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UNIT-I
Section A
1.The particular integral of D2y+4y=cosx is
1
1
(a)cosx (b)sinx (c) cosx
(d) sinx
3
3
2.If x is charge on the condenser,then the current I is
(a)1 (b)
𝑑𝑥
𝑑𝑡
(c)0 (d)
3.The solution of
𝑑2 𝑥
𝑑𝑡 2
𝑑2 𝑦
𝑑𝑦
𝑑𝑥
𝑑𝑥
2 +4
+ 4𝑦 = 0 𝑖𝑠
(a)y=Ae2x+Be-2x (b)y=e2x(Acos2x+Bsin2x) (c)y=(Ax+B) e-2x
(d)y=(Ax+B) e2x
4.The particular integral of(D2-3D+2)y= e2x is
1
1
3
4
(a) e2x (b)x e2x (c)xex (d) e2x
5.The complementary function of (D2+4)y=x2 is
(a)y=Ae2x+Be-2x (b)y=e2x(Acos2x+Bsin2x ) (c) )y=Ae3x+Be-2x (d)
Acos2x+Bsin2x
6. .The particular integral of
(a)
1 -3x
e
2
(b) -
1
3
xe-3x
(c)
𝑑2 𝑦
𝑑𝑥 2
1
4
+4
𝑑𝑦
𝑑𝑥
+ 3𝑦 = e-3x is
xe-3x (d) -
1
2
xe-3x
7. The solution of (D2-5D+6)y=0 is
(a)y=Ae2x+Be3x
(b)y=Ae2x+Be-3x (c)y=Ae-2x+Be3x
-3x
8.The particular integral of (D2+1)y=ex is
(d)y=Ae -2x+Be
(a)
1 -x
e
2
(b )
1 x
e
2
(c)1
(d) )
1 x
e
4
9.In a second order linear differential equation with constant
coefficients,the particular integral is given by
(a)f(D)*X (b)f(D)/X (c)X/f(D) (d)X+f(D)
10.The complementary function for a second order differential equation
is of the form c1emx+c2enx,the roots of the equation in the operator form
are
(a)real and equal (b)real and distinct
(c)imaginary (d)of type 𝛼 ± 𝛽
11.suppose if the roots of the auxillary equation are say -1 and -2 ,the
complementary function is of type
(b)(A+Bx) emx (c)(Ax2+Bx+c)emx (d)eαx(Acosβx
(a)Aemx+Benx
+Asinβx)
12.Which of the following cannot be represented by a differential
equation?
(a)motion of simple pendulum (b)motion of a car (c)oscillations of a
spring
(d)changing and discharging of a condenser through R and L
13. The C.F for the differential equation(D2-4D+3)y=sin2x is
(a)y=Ae -x+Be -3x (b)y=Ae x+Be -3x (c)y=Ae -x+Be 3x (d)y=Ae x+Be
3x
14.The particular integral of (D2+16)y=e-3x is
(a)
1 -3x
e
9
(b)
1
25
e-3x (c)A e3x+B e-3x (d)none of t
15.If x is the charge ,the differential equation for the circuit is
𝑑𝑖
𝑥
𝑑𝑡
𝑐
L +Ri+ =Esinwt where i is the current which is equal to
(a)
𝑑𝑥
𝑑𝑡
(b)
𝑑𝑅
𝑑𝑡
(c)
𝑑𝐿
(d)none of these
𝑑𝑡
16.The particular integral of (D2-2D+1)y=xex is
𝑥2
𝑥3 x
e
3
(a) ex (b)
2
(c)
𝑥3 x
e
2
(d)
𝑥3 x
e
6
17.The particular integral of (D2-4D+4)y=xe2x is
3 2x
(a)x e
18.
