Determinants — Class 12 solved problems

33 problems from this chapter, each solved step by step.

If A = [[1, 2], [3, 4]] find det(A) and adj(A).
The determinant of A is -2 and the adjoint of A is 4 & -2 -3 & 1.
The value of 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 is:
0
Using Cramer's rule, show that the following system of linear equations is consistent and hence solve it: x + y + z = 1, 2x + 3y + 2z = 2, x + y + 2z = 4.
x = -2, y = 0, z = 3
Let a R and A be a matrix of order 3 3 such that det(A)=-4 and A+I= 1 & a & 12 & 1 & 0 & 1 & 2, where I is the 3 3 identity. If det ((a+1)adj((a-1)A))=2^m3^n, m
16
Let A be a 3 × 3 matrix such that |adj (adj(adj A))| = 81. If S = n Z: |adj(adj A)|^(((n-1)²)/(2)) = |A|³n² - 5n - 4, then Σ_n S | A^(n² + n) | is equal to:
732
Find the equation of the line joining A(1,3) and B(0,0) using determinants and find k if D(k, 0) is a point such that area of triangle ABD is 3 sq units.
The equation of the line joining A(1,3) and B(0,0) is 3x - y = 0. The values of k for which the area of triangle ABD is 3 sq units are k=2 or k=-2.
If the system of linear equations 3x + y + β z = 3, 2x + α y - z = -3, x + 2y + z = 4 has infinitely many solutions, then the value of 22β - 9α is:
31
Let A = 2 & 2 + p & 2 + p + q 4 & 6 + 2p & 8 + 3p + 2q 6 & 12 + 3p & 20 + 6p + 3q. If det(adj(adj(3A))) = 2^m · 3^n, m, n N, then m + n is equal to:
24
Let A=[a_ij]=[ cc log_5 128 & log_4 5 log_5 8 & log_4 25 ]. If A_ij is the cofactor of a_ij, C_ij=Σ_k=1² a_ik A_jk, 1 ≤ i, j ≤ 2, and C=[C_ij], then 8|C| is equ
242
Let the system of equations x + 5y - z = 1, 4x + 3y - 3z = 7, and 24x + y + z =, with, R, have infinitely many solutions. Then the number of solutions of this s
12
Evaluate | ccx & x+1 x-1 & x |
1
Let A=[a_ij] be a 2 × 2 matrix such that a_ij 0,1 for all i and j. Let the random variable X denote the possible values of the determinant of the matrix A. Then
The variance of X is (3)/(8).
If the system of equations ( - 1)x + ( - 4)y + z = 5, x + ( - 1)y + ( - 4)z = 7, ( + 1)x + ( + 2)y - ( + 2)z = 9 has infinitely many solutions, then ² + is equa
12
Solve the system of equations using matrix method: x - y + 2z = 7, 3x + 4y - 5z = -5, 2x - y + 3z = 12.
x = -10, y = 1, z = 3
Evaluate | rr2 & 4 -1 & 2 |.
8
Evaluate =| ccc0 & sin α & -cos α -sin α & 0 & sin β cos α & -sin β & 0 | Solution Expanding along R_1, we get & =0| cc 0 & sin β -sin β & 0 |-sin α| cc -sin α
The value of the determinant is 0.
Let α, β (α ≠ β) be the values of m, for which the equations x+y+z=1, x+2y+4z=m, and x+4y+10z=m² have infinitely many solutions. Then the value of Σ_n=1^(10) (n
385 + 1 + (1)/(4) + (1)/(9) + (1)/(16) + (1)/(25) + (1)/(36) + (1)/(49) + (1)/(64) + (1)/(81) + (1)/(100) = 386 + (1)/(4) + (1)/(9) + (1)/(16) + (1)/(25) + (1)/
Let I be the identity matrix of order 3 3 and let A= & 2 & 34 & 5 & 67 & -1 & 2 with |A|=-1. Let B be the inverse of the matrix adj(A) adj(A²). Then | B + I| is
The problem cannot be solved to a specific numerical value without further information about or additional properties of the matrices involved. The expression s
Let A be a matrix of order 3 3 with A =5. If 2 adj (3A adj(2A)) =2^(α)3^(β)5^(), where α,β, N, then α+β+ is equal to:
27
The system of equations x + y + z = 6, x + 2y + 5z = 9, x + 5y + z = has no solution if:
The system has no solution if = 17 and ≠ 18.
Evaluate the determinant =| rrr1 & 2 & 4 -1 & 3 & 0 4 & 1 & 0 |.
The determinant = -52.
If the system of equations x + 2y - 3z = 2, 2x + y + 5z = 5, 14x + 3y + z = 33 has infinitely many solutions, then + is equal to:
12
If the system of equations 2x - y + z = 4, 5x + y + 3z = 12, 100x - 47y + z = 212 has infinitely many solutions, then - 2 is equal to
57
Let A be a square matrix of order 3 such that det(A) = -2 and det(3 adj(-6 adj(3 A))) = 2^(m+n) · 3^(mn), m > n. Then 4m + 2n is equal to:
38
If the system of linear equations: x + y + 2z &= 6 2x + 3y + az &= a + 1 -x - 3y + bz &= 2b where a, b R, has infinitely many solutions, then 7a + 3b is equal t
42
Let M and m respectively be the maximum and the minimum values of 1 + sin² x & cos² x & 4 sin 4x 1 + sin² x & cos² x & 4 sin 4x sin² x & cos² x & 1 + 4 sin 4x,
0
For some a, b, let f(x) = | ccc a + (sin x)/(x) & 1 & b a & 1 + (sin x)/(x) & b a & 1 & b + (sin x)/(x) |, x ≠ 0, lim_x → 0 f(x) = + a + b. Then ( + +)² is equa
16
intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines. 35. (a) If P = 1 & -1 & 0 2 & 3
= xy + yz + zx + xyz. If = 0, then x^(-1) + y^(-1) + z^(-1) = -1.
Find values of x for which | ll3 & x x & 1 |=| ll3 & 2 4 & 1 |.
x = ± 2√2
What is the determinant of the matrix 0 & 7 & 3 1 & 0 & 7 3 & 0 & 0? Simplify your answer. (determinant, matrix, linear algebra, cofactor expansion, square matr
The determinant of the matrix is 147.
Find the minor of element 6 in the determinant =| lll1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 |
The minor of element 6 is -6.
If the system of equations 2x + y + 3z = 5 3x + 2y - z = 7 4x + 5y + z = 9 has infinitely many solutions, then ( ² + ²) is equal to:
29
Let the system of equations: 2x + 3y + 5z &= 9, 7x + 3y - 2z &= 8, 12x + 3y - (4 +)z &= 16 - have infinitely many solutions. Then the radius of the circle centr
The radius of the circle is (7)/(5).

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