take the value（）