(b)
𝑒 𝑎𝑥
(𝐷−𝑎)2
𝑥 3 2x
e
6
(c)e2x(x+2)
=_________
𝑥
(a)
(d)xe-ax
ax
2
e
(b)
𝑥 2 ax
e
2
(d)none of these
(c)xe-ax (d)none of these
19.The solution of (D2+3D-40)y=0 is
(a)y=Ae-8x+Be5x (b) y=Ae8x+Be -5x
above
(c)y=Ae-8x+Be -5x (d)none of the
20.The solution of (D2+2D+5)y=xex is
(a)y=e-5/2 x(Acos
√17
√17
2
2
x+Bsin
−5
−5
2
2
x) (b)y=e√17/2 x(Acos( )x+Bsin( )x)
5x
(c)y=e (Acos17x+Bsin17x) (d)y=e
√17
(Acos5x+Bsin5x)
21.Particular integral of(D-1)2y=exsinx is
(a)xcosx (b)-xexsinx (c)xexsinx (d)excosx
22. Particular integral of(D+2)2y=xe-2x is
(a)
𝑥2
6
e
-2x
(b)
𝑥
6
e
-2x
(c)
𝑥3
6
e
23.The degree of the equation(
-2x
(d)
𝑥4
2
e -2x
𝑑2 𝑦 2 𝑑𝑦 3
) +( ) P+Qy=0
𝑑𝑥 2
𝑑𝑥
is
(a)one (b)two (c)three (d)none of these
24.If a second order differential equation is expressed in the
𝑑2 𝑦 𝑑𝑦
form
+ P(x)+Q(x)y=R(x) then it is called_________equation
𝑑𝑥 2 𝑑𝑥
(a) linear (b)nonlinear (c)homogeneous (d)none of these
25.The complementary function of (D2+D-2)y=sin2x is
(a)y=Ae-3x+Be-2x (b) y=Aex+Be-2x (c) y=Ae-x+Be-2x (d)(Ax+B) e2x
26.The particular integral of (D2-6D+9)y=e 9x is
(a)
𝑥2 x
e
2
(b)
𝑥 2 3x
e
2
(c)
𝑥
2
e2x (d)
𝑥 2 3x
e
2
Section B
1. Solve(D2+5D+6)y=5e-2x
2. Solve:(D2-8D+9)y=8sin5x
3.Solve (D2+3D+2)y = x2
4. Solve D2-4D-5)y = 4cos3x
5 .Solve (D2+16)y =cos4x
Section C
1. Solve (D2-4D-3)y=excos2x
2. Solve(D2+4)y=xsinx
3. Solve (D2-4D+3)y = sin 3x cos 2x
4. For an electric circuit with circuit constants L,R,C the charge q on a
𝑞
plate of the condenser is given by L𝑞̈ +R𝑞̇ + = E sin𝜔𝑡.The circuit is
𝑐
tuned to resonance so that 𝜔2=
1
𝐿𝐶
.If
4𝐿
R2<
𝑐
and q = i = 0 at t=0 .Show
that
Q=
𝐸
𝑅𝑊
[-cos 𝜔𝑡+𝑒
−𝑅𝑡
2𝐿
(cos pt+
𝑅
2𝐿𝑃
sin pt)
5. Solve (D2+3D+2)y =sinx +x2
UNIT-II
Section -A
1.Convergence towards the solution by Gauss-seidal method in
comparison with Gauss-Jacobi method is
(a)faster (b)slower (c)same (d)not comparable
𝑏
ℎ
2.The simpson`s rule is∫𝑎 𝑦 dx= (A+4B+2C),where A is
3
(a)sum of first and last ordinates (b)sum of the even ordinates (c)sum
of the odd ordinates (d)product of x and y
3.Gauss-Jordan method belongs to
(a)direct method (b)indirect method (c)bisection method (d)none of the
above
4.Simpson`s one –third rule on numerical integration is called a
(a)open formula (b)closed formula (c)Bessel`s formula (d)none of
these
5.Which of the following is correct?
(a)Gauss Elimination method is indirect method (b)Gauss-seidal
method is direct method
(c) Gauss-Jordan method is direct method (d) Gauss-Jacobi method is
direct method
𝑏
ℎ
6.The Trapezoidal rule is ∫𝑎 𝑦 dx= (A+2B) where B is
2
(a)sum of first and last ordinates (b)sum of the even ordinates (c)sum
of the odd ordinates (d)sum of the remaining ordinates
7.Which of the following is a wrong statement?
