Calculus : Algebra Test 2014
1.
Name:____________________
Solve the inequality 2x 2 £ 5x + 3
(3 marks)
1
Ans
(1 for some working and 2 for correct ans; 1 for each part)
- £ x£3
2
2. Solve the equation showing full working:
x +1 = 8- 3x +1
(4 marks)
Ans (x=8 or x=120) only x=8 is valid (2 for working, one if both ans and 2 if the
correct one selected)
3.
ln(3x - 2) - ln(x + 3) = 2lnk
Solve for x :
(3 marks)
Ans: x =
4. Write
Ans :
2 + 3k
3- k 2
2
(3 if everything correct; 2 minor mistake)
3+ 2 3
in the form a + b 3 , where a and b are real numbers
-4 + 3
(3 marks)
-18 11 3
(3 if all ok, 2 if minor mistake)
13
13
5. Find the solution set of : 2 2 x - 7.2 x+1 - 32 = 0
Ans (x=4)
(3 if all ok, 2 if minor mistake)
(3 marks)
6. If the equation kx 2 + 4x + 6 = 0 has two real solutions, determine the range of
allowable values of k
(3 marks)
Ans: k < 2 (3 if all ok, 2 if minor mistake)
3
-2 ± 5
7. Form the equation whose roots are : x =
and write the equation in the form
4
(3 marks)
y = ax 2 + bx + c
Ans: y =16x 2 +16x -1 (3 for correct eqn with full working, 2 if minor mistake)
8. Simplify
2 x+2.8 x-1
1
( )-2 x
4
Ans :(1/2) (3 correct answer with full working, 2 if minor mistake)
(3 marks)
x + 5 2 -3
+ =
3- x 3 x
9. Solve for x if
showing full working.
(4 marks)
Ans: (-3, -9) (4 for correct working with both ans, 3 if only one ans, 2 if minor
error, 0 if ans only)
10. Solve the following equation for x, in terms of p.
2 x-p =16 x
(2 marks)
Ans: x = -p
3
(2 all ok, 1 minor error)
11. Show that x 2 - 6x +10 is always positive
(2 marks)
Ans: using completing the square form and explanation
(2 if all ok, 1 if only completed square form and no explanation)
12. Solve : 5(3x-2) = 300 giving your answer correct to 4 s.f.
Ans: (1.848)
(2 if all ok, 1 ans only)
(2 marks)
13. Determine the value of p if x 4 - x 3 + px 2 - 2x +1 = 0 has a remainder of 17 when
divided by x - 2
(2 marks)
Ans: (p =3)
(2 using remainder thm or long division)
14. Factorise completely 3a3 + 24a6
(3 marks)
Ans : 3a3 (1+ 2a)(1- 2a + 4a2 ) (3 all ok, 2 incomplete factorization)
15. Factorise completely and hence solve the equation, g(x) = 2x 3 - x 2 -8x + 4 = 0
(3 marks)
Ans (-2, 2, 1/2) (Ans only 1, all 3 solns. with working 3, 2 for working with 2 solns.)
16. What is the least value of c if 3x 2 - 2x + c is never negative?
Ans (1/3)
(3 if all ok, minor error in working 2)
(3 marks)
17. If the quadratic equation 4x 2 - mx + 5 = 0 has one real root that is three times the
other, find the possible values of m. Leave your answer in root form.
(4 marks)
Ans: m = ±16
5
12
(Ans with full working 4, minor error 3)