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Test: 35 Year JEE Previous Year Questions: Matrices and Determinants

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34 Questions MCQ Test Mathematics For JEE | Test: 35 Year JEE Previous Year Questions: Matrices and Determinants

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Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 1

If a > 0 and discriminant of ax2+2bx+c is –ve, then

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 1

By R3 → R3 – (xR1 + R2);

= (ax2 + 2bx + c)(b2 – ac) = (+)(–) = -ve.

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 2

If the system of linear equations  x + 2ay +az = 0 ; x +3by+bz = 0 ; x +4cy+cz = 0 ; has a non - zero solution, then a, b, c.

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 2

For homogeneous system of equations to have non zero solution, Δ = 0

On simplification,

∴ a,b,c are  in Harmonic Progression.

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 3

If 1, ω,ω2 are the cube roots of unity, then

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 3

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 4

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 4

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 5

The only correct statement about the matrix A is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 5

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 6

If B is the inverseof matrix A, then α is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 6

Also since, B = A-1 ⇒  AB = I

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 7

If a1, a2 , a3, ......,an, .... are in G.P., then the value of the determinant

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 7

Let r be the common ratio, then

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 8

If A2 – A +I=0 , then the inverse of A is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 8

Given A2 - A +I= 0
A-1A2 - A-1A + A-1.I= A-1.0
(Multiplying A-1 on both sides)
⇒ A-1 +A-1 = 0 or A-1 = 1-A .

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 9

The system of equations

α x + y  + z = α – 1
x + α y + z = α – 1
x + y + α z = α – 1

has infinite solutions, if α is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 9

α x + y + z = α -1
x + α y + z = α – 1;
x + y + z α = α – 1

= α(α2 - 1) - 1(α - 1) + 1(1 - α)
= α (α-1)(α+1) - 1(α-1) - 1(α-1)
For infinite solutions, Δ = 0
⇒ (α - 1)[α2 + α - 1 - 1] =  0

⇒ (α - 1)[α2 + α - 2] = 0  ⇒ α = – 2,1;

But a ≠ 1 . ∴ α = – 2

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 10

If a2 + b+ c2 = – 2 and

then f (x) is a polynomial of degree

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 10

Applying, C1 → C1 + C2 + C3 we get

[As given that a2 + b2 + c2 = –2]
∴ a2 +b2+c2 + 2 = 0
Applying R1 → R1-R2 , R2 → R2-R3

f (x) = ( x - 1)2 Hence degree = 2.

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 11

If a1 , a2,a3 , ............, an , ...... are in G. P., then the determinant

is equal to

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 11

∵ a1, a2 ,a3 , ....... are in G.P..
∴ Using an = ar n -1 ,we get the given determinant, as

Operating C3 - C2 and C2-C1 and using

= 0 (two columns being identical)

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 12

If A and B are square matrices of size n × n such that A2 - B2 = (A -B)(A+B), then which of the following will be always true?

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 12

A2 - B2 = (A -B)(A+B)
A2 - B2 = A2 + AB - BA-B2 ⇒ AB = BA

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 13

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 13

Hence, AB = BA only when a = b
∴ There can be infinitely many B's for which AB = BA

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 14

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 14

Hence, D is divisible by both x and y

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 15

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 15

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 16

Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr(A), the sum of diagonal entries of a. Assume that A2 = I.

Statement-1 : If A ≠ I and A ≠ –I, then det(A) = –1

Statement-2 : If A ≠ I and A ≠ –I, then tr (A)  ≠ 0.

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 16

From these four relations,
a2 + bc = bc + d2 ⇒ a2 = d2
and b(a + d) = 0 = c( a + d) ⇒ a = – d
We can take a = 1, b = 0, c = 0, d  = –1 as one possible set of values, then

Clearly A ≠ I and A ≠ – I  and  det A = –1 ∴ Statement 1 is true.
Also if A ≠ I then tr(A) = 0
∴ Statement 2 is false.

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 17

Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx, and z = bx + ay. Then a2 + b2 + c2 + 2abc is equal to

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 17

The given equations are
–x + cy + bz = 0
cx –y + az = 0
bx + ay – z = 0
∵ x, y, z are not all zero

∴ The above system should not have unique (zero) solution

⇒ –1(1– a2) – c(– c – ab) + b(ac + b) = 0
⇒–1 + a2 + b2 + c2 + 2abc = 0
⇒ a2 + b2 + c2 + 2abc = 1

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 18

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 18

∵ All entries of square matrix A are integers, therefore all  cofactors should also be integers.
If det A = ± 1 then A–1 exists. Also all entries of A–1 are integers.

