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Question Number 119996 Answers: 1 Comments: 0

{ ((x^3 +y^2 =a)),((x^2 +y^3 =b)) :} [solve for:x,y,a≠b∈R]

${\begin{cases} x^{3} + y^{2} = a \\ x^{2} + y^{3} = b \end{cases} [s o l v e f o r : x, y, a \neq b \in R]$

Question Number 119979 Answers: 3 Comments: 1

Question Number 119939 Answers: 3 Comments: 0

Question Number 119921 Answers: 2 Comments: 0

solving the following system of equations { ((((3x−y)/(x−3y))=x^2 )),((((3y−z)/(y−3z))=y^2 )),((((3z−x)/(z−3x))=z^2 )) :}

$s o l v i n g t h e f o l l o w i n g s y s t e m o f e q u a t i o n s$ ${\begin{cases} \frac{3 x - y}{x - 3 y} = x^{2} \\ \frac{3 y - z}{y - 3 z} = y^{2} \\ \frac{3 z - x}{z - 3 x} = z^{2} \end{cases}$

Question Number 119894 Answers: 2 Comments: 0

Question Number 119849 Answers: 1 Comments: 0

Find all pair(x,y) of real numbers that are the solutions to the system { ((x^4 +2x^3 −y=−(1/4)+(√3))),((y^4 +2y^3 −x=−(1/4)−(√3))) :}

$F i n d a l l p a i r (x, y) o f r e a l n u m b e r s$ $t h a t a r e t h e s o l u t i o n s t o t h e s y s t e m$ ${\begin{cases} x^{4} + 2 x^{3} - y = - \frac{1}{4} + \sqrt{3} \\ y^{4} + 2 y^{3} - x = - \frac{1}{4} - \sqrt{3} \end{cases}$

Question Number 119848 Answers: 1 Comments: 0

Solve in real numbers the equation (x)^(1/(3 )) + ((x−1))^(1/(3 )) + ((x+1))^(1/(3 )) = 0

$S o l v e i n r e a l n u m b e r s t h e e q u a t i o n$ $\sqrt[3]{x} + \sqrt[3]{x - 1} + \sqrt[3]{x + 1} = 0$

Question Number 119832 Answers: 0 Comments: 0

evaluate: I = ∫_0 ^( 1) (((x+1)/x))^(x!) dx

$evaluate : I = \int_{0}^{1} {(\frac{x + 1}{x})}^{x!} d x$

Question Number 119831 Answers: 0 Comments: 1

evaluate: I = ∫_1 ^( ∞) ((1/x))^x dx

$evaluate : I = \int_{1}^{\infty} {(\frac{1}{x})}^{x} d x$

Question Number 119807 Answers: 1 Comments: 0

Let f be a real-valued function defined on the inte- rval [−1, 1]. If the area of the equilateral triangle with (0, 0) and (x, f(x)) as two vertices is (√3)/4, then f(x) is equal to (A) (√(1−x^2 )) (B) (√(1+x^2 )) (C) −(√(1−x^2 )) (D) −(√(1+x^2 ))

$Let f be a real - valued function defined on the inte -$ $rval [- 1, 1] . If the area of the equilateral triangle with$ $(0, 0) and (x, f (x)) as two vertices is \sqrt{3} / 4, then f (x)$ $is equal to$ $(A) \sqrt{1 - x^{2}} (B) \sqrt{1 + x^{2}}$ $(C) - \sqrt{1 - x^{2}} (D) - \sqrt{1 + x^{2}}$

Question Number 119800 Answers: 0 Comments: 2

Examples of functions such that f(x+y)=f(x)+f(y) for all x,y∈R

$Examples of functions such that$ $f (x + y) = f (x) + f (y) for all x, y \in R$

Question Number 119801 Answers: 0 Comments: 0

If f:R→R is a function such that f(0)=1 and f(x+f(y))= f(x)+y for all x, y∈R, then (A) 1 is a period of f (B) f(n)=1 for all integers n (C) f(n)=n for all integers n (D) f(−1)=0

$If f : R \to R is a function such that f (0) = 1 and f (x + f (y)) =$ $f (x) + y for all x, y \in R, then$ $(A) 1 is a period of f$ $(B) f (n) = 1 for all integers n$ $(C) f (n) = n for all integers n$ $(D) f (- 1) = 0$

Question Number 119797 Answers: 0 Comments: 0

Q1 Let M_2 be the set of square matrices of order 2 over the real number system and R={(A,B)∈M_2 ×M_2 ∣A=P^( T) BP for some non-singular P ∈M} Then R is (A) symmetric (B) transitive (C) reflexive on M_2 (D) not an equivalence relation on M_2 Q2 For any integer n, let I_n be the interval (n, n+1). Define R={(x, y)∈R∣both x, y ∈ I_n for some n∈Z} Then R is (A) reflexive on R (B) symmetric (C) transitive (D) an equivalence relation

$Q 1$ $Let M_{2} be the set of square matrices of order 2 over$ $the real number system and$ $R = {(A, B) \in M_{2} \times M_{2} ∣ A = P^{T} B P for some$ $non - singular P \in M}$ $Then R is$ $(A) symmetric$ $(B) transitive$ $(C) reflexive on M_{2}$ $(D) not an equivalence relation on M_{2}$ $Q 2$ $For any integer n, let I_{n} be the interval (n, n + 1) .$ $Define$ $R = {(x, y) \in R ∣ both x, y \in I_{n} for some n \in Z}$ $Then R is$ $(A) reflexive on R$ $(B) symmetric$ $(C) transitive$ $(D) an equivalence relation$

Question Number 119795 Answers: 4 Comments: 0

Solve in real numbers the system of equations { (((3x+y)(x+3y)(√(xy)) =14)),(((x+y)(x^2 +14xy+y^2 )= 36)) :}

