Advertisements
Advertisements
प्रश्न
Show that Δ = `|(x, "p", "q"),("p", x, "q"),("q", "q", x)| = (x - "p")(x^2 + "p"x - 2"q"^2)`
Advertisements
उत्तर
Applying C1 → C1 – C2, we have
Δ = `|(x - "p", "p", "q"),("p" - x, x, "q"),(0, "q", x)|`
= `(x - "p")|(1, "p", "q"),(-1, x, "q"),(0, "q", x)|`
= `(x - "p")|(0, "p" + x, 2"p"),(-1, x, "q"),(0, "q", x)|` Applying R1 → R1 – R2
Expanding along C1, we have
Δ = `(x - "p")("p"x + x^2 - 2"q"^2)`
= `(x - "p")(x^2 + "p"x - 2"q"^2)`
APPEARS IN
संबंधित प्रश्न
Find the value of x, if `|(2,4),(5,1)|=|(2x, 4), (6,x)|`.
Find the value of x, if `|(2,3),(4,5)|=|(x,3),(2x,5)|`.
Using the property of determinants and without expanding, prove that:
`|(x, a, x+a),(y,b,y+b),(z,c, z+ c)| = 0`
Let A be a square matrix of order 3 × 3, then | kA| is equal to
(A) k|A|
(B) k2 | A |
(C) k3 | A |
(D) 3k | A |
Without expanding at any stage, find the value of:
`|(a,b,c),(a+2x,b+2y,c+2z),(x,y,z)|`
Use properties of determinants to solve for x:
`|(x+a, b, c),(c, x+b, a),(a,b,x+c)| = 0` and `x != 0`
A matrix A of order 3 × 3 has determinant 5. What is the value of |3A|?
Let A = [aij] be a square matrix of order 3 × 3 and Cij denote cofactor of aij in A. If |A| = 5, write the value of a31 C31 + a32 C32 a33 C33.
A matrix of order 3 × 3 has determinant 2. What is the value of |A (3I)|, where I is the identity matrix of order 3 × 3.
A matrix A of order 3 × 3 is such that |A| = 4. Find the value of |2 A|.
If A is a 3 × 3 matrix, \[\left| A \right| \neq 0\text{ and }\left| 3A \right| = k\left| A \right|\] then write the value of k.
Which of the following is not correct in a given determinant of A, where A = [aij]3×3.
Solve the following system of linear equations using matrix method:
3x + y + z = 1
2x + 2z = 0
5x + y + 2z = 2
Using matrices, solve the following system of linear equations :
x + 2y − 3z = −4
2x + 3y + 2z = 2
3x − 3y − 4z = 11
If x, y ∈ R, then the determinant ∆ = `|(cosx, -sinx, 1),(sinx, cosx, 1),(cos(x + y), -sin(x + y), 0)|` lies in the interval.
The determinant ∆ = `|(sqrt(23) + sqrt(3), sqrt(5), sqrt(5)),(sqrt(15) + sqrt(46), 5, sqrt(10)),(3 + sqrt(115), sqrt(15), 5)|` is equal to ______.
If a1, a2, a3, ..., ar are in G.P., then prove that the determinant `|("a"_("r" + 1), "a"_("r" + 5), "a"_("r" + 9)),("a"_("r" + 7), "a"_("r" + 11), "a"_("r" + 15)),("a"_("r" + 11), "a"_("r" + 17), "a"_("r" + 21))|` is independent of r.
Prove tha `|("bc" - "a"^2, "ca" - "b"^2, "ab" - "c"^2),("ca" - "b"^2, "ab" - "c"^2, "bc" - "a"^2),("ab" - "c"^2, "bc" - "a"^2, "ca" - "b"^2)|` is divisible by a + b + c and find the quotient.
If x + y + z = 0, prove that `|(x"a", y"b", z"c"),(y"c", z"a", x"b"),(z"b", x"c", y"a")| = xyz|("a", "b", "c"),("c", "a", "b"),("b", "c", "a")|`
Let f(t) = `|(cos"t","t", 1),(2sin"t", "t", 2"t"),(sin"t", "t", "t")|`, then `lim_("t" - 0) ("f"("t"))/"t"^2` is equal to ______.
If A = `[(2, lambda, -3),(0, 2, 5),(1, 1, 3)]`, then A–1 exists if ______.
There are two values of a which makes determinant, ∆ = `|(1, -2, 5),(2, "a", -1),(0, 4, 2"a")|` = 86, then sum of these number is ______.
`|(0, xyz, x - z),(y - x, 0, y z),(z - x, z - y, 0)|` = ______.
If f(x) = `|((1 + x)^17, (1 + x)^19, (1 + x)^23),((1 + x)^23, (1 + x)^29, (1 + x)^34),((1 +x)^41, (1 +x)^43, (1 + x)^47)|` = A + Bx + Cx2 + ..., then A = ______.
The value of the determinant `abs ((1,0,0),(2, "cos x", "sin x"),(3, "sin x", "cos x"))` is ____________.
Find the minor of the element of the second row and third column in the following determinant `[(2,-3,5),(6,0,4),(1,5,-7)]`
If `Delta = abs((5,3,8),(2,0,1),(1,2,3)),` then write the minor of the element a23.
If `"abc" ne 0 "and" abs ((1 + "a", 1, 1),(1, 1 + "b", 1),(1,1,1 + "c")) = 0, "then" 1/"a" + 1/"b" + 1/"c" =` ____________.
Find the 5th term of expansion of `(x^2 + 1/x)^10`?
For positive numbers x, y, z, the numerical value of the determinant `|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` is
Value of `|(2, 4),(-1, 2)|` is
