TLF Uqp

Trigonometry University Questions
1 mark
1. tan(iθ) = a) −itanhθ b) i2 tanθ c) itan2 θ d) itanhθ

ex +e−x ex −e−x ex +e−x
2. sinhx is a) 2 b) 2 c) 2a d) 0
3. cosθ + isinθ is a) eiθ b) e−iθ c) e2θ d) e−2θ

4. The value of cosh2 x − sinh2 x is a) 2 b) 1 c) 0 d) −1
5. sin(ix) = a) isinhx b) −isinhx c) sinhx d) −sinhx

2 4 6
6. 1 − θ2! + θ4! − θ6! + . . . a) sinθ b) tanθ c) cosθ d) tan2 θ
1
7. tanhx = a) tanhx b) sechx c) cosechx d) cothx
8. Which is incorrect ?
1+tanh2 x
a) sinh 3x = 3sinh x + 4sinh3 x b) cosh 2x = 1−tanh2
3tanh x−tanh3 x
c) tanh 3x = 1+3tanh2 x d) cosh 3x = 4cosh3 x − 3cosh x
9. The value of cosh ( iπ

2 )= a) i b) −1 c) 1 d) 0
10. If z and w are complex numbers then z w =

a) ew logz b) ewz c) ez logw d)ew+z
11. If z is real then log z is a) real b) Imaginary c) Complex d) 0
12. cosh(ix) is a) cosx b) sinx c) coshx d) sinhx
5 mark
2 2
x y
1. Prove that sin2 A − cos2 A = 1 if sin(A + iB) = x + iy
2. Find the value of log(1 − i).

3. If coshu = secθ show that u = log( π4 + θ2 ).
u sin2x
4. If tan(x + iy) = u + iv prove that v = sin2y
c3 c5 5
5. Sum to infinity the series c cosα − 3 cos3α + 5 cos α . . .
6. Sum to infinity the series cosα + 12 cos(α + β) + 12 . 34 cos(α + 2β) + . . .

7. If cos(x + iy) = cosθ + isinθ then prove that cos2x + cosh2y = 2.
√
8. Prove that (i) sinh−1 x = loge (x + x2 + 1) (ii) tanh−1 x = 12 loge ( 1−x
1+x
).
9. Show that ii is real.
1
2tanh x
10. Prove that tanh2x = 1+tanh2 x .
c3 c5 5
11. Sum to infinity the series c cosα − 3 cos3α + 5 cos α . . .
12. Show that ii is real.

13. log(−3) 2
cos2θ cos4θ
14. Sum of the series 1 − 2! + 4! + ...
15. Find the sum to n terms of the series sin2 + sin22 + sin32 + . . .
16. Expand cos6 θ in cosine multiples of θ.

17. Expand sin4 θcos2 θ in cosine multiples of θ.
18. Find logsin(θ + iφ).

19. If logcos(α + iβ) = x + iy prove (i) x = 12 log( cos2α+cos2β
2 )
(ii) y = 12 [logsin(α + y) − logsin(α − y)].
20. Find log(1 + eiθ ).
8 mark
1. Prove that 2e2L = cosh2φ − cos2θ if logsin(θ + iφ) = L + iB.

u sin2x
2. Prove that v = sinh2y if tan(x + y) = u + iv.
3. Find value of (i) log(−3) (2) (ii) log(1 + i)

cos2θ cos4θ
4. (i) Sum of the series 1 − 2! + 4! + . . . (ii) Write short on Gregory’
series.
5. Find the value of sinh−1 x and tanh−1 x.
6. Find the sum to n terms of the series sin2 + sin22 + sin32 + . . .
7. Separate the real and imaginary parts of o (i) tan−1 (x+iy) (ii) sin−1 (cosθ+
isinθ)
8. If logsin(θ + iφ) = α + iβ then prove that 2e2α = cosh 2φ − cos2θ.
2
Laplace Transform University Questions
1 mark
√ √ √ √
π π 2 π
1. L(t1/2 ) = a) π b) s3/2
c) 2s3/2
d) s3/2
1
2. L−1 ( s−a )= a) 1 b) eat c) e−at d) cosat
3. L(1/t) = a) 1/s b) ∞ c) s d) none
4. L−1 ( 1s ) = a) 1
t b) ∞ c) 1 d) none
s s s
5. L[cos4t] = a) s2 −16 b) s2 +4 c) s2 +16 d) s2s+4
6. L−1 [ s2s−4 ] = a) cos 2t b) sin4t c) cosh 2t d) sinh 4t

1 1 1
7. L(x) = a) 1 b) s c) s2 d) s+1
a
8. L−1 ( s2 −a 2) = a) sin at b) cos at c) sinh at d) cosh at
s s s s2
9. L(cosat) = a) s+a b) s2 +a2 c) s+a2 d) s+a
1 1
10. L(1) = a) 0 b) 1 c) s d) s+a
1 2 s3 s3
11. L(t2 ) = a) s3 b) s3 c) 2 d) 2!
1 1 2
12. L−1 (1) = a)1 b) x c) s d) x2
13. L−1 ( s2s+4 ) = a) cos 3t b) cos 2t c) sin 2t d) tan 2t

3 s
14. L(sin 3t) = a) s2 +9 b) s2 −9 c) 3 d) 9
3 2 3 s
15. L(sinh 3t) = a) s2 +9 b) s2 +9 c) s2 −9 d) s2 −9
16. L(t2 + cos3t) is

1 s 2 3 2 s 3 1
a) s2 + s2 +9 b) s3 + s2 +9 c) s3 + s2 +9 d) s2 +9 + s2
t2 t4
17. L−1 ( s12 ) = a) t
2 b) 2 c) t3 d) 3
18. L(tsinh at) =

2s 2a 2sa 2
a) (s2 −a2 )2 b) (s2 −a2 )2 c) (s2 −a2 )2 d) (s2 −a2 )2
1 e3t t3 e−3t t3 e3t t2 e3t t4

19. L−1 ( (s−3) 5 ) is a) 24 b) 24 c) 24 d) 24
1 2! 3! 4!
20. L(t2 e3t ) = a) (s−3)3 b) (s−3)3 c) (s−3)3 d) (s−3)3
5 mark
1. Find L(te−t sint).
3

s+2
2. Find L−1 (s2 +4s+5)2 .
3. Find (i) L(cos3 2t) (ii) L(sin2 3t).