(a) Gauss Elimination method and Gauss-Jordan method are direct
method
(b) Gauss-seidal method and triangularisation method are direct
method
(c) Gauss-seidal method and Gauss-Jacobi methods are iterative type
(d) Gauss Elimination method and triangularisation method are
direct method
8. Which of the following is a direct method ?
(a) Gauss Elimination (b) ) Gauss-seidal (c) Gauss-Jacobi (d)none
9.In simpson`s rule,the number of intervals is always
(a)odd (b)even (c)odd or even (d)none
10. In simpson`s rule,the approximation is closer when it is
(a)large (b)smaller (c)large or small (d)none
11.Choose the wrong statement
(a) Gauss method the matrix is reduced to triangular matrix
(b) Gauss-Jordan method the matrix is reduced to diagonal matrix
(c) Gauss method is an elimination method
(d) Gauss-Jordan method makes use of iteration
12.Choose the correct answer
(a)The accuracy is the same in trapezoidal rule method and simpson`s
rule method
(b) The accuracy is greater in trapezoidal rule method than simpson`s
rule method
(c) The accuracy is greater in simpson`s rule method than trapezoidal
rule method
(d)none of the above statement is true
13.As soon a new value for a variable is bound by iteration,it is used
immediately in the following equation
(a) Gauss-seidal (b) Gauss-Jacobi (c) ) Gauss-Jordan
these
(d)none of
14.The order of error in trapezoidal formula is
(a)0 (b)h (c)h2(d)h3
15.Which one of the following is a iterative method to solve
simultaneous equation
(a) Gauss Elimination method(b) Gauss-seidal method (c) GaussJordan method (d)Triangularisation method
16. Simpson`s one –third rule will give exact result,if the entire curve
y=f(x) is itself a
(a)ellipse (b)circle (c)parabola (d)hyperbola
17.the error in simpson`s rule is of order
(a)h4 (b)h (c)h2(d)h3
18.The rate of convergence in Gauss-seidal method is roughly ---------- times than that at Gauss-Jacobi method
(a)one (b)two (c)three (d)four
19.When Gauss-Jordan method is used to solve AX=B,A is transform
into a------------- matrix
(a)upper triangular (b) lower triangular (c)diagonal (d)identity
20.Simpson`s three-eight rule can be applied only if the number of
subintervals is a
(a)multiple of 3 (b)multiple of 2 (c)multiple of 4 (d)none of these
21.If a set of numerical values of the integral f(x) is applied to
𝑏
∫𝑎 𝑓 (𝑥 ) 𝑑𝑥, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑎𝑡 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑖𝑠 𝑘𝑛𝑜𝑤𝑛 𝑎𝑠
(a)a numerical integration (b)quadrature (c)interpolation (d)none of
these
22.Condition for the convergence of Gauss-seidal method is that the
coefficient matrix is
(a)singular (b)non-singular (c)diagonally dominant (d)none of these
23. The condition for the convergence of Gauss-Jacobi
method
Is
(a)∑𝑛𝑗=1 |
𝑎𝑖𝑗
𝑎𝑖𝑖
| ≤ 1 for j≠i (b) ∑𝑛𝑗=1 |
𝑎𝑖𝑗
𝑎𝑖𝑖
| >1 for j≠I (c) ∑𝑛𝑗=1 |
𝑎𝑖𝑗
𝑎𝑖𝑖
| <1 for
j≠I (d)none of these
𝑏
ℎ
24. ∫𝑎 𝑦 dx= (y1+4(y2+y4+………+y2n)+2(y3+y5+……………+y2n3
1)+y2n+1)is
1
(a)Trapezoidal rule (b)simpson`s rule (c)Romberg rule (d)none of
3
these
25.which one is more reliable in the following?
