If A, then adj (3A2 + 12A) is equal to

Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr (A), the sum of diagonal entries of A. Assume that A₂= I.
Statement −1: If A ≠ I and A ≠ − I, then det A = − 1.
Statement −2: If A ≠ I and A ≠ − I, then tr (A) ≠ 0.

• Statement −1 is false, Statement −2 is true

• Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

• Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

• Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

• If det A = ± 1, then A^–1 exists but all its entries are not necessarily integers

• If detA ≠ ± 1, then A^–1 exists and all its entries are non-integers

• If detA = ± 1, then A^–1 exists and all its entries are integers

• If detA = ± 1, then A^–1 exists and all its entries are integers

Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx and z = bx + ay. Then a² + b² + c² + 2abc is equal to

• 2

• -1

• 0

• 0

If D = for x ≠ 0, y ≠ 0 then D is

• divisible by neither x nor y

• divisible by both x and y

• divisible by x but not y

• divisible by x but not y

16.

Let A =    If |A²| = 25, then |α| equals

• 5²

• 1

• 1/5

• 1/5

Let  a , b ∈ N. Then

• there cannot exist any B such that AB = BA

• there exist more than one but finite number of B’s such that AB = BA

• there exists exactly one B such that AB = BA

• there exists exactly one B such that AB = BA

If a₁, a₂, a₃,…, a_n,… are in G.P., then the determinant

• 1

• 0

• 4

• 4

If the system of linear equations
x + ky + 3z = 0
3x + ky – 2z = 0
2x + 4y – 3z = 0

has a non-zero solution (x,y,z), then xz/y² is equal to

• 30

• -10

• 10

• -30

If $\left|\begin{array}{ccc}x-4& 2x& 2x\\ 2x& x-4& 2x\\ 2x& 2x& x-4\end{array}\right| = (A +Bx)(x-A{)}^2$, then the ordered pair (A,B) is equal to

• (4,5)

• (-4,-5)

• (-4,3)

• (-4,5)