Question and Answers Forum

AlgebraQuestion and Answers: Page 229

Question Number 110596 Answers: 1 Comments: 2

Three real numbers a,b,c satisfying ab+c=10,bc+a=11,ca+b=14. Find (a−b)(b−c)(c−a)(a−1)(b−1)(c−1)

$$\mathrm{Three}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{satisfying} \\ $$$$\mathrm{ab}+\mathrm{c}=\mathrm{10},\mathrm{bc}+\mathrm{a}=\mathrm{11},\mathrm{ca}+\mathrm{b}=\mathrm{14}.\:\mathrm{Find} \\ $$$$\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{b}−\mathrm{c}\right)\left(\mathrm{c}−\mathrm{a}\right)\left(\mathrm{a}−\mathrm{1}\right)\left(\mathrm{b}−\mathrm{1}\right)\left(\mathrm{c}−\mathrm{1}\right) \\ $$

Question Number 110594 Answers: 1 Comments: 0

A polynomial satisfies f(x^2 −2)=f(x)f(−x). Assuming that f(x)≠0 for −2≤x≤2, what is the value of f(−2)+f(1)?

$$\mathrm{A}\:\mathrm{polynomial}\:\mathrm{satisfies} \\ $$$$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2}\right)=\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(−\mathrm{x}\right).\:\mathrm{Assuming}\:\mathrm{that} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\neq\mathrm{0}\:\mathrm{for}\:−\mathrm{2}\leqslant\mathrm{x}\leqslant\mathrm{2},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{f}\left(−\mathrm{2}\right)+\mathrm{f}\left(\mathrm{1}\right)? \\ $$

Question Number 110593 Answers: 1 Comments: 0

Find the maximum possible integer n such that (((n−1)(n^2 +n−3))/(n^2 +4)) is also an integer

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{possible}\:\mathrm{integer}\:\mathrm{n} \\ $$$$\mathrm{such}\:\mathrm{that}\:\frac{\left(\mathrm{n}−\mathrm{1}\right)\left(\mathrm{n}^{\mathrm{2}} +\mathrm{n}−\mathrm{3}\right)}{\mathrm{n}^{\mathrm{2}} +\mathrm{4}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{an} \\ $$$$\mathrm{integer} \\ $$

Question Number 110592 Answers: 3 Comments: 0

How many pairs of integers x and y satisfy the equation (1/x)+(1/y)=(1/(32))

$$\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}=\frac{\mathrm{1}}{\mathrm{32}} \\ $$

Question Number 110673 Answers: 2 Comments: 0

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

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

Question Number 110587 Answers: 1 Comments: 0

A palindrome is a number that remains the same when its numbers are reversed. The number n and n+192 are three−digit and four−digit palindromes respectively. What is the sum of the digits of m? Can this be solved mathematically?

$$\mathrm{A}\:\mathrm{palindrome}\:\mathrm{is}\:\mathrm{a}\:\mathrm{number}\:\mathrm{that} \\ $$$$\mathrm{remains}\:\mathrm{the}\:\mathrm{same}\:\mathrm{when}\:\mathrm{its}\:\mathrm{numbers} \\ $$$$\mathrm{are}\:\mathrm{reversed}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{n}\:\mathrm{and} \\ $$$$\mathrm{n}+\mathrm{192}\:\mathrm{are}\:\mathrm{three}−\mathrm{digit}\:\mathrm{and} \\ $$$$\mathrm{four}−\mathrm{digit}\:\mathrm{palindromes}\:\mathrm{respectively}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{m}?\: \\ $$$$ \\ $$$$\mathrm{Can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{solved}\:\mathrm{mathematically}? \\ $$

Question Number 110586 Answers: 1 Comments: 0

The length of the interval of solution a≤2x+5≤b is 15. What is b−a?

$$\mathrm{The}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{solution} \\ $$$$\mathrm{a}\leqslant\mathrm{2x}+\mathrm{5}\leqslant\mathrm{b}\:\mathrm{is}\:\mathrm{15}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{b}−\mathrm{a}? \\ $$

Question Number 110725 Answers: 2 Comments: 0

x^x^x^x^(...) =2 x=?

$${x}^{{x}^{{x}^{{x}^{...} } } } =\mathrm{2}\:\:\:\:\:\:\:{x}=? \\ $$

Question Number 110545 Answers: 1 Comments: 2

Question Number 110887 Answers: 1 Comments: 2

Question Number 110715 Answers: 1 Comments: 2

a,b,c,d are unit digits whose pairwise sums form an arithmetic progression. Given that a+b+c+d is even, find the common positive difference of the arithmetic progression.