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 19

Let A be a 2 × 2 matrix

Statement -2 : |adj A |= |A|

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 19

= | A | 2–1= |A|

∴ Both the statements are true and statement -2 is  a correct explantion for statement-1 .

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 20

Let a, b, c be such that b(a + c) ≠ 0 if

then the value of n is :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 20

(Taking transpose of second determinant)

⇒ n  should be an odd integer.

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 21

The number of 3 × 3 non-singular matrices, with four entries as 1 and all other entries as 0, is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 21

are 6 non-singular matrices because 6 blanks will be filled by 5 zeros and 1 one.
Similarly,  are 6 non-singular matrices.
So, required cases are more than 7, non-singular 3 × 3 matrices.

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 22

Let A be a 2 × 2 matrix with non-zero entries and let A2 = I , where I is 2 × 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.

Statement - 1 : Tr(A) = 0.
Statement -2 : |A| = 1.

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 22

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 23

Consider the system of linear equations ;

x1 + 2x2 + x3 = 3
2x1 + 3x2 + x3 = 3
3x1 + 5x2 + 2x3 = 1

The system has

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 23

⇒ Given system, does not have any solution.
⇒ No solution

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 24

The number of values of k for which the linear equations 4x + ky + 2z = 0 , kx + 4y + z = 0 and 2x + 2y + z = 0 possess a non-zero solution is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 24

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 25

Let A and B be two symmetric matrices of order 3.

Statement-1: A(BA) and (AB)A are symmetric matrices.
Statement-2: AB is symmetric matrix if matrix multiplication of A with B is commutative.

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 25

∴ A' = A, B' = B
Now (A(BA))' = (BA)'A'
=  (A'B')A' = (AB)A = A(BA)
Similarly ((AB)A)' = (AB)A
So, A(BA) and (AB)A are symmetric matrices.
Again (AB)' = B'A' = BA
Now if BA = AB, then AB is symmetric matrix.

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 26

If u1 and u2 are column matrices such that  then u1 + u2 is equal to :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 26

..(1)

We know,

Now, from equation (1), we have

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 27

Let P and Q be 3 x 3 matrices P ≠ Q. If P3 = Q3 and  P2Q = Q2P then determinant of (P2 + Q2) is equal to :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 27

Given P3 = Q3 ...(1)
and P2Q = Q2P ...(2)
Subtracting (1) and (2), we get
P3 – P2Q = Q3 – Q2P
⇒ P2 (P–Q) + Q2 (P – Q) = 0
⇒ (P2 + Q2) (P–Q) = 0 ⇒ |P2 + Q2| = 0 as P ≠ Q

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 28

If  is the adjoint of a 3 × 3 matrix A and |A| = 4, then α is equal to :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 28

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 29

If α, β ≠ 0, and f (n) = αnn and

then K is equal to:

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 29

Consider

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 30

If A is a 3 × 3 non-singular matrix such that AA' = A'A and B = A–1A', then BB' equals:

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 30

BB' = B(A-1 A')' = B(A ')'(A-1)' = BA (A–1)'
= (A -1 A')(A(A-1)')
= A–1A .A'.(A–1)' {as AA' = A'A}
= I(A–1A)'
= I.I = I2 = I

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 31

The set of all values of l for which the system of linear equations :

2x1 – 2x2 + x3 = λx1
2x1 – 3x2 + 2x3 = λx2
–x1 + 2x2 = λx3

has a non-trivial solution

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 31

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 32

a matrix satisfying the equation AAT = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to:

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 32

⇒ a + 4 + 2b = 0 ⇒ a + 2b = – 4 ...(i)
2a + 2 – 2b = 0 ⇒ 2a – 2b = – 2   ⇒ a – b = –1 ...(ii)
On solving (i) and (ii) we get
– 1 + b + 2b = – 4 ...(i)
b = – 1 and a = – 2
(a, b) = (–2, –1)

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 33

The system of linear equations

x + λy – z = 0
λx – y – z = 0
x + y – λz = 0

has a non-trivial solution for:

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 33

For trivial solution,

Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 34

then 5a + b is equalto :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Matrices and Determinants - Question 34

⇒ A–1A (adj A) = A–1A AT

⇒ 5a + b=5

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