IIT JAM Mathematics MCQ Test 5


30 Questions MCQ Test Mock Test Series for IIT JAM Mathematics | IIT JAM Mathematics MCQ Test 5


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This mock test of IIT JAM Mathematics MCQ Test 5 for Mathematics helps you for every Mathematics entrance exam. This contains 30 Multiple Choice Questions for Mathematics IIT JAM Mathematics MCQ Test 5 (mcq) to study with solutions a complete question bank. The solved questions answers in this IIT JAM Mathematics MCQ Test 5 quiz give you a good mix of easy questions and tough questions. Mathematics students definitely take this IIT JAM Mathematics MCQ Test 5 exercise for a better result in the exam. You can find other IIT JAM Mathematics MCQ Test 5 extra questions, long questions & short questions for Mathematics on EduRev as well by searching above.
QUESTION: 1

If , then 

Solution:

On operating [R1 → R1-R2  and R3→ R3-R2]

QUESTION: 2

Let  , then the rank of M is equal to

Solution:

then

If 

hence rank M = 2

QUESTION: 3

If  , then

Solution:

Given

i.e., AAT = I

AT = A-1

QUESTION: 4

If  , then the matrix A is equal to

Solution:

we have

QUESTION: 5

If A is a non-zero column matrix of order m * 1 and B is a non-zero row matrix of order 1 * n, then rank of AB is equal to

Solution:

Let  and B = [b11 b12 b13..... b1n] be two non-zero columns and row matrices respectively.

Since A, B are non-zero matrices.

∴ Matrix AB will be a non-zero matrix. The matrix AB will have atleast one non-zero element obtained by multiplying corresponding non-zero elements of A and B. All the two-rowed minors of AB clearly vanish. 

∴ rank of AB = 1

QUESTION: 6

If then

Solution:

Applying C1 → C1 - C2 + C3 - C4, we get

expanding with respect to first column

expanding with respect to the third column

= (a - 1)3 (a - 1)2 (a - 1) = (a - 1)6.

QUESTION: 7

If a 3 x 3 matrix A has its inverse equal to A, then A2 is equal to ,  ......

Solution:

given

A-1 = A

⇒ A2 = I

QUESTION: 8

The sum of the eigen values of the matrix  for real and negative values of x is

Solution:

Eigen values are given by the solution of equation 

Since x is real and negative , put x = -k, where k is positive constant 

∴ ( 3 - λ ) ( 1 - λ ) + 4 k = 0 

⇒ λ2 - 4λ + 3 + 4 k = 0

If λ2 and λ2 be the solutions of the above equation, then λ2 and λ are the eigen values.

Now,Sum of the eigen values = Sum of the roots of the above equation

i.e 

QUESTION: 9

Let U be a 3 * 3 complex Hermitian matrix which is unitary. Then the distinct eigen values of U are

Solution:

By the well known facts, we know that a Hermitian matrix can have only real eigenvalues and a unitary matrix can have eigenvalues of unit modules. If U be 3 * 3 complex Hermitian matrix which is unitary also then distinct eigenvalues of U are of unit modulus and real. They can be 

QUESTION: 10

Let the characteristics equation of a matrix M be λ2 - λ - 1 =0, then

Solution:

Given characteristic equation is λ2 - λ - 1 =0, By C - H theorem M2 - M - 1 = 0

⇒ M(M - 1) = i

⇒ M-1 = M - 1

QUESTION: 11

If  then find eigen values of the matrix I + A + A2, where I denotes the identity matrix

Solution:

A + I is a triangular matrix. We know that the e igen values of a triangular matrix are its diagonal elements. Eigen values of A2 + A + 2 are 3, 7, 13

QUESTION: 12

Let  An eigenvalue of A is 2. Find a basis for the corresponding eigen space.

Solution:

Row reduce the augmented matrix for (A - 2i)x = 0.

At this point, we are confident that 2 is indeed an eigenvalue of A because the equation (A - 2l)x = 0 have free variables. The general solution is

The eigen space, is

QUESTION: 13

An arbitrary vector X is an eigen vector of the matrix

Solution:

(1,1) Since the matrix is triangular, the eigen values are 1, a, b. If (x1 x2, x3) is an arbitrary eigen vector, (say) corresponding to eigenvalue 1.

We have, x1 = x1, ax2 = X2 which gives a = 1 and bx3 = x3 which gives b = 1. x2 & x3 being not zero, 

∴ (a, b) = (1, 1).

QUESTION: 14

The values of λ and μ for which the equations x + y + z = 3, x + 3y + 2z = 6 and x + λy + 3z = μ have

Solution:

Given x + y + z = 3, x + 3y + 2z = 6 and x + λy + 3z = μ

If Δ ≠  0 i.e. λ ≠ 0 the solution is unique.

Case 1 If λ ≠ 5 and μ s any real number, then unique solution exists 

Case 2 If  λ ≠ 5 ⇒ Δ = 0

If Δ≠ 0 , μ ≠ 9  the system has no solution.
Case 3 If λ = 5 and μ = 9, infinite solutions exist

QUESTION: 15

If the matrix B is obtained from the matrix A by interchanging two rows, then-

Solution:

Interchanging two rows (or columns) change the sign of determinant.

