D-23 / DC-23

TYPICAL QUESTIONS & ANSWERS

PART – I

OBJECTIVE TYPE QUESTIONS

Each Question carries 2 marks:

Choose the correct or best alternative in the following:

Q.1 The points 2i – j + k, i – 3j – 5k, 3i – 4j – 4k are the vertices of a triangle which is

(A) equilateral. (B) isosceles.

(C) right angled. (D) None of these.

Ans: C

Thus ∆ is right angled

Q.2 If then ordered pair (x, y) is

(A) (0, 2). (B) (0, 1).

(C) (1, 0). (D) (1, 1).

Ans: C

i.e. Pair (x, y) is (1, 0).

Q.3 If then is

(A) . (B) .

(C) . (D) .

Ans: D

…………….……………(1)

Similarly ……………….(2)

Adding equation (1) and (2) we get

Q.4 A vector of magnitude 2 along a bisector of the angle between the two vectors 2i - 2j + and i + 2j - 2 is

(A) . (B) .

(C) . (D) None of these.

Ans: A

Let and be unit vectors along a and b respectively. ,

Required vector .

Thus

Q.5 Let A and B be two matrices such that and AB =0. Then we must have

(A) B = 0. (B) B to be identity matrix.

(C) . (D) None of these.

Ans: D

Q.6 If then is

(A) 0. (B) 1.

(C) 2. (D) 3.

Ans: A

Since c₁ & c₂ are same

Q.7 exists only when n is

(A) zero. (B) –ve integer.

(C) +ve integer. (D) –ve rational.

Ans: C

, n is positive integer.

Q.8 The differential equation of the curve , where a and b are constants, is

(A) . (B) .

(C) . (D) .

Ans: D

Since y = a cos(x – b)

Q.9 If are vectors then is equal to

(A)

(B)

(C)

(D) none of above.

Ans: C

Q.10 If A, B are square matrices of the same size then

(A) (B)

(C) (D)

Ans: B

By definition

Q.11 If are two complex numbers then is

(A) (B)

(C) (D)

Ans: B

(Triangle inequality)

Q.12 The value of is equal to

(A) 3a²x (B) a²(3x - a)

(C) a²(3x + a) (D) 3ax²

Ans: C

Q.13 If I+A+A²+…+A^K=0, then A^-1 is equal to

(A) A^K (B) A^K-1

(C) A^K+1 (D) I+A

Ans: A

If (Characteristic equation of Matrix)

(Divided by A)

Q.14 If A is any real square matrix then A+A^t is

(A) Hermitian. (B) Skew-hermition.

(C) Symmetric. (D) Skew-symmetric.

Ans: C

Q.15 The Laplace transform L(tⁿ) is

(A) . (B) .

(C) . (D)

Ans: B

Q.16 The solution of differential equation is

(A) (B) .

(C) . (D)

Ans: C

A.E Roots are real and equal.

and P.I = 0

Q.17 The value of a₀ in the Fourier series is given by

(A) (B)

(C) (D) 0

Ans: A

By definition

Q.18 The inverse Laplace transform is

(A) (B)

(C) (D)

Ans: C

Q.19 Let and . Express in the form

a + bi, a , b R.

(A) (B)

(C) (D)

Ans: B

Q.20 The complex numbers , and satisfying are vertices of the a triangle which is

(A) acute-angled and isosceles (B) right-angled and isosceles

(C) obtuse-angled and isosceles (D) equilateral

Ans: D

Q.21 A unit vector parallel to 3i+4j-5k is

(A) (B)

(C) (D)

Ans: A

Q.22 Let = (1, 2, 0), = (-3, 2, 0), = (2, 3, 4). Then equals

(A) 33 (B) 30

(C) 31 (D) 32

Ans: D

Q.23 If is complex cube root of unity, and , then is equal to

(A) 0 (B) -A

(C) A (D) none of these

Ans: C

Q.24 If A and B are symmetric matrices, then AB + BA is a

(A) diagonal matrix (B) null matrix

(C) symmetric matrix (D) Skew-symmetric matrix

Ans: C

Q.25 The function is

(A) odd (B) even

(C) neither (D) none of these

Ans: B

Q.26 The function cos x + sin x + tan x + cot x + sec x + cosecx is

(A) both periodic and odd (B) both periodic and even

(C) periodic but neither even nor (D) not periodic

odd

Ans: C

Q.27 The Laplace Transform for sin at is

(A) (B)

