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AlgebraQuestion and Answers: Page 355

Question Number 21313 Answers: 0 Comments: 4

Let k be a real number such that the inequality (√(x − 3)) + (√(6 − x)) ≥ k has a solution then the maximum value of k is

$$\mathrm{Let}\:{k}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{inequality}\:\sqrt{{x}\:−\:\mathrm{3}}\:+\:\sqrt{\mathrm{6}\:−\:{x}}\:\geqslant\:{k}\:\mathrm{has}\:\mathrm{a} \\ $$$$\mathrm{solution}\:\mathrm{then}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:{k} \\ $$$$\mathrm{is} \\ $$

Question Number 21311 Answers: 0 Comments: 0

Let a and b be positive real numbers with a^3 + b^3 = a − b, and k = a^2 + 4b^2 , then (1) k < 1 (2) k >1 (3) k = 1 (4) k > 2

$$\mathrm{Let}\:{a}\:\mathrm{and}\:{b}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{with}\:{a}^{\mathrm{3}} \:+\:{b}^{\mathrm{3}} \:=\:{a}\:−\:{b},\:\mathrm{and}\:{k}\:=\:{a}^{\mathrm{2}} \:+\:\mathrm{4}{b}^{\mathrm{2}} , \\ $$$$\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{k}\:<\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:{k}\:>\mathrm{1} \\ $$$$\left(\mathrm{3}\right)\:{k}\:=\:\mathrm{1} \\ $$$$\left(\mathrm{4}\right)\:{k}\:>\:\mathrm{2} \\ $$

Question Number 21309 Answers: 0 Comments: 0

Suppose p is a polynomial with complex coefficients and an even degree. If all the roots of p are complex non-real numbers with modulus 1, prove that p(1) ∈ R iff p(−1) ∈ R.

$$\mathrm{Suppose}\:{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{with}\:\mathrm{complex} \\ $$$$\mathrm{coefficients}\:\mathrm{and}\:\mathrm{an}\:\mathrm{even}\:\mathrm{degree}.\:\mathrm{If}\:\mathrm{all} \\ $$$$\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{p}\:\mathrm{are}\:\mathrm{complex}\:\mathrm{non}-\mathrm{real} \\ $$$$\mathrm{numbers}\:\mathrm{with}\:\mathrm{modulus}\:\mathrm{1},\:\mathrm{prove}\:\mathrm{that} \\ $$$${p}\left(\mathrm{1}\right)\:\in\:{R}\:\mathrm{iff}\:{p}\left(−\mathrm{1}\right)\:\in\:{R}. \\ $$

Question Number 21308 Answers: 0 Comments: 0

Find all complex numbers z such that ∣z − ∣z + 1∣∣ = ∣z + ∣z − 1∣∣

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{complex}\:\mathrm{numbers}\:{z}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mid{z}\:−\:\mid{z}\:+\:\mathrm{1}\mid\mid\:=\:\mid{z}\:+\:\mid{z}\:−\:\mathrm{1}\mid\mid \\ $$

Question Number 21307 Answers: 0 Comments: 5

Let z_1 , z_2 , z_3 be complex numbers such that (i) ∣z_1 ∣ = ∣z_2 ∣ = ∣z_3 ∣ = 1 (ii) z_1 + z_2 + z_3 ≠ 0 (iii) z_1 ^2 + z_2 ^2 + z_3 ^2 = 0 Prove that for all n ≥ 2, ∣z_1 ^n + z_2 ^n + z_3 ^n ∣ ∈ {0, 1, 2, 3}.

$$\mathrm{Let}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{complex}\:\mathrm{numbers}\:\mathrm{such} \\ $$$$\mathrm{that} \\ $$$$\left(\mathrm{i}\right)\:\mid{z}_{\mathrm{1}} \mid\:=\:\mid{z}_{\mathrm{2}} \mid\:=\:\mid{z}_{\mathrm{3}} \mid\:=\:\mathrm{1} \\ $$$$\left(\mathrm{ii}\right)\:{z}_{\mathrm{1}} \:+\:{z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} \:\neq\:\mathrm{0} \\ $$$$\left(\mathrm{iii}\right)\:{z}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{2}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{3}} ^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all}\:{n}\:\geqslant\:\mathrm{2}, \\ $$$$\mid{z}_{\mathrm{1}} ^{{n}} \:+\:{z}_{\mathrm{2}} ^{{n}} \:+\:{z}_{\mathrm{3}} ^{{n}} \mid\:\in\:\left\{\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:\mathrm{3}\right\}. \\ $$