(a) simpson`s rule
these
(b) Trapezoidal rule (c)both (a) & (b) (d)none of
Section B
1. Solve the following equations by Gauss-elimination method
X+3y+3z=16,x+4y+3z=18,x+3y+4z=19
2. Solve the following equations by Gauss-Jacobi method,perform five
iterations only 27x+6y-z=85,6x+15y+2z=72,x+y+54z=110
3. Solve the following equations by Gauss elimination method
2x+y+4z = 12,8x-3y+2z = 20,4x+11y-z = 33
1 𝑑𝑥
4. Evalute ∫0
1+𝑥
by Trapezoidal rule taking h = 0.25
5. Solve the following equations by Gauss-elimination method
3X+4y+5z=18,2x-y+8z=13,5x-2y+7z=20
Section C
1.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
2. (Evaluate ∫0
1+𝑥 2
using simpson’s 1/3 rule taking h=1/6
3. Solve the following equations by Gauss elimination method
2x+y+4z = 12,8x-3y+2z = 20,4x+11y-z = 33
1 𝑑𝑥
4. Evalute ∫0
1+𝑥
by Trapezoidal rule taking h = 0.25
5. Solve the given system of equations by using Gauss-Seidal iteration
method
28x+4y-z=32 , x+3y+10z=-24, 2x+17y+4z=35
UNIT-III
Section-A
∞
1.If F[f(x)]=F(s),then∫−∞ |𝐹 (𝑥 )|2 dx is
∞
∞
∞
(a)∫−∞(𝑓(𝑥 ))2 ds (b) ∫−∞ |𝐹 (𝑠)|2 ds (c) ∫−0 |𝐹 (𝑠)|2 ds (d)none of
these
2.The complex fourier transform of f(x) is given by F[f(x)]=
(a)
1
∞
1
∞
1
∞
∫ 𝑓(𝑥) eisx dx (b) √2𝜋 ∫−∞ 𝑓(𝑥) dx (c) √2𝜋 ∫−∞ 𝑓(𝑥) cosnx dx
√2𝜋 −∞
(d)
1
∞
∫ 𝑓(𝑥) sin nx dx
√2𝜋 −∞
3. If F[f(x)]=F(s),then F[f(x-a)]=
(a)F(s-a) (b)F(s) (c)F(s+a) (d)0
4. If f(x) is an odd function of x,its fourier transform will be--------functions of s
(a)odd (b)even (c)analytic (d)circular
5.F[af(x)+bg(x)]=
(a)aF[f(x)]+bF[g(x)] (b)F[f(x)]+F[g(x)] (c)0 (d)1
6.The fourier cosine transform of f(t) =
2 ∞
2 ∞
2
(a)√ ∫0 ̅̅̅
𝑓𝑐 (s) cost ds (b) )√ ∫0 ̅̅̅
𝑓𝑐 (s) ds (c) )√ cost ds (d)1
𝜋
𝜋
𝜋
7. If F[f(x)]=F(s),then F[e2ax f(x)=
1
1
𝑎
𝑎
(a)F(s+a) (b)F(s-a) (c) F(s+a) (d) F(s-a)
8.The fourier cosine transform Fc[f(x)cosax]=
1
1
1
1
2
2
𝑎
2
(a)F(s+a) (b) {Fc(s+a)+Fc(s-a)} (c) {Fc( )-Fc(s-a)} (d) {Fc(s+a)-Fc(sa)}
9. The fourier sine transform Fs[f(x)sinax]=
1
1
1
1
2
2
𝑎
2
(a)F(s+a) (b) {Fs(s+a)+Fs(s-a)} (c) {Fs( )-Fs(s-a)} (d) {Fs(s+a)-Fs(sa)}
10. . If F[f(x)]=F(s),then F[f(x-a)]=
(a)e-iasF[f(x)] (b)
𝑒 −𝑖𝑎𝑠
|𝑎|
(c) eiasF[f(x)] (d) easF[f(x)]
11.If Fs(s) and Gs(s) are the fourier sine transforms of f(x) and g(x)
∞
respectively then∫0 𝑓 (𝑥 )𝑔(𝑥 )𝑑𝑥 =
∞
∞
(a)∫0
∞
Fs(s)Gs(s)ds (b) ∫−∞
Fs(s)Gs(s)ds (𝑐) ∫0
Fs(s)ds (d)0
12.F[eiaxf(x) ] is
1
𝑠
𝑎
𝑎
(a) F( ) (b)eias F(s) (c)F(s+a) (d)(-is)nF(s)
13.Parseval`s theorem is useful it f(x),the given function is defined as in
(a)[0,∞] (b) (0,1) (c) (-1,1) (d) (-∞, ∞)
14.The fourier transform of f(x) is given by
(a)
1
√
(d)
∞
1
∞
∞
1
𝑓(𝑡) eist dt (b)
𝑓(𝑡) eist dt (c)
𝑓(𝑡) eist dt
∫
∫
∫
−∞
−0
−∞
𝜋
√2𝜋
√2𝜋
1
∞
𝑓(𝑡) eist dt
∫
−∞
2𝜋
15.Which of the following is not a fourier transform?
