ANSWERS TO SOME OF THE EXERCISES

Remember that some of the numerical answers are only approximate, so that you need not agree with them exactly, but, of course, you should not be far out.

Chapter 2.

Exercise No.

6. (a) x² + 3xy + y²,

(b) 6x + 6y + 10z,

(c) 3a² + 12a + 12 = 3(a + 2)²,

(d) 2x − 3,

(e) a² − 2ab − b²,

(f) x²yz + xy²z + xyz² = xyz(x + y + z),

(g) 6a³b⁴,

(h) 2x⁶,

(i) − a² − 4x²,

(j) ½xy³,

(k) 3ab,

(l)

8. (a) 12,

(b) 12,

(c) 14,

(d) 5,

(e) 2,

(f) 6,

(g) 3,

(h) ½

(i) 3,

(j) 18,

(k) 1,

(l) 5,

(m) 2,

(n) 6a,

(o) 2a + b,

(p) a − b.

9. $285, $255.

10. $342, $171, $114.

11. 1½ hours from the time Tom starts.

12. 12.

13. For 6000 miles A requires 212 gallons and B 165 gallons.

14. 36 cents per peck.

Chapter 3.

7. 3x + 7y, 2a − 5b, 4a − 9b, a + 3, etc.

9. (x − 1)(x + 1), (a + b − c)(a + b + c), (a + b − c) (a − b + c), (a − b)(a + b)(a² + b²), (9 − x)(9 + x), (a − b − c)(a + b + c),(x + y − l)(x + y + 1), (x − y) (x + y)(x² + y²)(x⁴ + y⁴), (a + b − 1)(a + b + 1), (x + y − 2)(x + y + 2), 3(2x + 1).

10. (i) 90^°, (ii) 60^°, (iii) 50^°, (iv) 10^°, (v) 78^°. 12. 60^°, 49^°, 38½^°.

Chapter 4.

11. The difference between successive terms is (1 − f)/(n + 1 ).

15. 25, 125.

22. (a) n³ − (n − 1)³ = 3n² − 3n + 1, n(2n − 1), (b) n(3n − 2).

23. 4! = 24, 8! = 40320, 12! = 479001600, 16!

24. 56, 495, 4368.

25. 720.

26. 16, 26.

27. 5040, 720, 48, 24.

28. 15, 21.

Chapter 5.

11. 1.57 miles.

12. About 5, 910 yards.

13. 60.9 feet.

14. 13.65 miles.

15. 9 miles.

16. sin 2A = 2 sin A cos A,

sin 3A = −3 sin A cos² A − sin³ A = 3 sin A − 4 sin³ A,

cos 2A = cos² A − sin² A,

cos 3A = cos³ A − 3 cos A sin² A = 4 cos³ A − 3 cos A.

22. QK = 7,813 miles, KP = 5,117 miles, PQ = 12,962 miles.

23. AM = 2,996 miles, AZ = 4,863 miles, MZ = 1,890 miles.

25. 4,133 miles.

26. 42.4 miles.

27. 33^° 53½′ or 28^° 6½′.

Chapter 6.

2. (a) (a + 4)(a + 6), (b) (p + 2)(p + 3), (c) (x − 1)(x − 2),

(d) (m + 1)(m + 3), (e) (x − 2)(x − 8), (f) (f + 5)(f − 4),

(g) (t + 5)(t − 8), and so on.

3. (a) (x + 6)(x − 6), (b) (3x + 5)(3x − 5),

(c) 4(x + 5)(x − 5), (d) 25(2y + 1)(2y − 1), etc.

(i) (16t + 13s)(16t − 13s) , . . . , (o)(√3 + x)(√3 − x),

(p) (√2 + √3x)(√2 − √3x), etc.

4. (a) (x + 3)(3x + 1), (b) (2x + 5)(3x + 2),

(c) (2p + 1)(3p + 1),

(d) (t + s)(3t + 7), . . . , (f) (2q − 1)(3q − 2), . . . , (h) (4x² + 1)(5x² − 1), (i) (3 + 2x)(5 − 2x), (j) (2n − 3)(3n + 4), etc.

5. (a) (2a + 3x)(3a − x), (b) (3a − 5bc)(5a + 3bc),

(c) (a − 7b)(6a + 5b), (d) (a + b)(2a − 9b), etc.

6. (a) and (b) 1/(x + y), (c), (d) and (e) l/(x − y), (f) 3x/4z,

(g) (x + 1)/(x + 3), (h) (x + l)/(x + 2),

(i) (x + 1)/(2x + 5), (j) (3x + 7)/(x + 7), (k) b(a + 6) ÷ a, (l) (4a² + 2a + 1)/(2a − 1), (m) (2x − 3)/(x − 2).

7. (a) a(b + c)/bc, (b) ab, (c) −2x + 3y, (d) a²/(a − 2b),

(e) 5a/4(x + 1), (f) (a + 17b)/12, (g)a/4(x + 2y)

(h) y²(y − x)/(y + x).

8. (a) 2x/(x² − 1), (b) −2/(x² − 1), (c) 2b/(a² − b²),

(d) (a² + 2ab − b²)/(a² − b²), etc., (h) 2/(y − 4)(y − 6),

(i) 2x/(x + y), . . . , (k) 12/(t − 1)(t + 3)(t − 5), etc.

10. (a) 19, (b) a + 2b, (c) 12/13, (d) 2½, (e) −4/3, (f) 10/7.

14. (a) x − =10, y = 2, (b) 3, 4, (c) 6, 2, (d) 9, 15, (e) 11, 12, 13, (f) 3, 6, 1.

16. (i) n(2n − 1), (ii) ½n²(n + 1), (iii) n(3n² − 2),

(iv) 3n² − 3n + 1, (v) ½(3n² − 3n + 2), (vi) 2n² − 2n + 1.

18. (i) 1.1249, (ii) 0.9039, (iii) 1.5735, (iv) 128.7876.

Chapter 7.

4. 20 h. 28 m., 8:1 pm, 48t° W.

5. 46^° N.

7. September 23rd.

8. Thin waning crescent, 5:08 am, 11:08 am.

10. About 7,900.

11. North Devon.

12. About September 24th.

18. 57.7^°, 7:33 pm.

19. About 6:40 am, 5:20 pm, at Gizeh, 7 am, 5 pm at New York, 7:28 am, 4:32 pm at London.

Chapter 8.

2. (a) x² + y² = 16, (b) x² + y² − 4x − 6y = 3.

8. (a) 45^°, (b) 60^°.

9. Parallel and equally spaced.

11. y = C.

17. l = 100t².

19. (a) x = 0, y = 3 or x = 4, y = 0,

(b) x = 3, y = 4, or x = 4, y = 3, (c) x = 2, y = 1, or x = 3, y = 2 or x = −3, y = −2 or x = −2, y = − 1.

26. (i) 1 or −1.4, (ii) or −½, (iii) 1.2 or −1.

Chapter 9.

1. (i) 4118, (ii) 30288, (iii) 1992, (iv) 99.18.

2. 6227, 9692, 0.01950, 45.82, 1.009, 0.4181.

3. 2, 3, 3/2.

5. (i)125, 930, (ii) 0.968, (iii) 1.111.

6. (i) $265, (ii) between 7 and 8 years, (iii) $84.

7. 0.0010, 0.3466, 8.240.

Chapter 10.

8. 5x⁴ − 15x² + 10x − 4.

10. x = −1, y = 2, and x = 1, y = −2.

11. 8/π cubic feet.

12. 10x = 3y = ½c.

16. p = 11.41 + 500v^−1.4.

17. (i) −a² sin a²x, (ii) 12 cos 3x, (iii) na cos nx − nb sin nx.

20. A = 2x² + 3x + C; (i) 108, (ii) 216, (iii) 25.

21. 380.

24. 14.

26. 11,900 ft lb wt.

27. 100π cu in.

28. 369π cu in.

31. (i) x cos x + sin x, (ii) cos x sec² x − sin x tan x = cos x,

(iii) 6x² + 6x + 4.

32. (i) −l/(x + 1)², (ii) (x cos x − sin x)/x², (iii) −sin x/cos² x,

(iv) sec² x.

33. (i) ½ cos x/√sin x, (ii) na(ax + b)ⁿ⁻¹, (iii) a cos (ax + b),

(iv) (2ax + b)/(ax² + bx + c).

35. (i) y = A cos 2x + B sin 2x, (ii) y = Ae^2x + Be^−2x. Second part: (i) y – 5 cos 2x + 2 sin 2x (ii) y = ½(7e^2x + 3e^−2x).