Question Number 21294 Answers: 1 Comments: 0

Let z_1 , z_2 , z_3 be complex numbers, not all real, such that ∣z_1 ∣ = ∣z_2 ∣ = ∣z_3 ∣ = 1 and 2(z_1 + z_2 + z_3 ) − 3z_1 z_2 z_3 ∈ R. Prove that max(arg z_1 , arg z_2 , arg z_3 ) ≥ (π/6) . Where 0 < arg(z_1 ), arg(z_2 ), arg(z_3 ) < 2π.

$$\mathrm{Let}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{complex}\:\mathrm{numbers},\:\mathrm{not} \\ $$$$\mathrm{all}\:\mathrm{real},\:\mathrm{such}\:\mathrm{that}\:\mid{z}_{\mathrm{1}} \mid\:=\:\mid{z}_{\mathrm{2}} \mid\:=\:\mid{z}_{\mathrm{3}} \mid\:=\:\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{2}\left({z}_{\mathrm{1}} \:+\:{z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} \right)\:−\:\mathrm{3}{z}_{\mathrm{1}} {z}_{\mathrm{2}} {z}_{\mathrm{3}} \:\in\:{R}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{max}\left(\mathrm{arg}\:{z}_{\mathrm{1}} ,\:\mathrm{arg}\:{z}_{\mathrm{2}} ,\:\mathrm{arg}\:{z}_{\mathrm{3}} \right)\:\geqslant \\ $$$$\frac{\pi}{\mathrm{6}}\:.\:\mathrm{Where}\:\mathrm{0}\:<\:\mathrm{arg}\left({z}_{\mathrm{1}} \right),\:\mathrm{arg}\left({z}_{\mathrm{2}} \right),\:\mathrm{arg}\left({z}_{\mathrm{3}} \right) \\ $$$$<\:\mathrm{2}\pi. \\ $$

Question Number 21293 Answers: 1 Comments: 0

Let n be an even positive integer such that (n/2) is odd and let α_0 , α_1 , ...., α_(n−1) be the complex roots of unity of order n. Prove that Π_(k=0) ^(n−1) (a + bα_k ^2 ) = (a^(n/2) + b^(n/2) )^2 for any complex numbers a and b.

$$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{an}\:\mathrm{even}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such} \\ $$$$\mathrm{that}\:\frac{{n}}{\mathrm{2}}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{and}\:\mathrm{let}\:\alpha_{\mathrm{0}} ,\:\alpha_{\mathrm{1}} ,\:....,\:\alpha_{{n}−\mathrm{1}} \:\mathrm{be} \\ $$$$\mathrm{the}\:\mathrm{complex}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{of}\:\mathrm{order}\:{n}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({a}\:+\:{b}\alpha_{{k}} ^{\mathrm{2}} \right)\:=\:\left({a}^{\frac{{n}}{\mathrm{2}}} \:+\:{b}^{\frac{{n}}{\mathrm{2}}} \right)^{\mathrm{2}} \\ $$$$\mathrm{for}\:\mathrm{any}\:\mathrm{complex}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}. \\ $$

Question Number 21248 Answers: 0 Comments: 0

The locus of the centre of a circle which touches the given circles ∣z − z_1 ∣ = ∣3 + 4i∣ and ∣z − z_2 ∣ = ∣1 + i(√3)∣ is a hyperbola, then the length of its transverse axis is

$$\mathrm{The}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{which} \\ $$$$\mathrm{touches}\:\mathrm{the}\:\mathrm{given}\:\mathrm{circles}\:\mid{z}\:−\:{z}_{\mathrm{1}} \mid\:= \\ $$$$\mid\mathrm{3}\:+\:\mathrm{4}{i}\mid\:\mathrm{and}\:\mid{z}\:−\:{z}_{\mathrm{2}} \mid\:=\:\mid\mathrm{1}\:+\:{i}\sqrt{\mathrm{3}}\mid\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{hyperbola},\:\mathrm{then}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{its} \\ $$$$\mathrm{transverse}\:\mathrm{axis}\:\mathrm{is} \\ $$