(C) (D)

Ans: B

Q.28 The Inverse Laplace Transform for is

(A) (B)

(C) (D)

Ans: A

Q.29 The smallest positive integer n for which is

(A) 8 (B) 12

(C) 16 (D) None of these

Ans: D

Q.30 A square root of 3 + 4i is

(A) (B)

(C) (D) None of these

Ans: C

Q.31 Any vector a is equal to

(A) (B)

(C) (D)

Ans: A

Q.32 If a and b are two unit vectors inclined at an angle and are such that a + b is a unit vector, then is equal to

(A) (B)

(C) (D)

Ans: D

Q.33 The value of the determinant , where is an imaginary cube root of unity is

(A) (B) 3

(C) (D) 4

Ans: B

Q.34 The value of the determine is equal to

(A) -4 (B) 0

(C) 1 (D) 4

Ans: D

Q.35 The inverse of a diagonal matrix is

(A) not defined (B) a skew-symmetric matrix

(C) a diagonal matrix (D) a unit matrix

Ans: C

Q.36 The period of function sin 2x + cot 3x + sec 5x is

(A) (B)

(C) (D)

Ans: B

Q.37 The Laplace transform of is

(A) (B)

(C) (D)

Ans: A

Q.38 The solution of the differential equation is

(A) (B)

(C) (D)

Ans: C

Q.39 Modules of is

(A) (B)

(B) (D)

Ans: A

Let

Q.40 If then the value of cos x cos hy is

(A) –1 (B) 0

(C) 1/2 (D) 1

Ans: D

Q.41 The two non-zero vectors and are parallel if

(A) (B)

(C) (D)

Ans: A

Two non-zero vector and are parallel if = 0

Q.42 The volume of the parallelopipid with sides , A is

(A) 5 cubic units (B) 10 cubic units

(C) 15 cubic units (D) 20 cubic units

Ans: B

Volume of parallelepiped with sides

cubic units

Q.43 If then eigen value of A^–1 are

(A) (B) 1, 2, 3

(C) 0, 1, 2 (D)

Ans: A

Let A =

Eigen values of A are 1, 2, 3

eigen values of are

Q.44 The sum and product of the eigen values of are

(A) Sum = 5, Product = 7 (B) Sum = 7, Product = 5

(C) Sum = 5, Product = 5 (D) Sum = 7, Product = 7

Ans: B

Sum of Eigen value = 07

Product of Eigen value = 5

Q.45 If then the value of f(0) is

(A) 0 (B)

(C) (D)

Ans: C

Zero is the point of discontinuously

Q.46 The inverse Laplace transform of (s+2)^–2

(A) e^–2t (B) e^2t

(C) te^2t (D) te^-2t

Ans: D

by first shifting theorem

Q.47 The solution of the differential equation satisfying the condition y(0)=1, is

(A) (B)

(C) (D)

Ans: B

c.f =

putting x = 0, y(0) = 1

Putting

Q.48 Fourier Sine transform of 1/x is

(A) S (B) S/2

(C) S²/2 (D) –S²/2

Ans: C

Q.49 The complex numbers Z = x + iy, which satisfy the equation lie on

(A) the x-axis.

(B) the line y = 5.

(C) A circle passing through the origin.

(D) None of these.

Ans: A

i.e x-axis

Q.50 If , then

(A) (B)

(C) Z=0 (D) , with x real

Ans: B

Given

Q.51 If and are two unit vectors and is the angle between them, then is equal to

(A) (B) 0

(C) (D)

Ans: C

Given are Unit vector

Now

Q.52 A vector which makes equal angles with the vectors , and is

(A) (B)

(C) (D)

Ans: B

Let vector be

Let then

Q.53 If is a cube root of unity and , then

(A) x = 1 (B)

(C) (D) none of these

Ans: D

Q.54 If , then is equal to

(A) (a+b) (b+c) (c+a) (B) bc + ca + ab

(C) 2abc (D) none of these

Ans: D

= (a-b).a[a – c + a – c]

= 0

Q.55 If A is a skew-symmetric matrix and n is a positive integer, then is

(A) a symmetric matrix.