(a)f(x)=
1
∞
∞
1
∞
∞
∫ ∫ 𝑓(𝑡) eis(t-x) dt dx (b) f(x)=2𝜋 ∫−∞ ∫−∞ 𝑓(𝑡) cos s(t-x)
2𝜋 −∞ −∞
2
∞
∞
∞
dt dx (c) f(x)= ∫0 ∫0 𝑓(𝑡) sin st. sin sx dt dx (d)F(s)=∫0 𝑓(𝑡) e-st dt
𝜋
16.F[af(x)+bg(x)]=aFf(x)+bFg(x) is called
(a)shifting property (b)linearity property (c)similarity property (d) none
of the above
17.If f(x) and g(x) are any two functions,thenf*g=
∞
∞
(a)∫−∞ 𝑔(𝑡) dt (b) ∫−∞
(d)
𝑓(𝑥 − 𝑡)𝑔(𝑥 + 𝑡)dt (c)
∞
1
1
∞
∫ 𝑓(𝑡) g(x-t) dt
√2𝜋 −∞
∫ 𝑓(𝑡) g(x-t) dt
√2𝜋 0
18.The fourier transform of x𝑒
(a) 𝑒
−𝑠2
2
(b)is (c)is) 𝑒
−𝑠2
2
−𝑥2
2
is
(d)s
19.F[xnf(x)]=
(a)
𝑑𝑛
𝑑𝑠𝑛
n-1
(x ) (b)
𝑑𝑛
𝑑𝑠
𝑑
̅
̅ n (d) (-i)n )
𝑓
(s)
(b)
(𝑓(s))
𝑛
𝑑𝑠
𝑑𝑛
𝑑𝑠𝑛
̅
𝑓(s)
̅
̅
(a)f(-s) (b) 𝑓 (-s)
(c)− 𝑓(-s)
(d)f(s)
20.F[f(-x)]=
̅
21.If f(x) is an even function of x,its fourier transform 𝑓(s)
will be------------- function of s
(a)odd (b)even (c)multiple (d)inverse
22.The fourier transform of f(t)=e-a|t|,a>0 is
2
2
𝜋
𝜋
(a)√ a/a2+s2 (b)a/a2+s2 (c) )√
(d)
𝜋
2
∞
̅
23.If F[f(x)]= 𝑓 (s)
,then ∫−∞ |𝑓(𝑥 )|2 dx=
∞
̅ 2 ds (b) ) ∫∞ | 𝑓(s)|
̅ 2 ds (c) ∫∞ | 𝑓(s)|
̅ ds (d)0
(a) ∫−∞( 𝑓(s))
−∞
0
(a)-s 𝑓̅𝑐 (s) (b) 𝑓̅𝑐 (s)
24.Fs[f `(x)]=
25.The fourier sine transform of
𝑒 −𝑎𝑥
𝑥
is
2
𝑠
𝑠
2
𝑠
𝜋
𝑎
𝑎
𝜋
𝑎
(a)√ (b)tan-1( ) (c) (d) √ tan-1( )
Section B
(c)-s fc(s) (d) –s 𝑓̅𝑠 (s)
1. Find the fourier transform of f(x) = 𝑒
−𝑥2
2
2.Find the fourier sine transform of xe-ax
3.Find the fourier cosine transform of 𝑒 −𝑥
2
4.Find the fourier transform of f(x) defined by f(x)={x2,|x|≤a and
0,|x|>a
5. Find the fourier sine and cosine transform of e-ax
Section C
1. Find the fourier transform of the function defined by
𝑎2 − 𝑥 2 , |𝑥| ≤ 𝑎
f(x)= {
0
, |𝑥| > 𝑎
∞
𝑑𝑥
2.Using Parseval’s indentity evaluate ∫0
(𝑎2 +𝑥 2 )2
3. Find the fourier transform of f(x)={1-𝑥 2 ,|x|≤1 and 0,|x|>1
4.Find the fourier sine transform of f(x)=
1
𝑥(𝑎2 +𝑥 2 )
5. Find the fourier transform of the function defined by
𝑥2
|𝑥| ≤ 𝑎
f(x)= {
0
|𝑥| > 𝑎
UNIT IV
Section-A
1
∞
1.β(m,n)= (a)∫0 𝑥 𝑚−1 (1-x)n-1dx (m,n>0) (b)∫0 𝑥 𝑚−1 (1-x)n-1dx
(m,n>0)
∞
1
(c)∫−∞ 𝑥 𝑚−1 (1-x)n-1dx (m,n>0) (d)∫0 𝑥 𝑚 (1-x)n dx (m,n>0)
2.Гn =
(a)n! (b)n (c)n+1 (d)(n+1)!
