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AlgebraQuestion and Answers: Page 348

Question Number 19898 Answers: 2 Comments: 0

Simplify : i log (((x − i)/(x + i))).

$$\mathrm{Simplify}\::\:{i}\:\mathrm{log}\:\left(\frac{{x}\:−\:{i}}{{x}\:+\:{i}}\right). \\ $$

Question Number 19875 Answers: 1 Comments: 1

Question Number 19799 Answers: 0 Comments: 2

For a natural number b, let N(b) denote the number of natural numbers a for which the equation x^2 + ax + b = 0 has integer roots. What is the smallest value of b for which N(b) = 20?

$$\mathrm{For}\:\mathrm{a}\:\mathrm{natural}\:\mathrm{number}\:{b},\:\mathrm{let}\:{N}\left({b}\right)\:\mathrm{denote} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{numbers}\:{a}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} \:+\:{ax}\:+\:{b}\:=\:\mathrm{0}\:\mathrm{has} \\ $$$$\mathrm{integer}\:\mathrm{roots}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{smallest} \\ $$$$\mathrm{value}\:\mathrm{of}\:{b}\:\mathrm{for}\:\mathrm{which}\:{N}\left({b}\right)\:=\:\mathrm{20}? \\ $$

Question Number 19741 Answers: 0 Comments: 1

If z = x + iy and arg(((z − 2)/(z + 2))) = (π/6), then find the locus of z.

$$\mathrm{If}\:{z}\:=\:{x}\:+\:{iy}\:\mathrm{and}\:\mathrm{arg}\left(\frac{{z}\:−\:\mathrm{2}}{{z}\:+\:\mathrm{2}}\right)\:=\:\frac{\pi}{\mathrm{6}},\:\mathrm{then} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}. \\ $$

Question Number 19740 Answers: 1 Comments: 0

If ∣z^2 − 1∣ = ∣z∣^2 + 1, then z lies on

$$\mathrm{If}\:\mid{z}^{\mathrm{2}} \:−\:\mathrm{1}\mid\:=\:\mid{z}\mid^{\mathrm{2}} \:+\:\mathrm{1},\:\mathrm{then}\:{z}\:\mathrm{lies}\:\mathrm{on} \\ $$

Question Number 19739 Answers: 1 Comments: 0

If z = x + iy is a complex number satisfying ∣z + (i/2)∣^2 = ∣z − (i/2)∣^2 , then the locus of z is

$$\mathrm{If}\:{z}\:=\:{x}\:+\:{iy}\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number} \\ $$$$\mathrm{satisfying}\:\mid{z}\:+\:\frac{{i}}{\mathrm{2}}\mid^{\mathrm{2}} \:=\:\mid{z}\:−\:\frac{{i}}{\mathrm{2}}\mid^{\mathrm{2}} ,\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{is} \\ $$

Question Number 19738 Answers: 1 Comments: 1

Locus of the point z satisfying the equation ∣iz − 1∣ + ∣z − i∣ = 2 is

$$\mathrm{Locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:{z}\:\mathrm{satisfying}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mid{iz}\:−\:\mathrm{1}\mid\:+\:\mid{z}\:−\:{i}\mid\:=\:\mathrm{2}\:\mathrm{is} \\ $$

Question Number 19735 Answers: 1 Comments: 0

If z = λ + 3 + i(√(5 − λ^2 )), then the locus of z is a

$$\mathrm{If}\:{z}\:=\:\lambda\:+\:\mathrm{3}\:+\:{i}\sqrt{\mathrm{5}\:−\:\lambda^{\mathrm{2}} },\:\mathrm{then}\:\mathrm{the}\:\mathrm{locus} \\ $$$$\mathrm{of}\:{z}\:\mathrm{is}\:\mathrm{a} \\ $$

