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AlgebraQuestion and Answers: Page 352

Question Number 20684 Answers: 1 Comments: 0

If the equation x^2 + β^2 = 1 − 2βx and x^2 + α^2 = 1 − 2αx have one and only one root in common, then ∣α − β∣ is equal to

$${If}\:{the}\:{equation}\:{x}^{\mathrm{2}} \:+\:\beta^{\mathrm{2}} \:=\:\mathrm{1}\:−\:\mathrm{2}\beta{x}\:{and} \\ $$$${x}^{\mathrm{2}} \:+\:\alpha^{\mathrm{2}} \:=\:\mathrm{1}\:−\:\mathrm{2}\alpha{x}\:{have}\:{one}\:{and}\:{only} \\ $$$${one}\:{root}\:{in}\:{common},\:{then}\:\mid\alpha\:−\:\beta\mid\:{is} \\ $$$${equal}\:{to} \\ $$

Question Number 20671 Answers: 1 Comments: 0

The total number of positive integral solution(s) of the inequation ((x^2 (3x − 4)^3 (x − 2)^4 )/((x − 5)^5 (2x − 7)^6 )) ≤ 0 is/are

$${The}\:{total}\:{number}\:{of}\:{positive}\:{integral} \\ $$$${solution}\left({s}\right)\:{of}\:{the}\:{inequation} \\ $$$$\frac{{x}^{\mathrm{2}} \left(\mathrm{3}{x}\:−\:\mathrm{4}\right)^{\mathrm{3}} \left({x}\:−\:\mathrm{2}\right)^{\mathrm{4}} }{\left({x}\:−\:\mathrm{5}\right)^{\mathrm{5}} \left(\mathrm{2}{x}\:−\:\mathrm{7}\right)^{\mathrm{6}} }\:\leqslant\:\mathrm{0}\:{is}/{are} \\ $$

Question Number 20670 Answers: 1 Comments: 0

For the equation 3x^2 + px + 3 = 0, find the value(s) of p if one root is (i) square of the other (ii) fourth power of the other.

$${For}\:{the}\:{equation}\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{px}\:+\:\mathrm{3}\:=\:\mathrm{0}, \\ $$$${find}\:{the}\:{value}\left({s}\right)\:{of}\:{p}\:{if}\:{one}\:{root}\:{is} \\ $$$$\left({i}\right)\:{square}\:{of}\:{the}\:{other} \\ $$$$\left({ii}\right)\:{fourth}\:{power}\:{of}\:{the}\:{other}. \\ $$

Question Number 20652 Answers: 0 Comments: 4

Suppose x is a positive real number such that {x}, [x] and x are in a geometric progression. Find the least positive integer n such that x^n > 100. (Here [x] denotes the integer part of x and {x} = x − [x].)

$$\mathrm{Suppose}\:{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number} \\ $$$$\mathrm{such}\:\mathrm{that}\:\left\{{x}\right\},\:\left[{x}\right]\:\mathrm{and}\:{x}\:\mathrm{are}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{geometric}\:\mathrm{progression}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{least} \\ $$$$\mathrm{positive}\:\mathrm{integer}\:{n}\:\mathrm{such}\:\mathrm{that}\:{x}^{{n}} \:>\:\mathrm{100}. \\ $$$$\left(\mathrm{Here}\:\left[{x}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{part}\:\mathrm{of}\:{x}\right. \\ $$$$\left.\mathrm{and}\:\left\{{x}\right\}\:=\:{x}\:−\:\left[{x}\right].\right) \\ $$

Question Number 20726 Answers: 1 Comments: 0

Five distinct 2-digit numbers are in a geometric progression. Find the middle term.

$$\mathrm{Five}\:\mathrm{distinct}\:\mathrm{2}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{are}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{geometric}\:\mathrm{progression}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{middle} \\ $$$$\mathrm{term}. \\ $$

Question Number 20619 Answers: 1 Comments: 0

Solve the equation z^(n−1) = z^ (n ∈ N)

$${Solve}\:{the}\:{equation}\:{z}^{{n}−\mathrm{1}} \:=\:\bar {{z}}\:\left({n}\:\in\:{N}\right) \\ $$

Question Number 20631 Answers: 2 Comments: 0

Solve the inequality (x + 3)^5 − (x − 1)^5 ≥ 244.

$${Solve}\:{the}\:{inequality} \\ $$$$\left({x}\:+\:\mathrm{3}\right)^{\mathrm{5}} \:−\:\left({x}\:−\:\mathrm{1}\right)^{\mathrm{5}} \:\geqslant\:\mathrm{244}. \\ $$