(B) skew-symmetric matrix for even n only.

(C) diagonal matrix.

(D) symmetric matrix for even n only.

Ans: D

Q.56 The period of the function sin x + sin 2x + sin 3x is

(A) (B)

(C) (D)

Ans: D

then f(x) is periodic to Ө

Q.57 The Laplace transform of is

(A) (B)

(C) (D)

Ans: A

Putting

Q.58 The solution of the differential equation is

(A) (B)

(C) (D)

Ans: C

(m – 3(m – 2) = 0

m = 2, 3

P.I. =

Y = C.F. + P.I. =

Q.59 If and represent conjugate complex numbers then the value of x and y is

(A) . (B) .

(C) . (D) .

Ans: A

(1)

(2)

The conjugate of A is

But given

(3)

(4)

(5)

y = -4, 1

if y = -4 then by Eq. (4)

Q.60 Imaginary part of is

(A) – cos x cosh y (B) – cos x sinh y

(C) – sin x cosh y (D) – sin x sinh y

Ans: B

Imaginary point of

Imaginary part = -cos x sin hy

Q.61 Three vectors are coplanar, the value of their scalar triple product is

(A) 0 (B) 1

(C) –1 (D) i

Ans: A

Q.62 If is the angle between the vectors and such that then is

(A) (B)

(C) (D)

Ans: B

Q.63 The value of the determinant is

(A) 1 (B) 2

(C) –1 (D) 0

Ans: D

The value of

as two columns are similar

Q.64 If the product of two eigen values of the matrix is 16, then the third eigen value is

(A) 0 (B) 5

(C) 2 (D) –2

Ans: C

Since the product of two eigen value of the matrix is 16. check is by the options, the product of all the eigen value, should be equal to the value of the determinants.

In this question value of determinants is

6(9 – 1) + 2(-6 + 2) + 2(2 – 6)

48 – 8 – 8 = 48 – 16 = 32

Since two eigen value product = 16

Hence for product to be 32, third eigen value should be 2.

Q.65 If f(x) is defined in (0, L), then the period of f(x) to expand it as a half range sine series is

(A) L. (B) 0.

(C) 2L. (D) .

Ans: C

Q.66 The inverse Laplace transform is possible only when n is

(A) 0 (B) –ve integer

(C) –ve rational number (D) +ve integer

Ans: D

Q.67 The differential equation of a family of circles having the radius r and centre on the x axis is

(A) (B)

(C) (D)

Ans: A

The eq. of family of circle, having radius r, and centre on the x axis is

(1)

(2)

(3)

Putting the value from eq.(3) into the eq.(1)

Q.68 If y satisfies with then Laplace transform is

(A) (B)

(C) (D)

Ans: Correct option is not available; however the solution is:

Or solution is =

Ans is D; if y satisfies with

Q.69 If then is equal to

(A) .

(B) .

(C) .

(D) .

Ans. B

z₁ = r₁ (Cos q₁ + i Sin q₁)

z₂ = r₂ (Cos q₂ + i Sin q₂)

z₁ z₂ = r₁ r₂ (Cos q₁ + i Sin q₁) (Cos q₂ + i Sin q₂)

= r₁ r₂ [(Cos q₁ Cos q₂- Sin q₁ Sin q₂) + i (Cos q₁ Sin q₂ +Cos q₂ Sin q₁)]

= r₁ r₂ [Cos (q₁ +q₂) + i Sin (q₁ +q₂)]

Q.70 If is cube root of unity then is equal to

(A) 0. (B) 1.

(C) -1. (D) 3.

Ans. A

If w is cube root of unity then we know that 1+w+w²=0

Q.71 The roots of are

(A) 2, 3. (B) 3, 2.

(C) 4, -3. (D) 4, 3.

Ans. C

Given x²-x-12=0 Þ (x-4) (x+3) = 0 Þ x=4,-3

Q.72 If and then AB is equal to

(A) . (B) .

