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Test: Determinants - 1 - JEE MCQ


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25 Questions MCQ Test Mathematics (Maths) for JEE Main & Advanced - Test: Determinants - 1

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Test: Determinants - 1 - Question 1

Let a =  , then Det. A is

Detailed Solution for Test: Determinants - 1 - Question 1


Apply C2 → C2 + C3,

Test: Determinants - 1 - Question 2

Detailed Solution for Test: Determinants - 1 - Question 2

Apply , R1 → R1+R2+R3,



Apply , C3→ C- C1, C2C2 - C1,

=(a+b+c)3

Test: Determinants - 1 - Question 3

If A’ is the transpose of a square matrix A , then

Detailed Solution for Test: Determinants - 1 - Question 3

The determinant of a matrix A and its transpose always same.

Test: Determinants - 1 - Question 4

The roots of the equation det.   are

Detailed Solution for Test: Determinants - 1 - Question 4

⇒ (1-x)(2-x)(3-x) = 0 ⇒x = 1,2,3

Test: Determinants - 1 - Question 5

If A is a symmetric matrix, then At =

Detailed Solution for Test: Determinants - 1 - Question 5

If A is a symmetric matrix then by definition AT=A
Option A is correct.

Test: Determinants - 1 - Question 6

 is equal to 

Detailed Solution for Test: Determinants - 1 - Question 6


Apply , C1→C1 - C3, C2→C2-C3

= 10 - 12 = -2

Test: Determinants - 1 - Question 7

If A is a non singular matrix of order 3 , then |adj(A3)| =

Detailed Solution for Test: Determinants - 1 - Question 7

If A is anon singular matrix of order , then 

*Answer can only contain numeric values
Test: Determinants - 1 - Question 8

Let A B = [ B₁, B₂, B₃ ], where B₁, B₂, B₃ are column matrices, and 
If α = |B| and β is the sum of all the diagonal elements of B, then α³ + β³ is equal to


Detailed Solution for Test: Determinants - 1 - Question 8

x₁ = 1, y₁ = -1, z₁ = -1

x₂ = 2, y₂ = 1, z₂ = -2

x₃ = 2, y₃ = 0, z₃ = -1

α = |B| = 3
β = 1
α³ + β³ = 27 + 1 = 28

Test: Determinants - 1 - Question 9

Detailed Solution for Test: Determinants - 1 - Question 9

Correct option is D.

Test: Determinants - 1 - Question 10

The values of α, for which  lie in the interval

Detailed Solution for Test: Determinants - 1 - Question 10

 = 0
⇒ (2α + 3) { 7α / 6 } - (3α + 1) { -7 / 6 } = 0
⇒ (2α + 3) * (7α / 6) + (3α + 1) * (7 / 6) = 0
⇒ 2α² + 3α + 3α + 1 = 0
⇒ 2α² + 6α + 1 = 0
⇒ α = (-3 + √7) / 2 , (-3 - √7) / 2
Hence option (2) is correct.

*Answer can only contain numeric values
Test: Determinants - 1 - Question 11

Let for any three distinct consecutive terms a, b, c of an A.P, the lines ax + by + c = 0 be concurrent at the point P and Q(α, β) be a point such that the system of equations
x + y + z = 6,
2x + 5y + αz = β and
x + 2y + 3z = 4,
has infinitely many solutions. Then (PQ)² is equal to ____.


Detailed Solution for Test: Determinants - 1 - Question 11

∵ a, b, c and in A.P
⇒ 2b = a + c ⇒ a - 2b + c = 0
∴ ax + by + c passes through fixed point (1, -2)
∴ P = (1, -2)
For infinite solution,
D = D₁ = D₂ = D₃ = 0

⇒ α = 8

∴ Q = (8, 6)
∴ Q² = 113

Test: Determinants - 1 - Question 12

Detailed Solution for Test: Determinants - 1 - Question 12


Apply , C1 → C1 - C2, C2 → C2 - C3,

Because here row 1 and 2 are identical

Test: Determinants - 1 - Question 13

If f(x) =  then (1/5) f'(0) is equal to

Detailed Solution for Test: Determinants - 1 - Question 13


R₂ → R₂ - R₁, R₃ → R₃ - R₁

f(x) = 45
f'(x) = 0

Test: Determinants - 1 - Question 14

Consider the system of linear equations
x + y + z = 4μ,
x + 2y + 2λz = 10μ,
x + 3y + 4λ²z = μ² + 15
where λ, μ ∈ R.
Which one of the following statements is NOT correct?