Question Number 20552 Answers: 1 Comments: 0

The roots of the equation (3−x)^4 +(2−x)^4 =(5−2x)^4 are (a) all real (b) all imaginary (c) two real and two imaginary (d)none of the above .

$${The}\:{roots}\:{of}\:{the}\:{equation}\: \\ $$$$\:\left(\mathrm{3}−{x}\right)^{\mathrm{4}} +\left(\mathrm{2}−{x}\right)^{\mathrm{4}} =\left(\mathrm{5}−\mathrm{2}{x}\right)^{\mathrm{4}} \:{are} \\ $$$$\left({a}\right)\:{all}\:{real}\:\:\:\:\left({b}\right)\:{all}\:{imaginary} \\ $$$$\left({c}\right)\:{two}\:{real}\:{and}\:{two}\:{imaginary} \\ $$$$\left({d}\right){none}\:{of}\:{the}\:{above}\:. \\ $$

Question Number 20550 Answers: 1 Comments: 0

Find the equation of circle in complex form which touches iz + z^ + 1 + i = 0 and for which the lines (1 − i)z = (1 + i)z^ and (1 + i)z + (i − 1)z^ − 4i = 0 are normals.

$${Find}\:{the}\:{equation}\:{of}\:{circle}\:{in}\:{complex} \\ $$$${form}\:{which}\:{touches}\:{iz}\:+\:\bar {{z}}\:+\:\mathrm{1}\:+\:{i}\:=\:\mathrm{0} \\ $$$${and}\:{for}\:{which}\:{the}\:{lines}\:\left(\mathrm{1}\:−\:{i}\right){z}\:= \\ $$$$\left(\mathrm{1}\:+\:{i}\right)\bar {{z}}\:{and}\:\left(\mathrm{1}\:+\:{i}\right){z}\:+\:\left({i}\:−\:\mathrm{1}\right)\bar {{z}}\:−\:\mathrm{4}{i}\:=\:\mathrm{0} \\ $$$${are}\:{normals}. \\ $$

Question Number 20549 Answers: 1 Comments: 1

Show that if z_1 z_2 + z_3 z_4 = 0 and z_1 + z_2 = 0, then the complex numbers z_1 , z_2 , z_3 , z_4 are concyclic.

$${Show}\:{that}\:{if}\:{z}_{\mathrm{1}} {z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} {z}_{\mathrm{4}} \:=\:\mathrm{0}\:{and}\:{z}_{\mathrm{1}} \:+ \\ $$$${z}_{\mathrm{2}} \:=\:\mathrm{0},\:{then}\:{the}\:{complex}\:{numbers}\:{z}_{\mathrm{1}} , \\ $$$${z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} ,\:{z}_{\mathrm{4}} \:{are}\:{concyclic}. \\ $$

Question Number 20551 Answers: 1 Comments: 8

Find the minimum value of ∣a + bω + cω^2 ∣, where a, b and c are all not equal integers and ω(≠1) is a cube root of unity.

$${Find}\:{the}\:{minimum}\:{value}\:{of} \\ $$$$\mid{a}\:+\:{b}\omega\:+\:{c}\omega^{\mathrm{2}} \mid,\:{where}\:{a},\:{b}\:{and}\:{c}\:{are}\:{all} \\ $$$${not}\:{equal}\:{integers}\:{and}\:\omega\left(\neq\mathrm{1}\right)\:{is}\:{a}\:{cube} \\ $$$${root}\:{of}\:{unity}. \\ $$

Question Number 20488 Answers: 1 Comments: 1

Question Number 20430 Answers: 2 Comments: 1

Let a, b and c be such that a + b + c = 0 and P = (a^2 /(2a^2 + bc)) + (b^2 /(2b^2 + ca)) + (c^2 /(2c^2 + ab)) is defined. What is the value of P?

$$\mathrm{Let}\:{a},\:{b}\:\mathrm{and}\:{c}\:\mathrm{be}\:\mathrm{such}\:\mathrm{that}\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0} \\ $$$$\mathrm{and} \\ $$$${P}\:=\:\frac{{a}^{\mathrm{2}} }{\mathrm{2}{a}^{\mathrm{2}} \:+\:{bc}}\:+\:\frac{{b}^{\mathrm{2}} }{\mathrm{2}{b}^{\mathrm{2}} \:+\:{ca}}\:+\:\frac{{c}^{\mathrm{2}} }{\mathrm{2}{c}^{\mathrm{2}} \:+\:{ab}} \\ $$$$\mathrm{is}\:\mathrm{defined}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{P}? \\ $$

