Is the Determinants chapter in JEE Main syllabus 2025 the same as before?

Yes. According to the NTA official JEE Main 2025 syllabus, Determinants remains a core topic under Algebra. It covers properties of determinants, evaluation of 2×2 and 3×3 determinants, cofactors, minors, and the application to systems of linear equations. No major syllabus revision has affected this chapter

What is the most important concept in determinants for JEE Main?

The most tested concept is the condition for a system of linear equations to have no solution, a unique solution, or infinitely many solutions. This directly uses the condition D = 0 or D ≠ 0, combined with the values of D₁, D₂, and D₃. Mastering this single concept can secure 2–4 marks per paper.

How many questions from Determinants appear in JEE Main each year?

NTA typically sets 1 to 2 questions directly from determinants per JEE Main session. Since JEE Main is conducted in multiple sessions (January and April), students can expect 2 to 4 determinant-based questions across the full year. The topic is considered medium-weight in the Mathematics paper.

Can determinant questions appear in JEE Advanced too?

Yes. JEE Advanced tests determinants at a higher depth — expect questions involving abstract matrix properties, determinants of block matrices, or problems requiring multiple row/column operations in sequence. However, the JEE Main questions on this page are the correct foundation. Once these are solid, transition to JEE Advanced-level determinant problems. You can also explore the full NCERT Solutions library for chapter-wise theory support.

What is the difference between trivial and non-trivial solutions?

A trivial solution means x = y = z = 0 — the only solution. This occurs when D ≠ 0 for a homogeneous system. A non-trivial solution means at least one variable is non-zero. This requires D = 0. JEE Main asks for the values of a parameter (k or λ) that switch the system between these cases.

How do I prepare determinants from scratch for JEE Main?

Start with NCERT Class 12 Chapter 4 (Determinants), solve all examples and exercises, then move to previous year JEE Main questions sorted by sub-topic. Focus first on system-of-equations problems as they appear most frequently. Use eSaral's video lectures for visual explanation of row and column operations before attempting timed practice.

Determinant JEE Main Previous Year Questions & Solutions

Determinant - JEE Main Previous Year Question with Solutions

Determinant questions appear in almost every JEE Main session. Between 2009 and 2024, NTA has asked 1–2 questions per paper from this chapter, focusing on properties of determinants, solutions of linear equation systems (unique, infinite, no solution), and non-trivial solution conditions. Mastering these question types is essential for a 80+ score in JEE Main Mathematics.

eSaral Updated on: 28 May, 2026

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Q. Let $a, b, c$ be such that $\mathrm{b}(\mathrm{a}+\mathrm{c}) \neq 0 . \mathrm{If}$ If $\left|\begin{array}{ccc}{a} & {a+1} & {a-1} \\ {-b} & {b+1} & {b-1} \\ {c} & {c-1} & {c+1}\end{array}\right|+\left|\begin{array}{ccc}{a+1} & {b+1} & {c-1} \\ {a-1} & {b-1} & {c+1} \\ {-1} & {a} & {(-1)^{n+1} b} & {(-1)^{n} c}\end{array}\right|=0$ then the value of n is :- (1) Any odd integer (2) Any integer (3) Zero (4) Any even integer [AIEEE - 2009]

Ans. (1)

Q. Consider the system of linear equations : $\mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3}=3$ $2 \mathrm{x}_{1}+3 \mathrm{x}_{2}+\mathrm{x}_{3}=3$ $3 x_{1}+5 x_{2}+2 x_{3}=1$ The system has (1) Infinite number of solutions (2) Exactly 3 solutions (3) A unique solution (4) No solution [AIEEE - 2010]

Ans. (4) Here $D=0$ $\& \quad D_{1} \neq 0$ so we can say no solution

Q. The number of values of k for which the linear equations 4x + ky + 2z = 0 kx + 4y + z = 0 2x + 2y + z = 0 possess a non-zero solution is :- (1) 1 (2) zero (3) 3 (4) 2 [AIEEE - 2011]

Ans. (4)

