Math 121, Practice for Test 4
Hints and Answers
1. (a) (−3,1) (b) (3
5c, c) (c) No solution.
2. Let bbe the rate of the boat, and cbe the rate of the current. This leads to equations
4(b+c) = 120 and 4(b−c) = 120. Upon solving, one finds that the rate of the boat is 25mph
and the rate of the current is 5mph.
3. Let xbe the amount invested at 6% and let ybe the amount invested at 6.5%. Then
x+y= 25,000 and .06x+.065y= 1555. Solve these to find that $14,000 was invested at 6%
and $11,000 was invested at 6.5%.
4. (a) (2,−3,1) (b) (3c−1,2, c) (c) No solution.
5. Solve the equations 2a+b+c= 1, −a+ 2b+c= 12 and 3a+ 2b+c= 0 to find that
z=−3x+ 2y+ 5.
6. Let n= number of nickels, d=number of dimes and q= number of quarters, and solve the
equations .05n+.1d+.25q= 2, n= 2dand d=q+ 1 to find that there are 10 nickels, 5 dimes
and 4 quarters.
7. (a) (i) k6= 0, (ii) not possible, (iii) k= 0.
(b) k6= 0, (ii) k= 0, (iii) not possible.
8. (a) 3 4 −5
1−5−8(b)
3 4 −1−7
1−5 2 19
5 1 −2 5
The solutions will be the same as in 1(a) and 4(b).
9. (a) (5 −c, −2+3c, 0, c) (b) No solution (c) (0,13,0,5)
10. (a) AB =9 10 10
6−4 0 (b) Not possible (c) Not possible
(d) 5A−3C=−5 7 15
1 1 −1, (e) Not possible
11. (a) AB is a 3 ×1 matrix, BA doesn’t exist.
(b) Neither multiplications AB nor BA can be performed.
(c) AB is a 5 ×5 matrix, BA is a 3 ×3 matrix.
(d) AB is a 3 ×3 matrix BA is a 2 ×2 matrix.
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