A circle whose center is (2, 1) intercepts a line whose equation is 3x + 4y + 5 = 0 at point A and B. If the length of AB = 8, then the equation of the circle is ...
A. x^2+y^2-24x-2y-20=0
B. x^2+y^2-24x-2y-4=0
C. x^2+y^2-12x-2y-11=0
D. x^2+y^2-4x-2y+1=0
E. x^2+y^2-4x-2y+4=0
I don't know how to...
The remainder of p(x)=x^3+ax^2+4bx-1 divided by x^2+1 is –5a + 4b. If the remainder of p(x) divided by x + 1 is –a – 2, the value of 8ab is ...
A. -\frac34
B. -\frac12
C. 0
D. 1
E. 3
Dividing p(x) by x^2+1 by x^2+1 with –5a + 4b as the remainder using long division, I got (4bx – 1) – ((a – 1)x...
Given p and q are the roots of the quadratic equation ax^2-5x+c=0 with a\neq0. If p,q,\frac1{8pq} forms a geometric sequence and log_a18+log_ap=1, the value of c – a is ...
A. \frac13
B. \frac12
C. 3
D. 5
E. 7
Since p,q,\frac1{8pq} is a geometric sequence, then:
\frac{q}{p}=\frac{\frac1{8pq}}q...
In a bag there are m white balls and n red balls with mn = 200 and there are more white balls than red balls. If two balls are taken randomly at once and the probability of taking two different colored balls is \frac{40}{87} then the value of 2m + 3n is ...
A. 30
B. 45
C. 50
D. 70
E. 80
Okay...
Suppose the angles in triangle ABC is A, B, and C. If sin A + sin B = 2 sin C, the value of 2tan\frac12Atan\frac12B is ...
A. \frac83
B. \sqrt6
C. \frac73
D. \frac23
E. \frac13\sqrt3
Since A, B, and C are the angles of triangle ABC, then C = 180° – (A + B)
sin A + sin B = 2 sin C
sin A + sin B...
A quadratic function f(x)=x^2+2px+p has the minimum value of –p with p\neq0. If the curve's symmetrical axis is x = a, then a – f(a) = ...
A. –6
B. –4
C. 4
D. 6
E. 8
Because the curve's symmetrical axis is x = a, then:
-\frac{2p}{2(1)}=a
–p = a
a – f(a) = –p + (–p) = 0
I got zero. Is there...
The equation (a-1)x^2-4ax+4a+7=0 with a is a whole number has positive roots. If x_1>x_2 then x_2-x_1=...
A. –8
B. –5
C. –2
D. 2
E. 8
Since the equation has positive roots then x_1>0 and x_2>0 thus x_1+x_2>0 and x_1x_2>0
x_1+x_2>0
\frac{-(-4a)}{a-1}>0
x_1x_2>0
\frac{4a+7}{a-1}>0
However I...
From another forum. I though it looked simple enough, so I saved it in case I needed it. Turned out I couldn't solve it, so I tried to retrace from which forum it was from, but Google search showed no result. Thanks for your help, though. :)
3 friends, Alan, Brian and Chester, paint a house. If Alan had to paint it on his own, it would take him one hour more than the time it would take for all three to paint it together. If Brian had to paint it on his own, it would take him five hours more than the time it would take for all three...
The volume of a cuboid box with a square base is 2 litres. The production cost per unit of its top and its bottom is twice the production cost per unit of its lateral sides. Suppose the side length of its base is x and the height of the cuboid is h. The minimum production cost is reached when...
The point A is located on the coordinate (0, 5) and B is located on (10, 0). Point P(x, 0) is located on the line segment OB with O(0, 0). The coordinate of P so that the length AP + PB minimum is ...
A. (3, 0)
B. (3 1/4, 0)
C. (3 3/4, 0)
D. (4 1/2, 0)
E. (5, 0)
What I did:
f(x) = AP + PB...
If f(a) = 2, f'(a) = 1, g(a) = –1, and g'(a) = 2, the value of \lim_{x\to a}\frac{g(x)\cdot f(a)-g(a)\cdot f(x)}{x-a} is ...
A. 1
B. 3
C. 5
D. 7
E. 9
\lim_{x\to a}\frac{g(x)\cdot f(a)-g(a)\cdot f(x)}{x-a}=\lim_{x\to a}\frac{2g(x)+f(x)}{x-a}. How to determine the f(x) and g(x)? And when to use...
\lim_{x\to0}\frac{sin2x+sin6x+sin10x-sin18x}{3sinx-sin3x=}
A. 0
B. 45
C. 54
D. 192
E. 212
Either substituting or using L'Hopital gives \frac00. Is there any way to simplify it and make the result a real number?
Suppose you have an amount of 10000 money in any given currency. If you buy 2 things whose each costs 300 and 4 things whose each costs 700, how much money still left in you?
A friend asked me how to solve this question:
log_2(x+2)+log_{(x-2)}4=3
I said I had no idea because one is x + 2 and the other one is x - 2. If both are x + 2 or x - 2, I can do it. He said that if that's the case, even at his level he could solve it. This is what I've done so far regarding the...
@skeeter:
Ah, I see. Thanks for your help.
@Country Boy:
Sorry, I didn't know the proper term so I came up with what I had in mind at best, without realizing that the simple "above" is already the proper term.
If it's positive, it has two real roots. Okay, so D > 0.
D > 0
b^2-4ac>0
b^2>4ac
4ac<b^2
c<\frac{b^2}{4a}
A square number must be positive, so b^2 is positive. Since a > 0 then 4a > 0, so \frac{b^2}{4a} is still positive.
There's still a possibility that c is either negative or positive. What...
The graph's turning point of a quadratic function f(x)=ax^2+bx+c is over the X-axis. If the coordinate of the turning point is (p, q) and a > 0, the correct statement is ...
A. c is less than zero
B. c is more than zero
C. q is less than zero
D. q equals zero
Since the point (p, q) is over the...
The set of real numbers x at the interval [0, 2π ] which satisfy 2sin^2x\geq3cos2x+3 takes the form [a, b] ∪ [c, d]. The result of a + b + c + d is ...
a. 4π
b. 5π
c. 6π
d. 7π
e. 8π
What I've done thus far:
2sin^2x\geq3cos2x+3
2sin^2x\geq3(cos2x+1)
2sin^2x\geq3(cos^2x-sin^2x+sin^2x+cos^2x)...
If the sky is green, then a storm is coming.
If the storm comes, many things will be blown away, including faeces and roses. This, in turn, makes their smells mixed up.
Sorry, couldn't resist.
If f(x)=\frac{3x^2-5}{x+6} then f(0) + f'(0) is ...
A. 2
B. 1
C. 0
D. -1
E. -2
What I did:
If f(x)=\frac{u}{v} then:
u =3x^2-5 → u' = 6x
v = x + 6 → v' = 1
f'(x) =\frac{u'v-uv'}{v^2}=\frac{6x(x+6)-(3x^2-5)(1)}{(x+6)^2}
f(0) + f'(0) = \frac{3(0^2)-5}{0+6} +...
What's the area of the triangle? It's hard because the vertices aren't in the intersections of horizontal and vertical lines, so I have a hard time determining the side lengths, and it's also for Elementary Students Math Olympiads too.
My partner asked me about questions no. 8 and 9.
Number 8 asks about what is the area of the quadrilateral.
Number 9 asks about what number is below the number 25.
Those are questions for Elementary School Math Olympiads in my country but both of us were having a hard time figuring them out...