$S o l v e i n r e a l n u m b e r s t h e s y s t e m o f$ $e q u a t i o n s {\begin{cases} (3 x + y) (x + 3 y) \sqrt{x y} = 14 \\ (x + y) (x^{2} + 14 x y + y^{2}) = 36 \end{cases}$

Question Number 119774 Answers: 1 Comments: 0

Question Number 119757 Answers: 0 Comments: 0

For any integer n, let I_n be the interval (n, n+1). Define R={(x, y)∈R∣both x, y ∈ I_n for some n∈Z} Then R is (A) reflexive on R (B) symmetric (C) transitive (D) an equivalence relation

$For any integer n, let I_{n} be the interval (n, n + 1) .$ $Define$ $R = {(x, y) \in R ∣ both x, y \in I_{n} for some n \in Z}$ $Then R is$ $(A) reflexive on R$ $(B) symmetric$ $(C) transitive$ $(D) an equivalence relation$

Question Number 119713 Answers: 1 Comments: 3

If 3x+(1/(2x))=6 find 8x^3 +(1/(27x^3 ))

$If 3 x + \frac{1}{2 x} = 6 find {8 x}^{3} + \frac{1}{{27 x}^{3}}$

Question Number 119635 Answers: 1 Comments: 0

If a function f:R→R satisfies the relation f(x+1)+f(x−1)=(√3)f(x) for all x∈R then a period of f is (A) 10 (B) 12 (C) 6 (D) 4

$If a function f : R \to R satisfies the relation$ $f (x + 1) + f (x - 1) = \sqrt{3} f (x) for all x \in R$ $then a period of f is$ $(A) 10 (B) 12 (C) 6 (D) 4$

Question Number 119634 Answers: 2 Comments: 0

Q1 If f:R→R is defined by f(x)=[x]+[x+(1/2)]+[x+(2/3)]−3x+5 where [x] is the integral part of x, then a period of f is (A) 1 (B) 2/3 (C) 1/2 (D) 1/3 Q2 Let a<c<b such that c−a=b−c. If f:R→R is a function satisfying the relation f(x+a)+f(x+b)=f(x+c) for all x∈R then a period of f is (A) (b−a) (B) 2(b−a) (C) 3(b−a) (D) 4(b−a)

$Q 1$ $If f : R \to R is defined by$ $f (x) = [x] + [x + \frac{1}{2}] + [x + \frac{2}{3}] - 3 x + 5$ $where [x] is the integral part of x, then a period of f is$ $(A) 1 (B) 2 / 3 (C) 1 / 2 (D) 1 / 3$ $Q 2$ $Let a < c < b such that c - a = b - c . If f : R \to R is a$ $function satisfying the relation$ $f (x + a) + f (x + b) = f (x + c) for all x \in R$ $then a period of f is$ $(A) (b - a) (B) 2 (b - a)$ $(C) 3 (b - a) (D) 4 (b - a)$

Question Number 119580 Answers: 3 Comments: 0

Given k ∈ N. 1) justify these relations: 3^(2k) +1≡2[8] and 3^(2k+1) +1≡4[8]. 2) Given (E): 2^n −3^m =1. n and m are unknowed. • Show that if m is even , (E) does not have solution. ■ Deduct from the first question 1) that the couple (2;1) is the only solution of (E).

$Given k \in N .$ $1) justify these relations :$ $3^{2 k} + 1 \equiv 2 [8] and 3^{2 k + 1} + 1 \equiv 4 [8] .$ $2) Given (E) : 2^{n} - 3^{m} = 1 . n and m are unknowed .$ $∙ Show that if m is even, (E) does not have$ $solution .$ $◼ Deduct from the first question 1) that the$ $couple (2; 1) is the only solution of (E) .$

Question Number 119576 Answers: 2 Comments: 0

Question Number 119540 Answers: 1 Comments: 0

Question Number 119538 Answers: 1 Comments: 1

Question Number 119490 Answers: 0 Comments: 0

Let a<c<b such that c−a=b−c. If f:R→R is a function satisfying the relation f(x+a)+f(x+b)=f(x+c) for all x∈R then a period of f is (A) (b−a) (B) 2(b−a) (C) 3(b−a) (D) 4(b−a)

$Let a < c < b such that c - a = b - c . If f : R \to R is a$ $function satisfying the relation$ $f (x + a) + f (x + b) = f (x + c) for all x \in R$ $then a period of f is$ $(A) (b - a) (B) 2 (b - a)$ $(C) 3 (b - a) (D) 4 (b - a)$

Question Number 119483 Answers: 1 Comments: 0

Let a>0 and f:R→R a function satisfying f(x+a)=1+[2−3f(x)+3f(x)^2 −f(x)^3 ]^(1/3) for all x∈R. Then a period of f(x) is ka where k is a positive integer whose value is (A)1 (B)2 (C)3 (D)4

$Let a > 0 and f : R \to R a function satisfying$ $f (x + a) = 1 + {[2 - 3 f (x) + 3 f {(x)}^{2} - f {(x)}^{3}]}^{1 / 3}$ $for all x \in R . Then a period of f (x) is k a where k is$ $a positive integer whose value is$ $(A) 1 (B) 2 (C) 3 (D) 4$

Question Number 119396 Answers: 3 Comments: 1

If the roots of the equation 24x^4 −52x^3 +18x^2 +13x−6=0 are α , −α , β and (1/β). Find the value of α and β.

$I f t h e r o o t s o f t h e e q u a t i o n$ $24 x^{4} - 52 x^{3} + 18 x^{2} + 13 x - 6 = 0 a r e$ $α, - α, β a n d \frac{1}{β} . F i n d t h e v a l u e o f$ $α a n d β .$

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