Download Mathematics I-ECS - E

Nehru Arts & Science College
ECS Question Paper
Unit I
Section A
1.If 𝑟⃗ = x 𝑖⃗+y𝑗⃗+z𝑘⃗⃗ then ∇.𝑟⃗ is
(a) 3 (b) 0 (c)-3 (d) none of these
2. A vector 𝐹⃗ is irrotational when
(a) ∇.𝐹⃗ = 0 (b)∇2𝐹⃗ = 0 (c) ∇ × 𝐹⃗ = 0 (d) ∇𝐹⃗ = 0
3.If 𝐴⃗ = x 𝑦𝑖⃗+yz𝑘⃗⃗ then ∇.⃗⃗⃗⃗
𝐴 at (2,2,3) is
(a) 3 (b) 6 (c) 7 (d) 4
4.∇ 𝑟 = ------------------(a) 0 (b)
⃗⃗
x 𝑖⃗+y𝑗⃗+z𝑘
𝑟
(c)
1
𝑟
(d) x 𝑖⃗+y𝑗⃗+z𝑘⃗⃗
5. The derivative of a constant vector is
(a) -1 (b) 0 (c) 1 (d) none
6.A vector is said to be _____________- if its divergence is zero
(a) irrotational (b) solenoidal (c) gradiant (d) curl
7.∇ (uv) is equal to_______________
(a) u∇v+v∇u (b) u∇u+v∇v (c) ∇u+∇v (d) none
8.div(grad 𝛷) is__________(a 0 (b) 1 (c) -1 (d) ∇2𝛷
9. If 𝛷 =xyz2 then ∇𝛷 at (1,0,3) is
(a) 4𝑗⃗ (b) 2𝑖⃗ -3𝑗⃗ (c) 9𝑗⃗ (d) 𝑖⃗-𝑗⃗-2𝑘⃗⃗
10. If 𝐹⃗ = (x+y+1) 𝑖⃗+𝑗⃗+(-x-y)𝑘⃗⃗ then ∇ × 𝐹⃗ is
(a) 𝑖⃗+𝑗⃗+ 𝑘⃗⃗ (b) - 𝑖⃗+𝑗⃗+𝑘⃗⃗ (c) 2𝑖⃗+𝑗⃗ (d) - 𝑖⃗ +𝑗⃗-𝑘⃗⃗
11.div curl 𝐹⃗ is_____________
(a)0 (b) 1 (c) )∇2𝐹⃗ (d)∇ × (∇ × 𝐹⃗ )
12.If 𝐹⃗ =xz2 𝑖⃗ +xy2𝑗 +yz2 𝑘⃗⃗ then ∇ × 𝐹⃗ at (1,2,1) is
(a) 𝑖⃗-2𝑗⃗+4𝑘⃗⃗ (b) 0 (c) ) 𝑖⃗+2𝑗⃗+4𝑘⃗⃗ (d) ) − 𝑖⃗+2𝑗⃗+4𝑘⃗⃗
13. .If 𝑟⃗ = x 𝑖⃗+y𝑗⃗+z𝑘⃗⃗ then ∇ × 𝑟⃗ is
(a) 𝑟⃗ (b) 3 (c) 𝐴⃗ (d) 0
14.The value of curl grad 𝛷 ______________
(a) ∇ 𝛷 (b) ∇. ∇ 𝛷 (c ) 1 (d) 0
15.Let u and v be two scalar point functions ∇(u+v) is __________
(a) u∇v (b) v∇u (c) ∇u+∇v (d) u∇v+v∇u
16.A vector 𝐹⃗ is solenoidal when
(a) ∇.𝐹⃗ = 0 (b)∇2𝐹⃗ = 0 (c) ∇ × 𝐹⃗ = 0 (d) ∇𝐹⃗ = 0
17.If 𝐹⃗ = xz3 𝑖⃗-2xyz𝑗⃗+xz𝑘⃗⃗ then div 𝐹⃗ is ________
(a) x3-2yz+x
(b) y3-2xz+x (c) z3-2xz+x (d) 0
18. If 𝛷 = xyz then ∇ 𝛷 =_________
⃗⃗⃗⃗⃗ (b) 2x 𝑖⃗+2y𝑗⃗+2z𝑘⃗⃗ (c) x 𝑖⃗+y𝑗⃗+z𝑘⃗⃗
(a) x𝑦 𝑖⃗+y𝑧𝑗
⃗⃗⃗⃗+z𝑥𝑘
19. Given 𝛷(x,y,z)=x+y+z , ∇ 𝛷 is
(a) 3 (b) 0 (c) x 𝑖⃗+y𝑗⃗+z𝑘⃗⃗ (d) 𝑖⃗+𝑗⃗+ 𝑘⃗⃗
(d) yz 𝑖⃗+xz𝑗⃗+xy𝑘⃗⃗
20. If V is the volume bounded by a closed surface S and 𝑓⃗ is a vector valued function having
continuous partial derivatives then ∬ 𝑓⃗ .𝑛̂ ds = ∭ ∇ . 𝑓⃗ dv this theorem is known as __________𝑠
𝑟
(a) Green’s theorem (b) stoke’s theorem (c) laplace theorem (d) Gauss’s divergence theorem
