Problems
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An airplane flies against the wind from A to B in 8 hours. The same airplane returns from B to A, in the same direction as the wind, in 7 hours. Find the ratio of the speed of the airplane (in still air) to the speed of the wind.
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Find the area between two concentric circles defined by
x2 + y2 -2x + 4y + 1 = 0
x2 + y2 -2x + 4y - 11 = 0
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Find all values of parameter m (a real number) so that the equation 2x2 - m x + m = 0 has no real solutions.
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The sum an integer N and its reciprocal is equal to 78/15. What is the value of N?
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m and n are integers so that 4m / 125 = 5n / 64. Find values for m and n.
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Simplify: 3n + 4001 + 3n + 4001 + 3n + 4001
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P is a polynomial such that P(x2 + 1) = - 2 x4 + 5 x2 + 6. Find P(- x2 + 3)
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For what values of r would the line x + y = r be tangent to the circle x2 + y2 = 4?
Solutions to the Above Problems
Let x = speed of airplane in still air, y = speed of wind and D the distance between A and B. Find the ratio x / y
Against the wind: D = 8(x - y), with the wind: D = 7(x + y)
8x - 8y = 7x + 7y, hence x / y = 15
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Rewrite equations of circles in standard form. Hence equation x2 + y2 -2x + 4y + 1 = 0 may be written as
(x - 1)2 + (y + 2) 2 = 4 = 22
and equation x2 + y2 -2x + 4y - 11 = 0 as
(x - 1)2 + (y + 2) 2 = 16 = 42
Knowing the radii, the area of the ring is π (4)2 - π (2)2 = 12 π
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The given equation is a quadratic equation and has no solutions if it discriminant D is less than zero.
D = (-m)2 - 4(2)(m) = m2 - 8 m
We nos solve the inequality m2 - 8 m < 0
The solution set of the above inequality is: (0 , 8)
Any value of m in the interval (0 , 8) makes the discriminant D negative and therefore the equation has no real solutions.
Write equation in N as follows
N + 1/N = 78/15
Multiply all terms by N, obtain a quadratic equation and solve to obtain N = 5.
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4m / 125 = 5n / 64
Cross multiply: 64 4m = 125 5n
Note that 64 = 43 and 125 = 53
The above equation may be written as: 4m + 3 = 5n + 3
The only values of the exponents that make the two exponential expressions equal are: m + 3 = 0 and n + 3 = 0, which gives m = - 3 and n = - 3.
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3n + 4001 + 3n + 4001 + 3n + 4001 = 3(3n + 4001) = 3n + 4002
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P(x2 + 1) = - 2 x4 + 5 x2 + 6
Let t = x2 + 1 which also gives x2 = t - 1
Substitute x2 by t - 1 in P to obtain: P(t) = - 2 (t - 1)2 + 5 (t - 1) + 6 = -2 t 2 + 9t - 1
Now let t = - x2 + 3 and substitute in P(t) above to obtain
P(- x2 + 3) = -2 (- x2 + 3) 2 + 9 (- x2 + 3) - 1 = -2 x 4 + 3 x 2 + 8
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Solve x + y = r for y: y = r - x
Substitute in the equation of the circle:
x2 + (r - x)2 = 4
Expand: 2 x2 -2 r x + r 2 - 4 = 0
If we solve the above quadratic equation (in x) we will obtain the x coordinates of the points of intersection of the line and the circle. The 2 points of intersection "become one" and therefore the line and the circle become tangent if the discriminant D of the quadratic equation is zero. Hence
D = (-2r)2 - 4(2)(r2 - 4) = 4(8 - r2) = 0
Solve for r to obtain: r = 2 √2 and r = - 2√2
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