PSAT October 15, 2016
Section 4 Question 20
If y=x-2 and z=x+2, what is y^2z in terms of x?
A) x^3-2x^2-4x+8
B) x^3-2x^2-4x-8
C) x^2+x-2
D) x^2-4
CAS Solution
expand((x-2)^2*(x+2),x)
returns
x^3-2x^2-4x+8
PSAT October 15, 2016
Section 4 Question 11
y=x
y=0.5x+3
The system of equations above has solution (x,y). What is the value of x?
A) 1.5
B) 2.5
C) 3.0
D) 6.0
CAS Solution
solve(y=x
y=0.5x+3, x,y)
returns
x=6 and y=6
BONUS
Here is a NON-calculator hack for a NON-calculator problem:
PSAT October 15, 2016
Section 3, Question 7
y^2-6y-2=0
Which of the following pairs of values are the solutions for the equation above?
A) -3 + sqrt(11), -3-sqrt(11)
B) -3 + 2sqrt(11), -3-2sqrt(11)
C) 3 + sqrt(11), 3-sqrt(11)
D) 3 + 2sqrt(11), 3-2sqrt(11)
Non-Calculator Hack Solution:
Sum of solutions = -b/a = 6/1=6 Answer is C or D
Product of solutions = c/a=-2
C) product of solutions: 3^2-(sqrt(11))^2=9-11=-2
Answer: C
Bonus Hack 2
There seem to be a lot of questions that ask for the positive DIFFERENCE between the two roots of a quadratic.
Difference=sqrt(discriminant)/a
This shortcut is especially useful to find the difference between the roots in the case of quadratics that cannot easily be factored.
Practice Test 8 Section 4 Question 35 (Grid-In)
The graph of the function f, defined by f(x)=-1/2 (x-4)^2 + 10, is shown in the xy-plane above. If the function g (not shown) is defined by g(x)=-x+10, what is one possible value of a such that f(a)=g(a)?
CAS Solution
solve(-1/2 (x-4)^2 + 10=-x+10,x)
returns x=2 or x=8
Practice Test 8 Section 4 Question 27
In the xy-plane, the point (p, r) lies on the line with equation y=x+b, where b is a constant. The point with coordinates (2p, 5r) lies on the line with equation y=2x+b. If p is not equal to 0, what is the value of r/p ?
A.2/5 B. 3/4 C. 4/3 D. 5/2
CAS Solution
solve (r=p+b
5r=2(2p)+b, r,b)
returns r=-3b and p=-4b
Therefore r/p=3/4
Practice Test 8 Section 4 Question 14
Which of the following is a value of x for which the expression -3/(x^2+3x-10) is undefined?
A. -2 B. -2 C. 0 D. 2
CAS Solution
solve (x^2+3x-10=0,x)
returns x=-5 or x=2
Practice Test 8 Section 4 Question 13
If a^(−1/2) =x, where a>0, what is a in terms of x?
A. sqrt(x) B.-sqrt(x) C. 1/x^2 D. -1/x^2
CAS Solution
solve(a^(−1/2) =x,a)
returns a=1/x^2
Practice Test 8 Section 4 Question 8
x+1= 2/(x+1)
In the equation above, which of the following is a possible value of x+1?
A. 1 - sqrt 2 B. sqrt 2 C. 2 D. 4
CAS Solution
solve (x+1= 2/(x+1),x)
returns x=-(sqrt 2+1) or x = sqrt 2 - 1
Therefore x+1 could be sqrt2.
Maine School Day April 2017
Section 4, Question 24
A polynomial p^4 + 4p^3 +3p^2 -4p-4 can be written as (p^2-1)(p+2)^2. What are all the roots of the polynomial?
A) -2 and 1
B) -2,1,4
C)-2. -1, 1
D) -1,1,2
CAS Solution
solve(p^4 + 4p^3 +3p^2 -4p-4 =0,p)
returns p=-2 or p=-1 or p=1
Answer: C
Question 34 (Grid-In)
If x does not equal -1, what is the value of (1/x+1)(2x+2)?
CAS Solution
(1/x+1)(2x+2) Enter
returns 2
- If 2sqrt(2x)=a, what is the value of 2x in terms of a? A) a/2 B) a^2/4 C) a^2/2 D)4a^2
CAS Solution
solve(2sqrt(2x)=a and z=2x, z,x)
returns x=a^2/8 and z=a^2/4
Answer: B
- If x^4-y^4=-15 and x^2-y^2=-3, what is the value of x^2+y^2? A)5 B)4 C) 2 D)1
CAS Solution
solve(x^4-y^4=-15 and x^2-y^2=-3 and z= x^2+y^2,x,y,z)
returns x=-1 and y= -2 and z=5 or x=1 and y=2 and z=5.
Answer: A
Grid-In 36. A fashion buyer for a large retail store purchased 315 items directly from the manufacturer for a total
of $6000. Some of the items were dresses purchased for $25 each, and the rest were shirts purchased for
$10 each. How many more dresses than shirts did the buyer purchase?
CAS Solution
solve(d+s=315 and 25d+10s=6000, d, s)
returns d=190 and s=125
Answer: 190-125=65
- At a snack bar, each medium drink costs $1.85 and each large drink costs c more dollars than a medium drink. If 5 medium drinks and 5 large drinks cost a total of $20.50, what is the value of c ? A) 0.45 B) 0.40 C) 0.30 D) 0.25
CAS Solution
solve(5(1.85)+5(c+1.85)=20.50,c)
returns c=0.4
Answer B
- x - 2 = sqrt(x+10) Which of the following values of x is a solution to the equation above? A)-1 B)1 C)4 D)6
CAS Solution
solve(x - 2 = sqrt(x+10),x)
returns x=6
Answer: D
Thank you @Plotinus
May 2017 US Section 4 Question 9
The sum of two different numbers x and y is 70, and the difference when the smaller number is subtracted from the
larger number is 30. What is the value of xy?
A. 100 B. 210 C. 1000 D. 2,100
CAS Solution
solve(x+y=70
x-y=30
z=x*y, x,y,z)
returns x=50 and y=20 and z=1000
Answer: C
Question 18
x^2 + y = 7
x - y = 5
Which value is a y-coordinate of a solution to the system of equations above?
A. -8 B. -3 C. -2 D. 6
CAS Solution
solve(x^2 + y = 7
x - y = 5,x,y)
returns x = -4 and y = -9 or x = 3 and y = -2
Answer: C
Question 24 (CB Rating: Hard)
For 5 consecutive even integers, the sum of the first and the third is 20 less than 3 times the fourth integer. What is the fifth integer?
A. 12 B. 14 C. 16 D. 26
CAS Solution
The integers are x,x+2,x+4,x+6, and x+8
solve(x+x+4=3(x+6)-20,x)
returns x = 6
The fifth integer is x + 8 =14
Answer B.