Question Number 20372 Answers: 2 Comments: 0

The two roots of an equation x^3 − 9x^2 + 14x + 24 = 0 are in the ratio 3 : 2. Find the roots.

$$\mathrm{The}\:\mathrm{two}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{an}\:\mathrm{equation}\:{x}^{\mathrm{3}} \:−\:\mathrm{9}{x}^{\mathrm{2}} \\ $$$$+\:\mathrm{14}{x}\:+\:\mathrm{24}\:=\:\mathrm{0}\:\mathrm{are}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{3}\::\:\mathrm{2}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{roots}. \\ $$

Question Number 20367 Answers: 1 Comments: 0

Let f(x) = x^3 + 3x^2 + 9x + 6sinx, then find the number of real roots of the equation (1/(x − f(1))) + (2/(x − f(2))) + (3/(x − f(3))) = 0.

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{9}{x}\:+\:\mathrm{6sin}{x},\:\mathrm{then} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation} \\ $$$$\frac{\mathrm{1}}{{x}\:−\:{f}\left(\mathrm{1}\right)}\:+\:\frac{\mathrm{2}}{{x}\:−\:{f}\left(\mathrm{2}\right)}\:+\:\frac{\mathrm{3}}{{x}\:−\:{f}\left(\mathrm{3}\right)}\:=\:\mathrm{0}. \\ $$

Question Number 20368 Answers: 0 Comments: 2

If α is a real root of 2x^3 − 3x^2 + 6x + 6 = 0, then find [α] where [∙] denotes the greatest integer function.

$$\mathrm{If}\:\alpha\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{root}\:\mathrm{of}\:\mathrm{2}{x}^{\mathrm{3}} \:−\:\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{6}{x}\:+\:\mathrm{6}\:=\:\mathrm{0}, \\ $$$$\mathrm{then}\:\mathrm{find}\:\left[\alpha\right]\:\mathrm{where}\:\left[\centerdot\right]\:\mathrm{denotes}\:\mathrm{the} \\ $$$$\mathrm{greatest}\:\mathrm{integer}\:\mathrm{function}. \\ $$

Question Number 20308 Answers: 1 Comments: 0

For what value of k, (x + y + z)^2 + k(x^2 + y^2 + z^2 ) can be resolved into linear rational factors?

$$\mathrm{For}\:\mathrm{what}\:\mathrm{value}\:\mathrm{of}\:{k},\:\left({x}\:+\:{y}\:+\:{z}\right)^{\mathrm{2}} \:+ \\ $$$${k}\left({x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)\:\mathrm{can}\:\mathrm{be}\:\mathrm{resolved}\:\mathrm{into} \\ $$$$\mathrm{linear}\:\mathrm{rational}\:\mathrm{factors}? \\ $$

Question Number 20307 Answers: 2 Comments: 0

Show that a(b − c)x^2 + b(c − a)xy + c(a − b)y^2 will be a perfect square if a, b, c are in H.P.

$$\mathrm{Show}\:\mathrm{that}\:{a}\left({b}\:−\:{c}\right){x}^{\mathrm{2}} \:+\:{b}\left({c}\:−\:{a}\right){xy}\:+ \\ $$$${c}\left({a}\:−\:{b}\right){y}^{\mathrm{2}} \:\mathrm{will}\:\mathrm{be}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}\:\mathrm{if}\:{a}, \\ $$$${b},\:{c}\:\mathrm{are}\:\mathrm{in}\:\mathrm{H}.\mathrm{P}. \\ $$

Question Number 20298 Answers: 1 Comments: 0

Question Number 20297 Answers: 1 Comments: 0

Prove that the expression ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 can be resolved into two linear rational factors if Δ = abc + 2fgh − af^2 − bg^2 − ch^2 = 0

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{expression}\:{ax}^{\mathrm{2}} \:+\:\mathrm{2}{hxy} \\ $$$$+\:{by}^{\mathrm{2}} \:+\:\mathrm{2}{gx}\:+\:\mathrm{2}{fy}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{resolved}\:\mathrm{into}\:\mathrm{two}\:\mathrm{linear}\:\mathrm{rational}\:\mathrm{factors} \\ $$$$\mathrm{if}\:\Delta\:=\:{abc}\:+\:\mathrm{2}{fgh}\:−\:{af}^{\mathrm{2}} \:−\:{bg}^{\mathrm{2}} \:−\:{ch}^{\mathrm{2}} \:=\:\mathrm{0} \\ $$