21. If 𝑓⃗ is a constant vector then curl 𝑓⃗ is______(a) 0 (b) 1 (c) neither o nor 1 (d) none
22. A vector is said to be ____________ if its curl is zero
(a)solenoidal (b) divergence (c) scalar (d) irrotational
23.If 𝐹⃗ = 𝑖⃗+𝑗⃗+ 𝑘⃗⃗ then div 𝐹⃗ is __________
(a) 1 (b) 0 (c) 2 (d)4
24. The region in which the physical property is specified is called a _________-(a) scalar (b) vector (c) field (d) del
25.Fields are of two kinds _______- and __________
(a) scalar, vector (b) scalar,del (c) vector,curl (d) curl, divergence
Section B
1. Solve (D2+3D+2)y = x2
2. Solve D2-4D-5)y = 4cos3x
3. If 𝐹̅ = 𝑥𝑧 3 𝑖̅ − 2𝑥 2 𝑦𝑗̅ + 2𝑦𝑧 4 𝑘̅ find ∇, 𝐹̅ and ∇ × 𝐹̅ at the point (1, -1, 1)
4. Find the directional derivative ∅ = 𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥 in the direction of the vector
0)
𝑖̅ + 2𝑗̅ + 2𝑘̅ at (1, 2,
5.Compute the directional derivative of ∅ = 𝑥 2 + 𝑦 2 + 4𝑥𝑦𝑧 at (1, -2, 2) in the direction 2𝑖̅ − 2𝑗̅ + 𝑘̅
6. If 𝐹̅ = 𝑥 2 𝑧𝑖̅ − 2𝑦 3 22 𝑗̅ + 𝑥𝑦 2 2𝑘̅ then find 𝑑𝑖𝑣𝐹̅ and 𝑐𝑢𝑟𝑙𝐹̅ at (1, -1, 1)
7. Find the directional derivative of ∅ = 𝑥 + 𝑥𝑦 2 + 𝑦𝑧 3 at (0.1, 1) in the direction of 2𝑖̅ + 2𝑗̅ − 𝑘̅
8. If 𝐹̅ = (3𝑥 2 − 3𝑦𝑧)𝑖̅ + (3𝑦 2 − 3𝑥𝑧)𝑗̅ + (3𝑧 2 − 3𝑥𝑦)𝑘̅. Find ∇. 𝐹̅ and ∇ × 𝐹̅
9. If 𝑟⃗ = 𝑥𝑖⃗ + 𝑦𝑗⃗ + 𝑧𝑘⃗⃗ then prove that ∇𝑟 𝑛 = 𝑛𝑟 𝑛−2 𝑟̅
10.𝐹̅ = (3𝑥 2 − 3𝑦𝑧)𝑖̅ + (3𝑦 2 − 3𝑥𝑧)𝑗̅ + (3𝑧 2 − 3𝑥𝑦)𝑘̅. Find ∇. 𝐹̅ and ∇ × 𝐹̅ at
(1, 1, 1)
Section C
1. Find the directional derivatives 𝜑=2xy+5yz+zx at the point (1,2,3) in the directional of
3𝑖⃗ -5𝑗⃗ + 4𝑘⃗⃗
2. Solve (D2-4D+3)y = sin 3x cos 2x
3. For an electric circuit with circuit constants L,R,C the charge q on a plate of the condenser is given by
𝑞
1
4𝐿
L𝑞̈ +R𝑞̇ + 𝑐 = E sin𝜔𝑡.The circuit is tuned to resonance so that 𝜔2=𝐿𝐶 .If R2< 𝑐 and q = i = 0 at t=0 .Show
that
𝐸
−𝑅𝑡
𝑅
Q = 𝑅𝑊[-cos 𝜔𝑡+𝑒 2𝐿 (cos pt+2𝐿𝑃 sin pt)
4. Prove that 𝐹̅ = (2𝑥 + 𝑦𝑧)𝑖̅ + (4𝑦 + 𝑧𝑥)𝑗̅ − (6𝑧 − 𝑥𝑦)𝑘̅ is solenoidal as well as irrotational
5. Prove that ∇. (𝑟 𝑛 𝑟̅ ) = (𝑛 + 3)𝑟 𝑛
∞
6.Verify Green’s theorem in the plane for ∫𝑐 (𝑥 2 dx+ xydy), where C is the curve in the xy plane given
by x=0, y= 0, x= a, y= a.