Question Number 19733 Answers: 1 Comments: 0

If ∣z + 1∣ = (√2)∣z − 1∣, then the locus described by the point z in the argand diagram is a

$$\mathrm{If}\:\mid{z}\:+\:\mathrm{1}\mid\:=\:\sqrt{\mathrm{2}}\mid{z}\:−\:\mathrm{1}\mid,\:\mathrm{then}\:\mathrm{the}\:\mathrm{locus} \\ $$$$\mathrm{described}\:\mathrm{by}\:\mathrm{the}\:\mathrm{point}\:{z}\:\mathrm{in}\:\mathrm{the}\:\mathrm{argand} \\ $$$$\mathrm{diagram}\:\mathrm{is}\:\mathrm{a} \\ $$

Question Number 19734 Answers: 1 Comments: 0

If the imaginary part of ((2z + 1)/(iz + 1)) is −2, then the locus of the point representing z in the complex plane is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{imaginary}\:\mathrm{part}\:\mathrm{of}\:\frac{\mathrm{2}{z}\:+\:\mathrm{1}}{{iz}\:+\:\mathrm{1}}\:\mathrm{is}\:−\mathrm{2}, \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:\mathrm{representing} \\ $$$${z}\:\mathrm{in}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{plane}\:\mathrm{is} \\ $$

Question Number 19732 Answers: 1 Comments: 0

The locus of z given by ∣((z − 1)/(z − i))∣ = 1 is

$$\mathrm{The}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{given}\:\mathrm{by}\:\mid\frac{{z}\:−\:\mathrm{1}}{{z}\:−\:{i}}\mid\:=\:\mathrm{1}\:\mathrm{is} \\ $$

Question Number 19730 Answers: 1 Comments: 0

If z = x + iy and ∣z − 2i∣ = 1, then (1) z lies on x-axis (2) z lies on y-axis (3) z lies on a circle (4) None of these

$$\mathrm{If}\:{z}\:=\:{x}\:+\:{iy}\:\mathrm{and}\:\mid{z}\:−\:\mathrm{2}{i}\mid\:=\:\mathrm{1},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{z}\:\mathrm{lies}\:\mathrm{on}\:{x}-\mathrm{axis} \\ $$$$\left(\mathrm{2}\right)\:{z}\:\mathrm{lies}\:\mathrm{on}\:{y}-\mathrm{axis} \\ $$$$\left(\mathrm{3}\right)\:{z}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{a}\:\mathrm{circle} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{None}\:\mathrm{of}\:\mathrm{these} \\ $$

Question Number 19729 Answers: 1 Comments: 0

2x + 9y^2 = 4 2x^2 − 45y^2 + xy = 0 Find the value of xy

$$\mathrm{2}{x}\:+\:\mathrm{9}{y}^{\mathrm{2}} \:=\:\mathrm{4} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} \:−\:\mathrm{45}{y}^{\mathrm{2}} \:+\:{xy}\:=\:\mathrm{0} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{xy} \\ $$

Question Number 19700 Answers: 1 Comments: 0

What is the maximum possible value of k for which 2013 can be written as a sum of k consecutive positive integers?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of} \\ $$$${k}\:\mathrm{for}\:\mathrm{which}\:\mathrm{2013}\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{a} \\ $$$$\mathrm{sum}\:\mathrm{of}\:{k}\:\mathrm{consecutive}\:\mathrm{positive}\:\mathrm{integers}? \\ $$

Question Number 19698 Answers: 1 Comments: 0

Let f(x) = x^3 − 3x + b and g(x) = x^2 + bx − 3, where b is a real number. What is the sum of all possible values of b for which the equations f(x) = 0 and g(x) = 0 have a common root?