Q. If the trivial solution is the only solution of the system of equations x – ky + z = 0 kx + 3y – kz = 0 3x + y – z = 0 Then the set of all values of k is: (1) {2, –3} (2) R – {2, –3} (3) R – {2} (4) R – {–3} [AIEEE - 2011]

Ans. (2) Here for trival solution $D \neq 0$ So $\mathrm{D}=\left|\begin{array}{ccc}{1} & {-\mathrm{k}} & {1} \\ {\mathrm{k}} & {3} & {-\mathrm{k}} \\ {3} & {1} & {-1}\end{array}\right|=0$ $\Rightarrow \mathrm{D}=2 \mathrm{k}^{2}-12+2 \mathrm{k}=0 \Rightarrow \mathrm{k}=-3,2$ so $\mathrm{R}-\{-3,2\}$

Q. The number of values of k, for which the system of equations : (k + 1)x + 8y = 4k kx + (k + 3)y = 3k – 1 has no solution, is – (1) infinite (2) 1 (3) 2 (4) 3 [JEE(Main)-2013]

Ans. (2) $\frac{k+1}{k}=\frac{8}{k+3}=\frac{4 k}{3 k-1}$ (1) = (2) $\Rightarrow \quad k^{2}-4 k+3=0$ k = 1, 3 for k = 1 (2) = (3) for $\mathrm{k}=3 \quad(2) \neq(3)$ k = 3

Q. If $\alpha, \beta \neq 0,$ and $f(\mathrm{n})=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$ and $\left|\begin{array}{ccc}{3} & {1+f(1)} & {1+f(2)} \\ {1+f(1)} & {1+f(2)} & {1+f(3)} \\ {1+f(2)} & {1+f(3)} & {1+f(4)}\end{array}\right|=\mathrm{K}(1-\alpha)^{2}(1-\beta)^{2}(\alpha-\beta)^{2}$ then K is equal to : (1) $\alpha \beta$ (2) $\frac{1}{\alpha \beta}$ (3) 1 (4) –1 [JEE(Main)-2014]

Ans. (3)

$\therefore k=1$

Q. The set of all values of $\lambda$ for which the system of linear equations : $2 \mathrm{x}_{1}-2 \mathrm{x}_{2}+\mathrm{x}_{3}=\lambda \mathrm{x}_{1}$ $2 \mathrm{x}_{1}-3 \mathrm{x}_{2}+2 \mathrm{x}_{3}=\lambda \mathrm{x}_{2}$ $-\mathrm{x}_{1}+2 \mathrm{x}_{2}=\lambda \mathrm{x}_{3}$ has a non-trivial solution (1) contains two elements (2) contains more than two elements (3) is an empty set (4) is a singleton [JEE(Main)-2015]

Ans. (1)

Q. The system of linear equations $\mathrm{x}+\lambda \mathrm{y}-\mathrm{z}=0$ $\lambda \mathrm{x}-\mathrm{y}-\mathrm{z}=0$ $\mathrm{x}+\mathrm{y}-\lambda \mathrm{z}=0$ has a non-trivial solution for : (1) exactly three values of $\lambda$ (2) infinitely many values of $\lambda$ (3) exactly one value of $\lambda$ (4) exactly two values of $\lambda$ [JEE(Main)-2016]

Ans. (1) $\left|\begin{array}{ccc}{1} & {\lambda} & {-1} \\ {\lambda} & {-1} & {-1} \\ {1} & {1} & {-\lambda}\end{array}\right|=0 \quad \Rightarrow \quad \lambda=0,1,-1$

Q. 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 : (1) a singleton (2) an empty set (3) an infinite set (4) a finite set containing two or more elements [JEE(Main)-2017]

Ans. (1) $D=\left|\begin{array}{lll}{1} & {1} & {1} \\ {1} & {a} & {1} \\ {a} & {b} & {1}\end{array}\right|=0 \Rightarrow a=1$ and at a = 1 $\mathrm{D}_{1}=\mathrm{D}_{2}=\mathrm{D}_{3}=0$ But at a = 1 and b = 1 $\left.\begin{array}{ll}{\text { First two equations are }} & {x+y+z=1} \\ {\text { and third equation is }} & {x+y+z=0}\end{array}\right] \Rightarrow$ There is nosolution. $\mathrm{b}=\{1\} \Rightarrow$ it is a singleton set