Z ∞
2sin t − 3sinh t
4. Evaluate e−2t dt.
0 t
5. Find L[sin3tsin2t].
1
6. Find L−1 [ s(s+3) ].
7. Find L(xe−x cosx).

8. Find L(sin2 2t) and L(t2 e−3t ).

−1 s
9. Find L s2 a2 +b2 .

−1 s+1
10. Find L log( s−1 ) .
t
11. Find L( 1−e
t ).
12. Find L( 1−t sinat

t ) and L( t ).

−1 1
13. Find L (s+1)(s2 +2s+2) .
2
14. Find (i) L(sin3 2t) and L( sint t ).

−1 1
15. Find L (s2 +a2 )2 .
d
16. Prove that L[tf (t)] = − ds F (s).
8 mark
Z ∞ Z ∞
1. Using Laplace Transform evaluate (i) e−2t sin3t dt (ii) te−3t cost dt.
0 0
2. Using Laplace Transform solve y + 2y + 5y = 4e , given y = y 0 = 0

00 0 −t
when t = 0.
3. Using Laplace Transform y 00 − y 0 − 2y = 18e−t sin 3t, y(0) = 0, y 0 (0) = 3.
4. Using Laplace Transform solve (D2 + 4D + 13)y = 2e−x given that y(0) =
0, y 0 (0) = −1.

7s+2
5. Find (i) L(t2 e−t cos t) (ii) L−1 (s+2)(s2 −4s+3) .
4
6. Find (i) L[sin3 2t] (ii) L[t2 cos at].
7. Evaluate (i) L( 1−cos
x
x 1
) (ii) L−1 ( s(s+1)(s+2) ).
d2 y dy dy
8. Using Laplace Transform solve dt2 +2 dt −3y = sin t given that y = dt =0
d2 y
9. Solve dt2 + t dy 0
dt − y = 0 if y(0) = 0, y (0) = 1.
10. Solve using laplace transform (D2 + 4)y = t, y(0) = y 0 (0) = 0.

dy
11. Solve 3 dx
dt + dt + 4 dy
+ 2x = 1 and dx
dt + 3y = 0, x = y = 0, t = 0.
dt

00 2 0 −1 s+3
12. (i) Prove that L[f (t)] = s Lf (t)−sf (0)−f (0) (ii) Find L (s2 +6s+13)2 .
2
13. Solve t( ddt2y )−(2+t) dy
dt +3y = t−1 when y(0) = 0 using laplace transform.
5
Fourier Series University Questions
1 mark
1. The fourier coefficient a0 for f (x) = ex in (0, 2π) is

e2x−1 e2π −1 eπ −e−π
a) 0 b) 2π c) π d) π
2. An odd function is
a) xsinx b) x2 cosxc) x2 sinx d) sinx + cosx
R
λ+2π
3. The value of sin mxsin nxdx for m 6= n is
λ
a) 0 b) π c) 2π d) none
Rπ
4. The value of sin mxsin nxdx for m 6= n is
0
5. The value of a0 in the fourier series expansion of f (x) = x, −π < x < π is
a) 0 b) 1 c) π d) π 2
R
λ+2π
6. The value of cos mxcos nxdx for m 6= n is
x
7. If f (x) is an odd function in the interval −π < x < π then
Rπ
a) an = π2 f (x)cosnxdx, bn = 0 b) an = 0, bn = 0
0
2
Rπ
c) an = 0, bn = π f (x)sinnxdx d) none
0
5 mark
(
−x −π < x < 0
1. Find the fourier series for f (x) = .
x 0≤x<π
2. Find sine series for f (x) = k in 0 < x < π.

3. Express f (x) = x, −π < x < π as fourier series with period 2π
4. Find sine series for f (x) = c in the range 0 to π.
5. Express f (x) = c − x when 0 < x < c as half range cosine series with
period 2c.
(
0 0<x<l
6. In (0, 2l), f (x) = Expand f (x) as fourier series of period
a l < x < 2l
2π.
6
7. Find fourier series with period 3 represent f (x) = 2x − x2 in the range
(0,3).
(
x 0 < x < π2
8. Find sine series for f (x) =
0 π2 ≤ x < π
8 mark
π2
∞
P (−1)n cosnx
1. Show that x2 = 3 +4 n2 in the interval −π ≤ x ≤ π and
n=1
1 1 1 π2 1 1 1 π2
deduce that (i) 12 + 22 + 32 + ∙∙∙ = 6 .(ii) 12 − 22 + 32 − ∙∙∙ = 12 .

x e2x −1 1
2. Show that in the interval 0 < x < 2π the fourier series for e is π 2 +

P∞ P
∞
( cos nx
n2 +1 ) − ( nsin nx
n2 +1 ) .
n=1 n=1
2
3. If f (x) = x(2π −x) in 0 < x < 2π prove that f (x) = 2π3 −4( cosx cos2x
12 + 22 +
cos3x
32 + . . . )
(
0 0 < x < π2
4. Find a cosine series in the range 0 to π for f (x) = .
π − x π2 < x < π
πx
5. Expand 8 (π− x) in a sine series valid when 0 < x < π.
(
x 0<x<π
6. Find fourier series in the range 0 to 2π for f (x) =
2π − x π < x < 2π
7
Code No: 41157 Sub. Code: JSMA4A
B.Sc (CBCS) DEGREE EXAMINATION

