MTH401 Current Final Term Spring 2011 Complete Paper with MCQ S
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CoooL Virtual University Students google group MTH401 Current final term Spring 2011 complete paper with MCQ’s
The annihilator operator of the function y= e-6x is (D - 6)2 (D - 6)3 D - 6 D + 6
After converting the given differential equation 4y// + 64 y = sec3 x into standard form, the function f(x) is (Sec3x)/4 (Sec3x)/64 None of them Secx
Wronskian of the function yc = c1 + c 2cos x + c 3 sin x is 0
1
2
3
In the infinite series of (x-a) which can be written as
n 2 cn ( x- a ) = c0 + c 1 ( x - a ) + c 2 ( x - a ) + ... the number a is called the n=0 Radius of power series Centre of power series Base of power series None of them [email protected]
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n Suppose that a power series an ( x- a ) is represented by a function “f”whose domain is n=1 the interval of the convergence of the power series. That function “f” is continuous, differentiable and integrable on
(a + R, a - R) (R -a, R + a) (a - R, a + R) None of them
The interval of convergence for the function secx is
(-p , + p ) p p (- , ) 2 2 p ( ,p ) 2 None of them
dy The solution of the linear first order differential equation -2xy = 0 is dx 2 y= ex x2n y = n=0 n! 2n 2 x Both y= ex & y = n=0 n! None of them [email protected]
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CoooL Virtual University Students google group 1 The quantity X= Lg - is called Cg Reactance of circuit Impedance of circuit Quasi of circuit None of them
d2 x dx For the equation of free damped motion +2l + w 2 x = 0 the roots are dt2 dt 2 2 2 2 2 2 m1 = -l + l - w & m1 = -l - l - w If l- w > 0 then system is said to be Under damped Over damped Critically damped Noneof them
The time interval between two successive maxima of x( t )= Ae-lt sin[w2 - l 2 + f ] is called Phase period Quasi-period Both the period None of them
d2 x dx The given differential equation +5 + 4x = 0 is dt2 dt Over damped Critically damped Under damped None of them
The standard unit for measurement of inductance is [email protected]
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CoooL Virtual University Students google group Volt Ohms Henry None of them
Which of the rule in matrices under multiplication does not hold true? Commutative law Associative law Identity law None of them
轾x y z a 轾1 2 3 A=& B = 犏 p q r b matirx A B If 犏 犏 then the order of is 臌5 6 7 臌犏l m n o
2 4 2 3 3 3 None of them [email protected]
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CoooL Virtual University Students google group d 骣x 骣3- 7 骣 x 骣 4 The given system without the use of matrices 琪= 琪 琪 + 琪 sin t is dt 桫y 桫1 1 桫 y 桫 8 dx dy =3x - 7 y + 4sin 2 t ; = x + y + 8cos 2 t dt dt dx dy =3x - 7 y + 4sin t ; = x + y + 8cos t dt dt dx dy =3x - 7 y + 4sin t ; = x + y + 8sin t dt dt None of them
Suppose that {X1, X2, X3,…, Xn} is a set of n solutions vectors on an interval I, of a homogeneous system X/=AX. The set is said to be a fundamental set of solutions of the system on the interval I if the solution vectors are
Linearly dependent Linearly independent Homogeneous None of them
The coefficient matrix of the following homogeneous system of differential equation dx dy =3x + 2 y , = x + 2 y is dt dt 轾3 2 犏 臌2 2 轾3 1 犏 臌2 2 轾3 2 犏 臌1 2 None of them
轾3- 18 The matrix A = 犏 has an eigen value of multiplicity 臌2- 9 [email protected]
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2
3
4
轾1- 2 - 2 A =犏 -2 1 - 2 l= -1, - 1,5where l = - 1 The matrix 犏 has eigen values is a 臌犏2- 2 1 Single root of A triple root of A double root of A None of them
4- l 1 0 0 4-l 1 = 0 gives 0 0 4 - l l = 4 of multiplicity of 1 l = 4 of multiplicity of 2 l = 4 of multiplicity of 3 None of them
dy dx For the system of differential equations =2x , = 3 y the independent variable(s) is dt dt (are) x, t y, t x, y [email protected]
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CoooL Virtual University Students google group t
If L denote the linear differential operators with constant coefficients, then L1 L 4- L 2 L 3 represents the
L1 L 2
L4 L 3
L1 L 3
L4 L 2
L1 L 2
L3 L 4 None of them
Wronskian of x,x2 is x x2 x3 0
dy x+ y The general solution of differential equation = .is given by dx x
y ex = cx y ex = cy x ey = cx x - ey = cx [email protected]
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The form of the exact solution to dy 2 3y ex , y0 5 dx is
Ae1.5x Bxex Ae1.5x Bex Ae1.5x Bxex Ae1.5x Bex
If m and n are non negative integers and Pn ( x ) is a Legendre’s polynomial then
1
Pm( x ) P n ( x ) dx= 0 for m n -1 1
Pm( x ) P n ( x ) dx= 0 for m = n -1 1
Pm( x ) P n ( x ) dx= 0 for m < 0 -1 [email protected]
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Pm( x ) P n ( x ) dx= 0 for n < 0 -1
If A is a square matrix and its determinant is zero, then A is singular matrix. A is non singular matrix. A is scalar matrix. A is diagonal matrix.