7. Prove that ∇2rn = n(n+1)rn-2
8. Verify Gauss divergence theorem for the vector function 𝐹̅ = 4xz𝑖̅-y2𝑗̅+yz𝑘̅ over the cube bounded by
x=0, x= 1, y= 0, y= 1,z=0, z= 1.
9. . Verify divergence theorem for the vector function 𝐹̅ = 4x𝑖̅-2y2𝑗̅+z2 𝑘̅ over the region bounded by x2 +
y2 = 4 x=0, z=0, z= 3.
10. Verify Stoke’s theorem for 𝐹⃗ = x2𝑖⃗ + xy𝑗⃗ in the square region in the XOY-plane bounded by the lines
x=0,y=0,x=a,y=a.
UNIT - 2
Section A
1.In a skew symmetric matrix A=[aij] is
(a) aij = aji (b) aij= - aji (c) determinant of A=0 (d)none
2.The determinant of an orthogonal matrix is____________
(a)1 (b) 0 (c) -1 (d)± 1
3. The inverse of the inverse of a non-singular square matrix is _________(a) A-1 (b) A (c) A-2 (d) none
3 5
4. The inverse of the matrix A=(
) is
1 2
3 1
−3 −5
2 −5
−3 5
(a) (
) (b) (
) (c) (
) (d) (
)
5 2
−1 3
−1 −2
1 2
4 3
5. The sum of the characteristic value of (
) is
5 6
(a) 24 (b) 15 (c) 18 (d)10
6. A real matrix ‘A’ of order nxn is said to be orthogonal if
(a) |A|=0 (b) AT=A-1 (c) AAT =0 (d) A=A-1
7.The transpose of an unitary matrix is
(a) unitary (b) null matrix (c) an orthogonal matrix (d) none of these
3 2
8. The inverse of A= (
) is ___________7 5
1
−3 −2
5 2
5 −2
5 −2
(a) (
) (b) (
)
(c) (
)
(d) 29 (
)
−7 −5
7 3
−7 3
−7 3
1 1 1
9.The sum of the eigen value of the matrix A=1 2 2
is
1 2 3
(a) 6 (b) 14 (c) 3 (d) 8
10. A square matrix A is said to be unitary if (𝐴̅)T is______________(a) A-1
(b) AT (c) I (d) 0
11. A square matrix A=[aij] is said to be symmetric if aij is equal to _______(a) - aij (b) aji
(c) aii (d) ajj
1
2
1 2
12.The value of a for (
) =(
)
5+𝑎 5
3 5
1
3
(a)2 (b) (c)
(d) -2
2
5
1 2
2 5
13. If A= (
) and B = (
) then 2A-3B is
3 5
6 7
−2 −14
10 −14
0 13
5 12
(a) (
) (b) (
) (c) (
) (d) (
)
15 10
15
5
5 10
3 6
2 1
14.The characteristic equation of the matrix A=(
) is
0 3
2
2
2
2
(a) λ +5λ+6=0 (b) λ -5λ+6=0 (c) λ -5λ-6=0 (d) λ -λ+6=0
15.The product of two orthogonal matrices is
(a) unitary matrix (b) orthogonal matrix (c) diagonal matrix (d) symmetric matrix
16.A square matrix A is said to be singular if |A| is
(a) ≠0 (b) 0 (c) A-1
(d) 1
17.If I is a unit matrix and A is any matrix then IA=AI=___________
(a) I (b) A (c) 0 (d) A2
18.If A and B are orthogonal matrix of the same order then AB is __________
(a) not orthogonal matrix (b) unitary matrix (c) orthogonal matrix (d) hermitian matrix