November 2018
Fourth Semester
Mathematics-Main
Skill Based - Trigonometry, Laplace Transforms and Fourier Series
(For those who joined in July 2016 only)
Time: 3 hrs Maximum Mark: 75
Part-A—(10 x 1 = 10 marks)
Answer all the Questions
1. cosh( iπ
2 )= a) 0 b) 1 c) −1 d) i
2. If z is complex number then logz has
a) only one value b) finite number of value
c) infinite number of value d) not defined
3. The imaginary part of sin(x + iy) is
a) coscoshy b) sinxsinhy c) sinxcoshy d) cosxsinhy
4. If tanhx = sinθ then coshx =
a) sinθ b) cosθ c) secθ d) tanθ
6 3
5. L(x3 ) = a) s4 b) s4 c) s63 d) 6
s3

−1 1
6. L s(s+1) = a) e−x b) eax c) e−x d) ex
7. L−1 (F (s + a)) =———- L−1 (F (s))

a) e−x b) eax c) e−x d) ex
8. If y(0) = y 0 (0) = 0 then L(y 00 ) =
a) L(y) b) sL(y) c) s2 L(y) d) L(y 0 −y)
9. If f (x) is an odd function defined in (−π, π) then in the fourier series
expansion of f (x)
a) a1 = 0, an 6= 0 b) an = 0 c) b1 = 0, bn 6= 0 d) bn = 0
1 1 1
10. 12 + 32 + 52 + ...
π π2 π π2
a) 8 b) 8 c) 12 d) 12
8
Part-B—(5 x 5 = 25 marks)
Answer all the Questions choosing either (a) or (b)
11. a) Expand cos6 θ in cosine multiples of θ.

b) Expand sin4 θcos2 θ in cosine multiples of θ.
x2 y2
12. a) If sin(A + iB) = x + iy prove that (i) sin2 A − cos2 A =1
x2 y2
(ii) cosh2 B + sinh2 B = 1.
b) Find logsin(θ + iφ).

−1 cs+d
13. a) Find L (s+a)2 +b2 .

Rx
b) Prove that L−1 F (s) s = L−1 [F (s)]dx.
0
14. a) Solve dx
dt + ax = y; dy
dt + ay = x given x = 0, y = 1 when t = 0.
b) Solve y + 2y 0 + 5y = 3e−x given y(0) = 0, y 0 (0) = 3.
00
π sin2x sin4x sin6x

15. a) In (0, π) show that π − x = 2 + 1 + 2 + 3 + ...
b) Expand f (x) = x, −π < x < π in fourier series.
Part-C—(5 x 8 = 40 marks)
16. a) Expand cos5 θsin3 θ in series of sine multiples of θ.

1
b) Prove that cosn θ = 2n−1 [cos nθ+nC1 cos(n−2)θ+nC2 cos(n−4)θ+. . . ]
1
17. a) Find (i) log(1 + eiθ ) (ii) log( 1−e iθ ).
b) If logcos(α + iβ) = x + iy prove (i) x = 12 log( cos2α+cos2β

2 )
(ii) y = 12 [logsin(α + y) − logsin(α − y)].
18. a) If L(f (x)) = F (s) find (i) L(e−ax f (x)) (ii) L(e−4x x2 ) (iii) L(e−ax cosbx).

−1 s+2
b) Find L (s2 +4s+5)2 .
19. a) Solve y 00 − 2y 0 + y = xex given y(0) = y 0 (0) = 0.

b) Solve xy 00 − (2 + x)y 0 + 3y = x − 1 when y(0) = 0.
20. a) Expand f (x) = ex as fourier series in (0, 2π).

b) Expand y = cos 2x as series of sines in (0, π).
9
Code No: 41386 Sub. Code: SSMA4A

April 2019
Fourth Semester
Mathematics-Main
(For those who joined in July 2017 and afterwards)
1. When θ is expressed in radians then tanθ =

θ3 θ5 θ2 θ4 θ3 2θ 5
a) θ − 3! + 5! − . . . b) 1 − 2! + 4! − . . . c) θ + 3! + 15 + . . . d)
2. If cos0 = θ then θ =
a) 0.73 b) 0.51 c) 1.01 d) 0.23
3. The value of cosh2x =

a) cosh2 x − sinh2 x b) cosh2 x − sinh2 x c) 2sinhxcoshx d) 1 + 2cosh2 x
4. a) cos(ix) = a) icoshx b) coshx c) −icoshx d) −coshx
1 1 1 1
5. L(eat ) = a) s+a b) s2 +a2 c) s2 −a2 d) s−a
6. L−1 ( s12 ) = a) 0 b) 1 c) t d) eat

2 3 3 2
7. L(t2 eat ) = a) (s−a)2 b) (s−a)4 c) (s−a)3 d) (s−a)3
1
8. L−1 ( (s+a)2) =
a) te−at b) t2 eat c) teat d) t2 e−at
9. If f (x) is an odd function then f (x)cosnx is ———- function

a) positive b) negative c) odd d) even
10. An example of even function is

a) x b) |x| c) x + x2 d) x + x3
10
11. a) Solve approximately cos( π3 + θ) = 0.49.