3Г(n+1)=
(a))(n+1)! (b)nГn
1
(c)Гn
(d)1
𝜋
4.Г =
(a)√𝜋 (b)π (c)2π (d)
5.Г1 =
(a)1 (b)-1 (c)0 (d)undefined
2
∞
2
1
1
(a)∫0 𝑒 −𝑥 xn-1 dx (b)∫0 𝑒 −𝑥 xn-1 dx (c)∫0 𝑒 −𝑥 xn
6.Гn =
1
dx (d)∫0 𝑒 𝑥 xn+1 dx
Г7Г5
7.
Г12
is the value of --------- (a)β(7,5) only (b)β(5,7) only (c)β(7,5) or
β(5,7) (d)
35
12
8.1.3.5…..(2n-1)√𝜋 =
β(m,n)
β(m,n+1)
𝑚+𝑛
𝑚+𝑛
9.
= (a)
Г(n+1)
10.
𝑛
=
3
11.Г =
2
(a)2n√𝑛 +
(b)
β(m,n)
𝑛
(c)
1
2
β(m,n)
𝑚
(a)Гn (b)nГn+1 (c)nГn
(a)π (b)√
𝜋
2
𝜋
2
(c)√𝜋
1
(b)Г(n+ ) (c)2n (d)Гn
2
β(m+1,n+1)
(d)
𝑚+𝑛
(d)n
(d)1
1
1
12.∫0 𝑠𝑖𝑛2𝑚−1 𝜃 cos2n-1θdθ = (a) β(m,n) (b)β(m,n) (c) β(m,n)
2
4
(d) none of these
13.β(1,1)=
(a)1 (b)0 (c)-1 (d) none of the above
14.Г
−1
2
(a)√𝜋
=
𝜋
2
15.∫0 𝑠𝑖𝑛3θ cosθdθ =
1
1
1
2
2
2
(c)-2 Г (d) Г
(b)−√𝜋
1
1
1
2
2
2
(a) β(2,1) (b) β(3,1) (c)β(2,1) (d) β(4,2)
∞
16.The value of the integral ∫0 𝑒 −𝑥 x3 dx is
(a)Г2 (b)Г3 (c)Г1 (d)∞
𝜋
2
17.∫0 √𝑡𝑎𝑛𝜃 dθ =
𝜋
2
18..∫0 𝑠𝑖𝑛 θdθ =
n
1 3
1
4 4
4
(a) Г Г / 2 (b) Г
𝜋
(a)∫0 𝑐𝑜𝑠 𝑛 θ
(c)Г
3
4
(d)1/2
𝜋
2
∞
dθ (b)∫0 𝑐𝑜𝑠nθdθ (c)∫0 𝑐𝑜𝑠𝑛θ dθ
𝜋
2
(d)∫0 𝑐𝑜𝑠𝜃 𝑑𝜃
1
19.β(m, )=
2
(d) )Г
(a) 22m.β(m,m) (b)) 22m-1 .β(m,m) (c)) 22m+1 .β(m,m)
3
2
20..Гp.Г1-p =
(a)
𝜋
𝑠𝑖𝑛𝑝𝜋
(b)
𝑐𝑜𝑠𝑝𝜋
𝜋
1
1
𝜋
𝜋
2
2
𝑚
𝑚
21..β(m,m) .β(m+ ,m+ ) = (a) 21-4m(b)
22.The duplication formula .Г2m is
22𝑚−1
(b)
𝜋
22𝑚−1
(d)
𝜋
.ГmГm+
1
2
(c)
22𝑚
𝜋
(c)
𝑠𝑖𝑛𝑝𝜋
𝜋
(d)π
(c)π m (d)π 24m
(a).Гm+
1
2
.Гm
.Гm+1
1
23.The exact value of Г = (a)0.7724 (b)4.7724 (c)1.7724 (d)0.5772
2
24.Gauss`s pi function is defined interms of the gamma function by the
equation π(n)= --------
(a) Гn (b) Гn+1
(c) Гn-1
(d)n Гn-1
25.∫ ∫ ∫ 𝑥 a-1 yb-1zc-1 dxdydz= -----------(a) .