1 2
19.If A=(
) then the characteristic equation is
1 1
(a) λ2+2λ+1=0 (b) λ2-2λ+1=0 (c) λ2+2λ-1=0 (d) λ2-2λ-1=0
20.The inverse of an unitary matrix is__________
(a)orthogonal (b)unitary (c) scalar (d) null
21.If A is a square matrix and AT= A-1
then A is called as
(a) skew symmetric matrix (b) nilpotent matrix (c) orthogonal matrix (d) none
22.If Amn and Bpq are 2 matrices then the condition for multiplication is
(a) m=p (b) m=q (c) n=p (d) n=q
23. all the elements in the leading diagonal of a skew symmetric matrix are
(a)unique (b) zero (c) two (d) four
24. Which one of the following is an identity matrix?
1 0
5 0
(a) (
) (b) (
)
0 1
0 5
0
(c) (
1
1
)
0
1 1
(d) (
)
1 1
4 3
25.The characteristic roots of the matrix A=(
) is
5 6
(a) 1,2 (b) 2,9 (c) 1,9 (d) -1,-2
26. If 0 is a null matrix and A is any matrix then 0A=A0=___________
(a) I (b) 0 (c) A (d) A2
Section B
1. Solve the following equations by Gauss elimination method
2x+y+4z = 12,8x-3y+2z = 20,4x+11y-z = 33
1 𝑑𝑥
1+𝑥
2. Evalute ∫0
by Trapezoidal rule taking h = 0.25
1 2 3
3. Show that A=3 −2 1 satisfies the equation A3 -23A-40I=0
4 2 1
4. Prove that every square matrix can be expressed as sum of symmetric and skew symmetric matrices
and also prove that it is unique
5. Solve the following equations by Gauss-elimination method
X+3y+3z=16, x+4y+3z=18, x+3y+4z=19
6. Solve the following equations by Gauss-Jacobi method, perform five iterations only 27x+6y-z=85,
6x+15y+2z=72, x+y+54z=110
−1 2
2
1
7.Show that the matrix 𝐴 = 3 [ 2 −1 2 ] is orthoganal
2
2 −1
−2 2 −3
8.Find the eigen values of the matrix 𝐴 = [ 2
1 −6]
−1 −2 0
1 2
9.Find the inverse of 𝐴 = [3 8
4 9
10.Show that 𝐴 =
1
3
2
[3
−2
3
2
3
1
3
2
3
2
3
−2
]
3
−1
3
−1
2]
−1
Section C
1. Solve the following system of equation by Gauss-Jacobi method,correct to three decimal places
10x-5y-2z = 3,4x-10y+3z = -3,x+6y+10z = -3
1 𝑥2
1+𝑥 3
2. Evaluate ∫0
dx using Simpson’s rule with h = 0.25
3. Solve the given system of equations by using Gauss-Seidal iteration method
20x+y-2z=17,3x+20y-z=-18,2x-3y+20z=25
1 𝑑𝑥
1+𝑥 2
4. Evaluate ∫0
using simpson’s 1/3 rule taking h=1/6
7 6
5. Find the transpose,adjoint and reciprocal matrices of −1 2
3 3
2
4
8
6. (i) Prove that the product of two orthogonal matrix is orthogonal
(ii) Prove that the transpose of an orthogonal matrix is orthogonal
2
7. Find the eigen values and eigen vectors of the matrix 𝐴 = [0
1
0 1
2 0]
0 2
8. Solve the following system of equation by matrix inversion method x+2y-z, 3x-4y+2z=1, -x+3y-z=4
3 −4 4
9. Find the eigen value and eigen vector of the matrix [1 −2 4]
4 −1 3
10. Solve the following system of equation by matrix inversion method x+y+2z=4, 2x-y+3z, 3x-y-z=2
UNIT - 3
Section A
1.If L[f(t)]=F(s) then L[eat f(t)]=_______
(a)F(s-a) (b) F(s+a) (c) F(sa ) (d) F(s/a)
𝑠
2.L-1(( 𝑠2 +1)2)=________
1
2𝑎
1
2
(a) (t sinat) (b) (t sin t) (c) t cos 2t (d)t2 sin t
3. The laplace transform of sinat is__________
1
𝑎
𝑎
(a) 𝑠−𝑎 (b) 𝑠2 +𝑎2 (c) 𝑠2 −𝑎2
4.The inverse laplace transform of
2
3
5
(a) t (b) t (c) t (d)t
5.L(e2t sin t)=________
𝑠
(d) 𝑠2 −𝑎2
3!
𝑠4
is
1
2
1
(a) 𝑠2 +4𝑠+5 (b) 𝑠2 +1
1
(c) 𝑠2 −2𝑠
𝑠
(d) 𝑠2 −4𝑠+5
6.L-1(( 𝑠−1)2) =________
1
(a) 𝑠2 (b) et t (c) e-t t (d)e-2t
7. The laplace transform of 53 is
(a)
125
𝑠
53
3!
(b) 53 (c) not defined (d) 𝑠4
8.The inverse laplace transform of L(et) is
1
1
(a) s−1 (b) et (c) s+1 (d) not defined
9.L(e-2t t)=____________
1
1
1
s−2
(a) 𝑠2 −2 (b) ( 𝑠+2)2 (c)
(d)
2
s+1
1
10.L-1(𝑠2 +9)=_____𝟏
𝟑
(a)cos 3t (b) sin at (c)
11.