π sinx+cos2x
b) Evaluate x → 2 cos2 x .
1 cosec θ2 π
12. a) Prove log( 1−e iθ ) = log( 2 ) + i(2nπ + 2 − θ2 ).
b) Find the sum to infinity the series 1+cosθcosθ+cos2 θcos2θ+cos3 θcos3θ+
...
R∞
13. a) Evaluate te−3t costdt.
0

−1 1
b) Find L s(s2 +a2 ) .
d2 y dy dy 2
14. a) Solve dt2 − dt = sint subject to y = 2, dt = 0, ddt2y = 1 when t = 0.
2
b) Solve t ddt2y − (2 + t) dy
dt + 3y = t − 1 when y(0) = 0.
15. a) Determine fourier expansion of function f (x) = x where −π ≤ x ≤ π.
b) Obtain cosine series for f (x) = ex in 0 < x < π.
sin 7θ
16. a) Expand sin7θ in powers of cosθ and sinθ. Hence Prove that sinθ =
7 − 56sin2 θ + 112sin4 θ − 64sin6 θ.
tanθ 2524
b) If θ = 2523 show that θ is approximately equal to 1◦ 580 .
17. a) If sin(θ + iφ) = tanα + isecα then show that cos2θcosh2φ = 3.
sinθ sin2 θ
b) sum to infinity the series coshθ + 1! cosh2θ − 2! cosh3θ + ...
18. a) Find (i) L(sin2 2t) (ii) L( sinat
t ).

1
b) Find (i) L−1 s(s+a) (ii) L−1 (s+1)(s21+2s+2) .
dy
19. a) Solve the simultaneous equation x dx
dt + dt + 2x = 1, dx
dt + t dy
dt + 3y = 0
given x = 0, y = 0 when t = 0.
dy
R1
b) Determine which satisfy the equation dt + 3y + 2 ydt = 0 given
0
y(0) = 0.
(
π + 2x −π < x < 0
20. a) Find the fourier series of the function f (x) =
π − 2x 0<x<π
1 1 1 π2
Hence deduce that 12 + 32 + 52 + ∙∙∙ = 8 .
b) Find the half range cosine series of the function f (x) = x2 in 0 ≤ x < π
and hence find the sum of the series 1 − 212 + 312 − 412 + . . .
11
Code No: 40353 Sub. Code: JSMA4A

November 2019
Fourth Semester
Mathematics-Main
(For those who joined in July 2016 only)
1. cos 4θ + 4cos2θ + 3 =
a) 23 cos4 θ b) cos4 θ c) 24 cos4 θ d) 2cos4 θ
2. 16sin4 θ − 20sin2 θ + 5 =
sin5θ
a) sin5 θ b) sinθ c) sin4θ d) sin4θ
sinθ
3. cos(ix) =
a) cosx b) icosx c) cosh x d) icosh x
4. log(−1) =
a) i2nπ b) −i2nπ c) 0 d) i(2n + 1)π
5. L(tcost) =
1 s2 s2 −1 s2 −1
a) s2 +1 b) s2 +1 c) s2 +1 d) (s2 +1)2
t2 t4
6. L−1 ( s12 ) = a) t
2 b) 2 c) t3 d) 3
7. If y(0) = y 0 (0) = 0 then L(y 00 ) =

a) 0 b) 1 c) s2 L(y) d) sL(y)
1
8. L−1 ( s−2 )=
a) t − 2 b) e3t c) e−4t d) 2et
9. If f (x) is an even function then

a) f (x) = f (−x) b) f (x) = −f (−x) c) f (x) = f (x2 ) d) f (x) = f (f (x))
10. If f (x) is an odd function defined in (−l, l) then in the fourier series
expansion a0 = a) 0 b) l c) 2l d) π2
12
11. a) Expand cos 5θ in terms of powers of θ.

b) Prove 25 cos6 θ = cos 6θ + 6cos 4θ + 15cos2θ + 10.
√
12. a) Prove that cosh−1 (x) = loge (x + x2 − 1).
b) Separate real and imaginary parts of tan−1 (x + iy).
t
13. a) Find L−1 ( 1−e
t ).

b) Find L−1 (s+1)(s21+2s+2) .
d2 y
14. a) Solve dt2 + 2 dy
dt + 5y = 4e
−t
given that y(0) = y 0 (0) = 0 when t = 0.
d2 y
b) Solve dt2 + 4y = Asin kt given that y(0) = y 0 (0) = 0 when t = 0.
15. a) Find the fourier sine series of f (x) = x in 0 < x < 2.
(
π + 2x −π < x < 0
b) Find fourier expansion of f (x) =
π − 2x 0 ≤ x < π
sin 7θ
16. a) Prove that sinθ = 7 − 56sin2 θ + 112sin4 θ − 64sin6 θ.
5 3
b) Expand cos θsin θ with the series of sines of multiples of θ.
17. a) If sin(θ + iφ) = tanα + isecα then show that cos2θcosh2φ = 3.
sinθ sin2θ
b) Prove that 1! + 2! + ∙ ∙ ∙ = ecosθ sin(sinθ).
R∞
d
18. a) Prove (i) L(tf (t)) = − ds F (s) (ii) L( f (t)
t )= f (s)ds
0

s+2 s
b) Find (i) L−1 (s2 +4s+5)2 (ii) L−1 (s+3)2 +4
d2 y
19. a) Using Laplace Transform solve dt2 + 2 dy
dt − 3y = sin t given that
y = dy
dt = 0
dy d2 x
b) Solve dx
dt − dt − 2x + 2y = 1 − 2t and dt2 + 2 dy
dt + x = 0 given that
dx
x = 0, y = 0, dt = 0 when t = 0.
20. a) Find the fourier series of f (x) = x2 in (−π, π).
b) Find the fourier cosine series of f (x) = π − x in (0, π).
13

