## CSETMathGuru: THE Site for Single Subject Math

**Are you prepared for Subtest I Algebra?**

Then, have a stab at the following!

Assess yourself: You should do OK on the Test if you could solve 3-4 of the 5 Qs below, taking about 20 minutes/Q!

1.Find:

a) the equation of the parabola passing through (0,5), (2,5) and (-2,21).

b) the equation of the line with x-intercept 5/8 making a triangular area of 25/16 in the 4th Quadrant.

c) at what point(s) does the line intersect the parabola?

2. If 1, 1/2 and -3 are x-intercepts of a polynomial function f(x) = ax³ + bx² + cx + 3, find f(x) and graph it.

3. If f(x) = x² - 7x + k, find ALL value(s) of k such that f(x) has

a) No x-intercepts

b) Exactly 1 x-intercept

c) 2 x-intercepts

4. On the same graph, plot the following functions:

a) f(x) = x² - 2x - 8

b) g(x) = r(f(x))

c) h(x) = 1/f(x)

Note: r ~ Square Root

5. Graph the rational function: f(x) = (x² + 2x - 15)/x(x² - 7x + 12) stating its

a) Domain

b) x- and y-intercepts

c) Asymptotes

6. A logistic growth model of the form y = A/(B + C•e –Kx) can be used to describe the proportion of households,

*y*, that owns a VCR at year,

*x*, where A, B, C and K are constants. Express

*x*as a function of

*y*.

7. For what value(s) of K will the parabola y2 – 6y + x + K = 0 have exactly 1 y-intercept? Find the intercept.

8. For very large values of x, the graph of f(x) = x2 / (x + 1) behaves as that of a line y =

*m*x +

*b*. Find

*m*and

*b*.

9. The maximum safe load for a horizontal beam varies jointly width of the beam and the square of the thickness of the beam and inversely with its length. How will the thickness of the beam have changed if the maximum safe load were halved while the width of the beam was doubled and its length, increased by a factor of 3?

10. A swimming pool, rectangular in shape, is surrounded by a walkway of uniform width, say

*x*. If the outer dimensions of the walkway are 16m by 10m and the area of the pool alone is 112m2, find

*x*.

11. Find the Greatest Common Factor and Least Common Multiple of 5160, 5640, 4920, 4680 and 3720. [Express the LCM as a product of primes.]

12. What is the point of intersection of f(x) = ½ x + 3/2 and its inverse?

13. What shape does the graph |y| + |x|

__<__3 resemble?

14. What does the expression: (x3/y3 – 1)sqrt(x/y – 1)

**-1**/(x2/y2 – 1) simplify to?

15. ΔCAB is right-angled at A with AC = 8 and AB = 12. Points F, E and D lie on sides AB, BC and CA respectively such that AFED is a rectangle. If AF ≈

*x*, find an expression for the Area of rectangle AFED in terms of

*x*.

16. A company borrows $500,000 to expand its product line. Some of the money is borrowed at 7%, some at 8% and some at 10%. If the annual interest is $63,000 and the amount borrowed at 7% is 4 times that borrowed at 10%, express the system of equations to be solved in Matrix Form.

17. The number of bacteria, in thousands, in a Petri dish is given by the function:N(T) = 15T2 – 100T + 1000, 5

__<__T

__<__20 where T denotes the temperature in Celsius. The temperature of the Petri dish is a function of time,

*t*, in hour

T(

*t*) = 3

*t*+ 7, 0

__<__t

__<__5

a) Find the number of bacteria in the Petri dish after 3 hours have elapsed.

b) If 1,500,000 bacteria are found in the Petri dish, how many hours must have elapsed?

18. A parabola its

*x*-intercepts at –1 and 3. If the minimum value of the function is –2 find its equation.

19. Determine the constant term of the polynomial with integer coefficients of lowest degree with roots +/- 1 and (-3 +

*sqrt*7)/2, and whose leading coefficient is 3.

20. Find the magnitude and direction of the vector v = -sqrt i + j.

21. A computer manufacturer determines that it can sell 4000 machines at $750, and that for each $25 that the price is increased 20 fewer computers are sold. Let

*n*represent the number of $25 increases in the price. Express the total revenue of computer sales as a function of

*n*and find how much the manufacturer should charge to maximize his revenue.

22. Find the number of integers between 50 and 500 [

*inclusive*] divisible by 5

*or*7.

23. If $4500 compounded at 6.5% monthly for

*t*years grows to $6800, write an expression for

*t*using logs.

24. A square sheet of paper of side, A, is snipped at a distance of C units from the corners to produce a regular octagon of side, B units. Find B and C in terms of A.

25. There are 2 vectors

**a**and

**b**such that a = <-(sqrt 2)/2, -(sqrt 2)/2> and

**b**= -6

**a**. Find the magnitude and direction of b.

26. If point (3, 2) lies on the graph of the inverse of f(x) = 2x3 + x + A, find A.

Tips!

For #5

a), let the equation be y = ax² + bx + c. Substitute the given points and solve for

*a, b*and

*c*.

b), sketch a figure as described and denote the y-intercept as (0,

*k*) say. Find

*k*by using the information about the area of the triangle, A = ½ bh ≈ ½ (x-intercept)∙(y-intercept); then, find the equation of the line.

c) solve a) and b) simultaneously!

For #4, since 1, 1/2 and -3 are the x-intercepts, (1, 0), (½, 0) and (-3, 0) lie on the polynomial f(x). Plug in to solve the 3 equations in 3 unknowns

*a, b*and

*c*simultaneously. Alternately, and more quickly, y =

*p*(x - 1)(x - ½)(x+3) is the general form of the cubic polynomial with the given roots / x-intercepts. Since the constant term is 3,

*p*∙(-1)∙(- ½)∙(3) = 3 =>

*p*= 6. So the polynomial is readily: y = 6(x - 1)(x - ½)(x + 3) which can be easily graphed.

For #3, f(x) = x² – 7x +

*k*to have NO, ONE and TWO x-intercepts respectively, the discriminant, D = b2 – 4ac for x² – 7x + k should be <, = and > ZERO, respectively. Set the discriminant to each and find the value(s) of k!

For #2

a) and c) are straightforward [graphing a parabola and a simple rational expression, respectively].

b), g(x) = √x² – 2x - 8 = √(x – 4) (x + 2). Use this to determine the domain of g(x) ie. the values for which g(x) is defined (which would be x < -4 or x > 2). Then, simply construct an x-y table of, say 6 values, for

*x*values satisfying the Domain. Make approximations to simplify for

*y*- since a calculator is not permitted! - and sketch.

For#1 is a straightforward graphing of a Rational Function! Observe that (x - 3) factors out leaving a HOLE at x = 3.