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AlgebraQuestion and Answers: Page 327

Question Number 32208 Answers: 0 Comments: 0

Question Number 32203 Answers: 0 Comments: 5

Number of solutions of the equation z^3 +(([3(z^− )^2 ])/(∣z∣))=0 where z is a complex no.

$N u m b e r o f s o l u t i o n s o f t h e e q u a t i o n$ $z^{3} + \frac{[3 {(\bar{z})}^{2}]}{∣ z ∣} = 0 w h e r e z i s a c o m p l e x n o .$

Question Number 32184 Answers: 1 Comments: 0

If one vertex of the triangle having maximum area that can be inscribed in the circle ∣z−i∣=5 is 3−3i, then find other vertices of triangle.

$I f o n e v e r t e x o f t h e t r i a n g l e h a v i n g$ $m a x i m u m a r e a t h a t c a n b e i n s c r i b e d$ $i n t h e c i r c l e ∣ z - i ∣= 5 i s 3 - 3 i, t h e n$ $f i n d o t h e r v e r t i c e s o f t r i a n g l e .$

Question Number 32181 Answers: 1 Comments: 1

Intercept made by the circle zz^− +a^− z+az^− +r=0 on the real axis on complex plane is :−

$I n t e r c e p t m a d e b y t h e c i r c l e$ $z \bar{z} + \bar{a z} + a \bar{z} + r = 0 o n t h e r e a l a x i s o n$ $c o m p l e x p l a n e i s : -$

Question Number 32160 Answers: 1 Comments: 0

If z=cosθ+isinθ is a root of equation a_0 z^n +a_1 z^(n−1) +a_2 z^(n−2) +.....+a_(n−1) z+a_n =0 then prove that: i) a_0 +a_1 cos θ+a_2 cos 2θ+.....+a_n cos nθ=0 ii) a_1 sin θ + a_2 sin 2θ+....+a_n sin nθ=0.

$I f z = c o s θ + i s i n θ i s a r o o t o f e q u a t i o n$ $a_{0} z^{n} + a_{1} z^{n - 1} + a_{2} z^{n - 2} + . . . . . + a_{n - 1} z + a_{n} = 0$ $t h e n p r o v e t h a t :$ $i) a_{0} + a_{1} \cos θ + a_{2} \cos 2 θ + . . . . . + a_{n} \cos n θ = 0$ $i i) a_{1} \sin θ + a_{2} \sin 2 θ + . . . . + a_{n} \sin n θ = 0 .$

Question Number 32159 Answers: 1 Comments: 2

Express the following in a+ib form: (((cos x+isin x)(cos y+isin y))/((cosa+isin a)(cosb+isinb))).

$E x p r e s s t h e f o l l o w i n g i n a + i b f o r m :$ $\frac{(\cos x + i \sin x) (\cos y + i \sin y)}{(c o s a + i \sin a) (c o s b + i s i n b)} .$

Question Number 32142 Answers: 0 Comments: 0

Question Number 32132 Answers: 0 Comments: 0

∫(1/((x+1)ln(x)))dx=?

$\int \frac{1}{(x + 1) l n (x)} d x = ?$

Question Number 32110 Answers: 1 Comments: 0

If y=1+x^2 +x^3 and x=1+α, where α is small, show that y≈3+5α. Hence, find the increase in y when x is increased from 1 to 1.02

$If y = 1 + x^{2} + x^{3} and x = 1 + α, where α is small, show$ $that y \approx 3 + 5 α . Hence, find the increase in y when$ $x is increased from 1 to 1.02$

Question Number 32099 Answers: 1 Comments: 0

Question Number 32094 Answers: 0 Comments: 0

Find the ordinary argument (arg z) and the principal argument (Arg z) of z=(i/(−2−2i))

$F i n d t h e o r d i n a r y a r g u m e n t$ $(a r g z) a n d t h e p r i n c i p a l a r g u m e n t$ $(A r g z) o f z = \frac{i}{- 2 - 2 i}$

Question Number 32049 Answers: 2 Comments: 0

If ∣z−6−8i∣≤4 then Minimum value of ∣z∣ is A) 4 B) 5 C) 6 D) 8.

$I f ∣ z - 6 - 8 i ∣⩽ 4 t h e n M i n i m u m v a l u e$ $o f ∣ z ∣ i s$ $A) 4$ $B) 5$ $C) 6$ $D) 8 .$

Question Number 32047 Answers: 0 Comments: 1

Question Number 32002 Answers: 1 Comments: 0

If z^3 =z^ prove then ∣z∣=1.

$I f z^{3} = \bar{z} p r o v e$ $t h e n ∣ z ∣= 1 .$

Question Number 31991 Answers: 0 Comments: 1

g_n =(√(g_(n−1) +g_(n−2) )) g_1 =1 g_2 =3 g_n =..