(C) . (D) .

Ans. A

Given

Q.73. If A and B are invertible matrices of the same size then is equal to

(A) AB. (B) BA.

(C) . (D) .

Ans. C

Given

A^-1A = I, B^-1B = I

Now (AB) (B^-1 A^-1) = AIA^-1 = AA^-1 = I --......................---------------(1)

Also (B^-1 A^-1) (AB) = B^-1 (A^-1A) B = B^-1 IB = B^-1 B=I-----------------(2)

from 1 and 2, we get (AB)^-1 = B^-1 A^-1

Q.74 If A and B are the points (3, 4, 5) and (6, 8, 9) then the vector is

(A) . (B) .

(C) . (D) .

Ans. A

Given A ( 3,4,5) and B (6,8,9)

Q.75 The function f (x) = Sin x is

(A) non periodic. (B) periodic with period .

(C) periodic with period . (D) periodic with period .

Ans. C

We know that the function ¦(x) = Sin x is periodic and period is 2p

Q.76 The Laplace transform of Sinh (at) is

(A) . (B) .

(C) . (D) .

Ans. B

By definition

PART – II

NUMERICALS

Q.1 If the complex numbers be the vertices of an equilateral triangle, prove that . (7)

Ans:

Given that Z₁, Z₂, Z₃ be the vertices of an equilateral triangle.

i.e. …………….(1)

And ……………(2)

Dividing (1) by (2) we get

Q.2 If the roots of represent vertices of a triangle in the Argand plane, then find area of the triangle. (7)

Ans:

Root of above equation are the vertices of ∆

i, -i+1, -i-1

Q.3 Reduce to the modulus amplitude form. (7)

Ans:

Q.4 Prove that . (7)

Ans:

L.H.S.=

=R.H.S. Hence proved.

Q.5 If a square matrix A satisfies a relation Prove that exists and that being an identity matrix. (7)

Ans:

Given that a square matrix A satisfies a relation . By Cayley Hamilton Theorem

Thus Exists

Q.6 Show that any square matrix can be written as the sum of two matrices, one symmetric and the other anti-symmetric. (7)

Ans:

Let A be a square matrix

Now

= ………..(1) is a symmetric matrix

Also

= ……..(2) is a skew-symmetric

Also

= symmetric matrix + skew-symmetric (from (1) and (2) )

Q.7 Show that x = 2 is one root of the determinant and find other two roots. (6)

Ans:

Given , when x = 2, then

As two rows are same

Thus x – 2 is a root of given equation.

Now calculate other two Roots

Applying

x = 1, x = 2, x = -3

Thus other Roots are 1, -3

Q.8 Show that . (8)

Ans:

To prove

L.H.S. =

Applying we get

Q.9 If and be any two vectors, then show that

(i) .

(ii) . (7)

Ans:

(i) LHS =

= { = }

Hence Proved

(ii)

L.H.S =

= Hence Proved

Q.10 Forces of magnitudes 5, 3, 1 units respectively, act in the directions respectively on a particle. If the particle is displaced from the point to the point , find the work done by the resultant force.

(7)

Ans:

Force

= 123 + 108

= 231

Q.11 Verify that satisfies its characterstic equation and then find . (6)

Ans:

Characteristic Equation =

By Clayey Hamilton theorem

Now we have

This verifies the characteristic equation.

Now

Multiplying by A^-1

Q.12 Test for the consistency and solve the system of equations.

. (8)

Ans:

Test for consistency

, AX=B

Let

Now R(A) = R(C) = 2 < 3

System is consistent but infinity many solution.

Z = k, 11y – Z = 3

5x + 3y + 7z = 4

Q.13 Show that the area of the parallelogram with diagonals and is . (7)

Ans:

Let PQRS be a parallelogram with diagonal and they intersect at T

Area of parallalogram PQRS =

= Hence proved.

Q.14 Find the area of the triangle whose vertices are . (7)

Ans:

Let O be origin,

Area of Δ ABC =

Q.15 Find a Fourier series that represents the periodic function f (x) = , . (14)

Ans:

Let …………..(1)

= ( is odd function)

And

= ( is odd function)

Putting value of in (1) we get

Q.16 Find the Laplace transform of . (7)

Ans:

Now we have

= Ans.