November 2019
Fourth Semester
Mathematics-Main
(For those who joined in July 2017 onwards)
1. The coefficient of cosn θ in the expansion of cos nθ is

a) 2n b) 2n−1 c) 2n+1 d) 2n − 1
2. The coefficient of cosθ in the expansion of 2n−1 cosn θ is
a) nC n−1 b) 12 nC n−1 c) nCn/2 d) 12 nCn/2
2 2
3. tan(ix) =
a) tanh x b) itanh x c) 1i tanh x d) −itanh x
4. log 1=
a) 1 b) 0 c) i2nπ d) nπ
5. L(e2t ) =
1 1 1
a) s+2 b) s c) s−2 d) 1
a
6. L−1 ( s2 +a 2) =
a) cos at b) sin at c) cosh at d) sinh at

7. L(t) =
1 1
a) 0 b) 1 c) s d) s2
1
8. L−1 ( s−3 )=
a) e3t b) e−3t c) te3t d) te−3t
9. Fourier coefficient of a3 for f (x) = x3 in (−π, π) is
π2 2π 2 2π 3
a) 0 b) 3 c) 3 d) 3
10. The fourier coefficient a0 for f (x) = ex in (0, 2π) is

e2π −1 e2π −1 eπ −e−π
a) 0 b) 2π c) π d) π
14
11. a) Write cos 8θ in terms of sin θ.

b) Expand sin4 θcos2 θ in a series of cosines of multiples of θ.
x2 y2
12. a) Prove that sin2 A − cos2 A = 1 if sin(A + iB) = x + iy
b) Separate into real and imaginary parts of tan−1 (x + iy).
13. a) Find the value of L(te−tcost ).
b) Find the value of L( 1−cost
t ).
14. a) Find L−1 ( (s+1)(s21+2s+2) ).

b) Using laplace transform solve y 0 − 5y = 0, y(0) = 2.
(
−1 −π < x < 0
15. a) Find the fourier series for f (x) =
1 0<x<π
b) Find the the sine series for f (x) = x in (0, π).
16. a) Prove that cos π9 cos 2π 4π 1

9 cos 9 = 8 .
b) Give the expansion of cosn θ when n is positive integer.
17. a) Prove that u = log tan( π4 + θ2 ) if and only if cosh u = secθ.

b) Find sin α + sin(α + β) + ∙ ∙ ∙ + sin(α + (n − 1)β).
−t
−1
18. a) Find (i) L[ sint at ] (ii) L[ e 1 ].
1 s
b) Find (i) L−1 [ s(s+1)(s+2) ] (ii) L −1
[ (s+2) 2 ].
19. a) Using laplace transform solve y 00 +4y 0 +1y = 2e−t , y(0) = 0, y 0 (0) = −1.
b) Using laplace transform solve dx
dt + y = sint; dy
dt + x = cost given
x(0) = 2, y(0) = 0.
π2
P
∞
20. a) Show that x2 = 3 +4 (−1)n cosn2nx in the interval −π < x < π.
n=1
b) Find the fourier cosine series for the function f (x) = π − x in (0, π).
15

November 2020
Fourth Semester
Mathematics-Main
1. When θ is expressed in radians then cosθ =

3 5 2 4 3 5 2 3
a) θ− θ3! + θ5! −. . . b) 1− θ2! + θ4! −. . . c) θ+ θ3! + 2θ 2θ 3θ
15 +. . . d) θ+ 2! + 3! +. . .
2. sinθ( π6 + θ) =
a) 0.73 b) 1.001 c) 0.35 d) 0.51
3. The value of cosh( iπ
2 )=
a) 0 b) 1 c) −1 d) i
4. The value of tan(ix) =
a) tanh x b) itanh x c) 1i tanh x d) −itanh x
5. L(t) =
1 1 1 1
a) s b) s2 c) s3 d) s4
s
6. L−1 ( s2 +a 2) =
a) cosh at b) sinh at c) cosat d) sinat

7. L(te−at ) =
1 1 1 2
a) s2 +a2 b) (s+a)2 c) (s+a)3 d) (s+a)3
1
8. L−1 ( (s−a)2) =
a) eat b) cosh at c) te−at d) teat

9. An example of odd function is
a) xsinx b) x2 cosx c) x2 sinx d) sinx + cosx
10. For any even integer the value of cos nπ is
a) 1 b) 0 c) −1 d)(−1)n
16
11. a) Prove that sin5 θ = ( 214 )[sin5θ − 5sin3θ + 10sinθ].

θ2 θ4
b) If θ is small prove that θcotθ = 1 − 3 − 45 .
12. a) Prove that (cosh x + sinh x)n = cosh nx + sinh nx.

cosθ cos2θ cos3θ
b) Sum the series to infinity 1 + cosθ + cos2 θ + cos3 θ + ...
13. a) Find L(sin2 2t).
1
b) Find L−1 ( s(s+1)(s+2) ).
d2 y
14. a) Solve the equation dt2 + 4 dy dy
dt − 5y = 5 given that y(0) = 0, dt = 2 when
t = 0.
d2 y
b) Solve the equation + t dy
dt2
0
dt − y = 0 if y(0) = 0, y (0) = 1.
(
π + 2x −π < x < 0
15. a) Find the fourier series of the function f (x) =
π − 2x 0 < x < π
b) Find the fourier sine series for f (x) = k in 0 < x < π.
16. a) Prove that cos8θ = 128cos8 θ − 256cos6 θ + 160cos4 θ − 32cos2 θ + 1.