$g_{n} = \sqrt{g_{n - 1} + g_{n - 2}}$ $g_{1} = 1$ $g_{2} = 3$ $g_{n} = . .$

Question Number 31990 Answers: 1 Comments: 3

a_n =2a_(n−1) +3a_(n−2) a_0 =1 a_1 =2 a_n =...

$a_{n} = 2 a_{n - 1} + 3 a_{n - 2}$ $a_{0} = 1$ $a_{1} = 2$ $a_{n} = . . .$

Question Number 31963 Answers: 0 Comments: 0

find Re (((1+e^(iα) )/(1+e^(iβ) ))) and Im ( ((1+e^(iα) )/(1+e^(iβ) )) ) .

$f i n d R e (\frac{1 + e^{i α}}{1 + e^{i β}}) a n d I m (\frac{1 + e^{i α}}{1 + e^{i β}}) .$

Question Number 31927 Answers: 1 Comments: 0

The number of distinct real roots of equation x^4 −4x^3 +12x^2 +x−1=0.

$T h e n u m b e r o f d i s t i n c t r e a l r o o t s$ $o f e q u a t i o n x^{4} - 4 x^{3} + 12 x^{2} + x - 1 = 0 .$

Question Number 31915 Answers: 1 Comments: 3

If : (x^2 +x+2)^2 −(a−3)(x^2 +x+1)(x^2 +x+2) + (a−4)(x^2 +x+1)^2 =0 has at least one root , then find complete set of values of a.

$I f :$ ${(x^{2} + x + 2)}^{2} - (a - 3) (x^{2} + x + 1) (x^{2} + x + 2)$ $+ (a - 4) {(x^{2} + x + 1)}^{2} = 0 h a s a t l e a s t$ $o n e r o o t, t h e n f i n d c o m p l e t e s e t o f$ $v a l u e s o f a .$

Question Number 31895 Answers: 0 Comments: 3

Question Number 32372 Answers: 1 Comments: 0

Question Number 31868 Answers: 0 Comments: 0

Let a>b>1 be positive integers with b odd. Let n be a positive integer as well. If b^n divides a^n −1, prove that a^b > (3^n /n). Solution please. Thanks in advance!!

$L e t a > b > 1 b e p o s i t i v e i n t e g e r s w i t h b o d d .$ $L e t n b e a p o s i t i v e i n t e g e r a s w e l l . I f b^{n} d i v i d e s$ $a^{n} - 1, p r o v e t h a t a^{b} > \frac{3^{n}}{n} .$ $S o l u t i o n p l e a s e . T h a n k s i n a d v a n c e!!$

Question Number 31865 Answers: 1 Comments: 6

Range of function : f(x)= 6^x +3^x +6^(−x) +3^(−x) +2.

$R a n g e o f f u n c t i o n :$ $f (x) = 6^{x} + 3^{x} + 6^{- x} + 3^{- x} + 2 .$

Question Number 31864 Answers: 1 Comments: 0

Let S_n = Σ_(k=1) ^(4n) (−1)^((k(k+1))/2) k^2 . Then S_n can take the value(s) 1) 1056 2) 1088 3) 1120 4) 1332.

$L e t S_{n} = \underset{k = 1}{\sum^{4 n}} {(- 1)}^{\frac{k (k + 1)}{2}} k^{2} .$ $T h e n S_{n} c a n t a k e t h e v a l u e (s)$ $1) 1056$ $2) 1088$ $3) 1120$ $4) 1332 .$

Question Number 31804 Answers: 1 Comments: 0

A quadratic equation p(x)=0 having coefficient of x^2 unity is such that p(x)=0 and p(p(p(x)))=0 have a common root then, prove that : p(0)×p(1)=0.

$A q u a d r a t i c e q u a t i o n p (x) = 0 h a v i n g$ $c o e f f i c i e n t o f x^{2} u n i t y i s s u c h t h a t$ $p (x) = 0 a n d p (p (p (x))) = 0 h a v e a$ $c o m m o n r o o t t h e n,$ $p r o v e t h a t : p (0) \times p (1) = 0 .$

Question Number 31771 Answers: 1 Comments: 1

Consider a sequence in the form of groups (1),(2,2),(3,3,3),(4,4,4,4), (5,5,5,5,5),............ then the 2000th term of the above sequence is : ?

$C o n s i d e r a s e q u e n c e i n t h e f o r m o f$ $g r o u p s (1), (2, 2), (3, 3, 3), (4, 4, 4, 4),$ $(5, 5, 5, 5, 5), . . . . . . . . . . . .$ $t h e n t h e 2000 t h t e r m o f t h e a b o v e$ $s e q u e n c e i s : ?$

Pg 322 Pg 323 Pg 324 Pg 325 Pg 326 Pg 327 Pg 328 Pg 329 Pg 330 Pg 331

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