Question Number 21247 Answers: 1 Comments: 0

If [ ] represents the greatest integer function and f(x) = x − [x] then number of real roots of the equation f(x) + f((1/x)) = 1 are infinite. True/False

$$\mathrm{If}\:\left[\:\right]\:\mathrm{represents}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer} \\ $$$$\mathrm{function}\:\mathrm{and}\:{f}\left({x}\right)\:=\:{x}\:−\:\left[{x}\right]\:\mathrm{then} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${f}\left({x}\right)\:+\:{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{1}\:\mathrm{are}\:\mathrm{infinite}. \\ $$$$\boldsymbol{\mathrm{True}}/\boldsymbol{\mathrm{False}} \\ $$

Question Number 21236 Answers: 1 Comments: 0

If (1/a) + (1/(2a)) + (1/(3a)) = (1/(b^2 − 2b)) a and b are positive integers Find minimum value of a + b

$$\mathrm{If}\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{a}}\:+\:\frac{\mathrm{1}}{\mathrm{3}{a}}\:=\:\frac{\mathrm{1}}{{b}^{\mathrm{2}} \:−\:\mathrm{2}{b}} \\ $$$${a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{Find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{a}\:+\:{b} \\ $$

Question Number 21235 Answers: 1 Comments: 0

For any integer k, let α_k = cos (((kπ)/7)) + i sin (((kπ)/7)), where i = (√(−1)). The value of the expression ((Σ_(k=1) ^(12) ∣α_(k+1) − α_k ∣)/(Σ_(k=1) ^3 ∣α_(4k−1) − α_(4k−2) ∣)) is

$$\mathrm{For}\:\mathrm{any}\:\mathrm{integer}\:{k},\:\mathrm{let}\:\alpha_{{k}} \:=\:\mathrm{cos}\:\left(\frac{{k}\pi}{\mathrm{7}}\right)\:+ \\ $$$${i}\:\mathrm{sin}\:\left(\frac{{k}\pi}{\mathrm{7}}\right),\:\mathrm{where}\:{i}\:=\:\sqrt{−\mathrm{1}}.\:\mathrm{The}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{expression}\:\frac{\underset{{k}=\mathrm{1}} {\overset{\mathrm{12}} {\sum}}\mid\alpha_{{k}+\mathrm{1}} \:−\:\alpha_{{k}} \mid}{\underset{{k}=\mathrm{1}} {\overset{\mathrm{3}} {\sum}}\mid\alpha_{\mathrm{4}{k}−\mathrm{1}} \:−\:\alpha_{\mathrm{4}{k}−\mathrm{2}} \mid}\:\mathrm{is} \\ $$

Question Number 21234 Answers: 1 Comments: 0

Let f(x) = ax^2 + bx + c, where a, b, c are real numbers. If the numbers 2a, a + b, and c are all integers, then the number of integral values between 1 and 5 that f(x) can take is

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c},\:\mathrm{where}\:{a},\:{b},\:{c} \\ $$$$\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{2}{a}, \\ $$$${a}\:+\:{b},\:\mathrm{and}\:{c}\:\mathrm{are}\:\mathrm{all}\:\mathrm{integers},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{integral}\:\mathrm{values}\:\mathrm{between}\:\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{5}\:\mathrm{that}\:{f}\left({x}\right)\:\mathrm{can}\:\mathrm{take}\:\mathrm{is} \\ $$

Question Number 21230 Answers: 1 Comments: 0

For each positive integer n, consider the highest common factor h_n of the two numbers n! + 1 and (n + 1)!. For n < 100, find the largest value of h_n .

$$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:{n},\:\mathrm{consider} \\ $$$$\mathrm{the}\:\mathrm{highest}\:\mathrm{common}\:\mathrm{factor}\:{h}_{{n}} \:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{numbers}\:{n}!\:+\:\mathrm{1}\:\mathrm{and}\:\left({n}\:+\:\mathrm{1}\right)!.\:\mathrm{For}\:{n}\:<\:\mathrm{100}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:{h}_{{n}} . \\ $$