sin 3t (d) 2 sin 3t
1
L-1( )=_____
𝑠−𝑎
(a)e-at (b) sin at (c) cos at
12.L(e-at) is ___________
1
1
(a) 𝑠−𝑎
(d) eat
1
(b) 𝑠+𝑎 (c) 𝑎−𝑠 (d) sa
13.L(t sin at) is __________
2𝑎𝑠
2𝑎𝑠
2𝑎𝑠
2𝑎𝑠
(a) 𝑠2 −𝑎2 (b) ( 𝑠2 −𝑎2)2 (c) ( 𝑠2 +𝑎2 )2 (d) 𝑠2 +𝑎2
14.The laplace transform of t2 is__________
(a)
1
𝑠
(b)
1
𝑠2
(c)
2
𝑠2
(d)
2
𝑠3
15.The inverse laplace transform of
2t
1
𝑠+2
is
-2t
(a) e (b) e (c) sin 2t (d) cos 2t
16.L(e-t sin t)=____________
(a)
1
𝑠2 +1
17. L-1(
(b)
1
𝑠2 +2𝑠+2
1
)
( 𝑠−1)3
(a)et t2 (b)
(c)
1
𝑠2 +2𝑠+1
(d)
=________
𝑒𝑡𝑡2
2
(c)
𝑒 −𝑡 𝑡 2
2
(d) t2
18.L(cos at) is____________
(a)
𝑠
𝑠2 +𝑎 2
𝑎
𝑎
(b) 𝑠2 +𝑎2 (c) 𝑠2 −𝑎2
𝑠
(d) 𝑠2 −𝑎2
4
19. L-1(𝑠2 +16)=_____
sin 4𝑡
4
(a)
(b) sin 4t (c)
cos 4𝑡
4
(d) cos 4t
5
20. L-1(𝑠3 )=_____
5
(a) 5t2
5
5
(b) 2t3 (c) 2t
(d) 2t2
21.L(sinh at)=______________
(a)
𝑠
𝑠2 +𝑎 2
𝑠
𝑠2 −𝑎 2
22. L-1 (
𝑎
𝑎
(b) 𝑠2 +𝑎2 (c) 𝑠2 −𝑎2
𝑠
(d) 𝑠2 −𝑎2
) =_____________
1
𝑠2 +4𝑠+1
(a) sinh at
(b) cosh at (c)
cos ℎ 𝑎𝑡
𝑎
(d)
sin ℎ 𝑎𝑡
𝑎
23.L(tn) =_____________
(a)
(𝑛+1)!
(b) sn+1
𝑠𝑛
(c)
𝑛!
𝑠𝑛+1
(d) s
t 2
24.L(e t ) =___________
1
2
2
(a) ( 𝑠−1)3 (b) ( 𝑠+1)3 (c) ( 𝑠−1)3
(c)
1
𝑠+1
1
25. L-1(𝑠2 +25) =_____
(a) cos25t
(b) sin5t (c)
𝑠𝑖𝑛5𝑡
5
(d) cos5t
Section B
1. Find the fourier transform of f(x) = 𝑒
−𝑥2
2
2. Find the fourier sine transform of xe-ax
3. State and prove that any two properties of Laplace transform
4. Find[e-t(3sinh2t-5cosh2t)]
5. Find the fourier cosine transform of 𝑒 −𝑥
2
6. Find the fourier transform of f(x) defined by f(x)={x2,|x|≤a and 0,|x|>a
7.Find 𝐿{𝑡𝑒 −𝑡 𝑠𝑖𝑛𝑡}
1
8.Find 𝐿 {(𝑠+1)(𝑠2 +2𝑠+2)}
𝑠+2
9. Find 𝐿−1 {(𝑠+1)(𝑠2 +2𝑠+2)}
10. Find
L
1
[
s2
( s  4)( s  1)
2
]
Section C
1. Find the fourier transform of the function defined by f(x)= {
∞
2. Using Parseval’s indentity evaluate ∫0
𝑑𝑥
(𝑎 2 +𝑥 2 )2
2
(𝑠+1)(𝑠+4)
3. Use convolution theorem find y if L(y)=
1
1
𝑠
4. Find: (i) L (2 sinat- 2 atcosat) (ii) L-1((𝑠+2)3)
5. Find the fourier transform of f(x)={1-𝑥 2 ,|x|≤1 and 0,|x|>1
𝑎2 − 𝑥 2 , |𝑥| ≤ 𝑎
0
, |𝑥| > 𝑎
1
6. Find the fourier sine transform of f(x)=𝑥(𝑎2 +𝑥2 )
2 0 1
7.Find the eigen value and eigen vectors of the matrix A  (0 2 1)
1 0 2
8. Solve the following system of equation by matrix inversion method x+2y-z=2, 3x-4y+2z=1, -x+3yz=4

 cos 2t  cos 3t 
2
 (ii) Find L t sin t
t


9. (i) Find L 

s2
10. Find the inverse Laplace transform by convolution theorem 2
( s  a 2 )( s 2  b 2 )
UNIT - 4
Section A
1.Which is an odd function?