θ3 2 θ5 4 θ7
b) Prove that when θ is small 16 sin3 θ = 2
3! −(1+3 ) 5! +(1+3 +3 ) 7! +. . .
17. a) Prove that u = logtan( π4 + θ2 ) if and only if cosh u = secθ.
b) If tan(θ + iϕ) = cosα + isinα prove that (i) θ = 12 nπ + 14 π
(ii) ϕ = 12 log tan( 14 π + 12 α).
Z∞ −t
e − e−2t
18. a) Evaluate ( )dt.
t
0

s+2
b) Find L−1 (s2 +4s+5) 2 .
dy dy
19. a) Solve simultaneous equation 3 dx dx
dt + dt + 2x = 1; dt + 4 dt + 3y = 0 given
x = y = 0 at t = 0.
dy dy 2
b) Solve simultaneous equation dx d x
dt − dt −2x+2y = 1−2t; dt2 +2 dt +x = 0
dx
with conditions x = y = 0; dt = 0 when t = 0.
20. a) Find the fourier series for f (x) = |sinx| in (−π, π) of period 2π.
b) Find the half range sine series for f (x) = x(π − x) in (0, π). Deduce
3
that 113 − 313 + 513 − ∙ ∙ ∙ = π32 .
17

April 2021
Fourth Semester
Mathematics-Main
1. 10 = ——- radian
π π
a) π b) 180 c) 90 d) 2π
√
2. sin h−1 x = ———— a) loge (x + x2 − 1)
√ √ √
b) ±loge (x+ x2 − 1) c) loge (x+ x2 + 1) d) ±loge (x+ x2 + 1)
3. When θ is expressed in radians, sinθ =
θ3 θ5 θ2 θ3
a) θ − 3! + 5! + ... b) 1 + θ − 2 + 3 − ...
θ2 θ3 θ3 2θ 5
c) 1 + θ + 2 + 3 + ... d) θ + 3 + 15 + ...
4. tan hx = —-
a) tan x b) tan ix c) i tan ix d) −i tan ix
5. L(e2x ) =
1 1 1
a) s+2 b) s−2 c) s d) 1
6. L−1 ( 1s ) =
1
a) 1 b) 0 c) x d) x
7. L(sinh ax) =
a a a a
a) s2 b) (s+a)2 c) s2 −a2 d) s2 +a2
8. L−1 ( a2 a2s+b2 ) =
a) acos bx b) a1 cos bx c) a2 cos( bx
a ) d) 1 bx
a2 cos( a )
9. f (x) is an even function then f (−x) =

a) f (x) b) −f (x) c) f (x2 ) d) −f (x2 )
18
10. In the interval (−π, π) the fourier coefficient an =
Rπ Rπ Rπ Rπ
a) π1 1
f (x)cos nxdx b) 2π f (x)cos nxdx c) π1 cos nxdx d) 1
2π cos nxdx
−π −π −π −π
11. a) Prove that cos nθ = cosn θ − nC2 cosn−2 θsin2 θ + . . .

b) Prove that 25 cos6 θ = cos6θ + 6cos4θ + 15cos2θ + 10.
1+tanh x
12. a) Prove that 1−tanh x = cosh 2x + sinh 2x
b) If cosh u = secθ prove that u = loge ( π4 + θ2 ).
13. a) Find L(t2 + cos2tcost + sin2 t).
b) Find L−1 [log( s+a
s+b )].
14. a) Using Laplace transform solve y 0 + 3y = e−2x given y(0) = 4.

1
b) Find L−1 [ s(s+1)(s+2) ].
15. a) Find the sine series for the function f (x) = k, 0 < x < π.
b) Find the cosine series for the function f (x) = π − x in interval (0, π).

1
b) Prove cosn θ = 2n−1 [cos nθ + nC1 cos(n − 2)θ + nC2 cos(n − 4)θ + . . . ]
1 cosec θ2 π
17. a) Prove that log ( 1−e iθ ) = log( 2 ) + i(2nπ + 2 − θ2 ).
b) Find the sum to infinity the series 1+cosθcosθ+cos2 θcos2θ+cos3 θcos3θ+
...
s
18. a) (i) Prove L[f 00 (x)] = s2 L[f (x)] − sf (0) − f 0 (0) (ii) Find L−1 [ (s+2) 2 ].
b) Find value of L−1 [ (s+2)1+2s

2 (s−1)2 ].
19. a) Solve using laplace transform y 00 +4y 0 +13y = 2e−x , y(0) = 0, y 0 (0) = −1
b) Solve laplace transform dx
dt + y = sint, dy
dt + x = cost, x(0) = 2, y(0) = 0
π2
P
∞ n
20. a) Show that x2 = 3 +4 [ (−1) ncos
2
nx
] in −π < x < π. Deduce 1
12 −
n=1
2
1 1 π
22 + 32 − ∙∙∙ = 12 .
b) Find the fourier expansion of f (x) = x in the interval (−π, π).
19
Code No: 20318E Sub. Code: SSMA4A

November 2021
Fourth Semester
Mathematics-Main
1. When θ is expressed in radians cos θ = ———-

θ2 θ4 θ3 θ5
a) 1 − + − ... b) θ − + − ...
2! 4! 3! 5!
θ3 2θ5 θ2
c) θ + + − ... d) 1 + θ + + ...
3 15 2!
sinθ 863
2. The approximate value of θ if = is ——— radians
θ 864
1 1 1 1
a) b) c) d)
11 12 13 14
3. sin(ix) = —— a) sinh x b) cosh x c) isinh x d) icosh x
4. log 1 = ——- a) nπ b) 2nπ c) inπ d) i2nπ
1 1 2 3
5. L(x) = ——- a) b) c) d)
S S2 S3 S4
1
6. Value of L−1 ( ) is a) e−ax b) eax c) sin ax d) cos ax
s−a
7. SL(f (x)) − f (0) is a) f (x) b) L(f (x)) c) L(f 0 (x)) d) 0
L−1 (F (s))
8. L−1 [f (s+a)] is a) eax L−1 [f (x)] b) e−ax L−1 [f (x)] c) d) aL−1 (F (s))
a
9. An example of even function is a) x b) x3 c) x2 d) sinx
10. An example of odd function is
a) xsinx b) x2 cosx c) x2 sinx d) sinx + cosx
20
11. a) Prove that 25 cos6 θ = cos 6θ + 6cos 4θ + 15cos 2θ + 10.

b) Prove that cos nθ = cosn θ − nC2 cosn−2 θsin2 θ + . . .
π θ
12. a) If cosh u = secθ prove that u = loge tan( + ).
4 2
1 + tanh x
b) Prove that = cosh2x + sinh2x.
1 − tanh x
13. a) Find L(xe−x cosx).
Zx
−1 1
b) Show that L F (s) = L−1 [F (s)]dx.
s
0
14. a) Using Laplace transform solve y 0 + 3y = e−2x given y(0) = 4.

cs + d
b) Find L−1 .
(s + a)2 + b2
15. a) Find the sine series for the function f (x) = π − x in 0 ≤ x ≤ π.
b) Find the cosine series for f (x) = ex in 0 < x < π.