Question Number 21219 Answers: 1 Comments: 0

(A) If ∣w∣ = 2, then the set of points z = w − (1/w) is contained in or equal to (B) If ∣w∣ = 1, then the set of points z = w + (1/w) is contained in or equal to Options for both A and B: (p) An ellipse with eccentricity (4/5) (q) The set of points z satisfying Im z = 0 (r) The set of points z satisfying ∣Im z∣ ≤ 1 (s) The set of points z satisfying ∣Re z∣ ≤ 2 (t) The set of points z satisfying ∣z∣ ≤ 3

$$\left(\mathrm{A}\right)\:\mathrm{If}\:\mid{w}\mid\:=\:\mathrm{2},\:\mathrm{then}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points} \\ $$$${z}\:=\:{w}\:−\:\frac{\mathrm{1}}{{w}}\:\mathrm{is}\:\mathrm{contained}\:\mathrm{in}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{If}\:\mid{w}\mid\:=\:\mathrm{1},\:\mathrm{then}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points} \\ $$$${z}\:=\:{w}\:+\:\frac{\mathrm{1}}{{w}}\:\mathrm{is}\:\mathrm{contained}\:\mathrm{in}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{Options}\:\mathrm{for}\:\mathrm{both}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}: \\ $$$$\left(\mathrm{p}\right)\:\mathrm{An}\:\mathrm{ellipse}\:\mathrm{with}\:\mathrm{eccentricity}\:\frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\left(\mathrm{q}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mathrm{Im}\:{z} \\ $$$$=\:\mathrm{0} \\ $$$$\left(\mathrm{r}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mid\mathrm{Im}\:{z}\mid \\ $$$$\leqslant\:\mathrm{1} \\ $$$$\left(\mathrm{s}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mid\mathrm{Re}\:{z}\mid \\ $$$$\leqslant\:\mathrm{2} \\ $$$$\left(\mathrm{t}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mid{z}\mid\:\leqslant\:\mathrm{3} \\ $$

Question Number 21168 Answers: 1 Comments: 0

Let f(x) = ∣x − 1∣ + ∣x − 2∣ + ∣x − 3∣, then find the value of k for which f(x) = k has 1. no solution 2. only one solution 3. two solutions of same sign 4. two solutions of opposite sign

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:\mid{x}\:−\:\mathrm{1}\mid\:+\:\mid{x}\:−\:\mathrm{2}\mid\:+\:\mid{x}\:−\:\mathrm{3}\mid, \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{for}\:\mathrm{which}\:{f}\left({x}\right) \\ $$$$=\:{k}\:\mathrm{has} \\ $$$$\mathrm{1}.\:\mathrm{no}\:\mathrm{solution} \\ $$$$\mathrm{2}.\:\mathrm{only}\:\mathrm{one}\:\mathrm{solution} \\ $$$$\mathrm{3}.\:\mathrm{two}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{same}\:\mathrm{sign} \\ $$$$\mathrm{4}.\:\mathrm{two}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{opposite}\:\mathrm{sign} \\ $$

Question Number 21137 Answers: 1 Comments: 0

If the minimum value of ∣z+1+i∣ + ∣z−1−i∣ + ∣2 − z∣ + ∣3 − z∣ is k then (k − 8) equals

$$\mathrm{If}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mid{z}+\mathrm{1}+{i}\mid\:+\:\mid{z}−\mathrm{1}−{i}\mid\:+\:\mid\mathrm{2}\:−\:{z}\mid\:+\:\mid\mathrm{3}\:−\:{z}\mid\:\mathrm{is} \\ $$$${k}\:\mathrm{then}\:\left({k}\:−\:\mathrm{8}\right)\:\mathrm{equals} \\ $$

Question Number 21111 Answers: 0 Comments: 0

STATEMENT-1 : z_1 ^2 + z_2 ^2 + z_3 ^2 + z_4 ^2 = 0 where z_1 , z_2 , z_3 and z_4 are the fourth roots of unity. and STATEMENT-2 : (1)^(1/4) = (cos0° + i sin0°)^(1/4) .

$$\mathrm{STATEMENT}-\mathrm{1}\::\:{z}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{2}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{3}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{4}} ^{\mathrm{2}} \:= \\ $$$$\mathrm{0}\:\mathrm{where}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{and}\:{z}_{\mathrm{4}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{fourth} \\ $$$$\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}. \\ $$$$\boldsymbol{\mathrm{and}} \\ $$$$\mathrm{STATEMENT}-\mathrm{2}\::\:\left(\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} \:=\:\left(\mathrm{cos0}°\:+\right. \\ $$$$\left.{i}\:\mathrm{sin0}°\right)^{\frac{\mathrm{1}}{\mathrm{4}}} . \\ $$