Q.17 Find the inverse Laplace transform of . (7)

Ans:

= Ans.

Q.18 Solve . (7)

Ans:

Solve

A.E.,

m = -2, -3

C.F =

P.I =

Y = C.F + P.I =

Q.19 Use Laplace transform method to solve , if x = 2 and at t = 0. (7)

Ans:

Taking Lapalace transformation on both sides

x =

Q.20 Express in the form x+iy. (8)

Ans:

Q.21 Write down all the values of . (8)

Ans:

Let 1 + I = r()

, n = 0, 1, 2, 3

= , n = 0, 1, 2, 3

Q.22 Using vector method prove that the altitudes of a triangle are concurrent. (8)

Ans:

Let ABC be any angle

Draw AD ┴ BC and BE ┴ AC

Let AD and BE intersect at O. Join CO

We shall prove that CF ┴ AB

Let ā, , be the position vector of A, B, C respectively with O.

……….(1)

Also

………..(2)

Adding (1) and (2) we get,

Hence altitude of a triangle is concurrent.

Q.23 Find a unit vector perpendicular to the plane of vectors and . (8)

Ans:

Unit vector perpendicular to

Q.24 Prove that (8)

Ans:

Now

And

Adding equation1, equation 2 & equation 3 we get

= (b.c)(c.d) - (c.a)(b.d) + (c.b)(a.d) – (a.b)(c.d) - + (a.c)(b.d) – (b.c)(a.d)

= 0. Hence proved.

Q.25 Find the angle between two vectors and if = . (8)

Ans:

Let Angle between and be Ө

given

Q.26 Let A be a square matrix. Prove that A can be written the sum of a symmetric and a skew-symmetric matrix. (8)

Ans:

Let A be a square matrix

Let

Now

is a symmetric matrix

Also is skew-symmetric

Thus A = symmetric matrix + skew-symmetric.

Q.27 State Cayley Hamitton theorem and use it to find the inverse of, if the inverse exists. (8)

Ans:

Every square matrix satisfying its characteristic Equation.

By using Cayley-Hamilton Theorem

Q.28 Prove that . (8)

Ans:

L.H.S =

= (b – a)(c – a)(c – b)(ab + bc + ca)

= R.H.S.

Q.29 Give condition under which we can find so that the following system of linear equations has a non-trivial solution.

(8)

Ans:

Given system of equation

is homogenous. For non trivial solution.

R(A) = R(C) < n here n = 3

Obviously R(A) = R(C) = 2 i.e.

R(A) must be 2.

and

Q.30 Find the Fourier series of the function defined by

(8)

Ans:

Let

Where

Now

a_n =

And

Putting value of a₀, a_n and b_n in (1)

Fourier series

Q.31 Find the Fourier series representing the function

(8)

Ans:

f(x) = x, 0 < x < 2π

Let Fourier series of f(x)

………….(1)

Where

Now,

= 0

Now,

Q.32 If F(t) is piecewise continuous and satisfies for all and for some constants a and M then

(8)

Ans:

We are given ………..(1)

Without loss of generality, assume that a is positive.

Let

Then G(t) is continuous.

Also

………….(2)

Now except for points where F(t) is discontinuous.

is piece-wise continuous on each finite interval.

We know that if F(t) is continuous for all t and of experimental order a as, and if is of class A, then

----------F(o)

Therefore ……….G(o)

= as G(o)=0

Q.33 Define Inverse Laplace Transform of a function F(t). Prove that

(8)

Ans:

Q.34 Solve . (8)

Ans:

The given equation is

Auxiliary equation is

C.F =

P.I =

Q.35 If a, b, c are real numbers such that and b + ic = (1 + a)z, where z is a complex number, then show that . (8)

Ans:

Q.36 Given that and where is a cube root of unity. Express in terms of A, B, C and . (8)

Ans:

On adding,

Q.37 Show that for all real , . (8)

Ans:

Q.38 For any four vectors prove that .