Question Number 20296 Answers: 2 Comments: 0

If (m_r , (1/m_r )) ; r = 1, 2, 3, 4 be four pairs of values of x and y satisfy the equation x^2 + y^2 + 2gx + 2fy + c = 0, then prove that m_1 .m_2 .m_3 .m_4 = 1.

$$\mathrm{If}\:\left({m}_{{r}} \:,\:\frac{\mathrm{1}}{{m}_{{r}} }\right)\:;\:{r}\:=\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4}\:\mathrm{be}\:\mathrm{four}\:\mathrm{pairs} \\ $$$$\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{and}\:{y}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation} \\ $$$${x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:\mathrm{2}{gx}\:+\:\mathrm{2}{fy}\:+\:{c}\:=\:\mathrm{0},\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:{m}_{\mathrm{1}} .{m}_{\mathrm{2}} .{m}_{\mathrm{3}} .{m}_{\mathrm{4}} \:=\:\mathrm{1}. \\ $$

Question Number 20259 Answers: 1 Comments: 0

Find the number of real roots of the equation f(x) = x^3 + 2x^2 + 2x + 1 = 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{2}{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$

Question Number 20118 Answers: 1 Comments: 0

The quadratic equations x^2 − 6x + a = 0 and x^2 − cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then, find the common root.

$$\mathrm{The}\:\mathrm{quadratic}\:\mathrm{equations}\:{x}^{\mathrm{2}} \:−\:\mathrm{6}{x}\:+\:{a}\:=\:\mathrm{0} \\ $$$$\mathrm{and}\:{x}^{\mathrm{2}} \:−\:{cx}\:+\:\mathrm{6}\:=\:\mathrm{0}\:\mathrm{have}\:\mathrm{one}\:\mathrm{root}\:\mathrm{in} \\ $$$$\mathrm{common}.\:\mathrm{The}\:\mathrm{other}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first} \\ $$$$\mathrm{and}\:\mathrm{second}\:\mathrm{equations}\:\mathrm{are}\:\mathrm{integers}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{ratio}\:\mathrm{4}\::\:\mathrm{3}.\:\mathrm{Then},\:\mathrm{find}\:\mathrm{the}\:\mathrm{common} \\ $$$$\mathrm{root}. \\ $$

Question Number 20116 Answers: 1 Comments: 0

If a and b (≠ 0) are the roots of the equation x^2 + ax + b = 0, then find the least value of x^2 + ax + b (x ∈ R).

$$\mathrm{If}\:{a}\:\mathrm{and}\:{b}\:\left(\neq\:\mathrm{0}\right)\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{x}^{\mathrm{2}} \:+\:{ax}\:+\:{b}\:=\:\mathrm{0},\:\mathrm{then}\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{least}\:\mathrm{value}\:\mathrm{of}\:{x}^{\mathrm{2}} \:+\:{ax}\:+\:{b}\:\left({x}\:\in\:{R}\right). \\ $$

Question Number 20115 Answers: 1 Comments: 0

The value of a for which the equation (1 − a^2 )x^2 + 2ax − 1 = 0 has roots belonging to (0, 1) is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:{a}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\left(\mathrm{1}\:−\:{a}^{\mathrm{2}} \right){x}^{\mathrm{2}} \:+\:\mathrm{2}{ax}\:−\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{has}\:\mathrm{roots} \\ $$$$\mathrm{belonging}\:\mathrm{to}\:\left(\mathrm{0},\:\mathrm{1}\right)\:\mathrm{is} \\ $$

Question Number 20054 Answers: 1 Comments: 0

If α and β (α < β) are the roots of the equation x^2 + bx + c = 0, where c < 0 < b, then (1) 0 < α < β (2) α < 0 < β < ∣α∣ (3) α < β < 0 (4) α < 0 < ∣α∣ < β

$$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\left(\alpha\:<\:\beta\right)\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{x}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:=\:\mathrm{0},\:\mathrm{where} \\ $$$${c}\:<\:\mathrm{0}\:<\:{b},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{0}\:<\:\alpha\:<\:\beta \\ $$$$\left(\mathrm{2}\right)\:\alpha\:<\:\mathrm{0}\:<\:\beta\:<\:\mid\alpha\mid \\ $$$$\left(\mathrm{3}\right)\:\alpha\:<\:\beta\:<\:\mathrm{0} \\ $$$$\left(\mathrm{4}\right)\:\alpha\:<\:\mathrm{0}\:<\:\mid\alpha\mid\:<\:\beta \\ $$

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