Detailed Solution for Test: Determinants - 1 - Question 14

x + y + z = 4μ,
x + 2y + 2z = 10μ,
x + 3y + 4λ²z = μ² + 15
Δ =  = = (2λ - 1)²
For unique solution Δ ≠ 0, 2λ - 1 ≠ 0, ( λ ≠ 1/2 )
Let Δ = 0, λ = 1/2
Δy = 0, Δx = Δz = 
= (μ - 15)(μ - 1)
For infinite solution, λ = 1/2, μ = 1 or 15

Test: Determinants - 1 - Question 15

If I3 is the identity matrix of order 3 , then 13−1 is

Detailed Solution for Test: Determinants - 1 - Question 15

Because , the inverse of an identity matrix is an identity matrix.

Test: Determinants - 1 - Question 16

Consider the system of linear equations
x + y + z = 5,
x + 2y + λ²z = 9,
x + 3y + λz = μ, where λ, μ ∈ R.
Then, which of the following statement is NOT correct?

Detailed Solution for Test: Determinants - 1 - Question 16

 = 0
⇒ 2λ² - λ - 1 = 0
λ = 1, -1/2
 = 0 = μ = 13
Infinite solution λ = 1 & μ = 13
For unique solution λ ≠ 1
For no solution λ = 1 & μ ≠ 13
If λ ≠ 1 and μ ≠ 13
Considering the case when λ = -1/2 and μ ≠ 13, this will generate no solution case.

Test: Determinants - 1 - Question 17

If the system of linear equations
x - 2y + z = -4
2x + αy + 3z = 5
3x - y + βz = 3
has infinitely many solutions, then 12α + 13β is equal to

Detailed Solution for Test: Determinants - 1 - Question 17

D = 
= 1(αβ + 3) + 2(2β - 9) + 1(-2 - 3α)
= αβ + 3 + 4β - 18 - 2 - 3α
For infinite solutions D = 0, D₁ = 0, D₂ = 0 and D₃ = 0
D = 0
αβ - 3α + 4β = 17 ...... (1)

⇒ 1(5β - 9) + 4(2β - 9) + 1(6 - 15) = 0
13β - 9 - 36 - 9 = 0
13β = 54, β = 54/13 put in (1)
(54/13)α - 3α + 4(54/13) = 17
54α - 39α + 216 = 221
15α = 5, α = 1/3
Now, 12α + 13β = 12 * (1/3) + 13 * (54/13)
= 4 + 54 = 58

Test: Determinants - 1 - Question 18

A square matrix A is invertible iff det A is equal to

Detailed Solution for Test: Determinants - 1 - Question 18

For a square matrix A to be invertible, its determinant must satisfy a specific condition:

  • Invertibility Condition: A matrix A is invertible if and only if the determinant of A, denoted as det(A), is non-zero.
Test: Determinants - 1 - Question 19

If f(x) =  for all x ∈ R, then 2f(0) + f'(0) is equal to

Detailed Solution for Test: Determinants - 1 - Question 19

f(0) =  = 0
f'(x) = 
∴  f'(0) = 
= 24 - 6 = 18
∴ 2f(0) + f'(0) = 42

Test: Determinants - 1 - Question 20

If the system of equations
2x + 3y - z = 5
x + αy + 3z = -4
3x - y + βz = 7
has infinitely many solutions, then 13αβ is equal to