Question Number 21108 Answers: 1 Comments: 0

Let A, B, C be three sets of complex numbers as defined below A = {z : Im z ≥ 1} B = {z : ∣z − 2 − i∣ = 3} C = {z : Re((1 − i)z) = (√2)}. Let z be any point in A ∩ B ∩ C and let w be any point satisfying ∣w − 2 − i∣ < 3. Then, ∣z∣ − ∣w∣ + 3 lies between (1) −6 and 3 (2) −3 and 6 (3) −6 and 6 (4) −3 and 9

$$\mathrm{Let}\:{A},\:{B},\:{C}\:\mathrm{be}\:\mathrm{three}\:\mathrm{sets}\:\mathrm{of}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\mathrm{as}\:\mathrm{defined}\:\mathrm{below} \\ $$$${A}\:=\:\left\{{z}\::\:\mathrm{Im}\:{z}\:\geqslant\:\mathrm{1}\right\} \\ $$$${B}\:=\:\left\{{z}\::\:\mid{z}\:−\:\mathrm{2}\:−\:{i}\mid\:=\:\mathrm{3}\right\} \\ $$$${C}\:=\:\left\{{z}\::\:\mathrm{Re}\left(\left(\mathrm{1}\:−\:{i}\right){z}\right)\:=\:\sqrt{\mathrm{2}}\right\}. \\ $$$$\mathrm{Let}\:{z}\:\mathrm{be}\:\mathrm{any}\:\mathrm{point}\:\mathrm{in}\:{A}\:\cap\:{B}\:\cap\:{C}\:\mathrm{and}\:\mathrm{let} \\ $$$${w}\:\mathrm{be}\:\mathrm{any}\:\mathrm{point}\:\mathrm{satisfying}\:\mid{w}\:−\:\mathrm{2}\:−\:{i}\mid\:< \\ $$$$\mathrm{3}.\:\mathrm{Then},\:\mid{z}\mid\:−\:\mid{w}\mid\:+\:\mathrm{3}\:\mathrm{lies}\:\mathrm{between} \\ $$$$\left(\mathrm{1}\right)\:−\mathrm{6}\:\mathrm{and}\:\mathrm{3} \\ $$$$\left(\mathrm{2}\right)\:−\mathrm{3}\:\mathrm{and}\:\mathrm{6} \\ $$$$\left(\mathrm{3}\right)\:−\mathrm{6}\:\mathrm{and}\:\mathrm{6} \\ $$$$\left(\mathrm{4}\right)\:−\mathrm{3}\:\mathrm{and}\:\mathrm{9} \\ $$

Question Number 21200 Answers: 1 Comments: 1

Suppose an integer x, a natural number n and a prime number p satisfy the equation 7x^2 − 44x + 12 = p^n . Find the largest value of p.

$$\mathrm{Suppose}\:\mathrm{an}\:\mathrm{integer}\:{x},\:\mathrm{a}\:\mathrm{natural} \\ $$$$\mathrm{number}\:{n}\:\mathrm{and}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:{p} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{7}{x}^{\mathrm{2}} \:−\:\mathrm{44}{x}\:+\:\mathrm{12}\:=\:{p}^{{n}} . \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:{p}. \\ $$

Question Number 21006 Answers: 0 Comments: 0

Let z_1 and z_2 be two distinct complex numbers and let z = (1 − t)z_1 + tz_2 for some real number t with 0 < t < 1. If arg(w) denotes the principal argument of a non-zero complex number w, then (1) ∣z − z_1 ∣ + ∣z − z_2 ∣ = ∣z_1 − z_2 ∣ (2) Arg (z − z_1 ) = Arg (z − z_2 ) (3) determinant (((z − z_1 ),(z^ − z_1 ^ )),((z_2 − z_1 ),(z_2 ^ − z_1 ^ ))) = 0 (4) Arg (z − z_1 ) = Arg (z_2 − z_1 )