1
b) Prove that cosn θ = n−1 [cos nθ+nC1 cos(n−2)θ+nC2 cos(n−4)θ+. . . ].
2
17. a) If ia+ib = a + ib prove that a2 + b2 = e−(4n+1)πb .
cosx cos 2x cos 3x
b) Find the sum to infinity the series 1 + + + +...
cosx 2!cos2 x 3!cos3 x

1 + 2s
18. a) Find the value of L−1 .
(s + 2)2 (s − 1)2

1 − cos x
b) Find L .
x
19. a) Solve by laplace transform y 00 + 4y 0 + 13y = 2e−x , y(0) = 0, y 0 (0) = −1.
dx dy
b) Solve by laplace transform +y = sint, +x = cost, x(0) = 2, y(0) =
dt dt
0.
20. a) Find the fourier expansion of f (x) = x in (−π, π).
X∞
π2 (−1)n cos nx
b) Show that x2 = +4 [ ] in −π < x < π. Deduce
3 n=1
n2
1 1 1 π2
− + − ∙ ∙ ∙ = .
12 22 32 12
21
Code No: 30354E Sub. Code: ASMA41

April 2022
Fourth Semester
Mathematics-Main
sin nθ
1. The coefficient of cosn−1 θ in the expansion of is
sinθ
a) 2n b) 2n−1 c) 2n − 1 d) 0
1
2. If x = cosθ + isinθ then the value of xn + is
xn
a) 2cosθ b) 2sinθ c) 2cos nθ d) 2sin nθ
3. The value of cosh2 x+sinh2 x is a) 1 b) 0 c) cosh 2x d) sinh 2x

√
4. The value of loge (x + x2 − 1) is a) cosh−1 x b) sinh−1 x c) 0 d) 1
1 1 s s
5. The value of L(eat ) is a) b) c) d)
s+a s−a s+a s−a
1 1
6. The value of L−1 [ ] is a) 0 b) t c) s d)
s2 s
s 2 − a2
7. The value of L−1 [ ] is
(s2 + a2 )2
a) tsin at b) tsin t c) tcos at d) tcos t
8. If L[f (t)] = F (s) then F 0 [s] is

a) L[f 0 (t)] b) L[tf (t)] c) L[−tf (t)] d) 0
9. The function sin x is periodic with period

a) π b) 2π c) 3π d) 0
10. Which one of the following is an odd function ?

a) x2 b) x2 + 1 c) ex + e−x d) sinx
22
11. a) Express cos 8θ in terms of sin θ.

b) Expand sin7 θ in a series of sines of multiples of θ.
12. a) Separate into real and imaginary parts tanh(1 + i).
b) Find Log(1 − i).
13. a) Find L[sin3 2t].
s
b) Find L−1 [ 2 ].
(s − 1)2
Z∞
14. a) Evaluate te−3t cos tdt.
0
1
b) Find L−1 [ ].
(s − 1)(s + 3)(s2 + 1)
15. a) Obtain the Fourier series for the function f (x) = ex from x = 0 to 2π.
b) Find the half range sine series for f (x) = 1 in [0, l].
π 2π 4π 1
16. a) Show that cos .cos .cos = .
9 9 9 8
b) Expand sin4 θcos2 θ in a series of cosines of θ.
c2 c3
17. a) Find sum to infinity the series csinα + sin2α + sin3α + . . .
2! 3!
b) Separate into real and imaginary parts tan−1 (x + iy).
Γ(n + 1) 2
18. a) Prove that L(tn ) = and hence deduce L[t2 ] = 3 .
sn+1 s
1
b) Find L−1 [ ].
(s + 1)(s2 + 2s + 2)
d2 y dy dy
19. a) Solve +2 − 3y = sin t given that y = = 0 when t = 0.
dt2 dt dt
dx dy
b) Solve + y = sin t, + x = cos t given that x(0) = 2, y(0) = 0.
dt dt
2 4 cos 2x cos 4x
20. a) Prove that |sin x| = − ( + + . . . ).
π π 3 15
1 1 π2
b) Obtain fourier expansion of f (x) = (π −x)2 , deduce 2 + 2 +∙ ∙ ∙ = .
1 2 6
23
Code No: 20077E Sub. Code: ASMA41

November 2022
Fourth Semester
Mathematics-Main
1. The value of (cosθ + isinθ)n is

a) 0 b) 1 c) cosθ + isinθ d) sinθ + icosθ
1
2. If cosθ + isinθ = x then the value of x + is
x
a) 2cosθ b) 2isinθ c) 2icosθ d) 2sinθ
3. The value of 2sinhx coshx is a) 0 b) 1 c) cosh2x d) sinh2x
√
4. The value of loge (x + x2 + 1) is
a) sin hx b) cos hx c) sinh−1 x d) cosh−1 x
1 1 2
5. L(1) = a) b) c) d) 0
s s2 s3