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:−\:\mathrm{3}{x}\:+\:{b}\:\mathrm{and}\:{g}\left({x}\right)\:=\:{x}^{\mathrm{2}} \:+ \\ $$$${bx}\:−\:\mathrm{3},\:\mathrm{where}\:{b}\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{What} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:{b}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{equations}\:{f}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{and}\:{g}\left({x}\right) \\ $$$$=\:\mathrm{0}\:\mathrm{have}\:\mathrm{a}\:\mathrm{common}\:\mathrm{root}? \\ $$

Question Number 22315 Answers: 0 Comments: 0

Prove that the greatest coefficient in the expansion of (x_1 +x_2 +x_3 +...+x_k )^n = ((n!)/((q!)^(k−r) [(q+1)!]^r )) , where n = qk + r, 0 ≤ r ≤ k − 1

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{coefficient}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +...+{x}_{{k}} \right)^{{n}} \\ $$$$=\:\frac{{n}!}{\left({q}!\right)^{{k}−{r}} \left[\left({q}+\mathrm{1}\right)!\right]^{{r}} }\:,\:\mathrm{where}\:{n}\:=\:{qk}\:+\:{r}, \\ $$$$\mathrm{0}\:\leqslant\:{r}\:\leqslant\:{k}\:−\:\mathrm{1} \\ $$

Question Number 19696 Answers: 1 Comments: 0

Let m be the smallest odd positive integer for which 1 + 2 + ... + m is a square of an integer and let n be the smallest even positive integer for which 1 + 2 + ... + n is a square of an integer. What is the value of m + n?

$$\mathrm{Let}\:{m}\:\mathrm{be}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{odd}\:\mathrm{positive} \\ $$$$\mathrm{integer}\:\mathrm{for}\:\mathrm{which}\:\mathrm{1}\:+\:\mathrm{2}\:+\:...\:+\:{m}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{square}\:\mathrm{of}\:\mathrm{an}\:\mathrm{integer}\:\mathrm{and}\:\mathrm{let}\:{n}\:\mathrm{be}\:\mathrm{the} \\ $$$$\mathrm{smallest}\:\mathrm{even}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{1}\:+\:\mathrm{2}\:+\:...\:+\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{square}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{integer}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{m}\:+\:{n}? \\ $$

Question Number 22313 Answers: 1 Comments: 2

Question Number 19690 Answers: 1 Comments: 0

If ∣z − (4/z)∣ = 2, then find the maximum value of ∣z∣.

$$\mathrm{If}\:\mid{z}\:−\:\frac{\mathrm{4}}{{z}}\mid\:=\:\mathrm{2},\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mid{z}\mid. \\ $$

Question Number 19786 Answers: 1 Comments: 0

For natural numbers x and y, let (x, y) denote the greatest common divisor of x and y. How many pairs of natural numbers x and y with x ≤ y satisfy the equation xy = x + y + (x, y)?

$$\mathrm{For}\:\mathrm{natural}\:\mathrm{numbers}\:{x}\:\mathrm{and}\:{y},\:\mathrm{let}\:\left({x},\:{y}\right) \\ $$$$\mathrm{denote}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{common}\:\mathrm{divisor}\:\mathrm{of} \\ $$$${x}\:\mathrm{and}\:{y}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{natural} \\ $$$$\mathrm{numbers}\:{x}\:\mathrm{and}\:{y}\:\mathrm{with}\:{x}\:\leqslant\:{y}\:\mathrm{satisfy}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{xy}\:=\:{x}\:+\:{y}\:+\:\left({x},\:{y}\right)? \\ $$

Question Number 19785 Answers: 1 Comments: 0

If x^((x^4 )) = 4, what is the value of x^((x^2 )) + x^((x^8 )) ?

$$\mathrm{If}\:{x}^{\left({x}^{\mathrm{4}} \right)} \:=\:\mathrm{4},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${x}^{\left({x}^{\mathrm{2}} \right)} \:+\:{x}^{\left({x}^{\mathrm{8}} \right)} ? \\ $$