$$\mathrm{Let}\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\mathrm{and}\:\mathrm{let}\:{z}\:=\:\left(\mathrm{1}\:−\:{t}\right){z}_{\mathrm{1}} \:+\:{tz}_{\mathrm{2}} \:\mathrm{for} \\ $$$$\mathrm{some}\:\mathrm{real}\:\mathrm{number}\:{t}\:\mathrm{with}\:\mathrm{0}\:<\:{t}\:<\:\mathrm{1}.\:\mathrm{If} \\ $$$$\mathrm{arg}\left({w}\right)\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{principal}\:\mathrm{argument} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{non}-\mathrm{zero}\:\mathrm{complex}\:\mathrm{number}\:{w},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:\mid{z}\:−\:{z}_{\mathrm{1}} \mid\:+\:\mid{z}\:−\:{z}_{\mathrm{2}} \mid\:=\:\mid{z}_{\mathrm{1}} \:−\:{z}_{\mathrm{2}} \mid \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Arg}\:\left({z}\:−\:{z}_{\mathrm{1}} \right)\:=\:\mathrm{Arg}\:\left({z}\:−\:{z}_{\mathrm{2}} \right) \\ $$$$\left(\mathrm{3}\right)\:\begin{vmatrix}{{z}\:−\:{z}_{\mathrm{1}} }&{\bar {{z}}\:−\:\bar {{z}}_{\mathrm{1}} }\\{{z}_{\mathrm{2}} \:−\:{z}_{\mathrm{1}} }&{\bar {{z}}_{\mathrm{2}} \:−\:\bar {{z}}_{\mathrm{1}} }\end{vmatrix}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Arg}\:\left({z}\:−\:{z}_{\mathrm{1}} \right)\:=\:\mathrm{Arg}\:\left({z}_{\mathrm{2}} \:−\:{z}_{\mathrm{1}} \right) \\ $$

Question Number 21005 Answers: 0 Comments: 0

If z_1 = a + ib and z_2 = c + id are complex numbers such that ∣z_1 ∣ = ∣z_2 ∣ = 1 and Re(z_1 z_2 ^ ) = 0, then the pair of complex numbers ω_1 = a + ic and ω_2 = b + id satisfy (1) ∣ω_1 ∣ = 1 (2) ∣ω_2 ∣ = 1 (3) Re(ω_1 ω_2 ^ ) = 0 (4) ∣ω_1 ∣ = 2∣ω_2 ∣

$$\mathrm{If}\:{z}_{\mathrm{1}} \:=\:{a}\:+\:{ib}\:\mathrm{and}\:{z}_{\mathrm{2}} \:=\:{c}\:+\:{id}\:\mathrm{are}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\:\mid{z}_{\mathrm{1}} \mid\:=\:\mid{z}_{\mathrm{2}} \mid\:=\:\mathrm{1}\:\mathrm{and} \\ $$$$\mathrm{Re}\left({z}_{\mathrm{1}} \bar {{z}}_{\mathrm{2}} \right)\:=\:\mathrm{0},\:\mathrm{then}\:\mathrm{the}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\omega_{\mathrm{1}} \:=\:{a}\:+\:{ic}\:\mathrm{and}\:\omega_{\mathrm{2}} \:=\:{b}\:+\:{id} \\ $$$$\mathrm{satisfy} \\ $$$$\left(\mathrm{1}\right)\:\mid\omega_{\mathrm{1}} \mid\:=\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\mid\omega_{\mathrm{2}} \mid\:=\:\mathrm{1} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Re}\left(\omega_{\mathrm{1}} \bar {\omega}_{\mathrm{2}} \right)\:=\:\mathrm{0} \\ $$$$\left(\mathrm{4}\right)\:\mid\omega_{\mathrm{1}} \mid\:=\:\mathrm{2}\mid\omega_{\mathrm{2}} \mid \\ $$

Question Number 20983 Answers: 1 Comments: 0

Find the number of ordered triples (a, b, c) of positive integers such that abc = 108.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ordered}\:\mathrm{triples} \\ $$$$\left({a},\:{b},\:{c}\right)\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that} \\ $$$${abc}\:=\:\mathrm{108}. \\ $$

Question Number 20935 Answers: 1 Comments: 0

If ∣z + ω∣^2 = ∣z∣^2 + ∣ω∣^2 , where z and ω are complex numbers, then (1) (z/ω) is purely real (2) (z/ω) is purely imaginary (3) zω^ + z^ ω = 0 (4) amp((z/ω)) = (π/2)

$$\mathrm{If}\:\mid{z}\:+\:\omega\mid^{\mathrm{2}} \:=\:\mid{z}\mid^{\mathrm{2}} \:+\:\mid\omega\mid^{\mathrm{2}} ,\:\mathrm{where}\:{z}\:\mathrm{and}\:\omega \\ $$$$\mathrm{are}\:\mathrm{complex}\:\mathrm{numbers},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:\frac{{z}}{\omega}\:\mathrm{is}\:\mathrm{purely}\:\mathrm{real} \\ $$$$\left(\mathrm{2}\right)\:\frac{{z}}{\omega}\:\mathrm{is}\:\mathrm{purely}\:\mathrm{imaginary} \\ $$$$\left(\mathrm{3}\right)\:{z}\bar {\omega}\:+\:\bar {{z}}\omega\:=\:\mathrm{0} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{amp}\left(\frac{{z}}{\omega}\right)\:=\:\frac{\pi}{\mathrm{2}} \\ $$

