Properties of Determinants is a subtopic of Matrix and Determinants included in JEE Main 2022 Mathematics Syllabus. Overall Matrices and Determinants will have a weightage of 6-7 percent in JEE Main examinations. Properties of Determinants cover the basic properties that define a determinant as well as important properties such as - Reflection property, All Zero Property, Proportionality, Scalar multiple Property, Sum Property, etc.
Though all three sections are equally weighted, mathematics is an important aspect of JEE Main 2022, with respect to competitiveness, as most of the applicants find it difficult to prepare for the topic. Properties of Determinants is a relatively easy topic than those of Calculus and Coo-ordinate Geometry; combined with the competitive significance of the mathematics section, even 1-2 questions from the topic, if successfully attempted, can make a significant difference in the rank of a candidate. So, for the benefit of the candidate key properties of determinants, along with solved previous year JEE Main question is given below
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In Linear algebra, a determinant is a unique number that can be ascertained from a square matrix. The determinants of a matrix say K is represented as det (K) or, |K| or det K. The determinant of a square matrix is a value ascertained by the elements of a matrix. In 2 × 2 matrices, the determinants are calculated by
Question:
Solution:
Related:
Some basic properties of determinants are given below:
det (A+B+C) + det C ≥ det (A+B) + det (B+C)
It implies that determinant remains unchanged under an operation of the term Ci ⟶ Ci + αCj + βCkj where j and k are not equivalent to i, or a Mathematical operation of the term Ri ⟶ Ri + αRj + βRk, where, j and k are not equivalent to i.
In the above determinants of the cofactor matrix, Cij denotes the cofactor of the elements aij in Δ.
A minor is defined as a value computed from the determinant of a square matrix which is obtained after crossing out a row and a column corresponding to the element that is under consideration. Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column in which element aij lies. Minor of an element aij is denoted by Mij.
The cofactor is defined as the signed minor. Cofactor of an element aij, denoted by Aij is defined by A = (–1)i+j M, where M is minor of aij.
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Properties of Determinants is a sub-topic of Matrices and Determinants. It can take up to 2-3 weeks to complete the chapter. The chapter is of high importance and candidates must not avoid it. With proper preparation and planning, Candidates can score well. Check JEE Main 2022 Mathematics preparation
Difficulty Level | Slightly Difficult |
---|---|
Weightage in JEE Main Examination | 6.67% |
Years Featuring Most Questions from the Topic |
|
Time Needed for Preparation : Optimistic Scenario | 1-2 Weeks (if Basic are clear) |
Time Needed for Preparation : Pessimistic Scenario | 3-5 Weeks or more (if basic need to be revised) |
Related Link: JEE Main 2022 Important Books
1.If P= is the adjoint of a 3×3 matrix A and |A|=4, then α is equal to :
Soln: b. |P|=1(12−12)−α(4−6)+ 3(4−6)=2α−6
Now, adjA=P
⇒|adjA|=|P|
⇒|A|²=|P|
⇒|P|=16
⇒2α−6=16
⇒α=11
2.If ????, ???? ⍯ 0, and f(n)=????n+????n and
If K(1-????)2(1-???? )2(????-????)2, then K is equal to:
Soln: A. Consider,
So, k=1.
3.The set of all values of λ for which the system of linear equations:
2x₁-2x₂+x₃=ƛx₁
2x₁-3x₂+2x₃=ƛx₂
-x₁+2x₂=ƛx₃
has a non-trivial solution
Soln: A. 2x₁-2x₂+x₃=ƛx₁
2x₁-3x₂+2x₃=ƛx₂
-x₁+2x₂=ƛx₃
(2-ƛ)x₁-2x₂+x₃=0
2x₁-(3-ƛ)x₂+2x₃=0
-x₁+2x₂-ƛx₃=0
For non-trivial solution, Δ=0
Hence ƛ has 2 values .
4.The system of linear equations
x+λy−z=0
λx−y−z=0
x+y−λz=0
has a non-trivial solution for:
Soln: B. For trivial solution,
⇒ −λ (λ+1)(λ+1)=0
⇒ λ=0, +1,−1
5.The number of distinct real roots of the equation,
Soln: C. Given,
∴ Number of solutions = 2
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6.If S is the set of distinct values of 'b' for which the following system of linear equations
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solution, then S is :
Soln: D.
⇒ 1 [a – b] – 1 [1 – a] + 1 [b – a2] = 0
⇒ (a - 1)2 = 0
⇒ a = 1
For a = 1, the equations become
x + y + z = 1
x + y + z = 1
x + by + z = 0
These equations give no solution for b = 1
⇒ S is singleton set.
7.then the ordered pair (A, B) is equal to
Soln: D.
By comparing both sides we get, A = −4 and B = 5
Q. How many questions will be asked from properties of determinants in JEE Main 2022?
Ans- Properties of determinants are a very important topic and are given high weightage. The expected number of questions asked from this topic in JEE Main 2022 is 2-3.
Q. What are the topics included under Matrices and Determinants in JEE Main 2022 Mathematics Syllabus ?
Ans- The Matrices and Determinants include Matrices: Algebra of matrices, types of matrices, and matrices of order two and three; Determinants: Properties of determinants, evaluation of determinants, the area of triangles using determinants; Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations; Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices .For more details visit : JEE Main Mathematics Syllabus
Q. What is the Scalar Property of Determinants?
Ans. Scalar property of a determinant dictates that :
“A determinant remains unaltered in its numerical value if the rows and columns are interchanged.”
Q.What is a cofactor of a Determinant?
Ans- A cofactor is defined as the signed minor. Cofactor of an element aij, denoted by Aij is defined by A = (–1)i+j M, where M is minor of aij.A minor is defined as a value computed from the determinant of a square matrix which is obtained after crossing out a row and a column corresponding to the element that is under consideration.
Q. How much time will be needed to prepare Determinants for JEE Main 2022?
Ans -Properties of Determinants is a subtopic of Matrices and Determinants. It can take up to 2-3 weeks to complete the chapter for JEE Main 2022. The chapter is of high importance and candidates must not avoid it.
*The article might have information for the previous academic years, please refer the official website of the exam.