Recent questions tagged algebra

2 votes
1 answer
43
If $r, s$ and $t$ are consecutive odd integers with $r < s < t$, which of the following must be true?$rs = t$$r + t = 2t – s$$r + s = t – 2$$r + t = 2s$
0 votes
1 answer
45
If three positive real numbers $x, y, z$ satisfy $y – x = z – y$ and $x y z = 4,$ then what is the minimum possible value of $y?$$2^{\frac{1}{3}}$$2^{\frac{2}{3}}$$2^...
0 votes
1 answer
49
Let $x$ and $y$ be positive integers such that $x$ is prime and $y$ is composite. Then,$y – x$ cannot be an even integer$xy$ cannot be an even integer.$\frac{x+y}{x}$ c...
0 votes
1 answer
50
If $a, a + 2$ and $a + 4$ are prime numbers, then the number of possible solutions for $a$ is:onetwothreemore than three
1 votes
0 answers
53
Let $b$ be a positive integer and $a = b^2 – b$. If b $\geq$ 4, then $a^2 – 2a$ is divisible by$15$$20$$24$None of these
2 votes
1 answer
54
Let $x, y$ be two positive numbers such that $x + y =1.$ Then, the minimum value of $\left(x+ \frac{1}{x} \right)^2 + \left(y + \frac{1}{y}\right)^2$ is$12$$20$$12.5$$13....
0 votes
1 answer
55
If $a, b, c$ and $d$ are four positive real numbers such that $abcd = 1,$ what is the minimum value of $(1 + a)(1 + b)(1 + c)(1 + d)?$$4$$1$$16$$18$
0 votes
1 answer
56
$m$ is the smallest positive integer such that for any integer $n m,$ the quantity $n^3 – 7n^2 + 11n – 5$ is positive. What is the value of $m?$$4$$5$$8$None of thes...
0 votes
1 answer
57
Let $x, y$ and $z$ be distinct integers, $x$ and $y$ are odd and positive, and $z$ is even and positive. Which one of the following statements cannot be true?$(x − z)^2...
1 votes
1 answer
60
Let $x, y$ and $z$ be distinct integers, that are odd and positive. Which one of the following statements cannot be true?$xyz^2$ is odd.$(x − y)^2 z$ is even.$(x + y �...
0 votes
0 answers
62
If $pqr=1$ then $\frac{1}{1+p+r^{-1}} + \frac{1}{1+q+r^{-1}} + \frac{1}{1+r+p^{-1}} $ is equivalent to$p+q+r$$\frac{1}{p+q+r}$$1$$p^{-1}+q^{-1}+r^{-1}$
0 votes
0 answers
63
If $x^2 + 5y^2 + z^2 = 2y(2x+z)$, then which of the following statements are necessarily true?$x=2y$$x=2z$$2x=z$Only IOnly IIOnly IIIOnly I and II
0 votes
0 answers
64
For three integers $x, y$ and $z, x+y+z=15,$ and $xy+yz+xz=3.$ What is the largest value which $x$ can take?$3 \sqrt{13}$$\sqrt{19}$$13 /3$$\sqrt{15}$
0 votes
0 answers
65
if $x,y,z$ are distinct positive real numbers then $\frac{x^2(y+z) + y^2(x+z) + z^2(x+y)}{xyz}$ would begreater than $4$greater than $5$greater than $6$None of these
0 votes
2 answers
68
The number of non-negative real roots of $2^x - x - 1 =0$ equals _______$0$$1$$2$$3$
1 votes
1 answer
69
Let $y=\dfrac{1}{2+ \dfrac{1}{3+ \dfrac{1}{2+ \dfrac{1}{3+ \dots } } } }$ what is the value of $y?$$\frac{\sqrt{13} +3} {2}$$\frac{\sqrt{13} -3} {2}$$\frac{\sqrt{15} +3} ...
0 votes
1 answer
70
If $\frac{a}{a+b} = \frac{b}{c+a} = \frac{c}{a+b} = r$, then $r$ cannot take any value except$1/2$$-1$$1/2$ or $-1$-$1/2$ or $-1$
0 votes
1 answer
71
The total number of integer pairs $(x,y)$ satisfying the equation $x+y=xy$ is$0$$1$$2$None of these
0 votes
0 answers
73
Let $x=\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-}}}} \dots$ to infinity. Then $x$ equals$3$$\frac{\sqrt{13} -1}{2}$$\frac{\sqrt{13} +1}{2}$$\sqrt{13}$
0 votes
1 answer
74
If $\text{R} = \dfrac{30^{65} – 29^{65}}{30^{64} + 29^{64}}$ then$0 < \text{R} \leq 0.1$$0.1 < \text{R} \leq 0.5$$0.5 < \text{R} \leq 1.0$$\text{R} 1.0$
0 votes
1 answer
75
The number of solutions of the equation $2x+y=40$ where both $x$ and $y$ are positive integers and $x \leq y$ is:$7$$13$$14$$18$$20$
0 votes
0 answers
76
What are the values of $x$ and $y$ that satisfy both the equations?$2^{0.7x} \cdot 3^{-1.25y} = 8\sqrt{6} / 27$$4^{0.3x} \cdot 9^{0.2y} = 8.(81)^{\frac{1}{5}}$$x=2, y=5$$...
0 votes
2 answers
77
If $a/b=1/3, b/c=2, c/d=1/2, d/e=3$ and $e/f=1/4$ then what is the value of $abc/def?$$3/8$$27/8$$3/4$$27/4$$1/4$
3 votes
2 answers
78
How many pairs of positive integers, $m, n$ satisfy $1/m +4/n=1/12$ where $n$ is an odd integer less than $60?$$6$$4$$7$$5$$3$