$$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\mathrm{are}\:\mathrm{unit}\:\mathrm{digits}\:\mathrm{whose} \\ $$$$\mathrm{pairwise}\:\mathrm{sums}\:\mathrm{form}\:\mathrm{an}\:\mathrm{arithmetic} \\ $$$$\mathrm{progression}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\:\mathrm{is} \\ $$$$\mathrm{even},\:\mathrm{find}\:\mathrm{the}\:\mathrm{common}\:\mathrm{positive} \\ $$$$\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{arithmetic} \\ $$$$\mathrm{progression}. \\ $$

Question Number 110490 Answers: 1 Comments: 0

e^(iπ) = −1

$$ \\ $$$${e}^{\mathrm{i}\pi} \:=\:−\mathrm{1} \\ $$

Question Number 110471 Answers: 1 Comments: 0

the sum of all x values (x^2 −6x+9)^((x−3)/(x−1)) = (x−3)^((x−4))

$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{x}\:\mathrm{values} \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{9}\right)^{\frac{{x}−\mathrm{3}}{{x}−\mathrm{1}}} \:=\:\left({x}−\mathrm{3}\right)^{\left({x}−\mathrm{4}\right)} \\ $$

Question Number 110463 Answers: 0 Comments: 2

Find the gcd(n−1,n+1)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{gcd}\left({n}−\mathrm{1},{n}+\mathrm{1}\right) \\ $$

Question Number 110436 Answers: 2 Comments: 0

If 2f(x) + f((1/x)) = 3x what is f(x)

$$\:\:\:\:\:\mathrm{If}\:\mathrm{2f}\left(\mathrm{x}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:=\:\mathrm{3x} \\ $$$$\:\:\:\:\:\mathrm{what}\:\mathrm{is}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$

Question Number 110434 Answers: 1 Comments: 1

The product of the four terms of an increasing arithmetic progression is a, their sum is b, and the sum of their reciprocal is c. Suppose that a,b,c form a geometric progression whose product is 8000, find the sum of the first and fourth term.

$$\mathrm{The}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\:\mathrm{four}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{increasing}\:\mathrm{arithmetic}\:\mathrm{progression}\:\mathrm{is}\:\mathrm{a}, \\ $$$$\mathrm{their}\:\mathrm{sum}\:\mathrm{is}\:\mathrm{b},\:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their} \\ $$$$\mathrm{reciprocal}\:\mathrm{is}\:\mathrm{c}.\:\mathrm{Suppose}\:\mathrm{that}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{form} \\ $$$$\mathrm{a}\:\mathrm{geometric}\:\mathrm{progression}\:\mathrm{whose} \\ $$$$\mathrm{product}\:\mathrm{is}\:\mathrm{8000},\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{and}\:\mathrm{fourth}\:\mathrm{term}. \\ $$

Question Number 110425 Answers: 2 Comments: 3

Which of the following is not a factor of x^6 −56x+55 A. x−1 B. x^2 −x+5 C. x^3 +2x^2 −2x−11 D. x^4 +x^3 +4x^2 −9x+11 E. x^5 +x^4 +x^3 +x^2 +x−55 Please show all workings clearly. Thanks.

$$ \\ $$$$\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{factor} \\ $$$$\mathrm{of}\:\mathrm{x}^{\mathrm{6}} −\mathrm{56x}+\mathrm{55} \\ $$$$\mathrm{A}.\:\mathrm{x}−\mathrm{1}\:\mathrm{B}.\:\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{5}\:\mathrm{C}. \\ $$$$\mathrm{x}^{\mathrm{3}} +\mathrm{2x}^{\mathrm{2}} −\mathrm{2x}−\mathrm{11}\:\mathrm{D}. \\ $$$$\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{3}} +\mathrm{4x}^{\mathrm{2}} −\mathrm{9x}+\mathrm{11}\:\mathrm{E}. \\ $$$$\mathrm{x}^{\mathrm{5}} +\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} +\mathrm{x}−\mathrm{55} \\ $$$$ \\ $$$$\mathrm{Please}\:\mathrm{show}\:\mathrm{all}\:\mathrm{workings}\:\mathrm{clearly}. \\ $$$$\mathrm{Thanks}. \\ $$

Question Number 110420 Answers: 1 Comments: 0

Prove that x^5 −3x^4 −17x^3 −x^2 −3x+17 cannot be factorized completely over the set of polynomials with integral coefficients.