QUESTION: 16

The rank of the Matrix  is

Solution:

(Operating R1 -» R1 + R3 and R2 -> R2 - 2R3)

When

∴ ρ(A) = 1

where p(A) = number of non-zero rows

when

∴ ρ(A) = 2

when

∴ ρ(A) = 2

*Multiple options can be correct
QUESTION: 17

Consider the matrix,  then

Solution:

, so eigen values are X = 1,1,4

Now  co rresponding to λ = 1 . 

(A - 1) x = 0

Clearly A has only one eigenvector corresponding to λ = 1.

⇒ A is not diagonalizable.

Characteristic eqwill be,  

*Multiple options can be correct
QUESTION: 18

Let V be the vector space of polynomials with degree not exceeding two. Let S = {1, x -1, x2 - 2x + 1} be a subset of V. Then which of the following(s) is/are true ?

Solution:

We know dim V = 3 and S is a linearly independent subset of V which has 3 elements = S form a basis for V

Now take (2x2- 5x + 6) = a(1) + b ( x - 1) + c(x2 - 2x + 1) = 3(1) + (-1) ( x -1)+ 2(x2 - 2x + 1) = 2x2- 5x + 6 = O ption (C) is correct

*Multiple options can be correct
QUESTION: 19

Let  and  be  ordered basis of  be a linear transformation such that  

The matrix of T w.r. to basis  β1 and β2 is  then which of the followings is /are correct ? 

Solution:

we have

 

*Multiple options can be correct
QUESTION: 20

If A be any matrix then AAT and ATA are

Solution:

Let A be real matrix , then take

(AAT)T = (AT)AT = AAT

⇒ AAT is Symmetric

now

(ATA)T = AT(AT)T = AT

⇒ ATA is Symmetric

*Answer can only contain numeric values
QUESTION: 21

If matrix  then the rank of the matrix A is _____________.


Solution:

Perform ing operations 

we have

Performing Operation wee have

Hence all minors of 3rd and higher orders of the equivalent matrix are zero and one minor  of order 2 whose value = 3 - 4 = -1 ≠0

Hence Rank of the equivalent matrix is Rank (A) = 2.

*Answer can only contain numeric values
QUESTION: 22

If A be an invertible matrix and Suppose that the inverse of 7 A is  then the determinant of matrix A is ___________.


Solution:

Given

we know AA-1 = 1

on Solving (1), (2), (3), (4) we get 

*Answer can only contain numeric values
QUESTION: 23

If  has the eigen values 3 and 9 then the sum o f the eigen values of A3 is __________.


Solution:

Characterstic eqn of matrix A is

eigen values of matrix A is λ = 3, 9

then eigen values of  A3 → 27 , 729

so  = 27+729 = 756

*Answer can only contain numeric values
QUESTION: 24

The number of distinct real roots of  in the interval   is __________.


Solution:

Hence

or

or

(2 cos x + sin x) (sin x - cos x)= 0

∴ tan x = - 2 , 1, But  

*Answer can only contain numeric values
QUESTION: 25

The sum of distinct characteristic roots of the matrix  is _______.


Solution:

The characteristic equation of A is 

or (2 - λ) [(6 - λ)( 4 - λ) - 8 ] = 0

or ( 2 - λ) (λ2 - 10 λ+ 16) = 0

or ( 2 - λ) (λ - 2 ) ( λ - 8 ) = 0.

There fore the characteristic roots of A are given by λ = 2, 2, 8.

*Answer can only contain numeric values
QUESTION: 26

If 3, - 2 are the eigen values of a non-singular matrix A and |A| = 4, then the sum of the eigen values o f adj(A) are ________ .


Solution:

since  and if λ is eigen value of A, then λ-1 is eigen value of A-1.

Thus for adj (A)X = (A-1 X) |A| = |A| λ-1

Thus, eigen value corresponding to λ = 3 is 4/3 and corresponding to λ = -2 is -4/2 = -2

Sum of eigen values = 4/3 - 2

= -2/3

= - 0.67

*Answer can only contain numeric values
QUESTION: 27

The system of linear equations

(4d - 1) x + y + z = 0, - y + z = 0, (4d - 1 )z = 0 has a non-trivial solution, if the value of d is ___ .


Solution:

The system of homogeneous linear equations has a non-trivial solution, if

*Answer can only contain numeric values
QUESTION: 28

Find the value of a for which the following system of equations over R is inconsistent.


Solution:

The augmented matrix is 

Adding the first row to the second row, and 2 times the first row to the third row, we make the entries other than the pivot one in the 1 st column zeros. The row equivalent matrix not looks like

Next, we add 3 times the row to the third row to make the entry below the pivot one in the second column zero. This produces

which is the row echelon form of the augmented matrix. 1st last column is a pivot column unless a = 5. In other words, if the real number a * 5, then the given system is inconsistent.

*Answer can only contain numeric values
QUESTION: 29

If V is a vector space over an infinite field F such that dim V = 2, then the number of distinct subspaces V has ________.


Solution:

Here, V has a subspace of dimensions 0, 1 and 2 thus there are three subspaces.

*Answer can only contain numeric values
QUESTION: 30

The dimension of the subspace of R3 spanned by (-3,0, 1), (1, 2, 1) and (3, 0, - 1 ) is ______ .


Solution:

Here, we can se e that vectors (-3 , 0, 1) and (3, 0, - 1 ) are linearly dependent. Hence, dim = 2.