−1 1
6. The value of L (s+a)2 is
a) e−at b) e−at t c) eat t d) eat

7. The value of L(te−at ) is
1 1 s s
a) b) c) d)
s+a (s + a)2 s+a (s + a)2

−1 2
8. The value of L (s−a)3 is a) teat b) t2 eat c) tet d) t2 et
9. The function tanx is periodic with period

a) 0 b) 2π c) π d) 3π
10. Which one of the following is an even function
a) x b) x3 c) sinx d) ex + e−x
24
sin 6θ
11. a) Express in terms of cosθ.
sin θ
b) Expand cos6 θ in series of cosines of multiples of θ.
√
12. a) Prove that sinh−1 x = loge (x + x2 + 1).
b) Find log(1 − i).
13. a) Find L[sin2 2t].
s
b) Find L−1 [ 2 ].
(s + a2 )2
Z∞
14. a) Evaluate e−2t sin 3tdt.
0
1 + 2s
b) Find L−1 [ ].
(s + 2)2 (s − 1)2
15. a) Express f (x) = x as fourier series in −π < x < π.
b) Obtain half range sine series of ex in [0, 1].
ah bk
16. a) Prove that − = a 2 − b2 .
cosθ sinθ
1
b) Show that sin3 θcos5 θ = .
27 (sin 8θ + 2sinh6θ2sin4θ − 6sin2θ)
π θ
17. If coshu = secθ show that u = logtan( + ).
4 2
b) Find general value of log(−3) (−2).
18. a) Find (i) L[cos at] (ii) L[sinhat].
s+1 2sin ht
b) Prove that L−1 [log ]= .
s−1 t
d2 y dy dy
19. a) Solve 2
+2 − 3y = sint given that y= = 0 when t = 0
dt dt dt
dx dy dx dy
b) Solve 3 + + 2x = 1, +4 + 3y = 0 given that x = y = 0 at
dt dt dt dt
t = 0.
20. a) Explain fourier series for odd and even function.

2 4 2x 4x
b) Prove that |sinx| = − cos + cos + ... .
π π 3 15
25

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Trigonometry University Questions

1. tan(iθ) = a) −itanhθ b) i2 tanθ c) itan2 θ d) itanhθ

3. cosθ + isinθ is a) eiθ b) e−iθ c) e2θ d) e−2θ

5. sin(ix) = a) isinhx b) −isinhx c) sinhx d) −sinhx

9. The value of cosh ( iπ

10. If z and w are complex numbers then z w =

11. If z is real then log z is a) real b) Imaginary c) Complex d) 0

12. cosh(ix) is a) cosx b) sinx c) coshx d) sinhx

2. Find the value of log(1 − i).

6. Sum to infinity the series cosα + 12 cos(α + β) + 12 . 34 cos(α + 2β) + . . .

9. Show that ii is real.

12. Show that ii is real.

16. Expand cos6 θ in cosine multiples of θ.

18. Find logsin(θ + iφ).

20. Find log(1 + eiθ ).

1. Prove that 2e2L = cosh2φ − cos2θ if logsin(θ + iφ) = L + iB.

3. Find value of (i) log(−3) (2) (ii) log(1 + i)

5. Find the value of sinh−1 x and tanh−1 x.

6. Find the sum to n terms of the series sin2 + sin22 + sin32 + . . .

6. L−1 [ s2s−4 ] = a) cos 2t b) sin4t c) cosh 2t d) sinh 4t

13. L−1 ( s2s+4 ) = a) cos 3t b) cos 2t c) sin 2t d) tan 2t

16. L(t2 + cos3t) is

18. L(tsinh at) =

1 e3t t3 e−3t t3 e3t t2 e3t t4

1. Find L(te−t sint).

3. Find (i) L(cos3 2t) (ii) L(sin2 3t).

7. Find L(xe−x cosx).

12. Find L( 1−t sinat

2. Using Laplace Transform solve y + 2y + 5y = 4e , given y = y 0 = 0

10. Solve using laplace transform (D2 + 4)y = t, y(0) = y 0 (0) = 0.

1. The fourier coefficient a0 for f (x) = ex in (0, 2π) is

2. Find sine series for f (x) = k in 0 < x < π.

B.Sc (CBCS) DEGREE EXAMINATION

Skill Based - Trigonometry, Laplace Transforms and Fourier Series

(For those who joined in July 2016 only)

Time: 3 hrs Maximum Mark: 75

7. L−1 (F (s + a)) =———- L−1 (F (s))

Answer all the Questions choosing either (a) or (b)

11. a) Expand cos6 θ in cosine multiples of θ.

π sin2x sin4x sin6x

Answer all the Questions choosing either (a) or (b)

16. a) Expand cos5 θsin3 θ in series of sine multiples of θ.

b) If logcos(α + iβ) = x + iy prove (i) x = 12 log( cos2α+cos2β

19. a) Solve y 00 − 2y 0 + y = xex given y(0) = y 0 (0) = 0.

20. a) Expand f (x) = ex as fourier series in (0, 2π).

B.Sc (CBCS) DEGREE EXAMINATION

Skill Based - Trigonometry, Laplace Transforms and Fourier Series

(For those who joined in July 2017 and afterwards)

Time: 3 hrs Maximum Mark: 75

1. When θ is expressed in radians then tanθ =

3. The value of cosh2x =

6. L−1 ( s12 ) = a) 0 b) 1 c) t d) eat

a) te−at b) t2 eat c) teat d) t2 e−at

9. If f (x) is an odd function then f (x)cosnx is ———- function

10. An example of even function is

B.Sc (CBCS) DEGREE EXAMINATION

Skill Based - Trigonometry, Laplace Transforms and Fourier Series

(For those who joined in July 2016 only)

Time: 3 hrs Maximum Mark: 75

7. If y(0) = y 0 (0) = 0 then L(y 00 ) =

9. If f (x) is an even function then

11. a) Expand cos 5θ in terms of powers of θ.