Free Practice for SAT, ACT and Compass Maths tests


Order from greatest to least
a) 25^{100}
b) 2^{300}
c) 3^{400}
d) 4^{200}
e) 2^{600}

Find all rational zeros of P(x) = x^{3}  7x + 6.

Round all real zeros in the graph to the nearest integer and find a polynomial function P of lowest degree, with the absolute value of the leading coefficient equal to 1, that has the indicated graph.
.

2  i, where i is the imaginary unit, is a zero of P(x) = x^{4}  4x^{3} + 3x^{2} + 8x  10. Find all zeros of P.

Find a, b and c so that the graph of the quadratic function f(x) = ax^{2} + bx + c has a vertex at (2 , 1) and passes through the point (0 , 3).

f(x) is a quadratic function such that f(1) = 3 and f(5) = 3. Find the x coordinate of the vertex of the graph of f.

Find a and b so that the rational function f(x) = (ax^{4} + bx^{3} + 3) / (x^{3}  2) has an oblique asymptote given by y = 2x  3

Solve for x the equation log_{9}(x^{3}) = log_{2}(8)

Find the value of log_{y}(x^{4}) if log_{x}(y^{3}) = 2

Solve for x the equation log_{x}(8e^{3}) = 3

If 16^{x} + 16^{x  1} = 10, find 2^{2x}.

If a^{2}  b^{2} = 8 and a*b = 2, find a^{4} + b^{4}.

What are the maximum value and minimum values of f(x) = 2sin(2x  π/3)  5 + 3

If x < 7, simplify 4  3 + x

A car travels from A to B at an average speed of 50 km/hour. At what average speed would it have to travel from B to A to average 60 km/hour for the whole trip?

If x^{2}  y^{2} = 12 and x + y = 6, find x and y.

f(x) is a function such that f(x) + 3f(8  x) = x for all real numbers x. Find the value of f(2).

f(x) is a function such that f(2x + 1) = 2f(x) + 1 for all real numbers x and f(0) = 2. Find the value of f(3).

Find b so that the line y = 2x + b is tangent to the circle x^{2} + y^{2} = 4.

What is the remainder of the division (x^{100}  x^{99}  x + 1) / (x^{2}  3x + 2)

Evaluate the number represented by the infinite series √(1/3 + √(1/3 + √(1/3 + ...))).

Show that the 3 by 3 system of equations given below has no solutions.
2x + y  3z = 5
5x + 3y + 2z = 7
3x  4y + z = 8
Solutions to the Above Problems

25^{100}
2^{300} = (2^{3})^{100} = 8^{100}
3^{400} = (3^{4})^{100} = 81^{100}
4^{200} = (4^{2})^{100} = 16^{100}
2^{600} = (2^{6})^{100} = 64^{100}
from greatest to least: 3^{400} , 2^{600}, 25^{100} , 4^{200} , 2^{300}

P(x) = x^{3}  7x + 6 : given
leading coefficient 1 and its factors are : +1,1
constant term is 6 and its factors are : +1,1,+2,2,+3,3,+6,6
possible rational zeros : +1,1,+2,2,+3,3,+6,6
test : P(1) = 0, P(2) = 0 and P(3) = 0
x = 1, x = 2 and x = 3 are the zeros of P(x).

From the graph, x = 3 is a zero of multiplicity 2, x = 0 is a zero of multiplicity 1 and x = 2 is a zero of multiplicity 2.
P(x) = x(x + 3)^{2}(x  2)^{2} : polynomial with real zeros hence with lowest degree.

)
if 2  i is a zero and the coefficients of the polynomial are real then 2 + i (the conjugate) is also a solution.
P(x) = (x  (2  i))(x  (2 + i))*q(x) = ((x  2)^{2} + 1)*q(x)
q(x) = P(x)/((x  2)^{2} + 1) = (x^{2}  2)
x = 2  i , x = 2 + i , x = √2 and x =  √2 are the 4 zeros of P(x).

f(x) = a(x + 2)^{2} + 1 : equation of parabola in vertex form
f(0) = 3 = 4a + 1
a = 1 : solve for a
f(x) = (x + 2)^{2} + 1 = x^{2}  4x  3
a = 1 , b = 4 and c = 3 : identify coefficients

f(x) = ax^{2} + bx + c
f(1) = 3 which give 3 = a + b + c
f(5) = 3 which gives 3 = 25a + 5b + c
24a + 4b = 0 : subtract equation B from equation C
x coordinate of vertex = b/2a = 3 : from above equation

The oblique asymptote is the quotient resulting from the long division of ax^{4} + bx^{3} + 3 by x^{3}  2
The quotient obtained is ax + b
ax + b = 2x  3
a = 2 and b = 3 : for two polynomials to be equal, the corresponding coefficients has to be equal.

log_{9}(x^{3}) = log_{2}(8) : given
log_{2}(2^{3}) = 3 : simplify right hand side of given equation.
log_{9}(x^{3}) = 3 : rewrite the above equation
log_{9}(x^{3}) = log_{9}(9^{3}) : rewite 3 as a log base 9.
x^{3} = 9^{3} : obtain algebraic equation from eqaution D.
x = 9 : solve above for x.