Question Number 19688 Answers: 1 Comments: 0

The vertices of a square are z_1 , z_2 , z_3 and z_4 taken in the anticlockwise order, then z_3 = (1) −iz_1 + (1 + i)z_2 (2) iz_1 + (1 + i)z_2 (3) z_1 + (1 + i)z_2 (4) (1 + i)z_1 + z_2

$$\mathrm{The}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\:\mathrm{are}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \\ $$$$\mathrm{and}\:{z}_{\mathrm{4}} \:\mathrm{taken}\:\mathrm{in}\:\mathrm{the}\:\mathrm{anticlockwise}\:\mathrm{order}, \\ $$$$\mathrm{then}\:{z}_{\mathrm{3}} \:= \\ $$$$\left(\mathrm{1}\right)\:−{iz}_{\mathrm{1}} \:+\:\left(\mathrm{1}\:+\:{i}\right){z}_{\mathrm{2}} \\ $$$$\left(\mathrm{2}\right)\:{iz}_{\mathrm{1}} \:+\:\left(\mathrm{1}\:+\:{i}\right){z}_{\mathrm{2}} \\ $$$$\left(\mathrm{3}\right)\:{z}_{\mathrm{1}} \:+\:\left(\mathrm{1}\:+\:{i}\right){z}_{\mathrm{2}} \\ $$$$\left(\mathrm{4}\right)\:\left(\mathrm{1}\:+\:{i}\right){z}_{\mathrm{1}} \:+\:{z}_{\mathrm{2}} \\ $$

Question Number 19687 Answers: 1 Comments: 0

Let z_1 , z_2 , z_3 be three vertices of an equilateral triangle circumscribing the circle ∣z∣ = (1/2). If z_1 = (1/2) + (((√3)i)/2) and z_1 , z_2 , z_3 are in anticlockwise sense then z_2 is

$$\mathrm{Let}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{three}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{circumscribing}\:\mathrm{the} \\ $$$$\mathrm{circle}\:\mid{z}\mid\:=\:\frac{\mathrm{1}}{\mathrm{2}}.\:\mathrm{If}\:{z}_{\mathrm{1}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\sqrt{\mathrm{3}}{i}}{\mathrm{2}}\:\mathrm{and}\:{z}_{\mathrm{1}} , \\ $$$${z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{in}\:\mathrm{anticlockwise}\:\mathrm{sense}\:\mathrm{then}\:{z}_{\mathrm{2}} \:\mathrm{is} \\ $$

Question Number 19646 Answers: 0 Comments: 1

Find the sum of all possible digits that comes at ten′s place for 3^n where n is any natural number.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{digits}\:\mathrm{that} \\ $$$$\mathrm{comes}\:\mathrm{at}\:\mathrm{ten}'\mathrm{s}\:\mathrm{place}\:\mathrm{for}\:\mathrm{3}^{{n}} \:\mathrm{where}\:{n}\:\mathrm{is} \\ $$$$\mathrm{any}\:\mathrm{natural}\:\mathrm{number}. \\ $$

Question Number 19638 Answers: 1 Comments: 0

Let P(x) is a polynomial such that P(1) = 1, P(2) = 2, P(3) = 3, and P(4) = 5. Find the value of P(6).

$$\mathrm{Let}\:{P}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{such}\:\mathrm{that} \\ $$$${P}\left(\mathrm{1}\right)\:=\:\mathrm{1},\:{P}\left(\mathrm{2}\right)\:=\:\mathrm{2},\:{P}\left(\mathrm{3}\right)\:=\:\mathrm{3},\:\mathrm{and} \\ $$$${P}\left(\mathrm{4}\right)\:=\:\mathrm{5}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{P}\left(\mathrm{6}\right). \\ $$

Question Number 19643 Answers: 1 Comments: 0

Find the real solution of the equation (√(17 + 8x − 2x^2 )) + (√(4 + 12x − 3x^2 )) = x^2 − 4x + 13.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{real}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\sqrt{\mathrm{17}\:+\:\mathrm{8}{x}\:−\:\mathrm{2}{x}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{4}\:+\:\mathrm{12}{x}\:−\:\mathrm{3}{x}^{\mathrm{2}} }\:=\:{x}^{\mathrm{2}} \\ $$$$−\:\mathrm{4}{x}\:+\:\mathrm{13}. \\ $$

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