An electronic component of an electronic circuit that has the ability to store charge and opposes any change of voltage in the circuit is called
Inductor Resistor Capacitor None of them
Operator method is the method of the solution of a system of linear homogeneous or linear non-homogeneous differential equations which is based on the process of systematic elimination of the
Dependent variables Independent variable Choice variable None of them Any linear differential equation of the form dn y d n-1 y dy a xn+ a x n-1 +⋯ + a x + a y = g( x ) where a , a , a ..., a areconstants. is ndxn n-1 dx n-1 1 dx 0 0 1 2 n called Homogeneous equation Polar equation Equi-dimensioanl equation None of them [email protected]
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CoooL Virtual University Students google group 骣3 4 For eigen values l = 5,5 of a matirx A = 琪 ,there exists ...... eigen vectors. 桫-1 7 infinite one two three
Ordinary points of (x2-64 )( x 2 - 36 ) yⅱ+ xy - y = 0 are 0,1 8,-8 6,-6 None of others.
// / A singular point x= x0 of the given equation a2( x ) y+ a 1 ( x ) y + a 0 ( x ) y = 0 is said to be a regular singular point if
(x- x0 ) P ( x ) is analytic at x0
(x- x0 ) Q ( x ) is analytic at x0 2 (x- x0 ) P ( x ) &( x - x 0 ) Q ( x ) are analytic at x0 None of them [email protected]
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Singular points of the equation (x2- 4) 2 y // + ( x - 2) y / + y = 0 are x= -2, 2 None of them x =2 x = - 2
轾1 0 The matrix A = 犏 has ...... 臌0 1 Real and unequal value Repeated & real eigen value Complex eigen value None of them
Let l be an eigen value of a non zero square matrix A. Then the equation det(A-l I ) = 0 is called Trivial equation Characteristics equation Non-trivial equation None of them
2 If y1=x is solution of the differential equation, then formula for finding Second solution of x2 y // -2 y = 0 is
e-2x y= x2 dx 2 x2 e-2x y= x dx 2 x2 -2 e x y= x2 dx 2 x4 [email protected]
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CoooL Virtual University Students google group 2 y= x2 dx 2 x4
For ysin2 x- y 2 cos x = c where y (0) = 3; the valueof cis ------. 9 -9 10 -10
A differential equation M( x, y) dx+ N( x , y) dy = 0 is said to be an exact if -----.
抖M N = 抖x y 抖M N 抖y x 抖M N = 抖y x 抖M N = 抖y y
dy The integrating factor of the differential equation +(Cotx ) y = cos2 ( x ) is ------dx m = ln |Sinx | m = ln |Cosx | m = Sinx m = Cosx
2 Marks [email protected]
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Give two examples of Bessel’s Differential Equation?
Define matrix and give condition of multiplication of two matrices.
骣1-2t 骣 3 6 t Find Wronskain of two vectors X1=琪 e, X 2 = 琪 e . 桫-1 桫 5
What is half life of a substance?
Marks 3
d3 y d 2 y dy Write the differential equation -4 + 5 = 4x in the form L(y) = g(x) where L is dx3 dx 2 dx a differential operator with constant coefficients
c 0 Why we take 0 in Frobenius method?
Write the homogenous system of differential equations dx dy t 2- 5x + = 5 e dt dt
dx dy t -x + = e dt dt in decoupled form. [email protected]
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Find eigen vector of
骣-3 1 A=琪 ,corresponding eigen value l = -2 . 桫2- 4
Marks 5
Deduce the Special Case of Logistic Equation “Epidemic spread”?
dX 轾e4t sin 2 t 2 if X = Find 犏 3 dt 臌5t+ 4 ln(1 + t )
State the procedure to solve a differential equation by using Frobenius Theorem.
Write down the system of differential equations
dx dy =6x + y , = x + 3 y - 9 t - 9 dt dt in form of X'= AX + F ( t )