Advertisements
Advertisements
प्रश्न
Determine the roots of the equation `|(1,4, 20),(1, -2, 5),(1, 2x, 5x^2)|` = 0
उत्तर
Now `|(1,4, 20),(1, -2, 5),(1, 2x, 5x^2)| = (5)|(1, 4, 4),(1, -2, 1),(1, 2x, x^2)|`
Taking 5 as a common factor from C3
= `(5)(2)|(1, 2, 4),(1, -1, 1),(1, x, x^2)|`
Taking 2 as a common factor from C2
= `10|(1, 2, 4),(1, -1, 1),(1, x, x^2)|`
= `10|(0, 3,3),(0, -1 - x, 1 - x^2),(1, x, x^2)| {:("R"_1 -> "R"_1 - "R"_2),("R"_2 -> "R"_2 - "R"_3):}`
Expanding along C1
`101[3(1 - x^2) - (- 1 - x)(3)]}`
= 10[3(1 – x2) + 3(1 + x)]
= 10[3(1 + x)(1 – x) + 3(1 + x)]
= 10[3(1 + x)(1 – x + 1)]
= 30(1 +x)(2 – x)
Given the determinant value is 0
⇒ 30(1 + x)(2 – x) = 0
⇒ 1 + x = 0 or 2 – x = 0
⇒ x = – 1 or x = 2
So, x = – 1 or 2.
APPEARS IN
संबंधित प्रश्न
Without expanding the determinant, prove that `|("s", "a"^2, "b"^2 + "c"^2),("s", "b"^2, "c"^2 + "a"^2),("s", "c"^2, "a"^2 + "b"^2)|` = 0
Show that `|("b" + "c", "bc", "b"^2"C"^2),("c" + "a", "ca", "c"^2"a"^2),("a" + "b", "ab", "a"^2"b"^2)|` = 0
Prove that `|(1 + "a", 1, 1),(1, 1 + "b", 1),(1, 1, 1 + "c")| = "abc"(1 + 1/"a" + 1/"b" + 1/"c")`
Show that `|(x + 2"a", y + 2"b", z + 2"c"),(x, y, z),("a", "b", "c")|` = 0
If `|("a", "b", "a"alpha + "b"),("b", "c", "b"alpha + "c"),("a"alpha + "b", "b"alpha + "c", 0)|` = 0, prove that a, b, c are in G. P or α is a root of ax2 + 2bx + c = 0
If a, b, c are pth, qth and rth terms of an A.P, find the value of `|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)|`
If A = `[(1/2, alpha),(0, 1/2)]`, prove that `sum_("k" = 1)^"n" det("A"^"k") = 1/3(1 - 1/4)`
Without expanding, evaluate the following determinants:
`|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|`
If λ = – 2, determine the value of `|(0, lambda, 1),(lambda^2, 0, 3lambda^2 + 1),(-1, 6lambda - 1, 0)|`
Show that `|(1, 1, 1),(x, y, z),(x^2, y^2, z^2)|` = (x – y)(y – z)(z – x)
Find the area of the triangle whose vertices are (0, 0), (1, 2) and (4, 3)
Identify the singular and non-singular matrices:
`[(0, "a" - "b", "k"),("b" - "a", 0, 5),(-"k", -5, 0)]`
Determine the values of a and b so that the following matrices are singular:
B = `[("b" - 1, 2, 3),(3, 1, 2),(1, -2, 4)]`
If cos 2θ = 0, determine `[(theta, costheta, sintheta),(costheta, sintheta, 0),(sintheta, 0, costheta)]^2`
Choose the correct alternative:
If A = `[(1, -1),(2, -1)]`, B = `[("a", 1),("b", -1)]` and (A + B)2 = A2 + B2, then the values of a and b are
Choose the correct alternative:
If `|(2"a", x_1, y_1),(2"b", x_2, y_2),(2"c", x_3, y_3)| = "abc"/2 ≠ 0`, then the area of the triangle whose vertices are `(x_1/"a", y_1/"a"), (x_2/"b", y_2/"b"), (x_3/"c", y_3/"c")` is
Choose the correct alternative:
If x1, x2, x3 as well as y1, y2, y3 are in geometric progression with the same common ratio, then the points (x1, y1), (x2, y2), (x3, y3) are
If Δ is the area and 2s the sum of three sides of a triangle, then
If f(x) = `|(cos^2x, cosx.sinx, -sinx),(cosx sinx, sin^2x, cosx),(sinx, -cosx, 0)|`, then for all x
If `|(1 + x, x, x^2),(x, 1 + x, x^2),(x^2, x, 1 + x)|` = ax5 + bx4 + cx3 + dx2 + λx + µ be an identity in x, where a, b, c, d, λ, µ are independent of x. Then the value of λ is ______.