Q. If $\left|\begin{array}{ccc}{x-4} & {2 x} & {2 x} \\ {2 x} & {x-4} & {2 x} \\ {2 x} & {2 x} & {x-4}\end{array}\right|=(A+B x)(x-A)^{2},$ then the ordered pair $(A, B)$ is equal to : (1) (–4, 3) (2) (–4, 5) (3) (4, 5) (4) (–4, –5) [JEE(Main)-2018]

Ans. (2) $\left|\begin{array}{ccc}{\mathrm{x}-4} & {2 \mathrm{x}} & {2 \mathrm{x}} \\ {2 \mathrm{x}} & {\mathrm{x}-4} & {2 \mathrm{x}} \\ {2 \mathrm{x}} & {2 \mathrm{x}} & {\mathrm{x}-4}\end{array}\right|=(\mathrm{A}+\mathrm{Bx})(\mathrm{x}-\mathrm{A})^{2}$ Put $x=0 \Rightarrow\left|\begin{array}{ccc}{-4} & {0} & {0} \\ {0} & {-4} & {0} \\ {0} & {0} & {-4}\end{array}\right|=A^{3} \Rightarrow A=-4$ $\left|\begin{array}{ccc}{\mathrm{x}-4} & {2 \mathrm{x}} & {2 \mathrm{x}} \\ {2 \mathrm{x}} & {\mathrm{x}-4} & {2 \mathrm{x}} \\ {2 \mathrm{x}} & {2 \mathrm{x}} & {\mathrm{x}-4}\end{array}\right|=(\mathrm{Bx}-4)(\mathrm{x}+4)^{2}$ $\left|\begin{array}{ccc}{1-\frac{4}{\mathrm{x}}} & {2} & {2} \\ {2} & {1-\frac{4}{\mathrm{x}}} & {2} \\ {2} & {2} & {1-\frac{4}{\mathrm{x}}}\end{array}\right|=\left(\mathrm{B}-\frac{4}{\mathrm{x}}\right)\left(1+\frac{4}{\mathrm{x}}\right)^{2}$ Put $\mathrm{x} \rightarrow \infty \quad \Rightarrow \quad\left|\begin{array}{lll}{1} & {2} & {2} \\ {2} & {1} & {2} \\ {2} & {2} & {1}\end{array}\right|=\mathrm{B} \Rightarrow \mathrm{B}=5$ ordered pair (A, B) is (–4, 5)

Q. If the system of linear equations $x+k y+3 z=0$ $3 x+k y-2 z=0$ $2 x+4 y-3 z=0$ has a non-zero solution $(\mathrm{x}, \mathrm{y}, \mathrm{z}),$ then $\frac{\mathrm{xz}}{\mathrm{y}^{2}}$ is equal to : (1) 10 (2) – 30 (3) 30 (4) –10 [JEE(Main)-2018]

Ans. (1)

Q. If the system of linear equations : $x+a y+z=3$ $\mathrm{x}+2 \mathrm{y}+2 \mathrm{z}=6$ $x+5 y+3 z=b$ has no solution, then :- (1) $a=-1, b=9$ (2) $a \neq-1, b=9$ (3) $a=1, b \neq 9$ (4) $a=-1, b \neq 9$ [JEE(Main)-2018]

Ans. (4)

Q. The number of values of k for which the system of linear equations, (k+2)x + 10y = k kx + (k+3) y = k – 1 has no solution is : (1) infinitely many (2) 1 (3) 2 (4) 3 [JEE(Main)-2018]

Ans. (2)

Comments

Amitv

Sept. 14, 2025, 6:35 a.m.

Hii

Prajwal

Sept. 12, 2023, 6:04 p.m.

Nice

Ritesh Nagode

Jan. 22, 2024, 6:35 a.m.