Math Assessment 1: Trigonometry, Calculus, Algebra and Integration

This document provides a complete solution set for Math Assessment 1. The assessment covers a wide range of mathematical concepts including arithmetic and geometric progressions, infinite series, binomial expansions, and sketching trigonometric graphs. It explores trigonometric identities, equations, and applications within right-angled triangles. Additionally, the solutions delve into calculus, addressing differentiation, integration, and the application of substitution methods. Further, the document includes solutions to differential equations, partial fraction decomposition, and parametric equations. The assignment also assesses understanding of limits, series convergence, and basic algebraic manipulations. The solutions offer detailed step-by-step explanations to ensure a comprehensive understanding of each problem.

Math Assessment
1

l. a) Find the sum of 8 terms of these series:
(i) 1+4+7
It is an AP series with AP with a = 1 and d = 3
The sum of n terms of an AP is
Sn = (n/2)[2a+(n-1)d].
Therefore, the sum of 8 terms is
S8 = (8/2)[2*1+(8–1)*3]
= (8/2)[2+7*3]
= (4)(23)
Answer= 92
(ii)l+4+16+…..
an=a1×rn−1
a1= 1
r=4/1=16/4=64/ 16=4
So, sum of 8 terms will be
1*48-1
4-1
= 65536-1
3
Ans= 21845
2

1. b) Find the sum to infinity Of these series:
(i) 10+5+2.5…
= a1 (1-rn) / (1-r)
= infinite series means rn= 0
So, 1-rn = 1
So, S∞ = a1 / (1-r ).
=10/(1-0.5)
=10/0.5
= 20
(i) 1-1/2+1/4…
= a1 (1-rn) / (1-r)
= infinite series means rn= 0
So, 1-rn = 1
So, S∞ = a1 / (1-r ).
=1/(1-(-0.5))
=1/1.5
= 0.667
2 a) Find the coefficient of x3 in the expansion of (l + 3x)5
=Tr+1= 5Cr (1)r (3x)5-3
=put r=3
T3+1=5C3 (1)3 (3x)2
=10 (1) (3x)2
=10 (3x)2
3

2 b) Find the coefficient of x5 in the expansion of (a-2x)7
=Tr+1= 7Cr (a)r (-2x)7-r
=put r=5
T5+1=7C5 (a)5 (-2x)2
=21 (a)5 (-2x)2
=21 a5 (4x2)3
3. Sketch the graph of
a)
y= sin (x-90)
4

y= 1+sin 2x
5

b) y= tan ^ 1/2 x
6

y= 1-tan ^1/2x
4) .
7

6) Sketch a right-angled triangle with the shorter sides 3 cm and 4 cm, and label one of the non-
right angles a.
5
3
4
a)Write as. fractions:
(i) tan a = 5/4
(ii) sin a = 5 /3
(iii) cos a = 4/3
(iv) sin2 a = 25/ 9
(v) cos2 a = 16/ 9
b) Verify that
(i) tan a = sin a / cos a
Ans: tan a= 15 / 12
(ii) (ii) sin2 a + cos2 a = 1
Ans: 25/ 9+ 16/ 9 = 41/ 9
8

7. Solve these trig equations in the range 0° to 360°:
a) Sin 0 = 0.8
0.01396218
b) tan 20 = —1.5
0.36397023
c) 3 sin 0 4 cos 0
(3cosθ−4sinθ) 2=0
3cosθ−4sinθ=0
C) sin2 0 = 2(cos 0 + 1)
(sin θ +2cosθ)2=12
sin2θ+4cos2θ+4sinθcosθ=1
sin2θ=1−cos2θ
cos2θ=1−sin2θ
2sinθ−cosθ=2
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8) f(x) = / sin x / for 0 < x < 27
/sin x/ > ½
10

/sin x/>+ x 1 > -43/2
9. A team of four is selected from a group of 4 girls and 5 boys.
a)How many different selections are possible?
Ans: 12
b)What is the probability that the team will be all girls?
Ans: 4/ 9
Assignment-8
1) Use the substituton
a)
i) \int \frac{x}{\sqrt{2x^2}-1}dx
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=\int \frac{1}{4\left(\sqrt{u}-1\right)}du
=\frac{1}{4}\cdot \int \frac{1}{\sqrt{u}-1}du
=\frac{1}{4}\cdot \int \frac{2\left(v+1\right)}{v}dv :
= \frac{1}{4}\cdot \:2\left(\sqrt{2x^2}-1+\ln \left|\sqrt{2x^2}-1\right|\right)
ii) \frac{d}{dx}\left(y=cos2x\right)
= \frac{d}{dx}\left(y\right)=-\sin \left(2x\right)\cdot \:2
\frac{d}{dx}\left(y=\:x\:in\:x\right)
iii)
\frac{d}{dx}\left(y=\:\frac{x^2+1}{x+1}\right)
= \frac{d}{dx}\left(y\right)=\frac{x^2+2x-1}{\left(x+1\right)^2}
b)
i) y=\frac{e^x}{x}
x=0, y=0
\frac{e^x}{x}, min (1, e)
3 Evaluate these integrat correct to 3
i)
 \int _o^{in\:2}\:\left(e^{2x}-1\right)dx
 \int _o^{in\cdot \:2}e^{2x}-1dx
 \frac{1}{2}\left(e^{4ni}-e^{2o}\right)-\left(2ni-o\right)
 =\frac{1}{2}\left(e^{4ni}-e^{2o}\right)-\left(2ni-o\right)
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