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Syllabus

When I divide it by 2, the remainder is 1.

When I divide it by 3, the remainder is 2.

When I divide it by 4, the remainder is 3.

When I divide it by 5, the remainder is 4.

When I divide it by 6, the remainder is 5.

When I divide it by 7, the remainder is 6.

When I divide it by 8, the remainder is 7.

When I divide it by 9, the remainder is 8.

When I divide it by 10, the remainder is 9.

Q.If 2-

iis a root of the equation ax^{2}^{}+12x+b=0 (where a and b are real), then the value of ab is equal to$10.Fourpoints{z}_{1,}{z}_{2,}{z}_{3,}{z}_{4}incomplexplanesuchthat\left|{z}_{1}\right|\frac{3}{5},\left|{z}_{2}\right|=1,\left|{z}_{3}\right|\le 1and{z}_{3}=\frac{{z}_{1}\left({z}_{1}-{z}_{4}\right)}{{\overline{)z}}_{1}{z}_{4}-1}.\phantom{\rule{0ex}{0ex}}then\left|{z}_{4}\right|canbeequalto-\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(A\right){\mathrm{log}}_{5}7\left(B\right){\mathrm{log}}_{3}2\left(C\right){\mathrm{log}}_{x}e\left(D\right){\mathrm{log}}_{e}\mathrm{\pi}$

Q.The centre of a regular hexagon is at the point z=

i,If one of its vertices is at 2+i,then the adjacent vertices of 2-+iare at the points$\left(A\right)\frac{\mathrm{\pi}}{3}\phantom{\rule{0ex}{0ex}}\left(B\right)-\frac{\mathrm{\pi}}{3}\phantom{\rule{0ex}{0ex}}\left(\mathrm{C}\right)\frac{7\mathrm{\pi}}{3}\phantom{\rule{0ex}{0ex}}\left(\mathrm{D}\right)\frac{5\mathrm{\pi}}{3}$

^{4}-4z^{3}+10z^{2}-12z+10$15.Ifa,b0andc0.Then.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(A\right)0\le \left|{z}_{1}-{\overline{)z}}_{2}\right|\le 2\phantom{\rule{0ex}{0ex}}\left(B\right)0\left|{z}_{1}-{\overline{)z}}_{2}\right|\le \sqrt{2}\phantom{\rule{0ex}{0ex}}\left(C\right)\sqrt{2}\left|{z}_{1}-{\overline{)z}}_{2}\right|\le 2\phantom{\rule{0ex}{0ex}}\left(D\right)\sqrt{2}\left|{z}_{1}-{\overline{)z}}_{2}\right|2\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}16.Letk=\left|{z}_{1}+{z}_{2}\right|+\left|a+ic\right|,thenvalueofk,is-\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(A\right)\sqrt{2}-1\left(B\right)1\left(C\right)\sqrt{2}+1\left(D\right)2\sqrt{2}$

z

^{3}i+z^{2}-z+i=0$1.Ifzsatisfies\left|z-1\right|\left|z+3\right|then\omega =2z+3-i,(wherei=\sqrt{-1})satisfies\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(a\right)\left|\omega -5-i\right|\left|\omega +3+i\right|\left(b\right)\left|\omega -5\right|\left|\omega +3\right|\phantom{\rule{0ex}{0ex}}\left(c\right)Im\left(i\omega \right)1\left(d\right)\left|arg\left(\omega -1\right)\right|\mathrm{\pi}/2$