(a) |sin x| (b) 2-3x4+sin2x
(c) sinh2x (d) none
2. f(x) is said to be an even function if __________
(a) f(x) = -f(-x) (b) f(x) = -f(x) (c) f(x) = f(-x) (d) f(x) = -1
3.f(x) = cos x is an ___________ function
(a) odd (b) even (c)special (d) none of these
4.If f(x) = ex,-𝜋<x<𝜋 then the fourier coefficient a0=____
1
2
(a) 𝜋(𝑒 𝜋 − 𝑒 −𝜋 )
(b) 𝜋(𝑒 𝜋 − 𝑒 −𝜋 )
1
(c) ) 𝜋(𝑒 𝜋 + 𝑒 −𝜋 )
1
(d) ) 2(𝑒 𝜋 − 𝑒 −𝜋 )
5. If f(x) = x in 0<x< 𝜋 then the half range cosine series coefficient a0=________
𝜋
(a) 2
(b)
𝜋2
2
(c) 𝜋
1
(d) 𝜋
6. Fourier series are especially suitable for the expansion of _________
(a) periodic function (b) special function (c) implicit function (d) none of these
7.If f(x) = sin x in 0<x< 𝜋 then the range fourier series coefficient a0 = _______
1
2
3
4
(a) 𝜋 (b) 𝜋 (c) 𝜋 (d) 𝜋
𝑐+2𝜋
8. ∫𝑐
sin 𝑛𝑥 𝑑𝑥 is ___________
(a) 1 (b) 0 (c) -1 (d) 2
9.If f(x) =x2 in (-𝜋, 𝜋) then the value of fourier coefficient bn=__________
(a)
2𝜋2
3
(b)
2(−1)𝑛
𝑛
(c) 0 (d)
𝜋2
2
10. If f(x) = c in (0, 𝑐) then the value of a0 in the half range cosine series is
1
(a) c (b) 𝑐
(c) 2c (d) 0
11. The period of the function f(x) = cos x is
(a) 𝜋 (b) 0 (c) 2 𝜋 (d) 3 𝜋
12. f(x) =
𝑒 𝑥 +𝑒 −𝑥
2
is __________
(a) even function (b) odd function (c) neither even nor odd (d) none
13. If f(x) is even function then the fourier coefficient ___________
(a) a0 =0 (b) an=0 (c) bn=0 (d) both an=0 and a0 =0
14.The fourier cosine series of a function f(x) in 0<x< 𝜋 is
∞
(a) f(x)=∑∞
𝑛=1 𝑎𝑛 cos nx (b) f(x)=∑𝑛=1 𝑏𝑛 sin nx
(c) f(x)=
𝑎0
2
+ ∑∞
𝑛=1 𝑎𝑛 cos nx
1
(d) 2 ∑∞
𝑛=1 𝑎𝑛 cos nx
15. If f(x) = |x| ,-𝜋<x<𝜋 then the fourier coefficient 𝑎𝑛 =__________
𝜋2
2
(a)
2
2
(b) 𝜋 2 (c) 𝑛2 𝜋 (d) 𝑛2 𝜋 [(-1)n -1]
16.If f(x)=1 , 0<x< 𝜋 then the half range sine series coefficient bn=_____
(a) 𝑛
2
𝜋
[1-(-1)n ] (b) 𝑛
2
𝜋
[1+(-1)n ]
(c) 𝑛
1
𝜋
[1-(-1)n ]
(d) 0
17.The fourier series of f(x) converges to f(x) at all points when f(x) is _____
(a) finite (b) continuous (c) discontinuity (d) infinite
𝑎
18. If f(x) is even function then ∫−𝑎 𝑓(𝑥)𝑑𝑥 = _________
𝑎
1
𝑎
𝑎
(a) ∫0 𝑓(𝑥)𝑑𝑥 (b) 2 ∫0 𝑓(𝑥)𝑑𝑥 (c)2∫0 𝑓(𝑥)𝑑𝑥
(d) 0
19.In a half range sine series the coefficient bn=_________
𝑙
2
(a) 𝜋 ∫0 𝑓(𝑥)
sin 𝑛𝜋𝑥
𝑑𝑥
𝑙
𝜋
(b) ∫0 sin 𝑥 𝑑𝑥 (c)
2
𝑙
𝑙
∫0 𝑓(𝑥)
sin 𝑛𝜋𝑥
𝑑𝑥
𝑙
𝜋
(d) ∫−𝜋 𝑓(𝑥)
sin 𝑛𝜋𝑥
𝑑𝑥
𝑙
20.In a fourier series with period 3 to represents f(x)=2x-x2 then a0 =_________
(a) 1
1
(b) 0
(c) 2 (d) 4
21.The graph of an odd function is _________ about the origin
(a) horizontal (b) vertical (c) symmetric (d) parallel
𝑙
22.If f(x) = kx in (0, 2) then a0 =____________
𝑙
(b) 𝑙
(a) 2
(c) 0 (d)
𝑘𝑙
2
23. If f(x) is any function of x , it can be expressed the sum of an __________function of x
(a) odd (b) even (c) circular (d) analytic
24. The values of the coefficient a0 , 𝑎𝑛 , bn are called _________ formula
(a) Taylor’s (b) simpson’s (c) euler’s (d) adam’s
𝑐+2𝜋
25. ∫𝑐
𝑠𝑖𝑛2 𝑛𝑥 𝑑𝑥 is ___________
(a) 0 (b) 1 (c) 𝜋 2
(d) 𝜋
Section B
𝑎; 0 ≤ 𝑥 ≤ 𝜋
4𝑎
1
1
1. Show that f(x)={
is equivalent to 𝜋 [sinx+3sin3x+5sin5x+…..]
