Advertisements
Advertisements
प्रश्न
If A = `[(1/2, alpha),(0, 1/2)]`, prove that `sum_("k" = 1)^"n" det("A"^"k") = 1/3(1 - 1/4)`
Advertisements
उत्तर
A = `[(1/2, alpha),(0, 1/2)]`
|A| = `|(1/2,alpha),(0, 1/2)|`
= `1/4 - 0`
= `1/4`
A2 = A × A
= `[(1/2, alpha),(0, 1/2)] [(1/2, alpha),(0, 1/2)]`
= `[(1/4, alpha),(0, 1/4)]`
|A2| = `|(1/4, alpha),(0, 1/4)|`
=`1/4 xx 1/4 - 0`
= `(1/4)^2`
= `1/4^2`
|Ak| = `1/4^"k"`
So `sum_("k" = 1)^"n" det("A"^"k") = 1/4 + 1/4^2 + 1/4^3 + ...... + 1/4^"n"`
Which is a G.P with a `1/4` and r = `1/4`
∴ Sn = `("a"(1 - "r"^"n"))/(1 - "r")`
= `(1/4[1 - (1/4)^"n"])/(1 - 1/4)`
= `(1/4[1 - 1/4^"n"])/(3/4)`
= `1/4 xx 4/3[1 - 1/4^"n"]`
= `1/3[1 - 1/4^"n"]`.
APPEARS IN
संबंधित प्रश्न
Prove that `|(sec^2theta, tan^2theta, 1),(tan^2theta, sec^2theta, -1),(38, 36, 2)|` = 0
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
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:
`|(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|
Find the area of the triangle whose vertices are (0, 0), (1, 2) and (4, 3)
Identify the singular and non-singular matrices:
`[(1, 2, 3),(4, 5, 6),(7, 8, 9)]`
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 Δ = `|("a", "b", "c"),(x, y, z),("p", "q", "r")|` then `|("ka", "kb","kc"),("k"x, "k"y, "k"z),("kp", "kq", "kr")|` is
Choose the correct alternative:
If ⌊.⌋ denotes the greatest integer less than or equal to the real number under consideration and – 1 ≤ x < 0, 0 ≤ y < 1, 1 ≤ z ≤ 2, then the value of the determinant `[([x] + 1, [y], [z]),([x], [y] + 1, [z]),([x], [y], [z] + 1)]`
If Δ is the area and 2s the sum of three sides of a triangle, then
If P1, P2, P3 are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then `cosA/P_1 + cosB/P_2 + cosC/P_3` is equal to
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 ______.
For f(x)= `ℓn|x + sqrt(x^2 + 1)|`, then the value of`g(x) = (cosx)^((cosecx - 1))` and `h(x) = (e^x - e^-x)/(e^x + e^-x)`, then the value of `|(f(0), f(e), g(π/6)),(f(-e), h(0), h(π)),(g((5π)/6), h(-π), f(f(f(0))))|` is ______.
`|("b" + "c", "c", "b"),("c", "c" + "a", "a"),("b", "a", "a" + "b")|` = ______.
