Evaluate|x y x+y y x+y x x+y x y|
Solution and Explanation
Δ=|x y x+y y x+y x x+y x y|
Applying R1→R1+R2+R3,we have
Δ=|2(x+y) 2(x+y) 2(x+y) y x+y x x+y x y|
=2(x+y)|111 y x+y x x+y x y|
Applying C2→C2-C1 and C3→C3-C1,we have
Δ=2(x+y)|100 y x x-y x+y -y -x|
Expanding along R1, we have:
Δ=2(x+y)[-x2+y(x-y)]
=-2(x+y)(x2+y2-yx)
=-2(x3+y3)
Top Questions on Determinants
Let I be the identity matrix of order 3 × 3 and for the matrix $ A = \begin{pmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{pmatrix} $, $ |A| = -1 $. Let B be the inverse of the matrix $ \text{adj}(A \cdot \text{adj}(A^2)) $. Then $ |(\lambda B + I)| $ is equal to _______
- JEE Main - 2025
- Mathematics
- Determinants
Let $A = \{ z \in \mathbb{C} : |z - 2 - i| = 3 \}$, $B = \{ z \in \mathbb{C} : \text{Re}(z - iz) = 2 \}$, and $S = A \cap B$. Then $\sum_{z \in S} |z|^2$ is equal to
- JEE Main - 2025
- Mathematics
- Determinants
If $ y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1 \\27 & 28 & 27 \\1 & 1 & 1 \end{vmatrix} $, $ x \in \mathbb{R} $, then $ \frac{d^2y}{dx^2} + y $ is equal to
- JEE Main - 2025
- Mathematics
- Determinants
- The number of singular matrices of order 2, whose elements are from the set $ \{2, 3, 6, 9\} $ is:
- JEE Main - 2025
- Mathematics
- Determinants
- Let the system of equations be: $ 2x + 3y + 5z = 9, $ $ 7x + 3y - 2z = 8, $ $ 12x + 3y - (4 + \lambda)z = 16 - \mu, $ which has infinitely many solutions. Then the radius of the circle centered at $ (\lambda, \mu) $ and touching the line $ 4x = 3y $ is:
- JEE Main - 2025
- Mathematics
- Determinants
Questions Asked in CBSE CLASS XII exam
- A 1 cm segment of a wire lying along the x-axis carries a current of 0.5 A along the \( +x \)-direction. A magnetic field \( \mathbf{B} = (0.4 \, \text{mT}) \hat{j} + (0.6 \, \text{mT}) \hat{k} \) is switched on in the region. The force acting on the segment is:
- CBSE CLASS XII - 2025
- Magnetic Effects of Current and Magnetism
The correct IUPAC name of \([ \text{Pt}(\text{NH}_3)_2\text{Cl}_2 ]^{2+} \) is:
- CBSE CLASS XII - 2025
- Coordination Compounds
- Using integration, find the area of the region bounded by the line \[ y = 5x + 2, \] the \( x \)-axis, and the ordinates \( x = -2 \) and \( x = 2 \).
- CBSE CLASS XII - 2025
- Evaluation of Definite Integrals by Substitution
- Two conductors A and B of the same material have their lengths in the ratio 1:2 and radii in the ratio 2:3. If they are connected in parallel across a battery, the ratio \( \frac{v_A}{v_B} \) of the drift velocities of electrons in them will be:
- CBSE CLASS XII - 2025
- Current electricity
- Arun, Bashir and Joseph were partners in a firm sharing profits and losses in the ratio of 5 : 3 : 2. They admitted Daksh as a new partner who acquired his share entirely from Arun. If Arun sacrificed \( \frac{1}{5} \)th from his share to Daksh, Daksh's share in the profits of the firm will be :
- CBSE CLASS XII - 2025
- Partnership Accounts