−𝑎; 𝜋 ≤ 𝑥 ≤ 2𝜋
2. Find a0 and an in the fourier expansion of f(x)= {
1
1
𝑘𝑥; 0 ≤ 𝑥 ≤ 𝑙
0; 𝑙 ≤ 𝑥 ≤ 2𝑙
𝜋
1 3
4 4
𝛤 𝛤
3. Evaluate:∫0 𝑥 𝑛 (log𝑥)n dx (or) (b)show that ∫02 √𝑡𝑎𝑛𝜃 dθ =
2
4. Show that the triangle formed by the points √3+i,√2+√2i,1+√3I in the Argand plane is isosceles.
∞
5. Prove that β(m,n) = ∫0
𝑦 𝑛−1
dy
(1+𝑦)𝑚+𝑛
∞
∞
6. Evalute (i) ∫0 𝑥 4 e-x dx (ii) ∫0 𝑥 6 e-3x dx
7. Determine the Fourier expansion of the function f ( x)  x in    x  
8. Find the Fourier cosine series for the function f ( x)    x in the interval (0,𝜋)
9. Expand f(x)= x in    x  
10. Obtain the half range cosine series for f(x)=x in (0,𝜋)
Section C
1. Expand fourier series of periodicity 2π of f(x)=xsinx 0<x<2π
2. )(i) Explain Dirichlet conditions under the fpurier series
(ii) Find the fourier cosine series for f(x)= π-x in 0<x<π
1
2
3. Show that β(m, )=22m-1 β(m,m)
𝛤𝑚𝛤𝑛
𝛤𝑚+𝑛
4. Prove that β(m,n)=
1
5. Express ∫0 𝑥 𝑚 (1 − 𝑥 𝑛 )𝑝 dx interms of gamma function and evaluate
1
∫0 𝑥 5 (1 − 𝑥 3 )10 dx
ГmГn
6. Prove that β(m,n)= Г(m+n)
7. Find the fourier series of f(x)=𝑥 2 in the interval (-π<x<π). Deduce that
1 1
1
2



.......

12 23 32
12
8. Find the Fourier series of f(x) = 0 in -2<x<0
= x in 0<x<2
9.Obtain the Fourier series for f(x) = (  x) 2 is (  ,  )
10.Find the fourier series for f(x) = 𝑥 2 in -1< x < 1
UNIT - 5
Section A
1.A coin is tossed then the probability that the head
(a)
1
3
(b) 1 (c)
1
2
will appear is _______
(d) 0
2.From the binomial distribution with “ probability of success is _______”
𝑝
(a) 𝑞 (b) q (c) p-1 (d) p
3. The limits of probabilities are ______
(a) 0 and 1 (b) -1 and 1 (c) 0 and ∞ (d) none of these
4. Mean of binomial distribution is
(a) npq (b) p (c) np (d) √𝑛𝑝𝑞
5. If p(A)=0.28,p(B)=0.62 and p(A∩ 𝐵) = 0.12 then p(A∪B)= _____
(a) 0.90 (b) 1.02 (c) 0.82 (d) 0.78
6. Variance of the binomial distribution p(x)=𝑛𝑐𝑥 px qn-x is ________
(a) npq (b) np (c)n2pq (d) np2q2
7. The probability of a sure event is ___________
(a) 1 (b) not defined (c) <1 (d) >1
1
4
8.The mean of a binomial distribution Bi(10, ) is ________________
(a)
30
16
5
2
(b)
1
4
(c)
(d) 10
1
1
9. If A and B are disjoint event and p(A)= 3 and p(B)= 5 then p(A∪B)=______
1
2
8
1
15
(a) 8 (b) 15 (c) 15 (d)
10.For an impossible event the probability is
(a) -1 (b) 1 (c) 0 (d) 2
11.The standard deviation of the binomial distribution is _____________
(a) np (b) npq (c) √𝑛𝑝𝑞 (d) √𝑛𝑝
12.If A and B are any two events then p(A∪B)=______
(a)p(A)+p(B)-p(A∩ 𝐵) (b) p(A)+p(B) (c) p(A).p(B) (d) p(A)+p(B)+p(A∩ 𝐵)
13.If two coins are tossed then the probability of getting all head is _______
1
1
(a) 8 (b) 2 (c)
1
4
3
(d) 4
14. If A and B are mutually exclusive events and p(A)=
(a)
19
24
(b)
1
4
(c)
25
24
(d)
2
3
p(B)=
3
8
then p(A∪B)=______________
1
2
15.The chance of throwing three with an ordinary six faced die is ___________
(a)
1
6
(b)
1
2
(c)
1
3
(d) 0
16. Sum of the probabilities is __________
(a) 0 (b) 1 (c) -1 (d) 2
3
2
11
and p(A∪B)= 12 then p(A∩ 𝐵) is ___________
17.If p(A)= 4 , p(B)= 3
