Maths

The matrix $P=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill \\ \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ is a

(A) Square matrix

(B) Diagonal matrix

(C) Unit matrix

(D) None

(i) For any two complex numbers $z_1,z_2$ and any real numbers $a,b$ , $?a^2{z}_1+b^2{z}_2?-?b^2{z}_1+a^2{z}_2?=.....$

(ii) The value of $\sqrt[]{-25}\times \sqrt[]{-9}$ is .

(iii) The number $\frac{{\left(1-i\right)}^3}{1-i}$ is equal to .

(iv) The sum of the series $i+i^2+i^3+\dots$ up to 1000 terms is .

(v) Multiplicative inverse of $1+i$ is .

(vi) If $z_1$ and $z_2$ are complex numbers such that $z_1+z_2$ is a real number, then $z_2=....$

(vii)  $\left(z\right)+z\left(z\ne 0\right)$ is .

(viii) If  $?z+4?\le 3$ , then the greatest and least values of $?z+1?$ are . and .

(ix) If $\frac{z-2}{z+6}=\frac{\pi }{6},$ then the locus of $z$ is .

(x) If $?z?=4$ and $\left(z\right)=\frac{5\pi }{6}$ , then $z=\dots \dots \dots \dots$

(i) For any two complex numbers $z_1,z_2$ and any real numbers $a,b$ , $?a^2{z}_1+b^2{z}_2?-?b^2{z}_1+a^2{z}_2?=.....$

(ii) The value of $\sqrt[]{-25}\times \sqrt[]{-9}$ is .

(iii) The number $\frac{{\left(1-i\right)}^3}{1-i}$ is equal to .

(iv) The sum of the series $i+i^2+i^3+\dots$ up to 1000 terms is .

(v) Multiplicative inverse of $1+i$ is .

(vi) If $z_1$ and $z_2$ are complex numbers such that $z_1+z_2$ is a real number, then $z_2=....$

(vii)  $arg \left(z\right)+arg z\left(z\ne 0\right)$ is .

(viii) If $?z+4?\le 3$ , then the greatest and least values of $?z+1?$ are . and .

(ix) If $\frac{z-2}{z+6}=\frac{\pi }{6},$ then the locus of $z$ is .

(x) If $?z?=4$ and $\left(z\right)=\frac{5\pi }{6}$ , then $z=\dots \dots \dots \dots$

If a₁, a₂, a₃…. and b₁, b₂, b₃…. are A.P., and a₁ = 2, a₁₀ = 3, a₁b₁ = 1 – a₁₀b₁₀, then a₄b₄ is equal to…….

Let S = $2+\frac{6}{7}+\frac{12}{7^2}+\frac{20}{7^3}+\frac{30}{7^4}+....$  Then 4S is equal to

The number of points of intersection of $\left|z-\left(4+3i\right)\right|=2and\left|z\right|+\left|z-4\right|=6,z\in C,$  is

Kindly consider the following

 $\underset{x\to 0}{lim}\frac{{\left(1+x\right)}^n-1}{x}$ is

(a)  $n$

(b)   1

(c)   – $n$

(d)   0