PYQs Learn Notes Mocks Board Formulas Matrices & Determinants
Determinant equations and their roots det A ( x ) = 0 \det A(x) = 0 det A ( x ) = 0 x x x the variable in the entries Setting a determinant containing a variable to zero produces a polynomial equation, and its roots are usually read off by inspection or through the sum and product of roots.
2018–2026
years it has appeared
Used alongside How often each of these appears in the same solution as this one.
17 questions whose solution uses this Every one was read individually — a question is here because its solution invokes this, not because the stem mentions it.
Filter by exam All exams17 NDA14 CBSE3
Filter by past-year or practice All questions17 Past-year10 Practice7
Showing 17 of 17 questions — easiest first, then oldest paper first.
CBSE Class 12 · 2023 · 65/5/1 · Q5 Easy Add #1 · CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
If ∣ α 3 4 1 2 1 1 4 1 ∣ = 0 \begin{vmatrix} \alpha & 3 & 4 \\ 1 & 2 & 1 \\ 1 & 4 & 1 \end{vmatrix} = 0 α 1 1 3 2 4 4 1 1 = 0 , then the value of α \alpha α is Practice question Moderate Add #2 · NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
If 7 and 2 are the root of ∣ x 3 7 2 x 2 7 6 x ∣ = 0 \left| \begin{matrix}
x & 3 & 7 \\
2 & x & 2 \\
7 & 6 & x
\end{matrix} \right| = 0 x 2 7 3 x 6 7 2 x = 0 then the third root is : Practice question Moderate Add #3 · NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
If 5 5 5 and 7 7 7 are the roots of the equation ∣ x 4 5 7 x 7 5 8 x ∣ = 0 , \left| \begin{matrix}
x & 4 & 5 \\
7 & x & 7 \\
5 & 8 & x
\end{matrix} \right| = 0, x 7 5 4 x 8 5 7 x = 0 , then what is the third root? Practice question Moderate Add #4 · NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
If ∣ x + a b c a x + b c a b x + c ∣ = 0 \begin{vmatrix} x+a & b & c \\ a & x+b & c \\ a & b & x+c \end{vmatrix} = 0 x + a a a b x + b b c c x + c = 0 , then x = x = x = A 0 , − ( a + b + c ) 0,\ -(a+b+c) 0 , − ( a + b + c ) B C D Practice question Moderate Add #5 · NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
The roots of the equation ∣ x − 1 1 1 1 x − 1 1 1 1 x − 1 ∣ = 0 \begin{vmatrix} x-1 & 1 & 1 \\ 1 & x-1 & 1 \\ 1 & 1 & x-1 \end{vmatrix} = 0 x − 1 1 1 1 x − 1 1 1 1 x − 1 = 0 are Practice question Moderate Add #6 · NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
If x ≠ 0 x \ne 0 x = 0 , ∣ x + 1 2 x + 1 3 x + 1 2 x 4 x + 3 6 x + 3 4 x + 1 6 x + 4 8 x + 4 ∣ = 0 \begin{vmatrix} x+1 & 2x+1 & 3x+1 \\ 2x & 4x+3 & 6x+3 \\ 4x+1 & 6x+4 & 8x+4 \end{vmatrix} = 0 x + 1 2 x 4 x + 1 2 x + 1 4 x + 3 6 x + 4 3 x + 1 6 x + 3 8 x + 4 = 0 , then 2 x + 1 = 2x+1 = 2 x + 1 = NDA · Apr 2021 · Q18 Moderate Add #7 · NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
If the determinant ∣ x 1 3 0 0 1 1 x 4 ∣ = 0 \begin{vmatrix}x & 1 & 3\\0 & 0 & 1\\1 & x & 4\end{vmatrix} = 0 x 0 1 1 0 x 3 1 4 = 0 , then what is x x x equal to? CBSE Class 12 · 2025 · 65/4/2 · Q7 Moderate Add #8 · CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
If ∣ − 1 2 4 1 x 1 0 3 3 x ∣ = − 57 \begin{vmatrix} -1 & 2 & 4 \\ 1 & x & 1 \\ 0 & 3 & 3x \end{vmatrix} = -57 − 1 1 0 2 x 3 4 1 3 x = − 57 , the product of the possible values of x is : NDA · Sep 2026 · Q39 Moderate Add #9 · NDA → Mathematics → Matrices & Determinants → Linear Systems — Consistency, Cramer's Rule, Solution Space
If
x + a y + a 2 z = 1 x + ay + a^2 z = 1 x + a y + a 2 z = 1 , x + b y + b 2 z = 1 x + by + b^2 z = 1 x + b y + b 2 z = 1 , x + c y + c 2 z = 1 x + cy + c^2 z = 1 x + cy + c 2 z = 1
where a ≠ b ≠ c a \ne b \ne c a = b = c , then what is x + y + z x + y + z x + y + z equal to? CBSE Class 12 · 2026 · 65/2/1 · Q6 Moderate Add #10 · CBSE Class 12 → Mathematics → Determinants → Determinant and its Properties