Гa ГbГc
(b) Гa Гb Гc (c) Гa+b+c+1
Гa+b+c+1
(d) Гabc
Section B
1
1
1. Evaluate:∫0 𝑥 𝑛 (log )n dx
𝑥
𝜋
2
2.show that ∫0 √𝑡𝑎𝑛𝜃 dθ =
3. Prove that β(m,n) =
1 3
4 4
𝛤 𝛤
2
∞ 𝑦 𝑛−1
∫0 (1+𝑦)𝑚+𝑛 dy
∞
∞
4.Evalute (i) ∫0 𝑥 4 e-x dx (ii) ∫0 𝑥 6 e-3x dx
5. show that  (m, n)  2 sin 2m 1  cos 2n 1  d
Section C
1
1.Show that β(m, )=22m-1 β(m,m)
2
2.Prove that β(m,n)=
𝛤𝑚𝛤𝑛
𝛤𝑚+𝑛
1
3. Express ∫0 𝑥 𝑚 (1 − 𝑥 𝑛 )𝑝 dx interms of gamma function and
1
∫0 𝑥 5 (1 − 𝑥 3 )10 dx
4. evaluate
 /2
5. Evaluate 
0
tan  d
UNIT V
Section-A
1.arg(z1z2)= ------------ where z1 and z2 are two complex numbers
(a)arg z1+arg z2 (b) arg z1- arg z2
(c) arg z1.arg z2
(d)argz1/argz2
2.|z-i|=5 on the argan Ф diagram is a _____________(a)point (b)circle (c)a straight line (d)none of these
3.(1+i√3)n+.(1-i√3)n = -----------𝑛𝜋
(a)2n+1cos
3
(b) 2ncos
𝑛𝜋
3
𝑛𝜋
(c) 2n+1sin
4.If x=cos θ+isin θ,then xn+
1
𝑥𝑛
3
𝑛𝜋
(d) 2ncos
2
= ---------
(a)cosnθ+isinnθ (b) cosnθ-isinnθ (c)2 cosnθ (d) cosθ-isinθ
5.|z1+z2|≥ --------------- where z1,z2 are complex numbers
(a)|z1|+|z2| (b)|z1|-|z2|
(c)|z1| (d)|z2|
6.(cos θ+isinθ)n = --------------(a)cosnθ-isinnθ (b)cosnθ+isinnθ
(c) cosθ+isinθ
(d)
cosθ+isinθ
𝑛
7.The values of (cos θ+isinθ)3+(cos θ-isinθ)3 is
(a)2 (b) 1 (c)2cos2θ (d) 0
8.sin θ = ----------(c)θ (d) 1
(a)θ +
𝜃3
3!
+
𝜃5
5!
+…………. (b) θ -
𝜃3 𝜃5
3!
+
5!
-………….
9.cos θ = ----------- (a)1+
………(c) 1-
𝜃2
2!
cos3θ+isin3θ
10.
cos2θ+isin2θ
+
𝜃4
4!
𝜃2
2!
+
𝜃4
4!
+…………. (b) 1+θ +
𝜃2 𝜃4
2!