Hence prove that (8)

Ans:

Adding the three relation we get

Q.39 In let , . Then find the vector representing AB and OM, where M is the midpoint of AB. (4)

Ans:

Q.40 Prove that the straight line joining the mid-points of two non-parallel sides of a trapezium is parallel to the parallel sides and is half their sum. (12)

Ans:

let ABCD be the trapezium and let A be at origin

i.e. PQ is parallel to AB and half the sum of parallel sides.

Q.41 For reals A, B, C, P, Q, R find the value of determinant

(8)

Ans:

Q.42 Using matrix method find the values of and so that the system of equations:

has infinitely many solutions. (8)

Ans:

x = 3 – z

y = z – 2

z = arbitrary

i.e. infinite solution.

Q.43 Solve the system of equations

by using inverse of a suitable matrix. (8)

Ans:

Q.44 Using Cayley-Hamilton theorem find for . (8)

Ans:

Q.45 State whether the function f(x) having period 2 and defined by

is even or odd. Find its Fourier Series. (16)

Ans:

Q.46 Find the Laplace transform of . (8)

Ans:

Recall the first shift theorem

Q.47 Solve (8)

Ans:

A.E =

C.F =

P.I =

Y = CF + P.I

Y = Ans.

Q.48 Find the Inverse Laplace transform for . (8)

Ans:

Q.49 Solve the differential equation

given that y = -0.9 and , when x=0 (8)

Ans:

C.F =

P.I. =

Q.50 Using the Laplace transform solve the differential equation

with initial conditions . (8)

Ans:

Q.51 If n is a positive integer, prove that . (8)

Ans.

………….(1)

………….(2)

………….(3)

from (2) and (3), r = 2,

------>(4)

put the value of r and q in eq n(y) we have

Q.52 Find all the values of and show that the product of all these values is 1. (8)

Ans:

…………(1)

……………(2)

…………….(3)

where m = 0, 1, 2, 3.

\The continued product of these roots

$Oval: \cosp = -1 and sinp = 0$

Q.53 If the roots of represent vertices of a triangle in the Argand plane, then find area of the triangle. (8)

Ans:

Roots are z = i, -i + 1, -i -1,

= 2.

Q.54 Find the value of if

. (8)

Ans:

= i(-1) + 7j + 5k

Now,

Q.55 Prove that the sum of all the vectors drawn from the centre of a regular octagon to its vertices is the zero vector. (8)

Ans:

Let ABCDEFGH be a regular octagon

And O the centre of this octagon, O is

the mid-point of diagonals AE, BF, CG and DH.

Now,

Q.56 Find the moment about the point of the force represented in magnitude and position by , where the point A and B have the co-ordinates and respectively. (8)

Ans:

Q.57 Show that . (8)

Ans: Multiplying C₁, C_2,& C₃by a, b and c respectively, we get

Q.58 Write the following system of equations in the matrix form AX = B and solve this for X by finding A^-1.

(8)

Ans:

Writing the given equations in matrix form, we have

Q.59 Using matrix methods, find the values of and so that the system of equations

has (i) unique solution and (ii) has no solution (8)

Ans: Ax = B

(ii) For no solution

C(A) ≠ C(C)

If -l-5 = 0 C( A ) = 2

And m-9 ≠ 0 C( C ) = 3

For unique solution l ≠ -5, m ≠ 9

For no solution l = -5, m ≠ 9

Q.60 Verify Cayley Hamilton theorem for the matrix

Use Cayley Hamilton theorem to evaluate A^-1 and hence solve the equations

(8)

Ans:

Q.61 Find the Fourier series for the functions

(16)

Ans:

Q.62 Find the Laplace transform (8)

Ans:

Q.63 Find the inverse Laplace transform (8)

Ans:

Q.64 Solve the differential equation

(8)

Ans:

Auxiliary equation is

It is a case of failure.

Y = C.F. + P.I.