Detailed Solution for Test: Determinants - 1 - Question 20

Using family of planes
2x + 3y - z - 5 = k₁(x + αy + 3z + 4) + k₂(3x - y + βz - 7)
2 = k₁ + 3k₂, 3 = k₁α - k₂, -1 = 3k₁ + βk₂, -5 = 4k₁ - 7k₂
On solving we get
k₂ = 13/19, k₁ = -1/19, α = -70, β = -16/13
13αβ = 13(-70)(-16/13) = 1120

Test: Determinants - 1 - Question 21

If the entries in a 3 x 3 determinant are either 0 or 1 , then the greatest value of this determinant is :

Detailed Solution for Test: Determinants - 1 - Question 21


Test: Determinants - 1 - Question 22

The system of equations given is:x + 2y + 3z = 34x + 3y - 4z = 48x + 4y - λz = 9 + μThe question asks for the ordered pair (λ, μ) when the system has infinitely many solutions.

Detailed Solution for Test: Determinants - 1 - Question 22

x + 2y + 3z = 3 .... (i)
4x + 3y - 4z = 4 .... (ii)
8x + 4y - λz = 9 + μ .... (iii)
(i) × 4 - (ii) → 5y + 16z = 8 .... (iv)
(ii) × 2 - (iii) → 2y + (λ - 8)z = -1 - μ .... (v)
(iv) × 2 - (iii) × 5 → (32 - 5(λ - 8))z = 16 - 5( - 1 - μ)
For infinite solutions → 72 - 5λ = 0 → λ = 72/5
21 + 5μ = 0 → μ = -21/5
⇒ (λ, μ) ≡ (72/5, -21/5)

Test: Determinants - 1 - Question 23

Let S₁ and S₂ be respectively the sets of all a ∈ ℝ - {0} for which the system of linear equations
ax + 2ay - 3az = 1
(2a + 1)x + (2a + 3)y + (a + 1)z = 2
(3a + 5)x + (a + 5)y + (a + 2)z = 3
has unique solution and infinitely many solutions. Then

Detailed Solution for Test: Determinants - 1 - Question 23

Δ = ​​​​​​
= a(15a² + 31a + 36) = 0 ⇒ a = 0
Δ ≠ 0 for all a ∈ ℝ - {0}
Hence S₁ = ℝ - {0}, S₂ = ∅

Test: Determinants - 1 - Question 24

In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then it can be decomposed into n determinants, where n has value

Detailed Solution for Test: Determinants - 1 - Question 24

To determine the number of decomposed determinants, we start by considering the linearity property of determinants over columns. Each column in the given third-order determinant is a sum of terms: the first column has 2 terms per element, the second column has 3 terms per element, and the third column has 4 terms per element.

  1. First Column (2 terms per element): Each element in the first column can be split into two terms, leading to 2 determinants.

  2. Second Column (3 terms per element): Each element in the second column can be split into three terms. For each of the 2 determinants from the first column, splitting the second column results in 2×3=6 determinants.

  3. Third Column (4 terms per element): Each element in the third column can be split into four terms. For each of the 6 determinants from the previous step, splitting the third column results in 6×4=24 determinants.

Thus, the total number of decomposed determinants is 2×3×4=24 

The value of n is 24 

Test: Determinants - 1 - Question 25

The given system of equations is:
ax + 2y + z = 1
2ax + 3y + z = 1
3x + ay + 2z = β
For some α, β ∈ ℝ. The question asks which of the following is NOT correct.

Detailed Solution for Test: Determinants - 1 - Question 25

The given system is represented in terms of determinants:
D = | α 2 1 |
| 2α 3 1 |
| 3 α 2 |
which gives D = 0 when α = -1 or α = 3.
Dx = | 2 1 1 |
| 3 1 1 |
| α 2 β |
which gives
Dx = 0 when β = 2.|
Dy = | α 1 1 || 2α 3 1 |
| 3 2 β |
which gives Dy = 0 (determinant is zero).
Dz = | α 2 1 |
| 2α 3 1 |
| 3 α β |
which gives Dz = 0.
The solution states that when β = 2 and α = -1, the system has an infinite solution.

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