(a)
4
5
(b)
1
2
(c)
1
3
(d)
1
5
18.A set of event is said to be ______ if the occurrence of any one of them does not affect the
occurrence of the other.
(a) dependent (b) independent (c) mutually exclusive (d) sure
19.Outcome are said to be ______ when no other outcome is possible from the random experiment.
(a) exhaustive outcomes (b) equally likely outcomes (c)mutually exclusive outcomes (d) favourable
outcomes
20. 𝑛𝑐0 = __________
1
(a) 0 (b) 2 (c) 1 (d) 2
21. The probability that a leap year selected at random has 53 fridays is____
2
6
8
(a) 7 (b) 7 (c) 7 (d) 1
22. Addition theorem is applied in the case of __________ events
(a) independent (b) dependent (c) mutually exclusive (d) equally likely
23. If the mean and variance of a binomial variate are 8 and 6 then p=_______
1
1
4
(a) 8 (b) 2 (c) 4 (d) 3
24.Suppose a die is thrown then the probability of getting an even number is
1
(a) 2
3
4
(b) 2 (c) 3 (d) 1
25.multiplication theorem is applied in the case of_____________ events
(a) independent (b) dependent (c) mutually likely (d) favourable
Section B
1. Explain the terms(i) equaly likely events (ii) Mutually exclusive events
2. State and prove addition rule of probability
3. Show that the triangle formed by the points √3+i,√2+√2i,1+√3I in the Argand plane is isosceles.
4. If
sin 𝜃 5045
=
𝜃
5046
show that θ=1⁰58’ approximately.
5 Express cos 6θ interms of cos θ.
6. Prove that (1+i√3)n +(1-i√3)n = 2n+1 cos
𝑛𝜋
3
7. Explain the different types of errors
8. State and prove addition theorem of probability
9. State and prove multiplication theorem of probability
10. Two unbiased dies are thrown. Find the probability that (i) both the dice show the same number (ii)
the total of the number on the dice is 8
Section C
1. Fit a binomial distribution to the following frequency distribution.calculate theoretical frequencies:
No.of successes: 0
1
2 3 4 5 6
Frequency
:13 25 52 58 32 16 4
2. Use the method of least squares to fit straight line to the following data:
X: 0 1
2
3
4
Y: 1 1.8 3.3 4.5 6
3. Show that the point representing the complex numbers 5+4i ,3-2i,-4-3i,-2+3i
4. Prove that
𝑐𝑜𝑠7𝜃
𝑐𝑜𝑠𝜃
form a parallelogram
= 64cos6θ-112cos4θ+56cosθ-7
5. The mean and variance of a binomial variable X with parameters 𝑛 and 𝑝 are 16 and 8 respectively.
Find (i) p (x=0) (ii) p(x=1)
6. Fit a straight line of the form y=a+bx for the following data:
X : 6 2 10 4 8
Y : 9 11 5 8 7
7. Fit a straight line of the form y=a+bx for the following data:
X: 2 3 5
8 10
Y : 5 6 10 18 21
8. A box contains 100 transistors, 20 of which are defective, 10 are selected for selection. Using binomial
distribution find the probability that (i) all 10 are defective (ii) all 10 are good (iii) atleast one is defective
9.(i)What is meant by binomial distribution? What are its features?
(ii) The mean of a binomial distribution is 20 and the variance is 16. Calculate n,p,q
10. Fit a straight line of the form y=a+bx for the following data:
X : 20 25 30
35 40 45
Y : 207 232 295 355 370 395