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{x}^{\mathrm{5}} −\mathrm{3x}^{\mathrm{4}} −\mathrm{17x}^{\mathrm{3}} −\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{17}\:\mathrm{cannot}\:\mathrm{be} \\ $$$$\mathrm{factorized}\:\mathrm{completely}\:\mathrm{over}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of} \\ $$$$\mathrm{polynomials}\:\mathrm{with}\:\mathrm{integral}\:\mathrm{coefficients}. \\ $$

Question Number 110419 Answers: 1 Comments: 2

Let t be a root of x^3 −3x+1=0, if ((t^2 +pt+1)/(t^2 −t+1)) can be written as t+c for some p,c ∈ Z, then p−c equals?

$$\mathrm{Let}\:\mathrm{t}\:\mathrm{be}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} −\mathrm{3x}+\mathrm{1}=\mathrm{0},\:\mathrm{if}\: \\ $$$$\frac{\mathrm{t}^{\mathrm{2}} +\mathrm{pt}+\mathrm{1}}{\mathrm{t}^{\mathrm{2}} −\mathrm{t}+\mathrm{1}}\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{t}+\mathrm{c}\:\mathrm{for} \\ $$$$\mathrm{some}\:\mathrm{p},\mathrm{c}\:\in\:\mathbb{Z},\:\mathrm{then}\:\mathrm{p}−\mathrm{c}\:\mathrm{equals}? \\ $$

Question Number 110399 Answers: 0 Comments: 5

The identity 2[16a^4 +81b^4 +c^4 ]=[4a^2 +9b^2 +c^2 ]^2 cannot result from which of the following equations? A. 6b=4a+2c B. 6a=9b+3c C. 6b=−4a+2c D.c= −2a−3b E. 6c=2b+3a

$$\mathrm{The}\:\mathrm{identity} \\ $$$$\mathrm{2}\left[\mathrm{16a}^{\mathrm{4}} +\mathrm{81b}^{\mathrm{4}} +\mathrm{c}^{\mathrm{4}} \right]=\left[\mathrm{4a}^{\mathrm{2}} +\mathrm{9b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} \right]^{\mathrm{2}} \\ $$$$\mathrm{cannot}\:\mathrm{result}\:\mathrm{from}\:\mathrm{which}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{equations}?\: \\ $$$$\mathrm{A}.\:\mathrm{6b}=\mathrm{4a}+\mathrm{2c}\:\mathrm{B}.\:\mathrm{6a}=\mathrm{9b}+\mathrm{3c}\:\mathrm{C}. \\ $$$$\mathrm{6b}=−\mathrm{4a}+\mathrm{2c}\:\mathrm{D}.\mathrm{c}=\:−\mathrm{2a}−\mathrm{3b}\:\mathrm{E}. \\ $$$$\mathrm{6c}=\mathrm{2b}+\mathrm{3a} \\ $$

Question Number 110374 Answers: 2 Comments: 0

If P(x) is a polynomial whose sum of coefficients is 3 and P(x) can be factorised into two polynomials Q(x),R(x) with integer coefficients, the sum of the coefficients Q(x)^2 +R(x)^2 is

$$\mathrm{If}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{whose}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{coefficients}\:\mathrm{is}\:\mathrm{3}\:\mathrm{and}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{factorised}\:\mathrm{into}\:\mathrm{two}\:\mathrm{polynomials} \\ $$$$\mathrm{Q}\left(\mathrm{x}\right),\mathrm{R}\left(\mathrm{x}\right)\:\mathrm{with}\:\mathrm{integer}\:\mathrm{coefficients}, \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{coefficients} \\ $$$$\mathrm{Q}\left(\mathrm{x}\right)^{\mathrm{2}} +\mathrm{R}\left(\mathrm{x}\right)^{\mathrm{2}} \:\mathrm{is} \\ $$

Question Number 110359 Answers: 1 Comments: 0

Let f(x)=∣x−2∣+∣x−4∣−∣2x−6∣, for 2≤x≤8. The sum of the largest and smallest values of f(x) is

$$\mathrm{Let} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mid\mathrm{x}−\mathrm{2}\mid+\mid\mathrm{x}−\mathrm{4}\mid−\mid\mathrm{2x}−\mathrm{6}\mid, \\ $$$$\mathrm{for}\:\mathrm{2}\leqslant\mathrm{x}\leqslant\mathrm{8}.\:\mathrm{The}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{largest}\:\mathrm{and} \\ $$$$\mathrm{smallest}\:\mathrm{values}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is} \\ $$

Question Number 110318 Answers: 1 Comments: 0

(a+b−c)^2 =??

$$\left({a}+{b}−{c}\right)^{\mathrm{2}} =?? \\ $$

Question Number 110299 Answers: 3 Comments: 0

Question Number 110294 Answers: 1 Comments: 0

∣2x+1∣−∣x−2∣ < 4 find the solution set

$$\mid\mathrm{2}{x}+\mathrm{1}\mid−\mid{x}−\mathrm{2}\mid\:<\:\mathrm{4}\: \\ $$$${find}\:{the}\:{solution}\:{set} \\ $$

Question Number 110269 Answers: 1 Comments: 4

Pg 224 Pg 225 Pg 226 Pg 227 Pg 228 Pg 229 Pg 230 Pg 231 Pg 232 Pg 233

Contact: info@tinkutara.com