Test 1
2000, Fall

Version A

Mixture. 10 points.

We have 6 kg 20% concentrated solution. How much of 12% solution should be added to decrease its concentration to 15%?

Algebraic rules. 3+3+4 points.

Compute and simplify the following expressions so that only positive exponents remain!

(((^3) (y^2) (1/(Sqrt[y])))/(x^2)), 	(1/(((((2 y)/(x^2))))^4)), 	-((8 (a^(-3/2)) (b^2) c)/((2 (b^3))/(a^4))).

Evaluation. 2+2+2+2+3+4 points.

Evaluate the following expressions!

((5^4)/((Sqrt[25]) 125)), 	((1/64)^(1/3)), 	(Subscript[log, 2]) (1/16), 	(10^(lg 3)), 		(3^(2 (Subscript[log, 3]) 4)), 	(((Subscript[log, (1/2)]) 3)/((Subscript[log, 2]) 9)).

Functions 1. 5 points.

Determine if the following curves represent graphs of functions, one-to-one functions!

Straight line. 15 points.

A straight line passing through the point (2, 1), intercepts the y-axis at y=2. Give the equation of the straight line. Find the equation if the straight line that passes through the point (1, 2) and perpendicular to the above line! Draw the graphs!

Functions 2. 5 points.

Find the common points of the functions!

y = 2x-1, 	y = (x^2)-4x-1.

Parabola. 15 points.

Determine the peak point equation and (if there exist real roots) the factorized equation of the parabola:

y= (x^2)-3x-4.

Give the coordinates of the peak point and the intersepts with the axes, and draw the graph of the parabola!

Transformations. 10 points.

Which transformation steps on the graph of the function (2^x) are required to get the graph of 1-(2^(-x))? Make sketch of the steps!

Inverse function. 15 points.

Find the inverse function of f(x) = (((x-2))^3) - 1! Plot the graphs of f(x) and (f^(-1))(x)!

Version B

Mixture. 10 points.

We have 3 kg,10% concentrated solution. How much of 20% solution should be added to increase the concentration to 15%?

Algebraic rules. 3+3+6 points.

Compute and simplify the following expressions so that only positive exponents remain!

(((x^2) y+x (y^2))/xy), 	((2 (x^2))/(((y^3) x)/(y^2))), 		((1-(x/(x-y)))/(1-((y-x)/(x+y)))).

Evaluation. 2+2+2+2+3+4 points.

Evaluate the following expressions!

(((5^3) (3^4))/((Sqrt[81]) 25)), 	((1/64)^(1/3)), 	(Subscript[log, 4]) (1/2), 	(10^lg4), 		(5^(2 (Subscript[log, 5]) 4)), 	(((Subscript[log, (1/3)]) 2)/((Subscript[log, 3]) 8)).

Elementary functions. 2+2+2+2+3+3+3 points.

Plot the graph of the following functions!

(x^2), 	(x^3), 	(Sqrt[x]), 	(x^(1/3)), 	2x-1, 	(3^(-x)), 	(Subscript[log, 2])x.

Transformations. 5+5+6 points.

Use the known transformations to plot the following functions!

(x/(x-2)), 	(x^2)-2x, 	1-(2^x).

Equations. 5 points.

Solve the following system of equation!

y+2x = -1, 	4x-3y = 2.

Parabola. 10 points.

Determine the peak point equation of the parabola y = (x^2)-3x-4! Give the peak point and the zero positions and draw the graph of the parabola!

Inverse function. 15 points.

Find the inverse function of f(x) = (((x-1))^5)+2! Plot the graphs of f(x) and (f^(-1))(x)!