Q.65 By using Laplace transform, solve the differential equation

(8)

Ans:

Taking inverse Laplace transform

Q.66 A rigid body is spinning with angular velocity 27 radians per second about an axis parallel to passing through the point . Find the velocity of the point of the body whose position vector is . (8)

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Q.67 Find the sides and angles of the triangle whose vertices are , and . (8)

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= (2,1,-1) – (1,-2,2)

= (3,-1,2) – (2,1,-1)

= (3,-1,2) – (1,-2,2)

cosC = = 0

Now

Q.68 Find the volume of the tetrahedron formed by the point (1,1,1) (2,1,3) (3,2,2,), (3,3,4). (8)

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Area of Trapezium =

Q.69 The centre of a regular hexagon is at the origin and one vertex is given by on the Argand diagram. Determine the other vertices. (8)

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(-2, 0)E x

O(0,0) B(2, 0)

, B(2,0),

, E(-2,0),

Q.70 Prove that the general value of which satisfies the equation

where m is any integer (8)

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Equating real part on both side

Q.71 Use De Moivre’s theorem to solve the equation x⁴–x³+x²–x+1=0 (8)

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Given that

Multiplying on both side by (x + 1)

Putting n = 0,1,2,3,4

But

Hence roots of are

But root → corresponding to (x + 1)

Root of the equation

Q.72 Show that

(8)

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R.H.S. hence proved.

Q.73 Express the following matrix as a sum of a symmetric matrix and a skew symmetric matrix.

. (8)

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Thus and

A = symmetric + skew symmetric

Q.74 Find the values of l, for which following system of equations has non-trivial solutions. Solve equations for all such values of l.

(8)

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AX = B

If system of equations has non-trivial solution then R(A) = R(C) < n = 3

Putting λ = 0

let

(infinite solution)

At λ = 3

= 2x + 10y + 6z = 0

Let

Q.75 Find the characteristic equation of the matrix and hence evaluate the matrix equation A⁸–5A⁷+7A⁶–3A⁵+A⁴–5A³+8A²–2A+I. (8)

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By using Cayley-Hamilton Theorem

Now,

Q.76 Expand in a Fourier Series.

Hence evaluate (16)

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Where,

Thus the fourier series is

Let x = 0,

0 =

Thus

Q.77 Simplify . (8)

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= putting

Q.78 Find all the values of . (8)

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Let

Putting n = 0,1,2,3,4

Q.79 If and are two complex numbers, prove that

If and only if is purely imaginary. (8)

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Prove that

is purely imaginary

First assuming that and prove that

is purely imaginary

Given

Let

(given)

Now we have

is purely imaging

Conversely assuming that is purely imaging and we shall prove that

(purely imaging i.e. Real part 0)

Q.80 A vector satisfies the equation . Prove that provided and are not perpendicular. (8)

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In question condition must be instead of

(given condition is wrong it should be or x.a = 0)

Q.81 Using vector methods prove that the diagonals of a parallelogram bisect each other. (8)

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In parallelogram =

position vector = position vector

mid point of = mid point of

Diagonal of bisect to each other.

Q.82 The constant forces 2i – 5j + 6k, -i+2j-k and 2i + 7j act on a particle which is displaced from position 4i – 3j – 2k to position 6i + j – 3k. Find the total work done. (8)

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Displacement = (6i + j – 3k) -

w = f.d =

= 6 + 16 – 5

= 17 N

Q..83 Show that

(8)

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Applying on L.H.S.

= = R.H.S. Hence proved.

Q.84 Write the following equations in the matrix form AX = B and solve for X by finding .

(8)

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Q.85 Test the consistency of the following equations and if possible, find the solution

(8)

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Given system of equation

Now

R(A) = R(C) < n

R(A) = R(c) = 2 < 3

Given system of equation is a consistent

Now we have

Let z = k, -y + 3z = 2

-y = 2 – 3k

y = 3k – 2

x + y – 3z = -1

x = -1 – 3k + 2 + 3k

x = +1

Different value of k, system has infinite solution.

Q .86 Obtain the characteristic equation of the matrix and use Cayley-Hamilton theorem to find its inverse. (8)

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, characteristic equation

i.e. ……………..(1)

by using Clayey-Hamilton Theorem, A satisfying (1)

Q.87 Find the Fourier series expansion for the function

. (16)

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Let …………….(1)

Now, we have

Now,

Q.88 Find the Laplace transform of . (8)

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we know that

Now,

Q.89 Find the Inverse Laplace transform of (8)

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Q.90 Solve the differential equation

. (8)

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m = 2, 3

C.F. =

P.I. =

y = C.F + P.I

Q.91 By using Laplace transform solve the differential equation

with initial conditions , when t = 0. (8)

Ans:

………………..(1)

Taking Laplace transform of equation (1)

By convolution

Q.92 Find the moment of the force about a line through the origin having direction of , due to a 30 Kg force acting at a point (– 4, 2, 5) in the direction of . (8)

Ans:

Let D be given line through the origin O and be the force through A(-4, 3, 5).

