Determinant Year Wise Questions PDF

# Determinant Year Wise Questions ## 2023 **(i) Show that** | 1 0 b | | a -c 0 | = 0 | 0 1 -b | [2 Marks] **(ii) Show that** | a a+b a+b+c | | 2a 3a+2b 4a+3b+2c | = 0 | 3a 6a+3b 10a+6b+3c | **OR** a, b, c are real numbers and...

# Determinant Year Wise Questions ## 2023 **(i) Show that** | 1 0 b | | a -c 0 | = 0 | 0 1 -b | [2 Marks] **(ii) Show that** | a a+b a+b+c | | 2a 3a+2b 4a+3b+2c | = 0 | 3a 6a+3b 10a+6b+3c | **OR** a, b, c are real numbers and | b+c c+a a+b | | c+a a+b b+c | ≠ 0 | a+b b+c c+a | Show that either a+b+c = 0 or abc = 0. [4-Marks] ## 2022 **(i) Show that** | 1 + a 1 1 | = abc (1 + 1/a + 1/b + 1/c) | 1 1 + b 1 | | 1 1 1 + c | **OR** Show that | 1 x x^2 | = (1 - x) ^ 3 | x 1 x | | x^2 x 1 | **(ii) If** | 3 2 4 | = | 2 x 3 | , find the value of x. | 1 2 7 | [2-Marks] ## 2020 **(i) Show that** | 1 logb logc | | loga 1 logc | = 0. (a>0, b>0, c>0) | loga logb 1 | [2-Marks] **(ii) If** x+y+z 20 and | x x^2 1+x^3 | | y y^2 1+y^3 | = 0, then show that | z z^2 1+z^3 | 1+xyz = 0 **OR** Show that | 1 x x^2 | | x 1 x | = (1 - x)^3 | x^2 x 1 | [4-Marks] ## 2019 **(i) If** | 2 3 | = | 3 x 3 |, find the value of x. | 4 5 | | 2 x 5 | [2-Marks] **(ii) Show that** | 1 + a 1 1 | = 2abc (1 + 1/a + 1/b + 1/c), | 1 1 + b 1 | (abc ≠ 0) | 1 1 1 + c | **OR** Using Cramer's rule, solve the questions: 3x + y + z = 10 x + y - z = 0 5x - 9y = 1 [4-Marks] ## 2018 **(i) Without expanding the determinant**, prove that | 1 a a^2 | = (a+b)(b-c)(c-a) | 1 b b^2 | | 1 c c^2 | [2-Marks] **(ii) If** | x-1 1 1 | = 0, find the value of x. | 1 x+1 -1 | | -1 1 x+1 | **OR** Prove that | bc b^2+bc c^2+bc | = (ab+bc+ca)² | a^2+ac -ac c^2+ac | | a^2+ab b^2+ab -ab | [4-Marks] ## 2017 **(i) Prove that** | 1 logy logz | | logx 1 logz | = 0. | logx logy 1 | [2-Marks] **(ii) Solve by Cramer's rule:** x + 3y = 4; y + 3z = 7; 4x + z = 6 [4-Marks] **(iii) Prove that** | 2ab^2 a b | = (a^2 + b^2)^2 | a^2 b 2ab | | b^2 2ab a | **OR** **Without expanding the determinant, show that the determinant** | a^2+10 ab ac | | ab b^2+10 bc | is dividable by 100. | ac be c^2+10 | [4-Marks] ## 2016 **(i) Evaluate:** | α β γ | | α^2 β^2 γ^2 | | β+γ γ+α α+β | [2-Marks] **(ii) If p, q, r are not in a geometric progression and** | 1 p p^2 | | 1 q q^2 | = 0, then prove that px^2 + 2qx + r = 0 | 1 r r^2 | **OR** **Show that** | a^2+1 ab ac| | ab b^2+1 bc | = 1 + a^2 + b^2 + c^2 | ca bc c^2+1 | [4-Marks] ## 2015 **(i) Without expanding, show that** | 9 9 12 | ≠ 0. | 1 -3 -4 | | 1 9 12 | [2-Marks] **(ii) Solve the following equations by Cramer's method:** x + y = 1; y + z = 2; x + z = 3. [4-Marks] **(iii) Show that** | a+b+2c a b | | c b+c+2a b | = 2 (a+b+c)^3 | c c+a+2b a | **OR** **Solve for x**: | x c+x b+x | | c+x x a+x | = 0 | b+x a+x x | [4-Marks]