log_{x}(y^{3}) = 2 : given
x^{2} = y^{3} : rewrite in exponential form
x^{4} = y^{6} : square both sides
x^{4} = y^{6} : rewrite the above using the log base y
log_{y}(x^{4}) = log_{y}(y^{6}) = 6

log_{x}(8e^{3}) = 3 : given
x^{3} = 8e^{3} = (2e)^{3} x = 2e

16^{x} + 16^{x  1} = 10 : given
4^{2x} + 4^{2x} / 16 = 10
4^{2x} = 160/17 : solve for 4^{2x} 4^{x} = 4 √(10) / √(17) : extract the square root
2^{2x} = 4^{x} = 4 √(10) / √(17)

a^{2}  b^{2} = 8 : given
a^{4} + b^{4}  2a^{2}b^{2} = 8^{2} : square both sides and expand.
a*b = 2 : given
a^{2}b^{2} = 2^{2} : square both sides.
a^{4} + b^{4}  2(4) = 8^{2} : substitute
a^{4} + b^{4} = 72

1 ≤ sin(2x  π/3) ≤ 1 : range of a sine function
2 ≤ 2sin(2x  π/3) ≤ 2 : multiply all terms of the double inequality by 2
2  5 ≤ 2sin(2x  π/3)  5 ≤ 2  5 : add 5 to all terms of the inequality.
7 ≤ 2sin(2x  π/3)  5 ≤ 3
3 ≤ 2sin(2x  π/3)  5 ≤ 7 : change the above using absolute value.
3 + 3 ≤ 2sin(2x  π/3)  5 + 3 ≤ 7 + 3 : add 3 to all terms of the double inequality.
The maximum value of f(x) is equal to 10 and the minimum value of f(x) is equal to 6.

If x < 7 then x <  3 and x + 3 < 0 and 3 + x = (3 + x)
4  3 + x = 4 + 3 + x = x + 7 =  (x + 7) =  x  7 : since x + 7 < 0

Let d be the distance between A and B
T1 = d / 50 : travel time from A to B
Let S be the speed from B to A
T2 = d/S : travel time from B to A
60 = 2d/(T1 + T2) : average speed for the whole trip
60 = 2d/(d/50 + d/S) : substitute T1 and T2
S = 75 km/hour : solve the above equation for S.

x^{2}  y^{2} = (x  y)(x + y) = 12 : given
6(x  y) = 12 : substitute x + y by 6
(x  y) = 2 : solve for x  y
(x  y) = 2 and x + y = 6 : 2 by 2 system.
x = 2 , y = 4 : solve above system.

f(x) + 3f(8  x) = x : given
f(2) + 3f(6) = 2 : x = 2 above
f(6) + 3f(2) = 6 : x = 6 above
f(6) = 6  3f(2) : solve equation C for f(6)
f(2) + 3(6  3f(2)) = 2 : substitute
f(2) = 2 : solve above equation.

f(2x + 1) = 2f(x) + 1 : given
f(3) = 2f(1) + 1 : x = 1 in A
f(1) = 2f(0) + 1 : x = 0 in A
f(3) = 11 : substitute

x^{2} + y^{2} = 4 : given
x^{2} + (2x + b)^{2} = 4 : substitute y by 2x + b
5x^{2} + 4bx + b^{2}  4 = 0
The number of points of intersection is given by the number of solutions of the above equation. The line and circle are tangent if the above quadratic equation has only one solution which means that the discriminant is equal to zero. Find the discriminant as a function of b and solve.
b = √2 and b = √2 : 2 solutions.

(x^{100}  x^{99}  x + 1) / (x^{2}  3x + 2)
Let P(x) = x^{100}  x^{99}  x + 1 , D(x) = x^{2}  3x + 2
The division of the two polynomials may be written as
P(x) = D(x) Q(x) + r(x) , where Q(x) is the quotient and r(x) is the remainder that will have a degree equal to one or lower. r(x) = a x + b
We now need to find a and b that define the remainder.
Note that D(x) may be factored as follows: D(x) = x^{2}  3x + 2 = (x  1)(x  2)
Hence: P(x) = (x  1)(x  2) Q(x) + a x + b
Using the zeros of D(x) to write:
P(1) = (1  1)(1  2) Q(1) + a (1) + b gives a + b = P(1)
P(2) = (2  1)(2  2) Q(2) + a (2) + b gives 2 a + b = P(2)
We now need to evaluate P(1) and P(2)
P(1) = 1^{100}  1^{99}  (1) + 1 = 0
First rewrite P(x) = x^{99}(x  1)  x + 1 ; Hence P(2) = 2^{99}(2  1)  2 + 1 = 2^{99}  1
We now have a system of equations to solve and find a and b.
a + b = 0 and 2 a + b = 2^{99}  1
a = 2^{99}  1 and b = 1  2^{99} remainder: r(x) = (2^{99}  1) x + 1  2^{99}

Let y = √(1/3 + √(1/3 + √(1/3 + ...))).
square both sides to obtain: y^{ 2} = 1/3 + √(1/3 + √(1/3 + √(1/3 + ...)))
We can write: y^{ 2} = 1/3 + y
Solve the above quadratic equation to obtain: y = (3 + √(21)) / 6 and y = (3  √21) / 6
y is positive hence the solution: √(1/3 + √(1/3 + √(1/3 + ...))) = y = (3 + √(21)) / 6
