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
Prove that `|(1 + "a", 1, 1),(1, 1 + "b", 1),(1, 1, 1 + "c")| = "abc"(1 + 1/"a" + 1/"b" + 1/"c")`
Prove that `|(1, "a", "a"^2 - "bc"),(1, "b", "b"^2 - "ca"),(1, "c", "c"^2 - "ab")|` = 0
Show that `|("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|` is divisiible by x4
If a, b, c, are all positive, and are pth, qth and rth terms of a G.P., show that `|(log"a", "p", 1),(log"b", "q", 1),(log"c", "r", 1)|` = 0
Find the value of `|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` if x, y, z ≠ 1
Without expanding, evaluate the following determinants:
`|(2, 3, 4),(5, 6, 8),(6x, 9x, 12x)|`
Without expanding, evaluate the following determinants:
`|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|`
If A and B are square matrices of order 3 such that |A| = –1 and |B| = 3, find the value of |3AB|
Show that `|("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + b")|` = 8abc
Show that `|("b" + "C", "a", "a"^2),("c" + "a", "b", "b"^2),("a" + "b", "c", "c"^2)|` = (a + b + c)(a – b)(b – c)(c – a)
Solve `|(4 - x, 4 + x, 4 + x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)|` = 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)
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)]`
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 f(x) = `|(cos^2x, cosx.sinx, -sinx),(cosx sinx, sin^2x, cosx),(sinx, -cosx, 0)|`, then for all x
A pole stands vertically inside a triangular park ΔABC. If the angle of elevation of the top of the pole from each corner of the park is same, then in ΔABC the foot of the pole is at the
What is the value of Δ if, Δ = `|(0, sin alpha, - cos alpha),(-sin alpha, 0, sin beta),(cos alpha, - sin beta, 0)|`
Let a, b, c, d be in arithmetic progression with common difference λ. If `|(x + a - c, x + b, x + a),(x - 1, x + c, x + b),(x - b + d, x + d, x + c)|` = 2, then value of λ2 is equal to ______.
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 ______.