+
4!
-
-…………. (d)0
= ------------ (a) cos5θ+isin5θ (b) cos5θ-isin5θ (c)
cosθ+isinθ (d) cosθ-isinθ
𝜋
𝜋
1
8
8
2
11.(cos + isin )8 = --------- (a)1 (b)-1 (c)0 (d)
𝑧
12.If z1and z2 are two complex numbers, then arg( 1) = -------𝑧2
(a)arg z1+arg z2 (b) arg z1- arg z2
(c) arg z1
(d)argz2
13.If z=3+4i ,the value of z 𝑧̅ is
(a)25 (b)7 (c) 1 (d)12
14.The complex conjugate -4-9i is
(a)4+9i
(b)-4-9i (c)4-9i (d)-4+9i
15.Express (3+2i)+(-7-i)in standard form
(a)-4+I (b)4+I (c)-4-I (d)-7+i
16.The value of e2iθ-e-2iθ is
(a)2i sin2θ (b) 2i cos2θ (c) 2sin2θ (d) 2cos2θ
17.The argument of z=1+I is
𝜋
𝜋
2
4
(a) (b)
(c)
3𝜋
4
(d)
𝜋
8
18.Choose the wrong statement
(a)The modulus of the product of 2 complex is the product of their
moduli
(b) The modulus of the sum of 2 complex numbers never exceeds the
sum of their moduli
(b) ) The modulus of the sum of 2 complex numbers is greater than the
sum of their moduli
(b) ) The modulus of the difference of 2 complex numbers is greater
than or equal to the difference of their moduli
19.cosnθ = --- (a)cosnθ−
𝑛(𝑛−1)
cosn-3θ+… (c) cosnθ sinθ+
(b)
2
cosn-1θ cos3θ (d)none of these
1
20.cos6θ= -----1
2!
𝑛(𝑛+1)
cosn-2θ sin2θ +….. (b)ncosn-1θsinθ n𝑐3
(a) (cos6θ+6cos4θ+15cos2θ+10)
32
(cos6θ+15cos2θ) (c)cos6θ (d)10
16
21The polar form of 1+i is
𝜋
𝜋
𝜋
𝜋
𝜋
𝜋
4
𝜋
4
𝜋
4
4
2
2
4
4
(a)cos +isin (b) 2(cos +isin ) (c)√2cos +isin )
√2(cos +isin )
𝜋
𝜋
4
4
𝜋
𝜋
(cos 3 +isin 3
(cos +isin
22.
)4
)3
= ------- (a)1 (b)0 (c)2 (d) 3
23.The amplitude form of 1+i√3 is
(a)tan-1 √3 (b)π (c)
𝜋
2
(d) -1
24.|1+i| = ------- (a)2 (b)1 (c)-1 (d)√2
25.(1+cosθ +isinθ)n +(1+ cosθ -isinθ)n = --------(a)2n+1cosn
𝜃
2
cosn
𝜃
2
(b)2n cos
𝜃
2
cosn
𝜃
2
Section B
(c) 1 (d) 0
(d)
1. Show that the triangle formed by the points √3+i,√2+√2i,1+√3I in
the Argand plane is isosceles
2.If
sin 𝜃 5045
𝜃
=
5046
show that θ=1⁰ 58’ approximately.
3. Express cos 6θ interms of cos θ
4.Prove that (1+i√3)n +(1-i√3)n = 2n+1 cos
𝑛𝜋
3
5.show that the points -45-5i , 1-i and 6+3i in the Argand diagram are
collinear.
Section C
1. Show that the point representing the complex numbers 5+4i ,3-2i,-43i,-2+3i
form a parallelogram.
2.Prove that
3. Prove that
4. If
sin 𝜃
𝜃
=
𝑐𝑜𝑠7𝜃
𝑐𝑜𝑠𝜃
sin 7𝜃
sin 𝜃
5045
5046
= 64cos6θ-112cos4θ+56cosθ-7
=7-56 sin2𝜃 +112sin4𝜃 -64sin6𝜃
prove that 𝜃 is 1⁰58’ nearly.
5.Expand cos8 𝜃 in terms of sin 𝜃