Moment of about

Thus the moment of about the line

Q.93 Prove that the right bisectors of the sides of a triangle intersect at its circum centre. (8)

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Let A,B,C be the vertices of ∆ ABC, the mid-point of the sides BC, CA and AB are D,E,F let ┴ at D and E to BC and CA respectively

interests the point ; then

……….(1)

And

……(2)

Adding (1) & (2) , we get

Ans.

Q..94 Show that the components of a vector along and perpendicular to in the plane of and are and . (8)

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Let and OM be the projection of on .

Component of along = OM

Also component of

= Ans.

Q.95 If show that and . (8)

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Also

(By Componendo and Devidendo)

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Q.96 If then . (8)

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Let

Now

Q.97 Show that the origin and the complex numbers represented by the roots of the equation , where a, b are real, form an equilateral triangle if . (8)

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Thus B

, hence they form an equilateral triangle.

Q.98 Prove that

. (8)

Ans:

Ans.

Q.99 Determine the values of when is orthogonal. (8)

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If A is orthogonal then

But

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Q.100 Find the values of k such that the system of equations, , has non-trivial solution. (8)

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The set of equation is

For a non-trivial solution ρ(A) = ρ(A : B) = 2

Thus

A =

-4(3 – 4k) – (1 – 2k)(k – 12) = 0

Q.101 Find the characteristic equation of the matrix . Hence find . (8)

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Characteristic equation is

Q.102 Find the Fourier series for . (16)

Ans:

Ans.

Q.103 Find . (8)

Ans:

Ans.

Q.104 Find the inverse Laplace transform of . (8)

Ans:

Ans.

Q..105 Using Laplace transformation, solve the following differential equation:

if x(0) = 1, . (8)

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Taking Laplace Inverse transform

Put we get

Q..106 If z is any complex number and is its complex conjugate then show that . (7)

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Also |z|² = ----------(2)

From (1) and (2),

Q..107 Find the square root of the complex number 3 + 4i. (7)

Ans:

and

= 9+16 = 25

from (1) and (3) x²=4, y²=1

from (2) xy is positive so if x=2, y=1 and x=-2, y=-1

Hence

Q..108 If then find (7)

Ans:

Given z=Cos q +i Sin qÞ zⁿ = Cos n q + i Sin n q,
z^-n =Cosnq-iSinnqTherefore zⁿ+z^-n= 2 Cos n q.

Q..109 If then show that . (7)

Ans:

Given

.................

Now

= Cos p + i Sin p = -1

Q..110. If a square matrix A is invertible then show that (transpose of A) is also invertible and . (7)

Ans:

Since A is invertible matrix, therefore |A| ¹0 Þ|A^T|¹0

Þ A^T is also invertible

Now AA^-1 = I = A^-1A Þ (AA^-1)^T = I = (A^-1A)^T Þ (A^-1)^T A^T= I = A^T(A^-1)^T

Þ (A^T)^-1 = (A^-1)^T

Q..111 Compute the inverse of the matrix . (7)

Ans:

and

, ,

Q..112. Evaluate where is a complex cube root of unity. (7)

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= 0, Since

Q..113 Show without evaluating that determinant . (7)

Ans:

C₂ à C₂+C₃

= (x+y+z) 0 [C₁ and C₂ are identical]

= 0

Q..114 Find the position vector of a point which divides the line joining two given points in three dimensional space. (7)

Ans:

Let the position vectors of points A and B are and respectively. Let P be the point which divides the line joining A and B in the ratio m:n and let be the position vector of P. Then where O is origin

Given

Now ,

From (i) we get

Q..115. Show that the vectors and form the sides of a right angled triangle. (7)

Ans: