Mathematics Homework Solution: Advanced Topics and Problems

This document contains the solutions to a comprehensive mathematics homework assignment covering a wide range of topics. The solutions include answers to multiple-choice questions, detailed step-by-step solutions to problems involving trigonometry (like finding angles and using sine rule), vectors (including vector operations and finding angles between vectors), and algebra (solving systems of equations and working with sequences). The assignment also addresses calculus concepts, such as finding the sum of geometric sequences, and includes solutions to problems involving conic sections (ellipses and parabolas) and polar coordinates. Furthermore, the solutions incorporate real-world applications, such as a word problem involving ticket sales, and proofs of trigonometric identities. The reference list includes a textbook on mathematical methods for physicists and a mathematics textbook.

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1. (A) -0.7265
2. (C) Side-Angle-Side; law of cosines
3. (C) ellipse
4. (B) f(x) = 3 – cosx
5. (B) There are infinitely many solutions. The solutions are (2t + 1, 4, t) for all real
numbers t.
6. (D) r = 1 – 2sinθ
7. (A) sum of geometric sequence; sum = 6.4416
8. (B) arithmetic sequence; 471
9. a. 300°
b. 132°
10. a. - √ 2
b. -1/ √3
11. a. 2π/6
b. π/6
12. x = 90°, 210°, 270°, 330°
13. a. π
b. 2π/3

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14. a. (-√3, 1)
b. (6 √2, 7π/4)
15. a. Parabola opens to the right
b. vertex : (-2, 3)
c. focus : (0, 3)
16. a. x + y = 2000
52x + 38y = 86780
b. x = 770, y = 1230
c. Out of the total 2000 tickets bought for the amusement park, 770 tickets rating
$52/ticket for adults and 1230 tickets rating $38/ticket for children were sold amounting to
total $86780.
17. We will use the definition of sine of angle i.e.,
sin(A) = Perpendicular
Hypotenuse
sin(73°) = h
15
h = 0.9563 x 15
h = 14.34 ft
Yes, the contractor will be able to reach a window that is 14 feet above ground level
18. a. Comparing with the standard equation : ( x−h)2
a2 + ( y−k)2
b2 = 1
Since b>a as 7>5
The semi-Major axis of this plot of the given equation is vertical.

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b. We know-
c2 = b2 – a2
c2 = 49 – 25
c = 2 √ 6, -2√ 6
Coordinates of Foci : (5, 3.899), (5, -5.899) or (5, 2 √ 6 - 1), (5, -2 √ 6 - 1)
19. cosθ = 12/13
sinθ = (1 – cos2θ)1/2
= (1 – 144/169)1/2
sinθ = ±5/13
Since θ lies in IV quadrant, we have to consider the negative value only
So, sinθ = -5/13
20. 1 – (sinx – cosx)2 = sin(2x)
LHS-

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= 1 – (sin2x + cos2x – 2sinxcosx)
= 1 – (1 – 2sinxcosx) (sin2x + cos2x = 1)
= 1 – 1 + 2sinxcosx
= sin(2x) (sin(2x) = 2sinxcosx)
= RHS
21. Using the sine rule-
a
sinA = b
sinB= c
sinC
a
sin 40 ° = b
sin 63 ° = 38
sin 77°
b
sin 63° = 38
sin 77 °
b = 34.75
22. a. u = <4,2>
v = <5,-10>
2v = <10,-20>
u + 2v = <4 + 10, 2 – 20>
= <14, -18>
b. |u| = √ 42+ 22
= 2 √5
c. u.v = (4)(5) + (2)(-10)
= 20 – 20
= 0
d. We know-
u.v = |u||v|cosθ

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0 = (2 √5)(5√5)cosθ
cosθ = 0
θ = 90°
23. x – y + 3z = 8 (1)
-2x + y + 8z = 1 (2)
x – y – 2z = 3 (3)
Subtracting Eq(3) from Eq(1)-
0x – 0y + 5z = 5 (4)
-2x + y + 8z = 1
x – y – 2z = 3
Adding Eq(3) and Eq(2)-
0x – 0y + 5z = 5
-x + 0y + 6z = 4 (5)
x – y – 2z = 3
From Eq(4), z = 1
From Eq(5), x = 2
From Eq(1), y = -3

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References-
Arfken G., Weber H. & Harris F.E.,(2012), Mathematical Methods for Physicists, 7th
edition. Cambridge: Academic Press
Sharma R.D. (2018). Mathematics (2019 edition). Dhanpat Rai Publication, New Delhi