Question Number 20934 Answers: 0 Comments: 1

If z satisfies ∣z − 1∣ < ∣z + 3∣, then ω = 2z + 3 − i satisfies (1) ∣ω − 5 − i∣ < ∣ω + 3 + i∣ (2) ∣ω − 5∣ < ∣ω + 3∣ (3) Im (iω) > 1 (4) ∣arg(ω − 1)∣ < (π/2)

$$\mathrm{If}\:{z}\:\mathrm{satisfies}\:\mid{z}\:−\:\mathrm{1}\mid\:<\:\mid{z}\:+\:\mathrm{3}\mid,\:\mathrm{then}\:\omega\:= \\ $$$$\mathrm{2}{z}\:+\:\mathrm{3}\:−\:{i}\:\mathrm{satisfies} \\ $$$$\left(\mathrm{1}\right)\:\mid\omega\:−\:\mathrm{5}\:−\:{i}\mid\:<\:\mid\omega\:+\:\mathrm{3}\:+\:{i}\mid \\ $$$$\left(\mathrm{2}\right)\:\mid\omega\:−\:\mathrm{5}\mid\:<\:\mid\omega\:+\:\mathrm{3}\mid \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Im}\:\left({i}\omega\right)\:>\:\mathrm{1} \\ $$$$\left(\mathrm{4}\right)\:\mid\mathrm{arg}\left(\omega\:−\:\mathrm{1}\right)\mid\:<\:\frac{\pi}{\mathrm{2}} \\ $$

Question Number 20933 Answers: 1 Comments: 0

If z is a complex number satisfying z + z^(−1) = 1, then z^n + z^(−n) , n ∈ N, has the value (1) 2(−1)^n , when n is a multiple of 3 (2) (−1)^(n−1) , when n is not a multiple of 3 (3) (−1)^(n+1) , when n is a multiple of 3 (4) 0 when n is not a multiple of 3

$$\mathrm{If}\:{z}\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number}\:\mathrm{satisfying} \\ $$$${z}\:+\:{z}^{−\mathrm{1}} \:=\:\mathrm{1},\:\mathrm{then}\:{z}^{{n}} \:+\:{z}^{−{n}} ,\:{n}\:\in\:{N},\:\mathrm{has} \\ $$$$\mathrm{the}\:\mathrm{value} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{2}\left(−\mathrm{1}\right)^{{n}} ,\:\mathrm{when}\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3} \\ $$$$\left(\mathrm{2}\right)\:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} ,\:\mathrm{when}\:{n}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of} \\ $$$$\mathrm{3} \\ $$$$\left(\mathrm{3}\right)\:\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} ,\:\mathrm{when}\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{0}\:\mathrm{when}\:{n}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3} \\ $$

Question Number 20932 Answers: 0 Comments: 0

If a, b, c are real numbers and z is a complex number such that, a^2 + b^2 + c^2 = 1 and b + ic = (1 + a)z, then ((1 + iz)/(1 − iz)) equals. (1) ((b − ic)/(1 − ia)) (2) ((a + ib)/(1 + c)) (3) ((1 − c)/(a − ib)) (4) ((1 + a)/(b + ic))

$$\mathrm{If}\:{a},\:{b},\:{c}\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{and}\:{z}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{complex}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that},\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \\ $$$$=\:\mathrm{1}\:\mathrm{and}\:{b}\:+\:{ic}\:=\:\left(\mathrm{1}\:+\:{a}\right){z},\:\mathrm{then}\:\frac{\mathrm{1}\:+\:{iz}}{\mathrm{1}\:−\:{iz}} \\ $$$$\mathrm{equals}. \\ $$$$\left(\mathrm{1}\right)\:\frac{{b}\:−\:{ic}}{\mathrm{1}\:−\:{ia}} \\ $$$$\left(\mathrm{2}\right)\:\frac{{a}\:+\:{ib}}{\mathrm{1}\:+\:{c}} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{1}\:−\:{c}}{{a}\:−\:{ib}} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{1}\:+\:{a}}{{b}\:+\:{ic}} \\ $$

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