If ∣ − 1 − 2 5 − 2 a − 1 0 4 2 a ∣ = − 86 \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86 − 1 − 2 0 − 2 a 4 5 − 1 2 a = − 86 , then the sum of all possible values of a is #11 · NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
If ∣ sin x cos x cos x cos x sin x cos x cos x cos x sin x ∣ = 0 \begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0 sin x cos x cos x cos x sin x cos x cos x cos x sin x = 0 , then the number of distinct real roots of this equation in the interval − π / 2 < x < π / 2 -\pi/2 < x < \pi/2 − π /2 < x < π /2 is #12 · NDA → Mathematics → Matrices & Determinants → Special Determinants — Trig, Complex, Roots of Unity, Polynomial
If x, y, z are all different and not equal to zero and ∣ 1 + x 1 1 1 1 + y 1 1 1 1 + z ∣ = 0 \begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{vmatrix} = 0 1 + x 1 1 1 1 + y 1 1 1 1 + z = 0 , then the value of x − 1 + y − 1 + z − 1 x^{-1} + y^{-1} + z^{-1} x − 1 + y − 1 + z − 1 is equal to A B x − 1 y − 1 z − 1 x^{-1}y^{-1}z^{-1} x − 1 y − 1 z − 1 C D NDA · Sep 2018 · Q19 Hard Add #13 · NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
If a + b + c = 0 a + b + c = 0 a + b + c = 0 , then one of the solutions of ∣ a − x c b c b − x a b a c − x ∣ = 0 \begin{vmatrix} a-x & c & b \\ c & b-x & a \\ b & a & c-x \end{vmatrix} = 0 a − x c b c b − x a b a c − x = 0 is A B x = 3 ( a 2 + b 2 + c 2 ) 2 x = \sqrt{\frac{3(a^2+b^2+c^2)}{2}} x = 2 3 ( a 2 + b 2 + c 2 ) C x = 2 ( a 2 + b 2 + c 2 ) 3 x = \sqrt{\frac{2(a^2+b^2+c^2)}{3}} x = 3 2 ( a 2 + b 2 + c 2 ) D NDA · Sep 2019 · Q12 Hard Add #14 · NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
What are the values of x x x that satisfy the equation ∣ x 0 2 2 x 2 1 1 1 1 ∣ + ∣ 3 x 0 2 x 2 2 1 0 1 1 ∣ = 0 \begin{vmatrix} x & 0 & 2 \\ 2x & 2 & 1 \\ 1 & 1 & 1 \end{vmatrix} + \begin{vmatrix} 3x & 0 & 2 \\ x^{2} & 2 & 1 \\ 0 & 1 & 1 \end{vmatrix} = 0 x 2 x 1 0 2 1 2 1 1 + 3 x x 2 0 0 2 1 2 1 1 = 0 ? NDA · Apr 2023 · Q10 Hard Add #15 · NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
What is the sum of the roots of the equation ∣ 0 x − a x − b 0 0 x − c x + b x + c 1 ∣ = 0 \begin{vmatrix}0&x-a&x-b\\0&0&x-c\\x+b&x+c&1\end{vmatrix}=0 0 0 x + b x − a 0 x + c x − b x − c 1 = 0 ? Consider the following for the next two (02) items that follow: Let Δ ( a , b , c , α ) = ∣ a b a α + b b c b α + c a α + b b α + c 0 ∣ \Delta(a,b,c,\alpha)=\begin{vmatrix}a&b&a\alpha+b\\b&c&b\alpha+c\\a\alpha+b&b\alpha+c&0\end{vmatrix} Δ ( a , b , c , α ) = a b a α + b b c b α + c a α + b b α + c 0 . NDA · Apr 2023 · Q52 Hard Add #16 · NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
If Δ ( 7 , 4 , 2 , α ) = 0 \Delta(7,4,2,\alpha)=0 Δ ( 7 , 4 , 2 , α ) = 0 , then α \alpha α is a root of which one of the following equations? A 7 x 2 + 4 x + 2 = 0 7x^{2}+4x+2=0 7 x 2 + 4 x + 2 = 0 B 7 x 2 − 4 x + 2 = 0 7x^{2}-4x+2=0 7 x 2 − 4 x + 2 = 0 C 7 x 2 + 8 x + 2 = 0 7x^{2}+8x+2=0 7 x 2 + 8 x + 2 = 0 D 7 x 2 − 8 x + 2 = 0 7x^{2}-8x+2=0 7 x 2 − 8 x + 2 = 0 NDA · Apr 2023 · Q51 Hard Add #17 · NDA → Mathematics → Matrices & Determinants → Determinant Properties, Operations, and Sums
If Δ ( a , b , c , α ) = 0 \Delta(a,b,c,\alpha)=0 Δ ( a , b , c , α ) = 0 for every α > 0 \alpha>0 α > 0 , then which one of the following is correct? A a , b , c a,b,c a , b , c are in AP B a , b , c a,b,c a , b , c are in GP C a , 2 b , c a,2b,c a , 2 b , c are in AP D a , 2 b , c a,2b,c a , 2 b , c are in GP Every identity this chapter tests, with its own question set.
All Matrices & Determinants formulas 