﻿ Engineering Entrance Exam Question and Answers | Determinants - Zigya

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# Determinants

#### Multiple Choice Questions

1.

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 y but not x

B.

divisible by both x and y C2 → C2 – C1 & C3 → C3 – C1 Hence D is divisible by both x and y.
138 Views

2.

The system of equations

x + y + z = 0

2x + 3y + z = 0

and x + 2y = 0

has

• a unique solution; x = 0, y = 0, z = 0

• infinite solutions

• no solution

• finite number of non-zero solutions

B.

infinite solutions

The given system of equations are

x + y + z = 0,

2x + 3y + z = 0,

and  x + 2y = 0

3.

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 false.

D.

Statement − 1 is true, Statement − 2 is false. 175 Views

4.

Let A = If |A2| = 25, then |α| equals

• 52

• 1

• 1/5

• 5

C.

1/5 143 Views

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 exist infinitely many B’s such that AB = BA

D.

there exist infinitely many B’s such that AB = BA AB = BA only when a = b
146 Views

6.

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 need not exist

C.

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

Each entry of A is integer, so the cofactor of every entry is an integer and hence each entry in the adjoint of matrix A is integer. Now detA = ± 1 and A–1 =(1/ det(A)) (adj A)

⇒ all entries in A–1 are integers

195 Views

7.

The simultaneous equations Kx + 2y - z = 1, (K - I)y - 2z = 2 and (K + 2)z = 3 have only one solution when :

• K = - 2

• K = - 1

• K = 0

• K = 1

B.

K = - 1

The system of given equations are

Kx + 2y - z = 1             ...(i)

(K - 1)y - 2z = 2           ...(ii)

and (K + 2)z = 3           ...(iii)

This system of equations has a unique solution, if

8.

If x = - 5 is a root of  = 0, then the other roots are :

• 3, 3.5

• 1, 3.5

• 1, 7

• 2, 7

B.

1, 3.5

9.

If a1, a2, a3,…, an,… are in G.P., then the determinant • 1

• 0

• 4

• 2

B.

0

C1 – C2, C2 – C3 two rows becomes identical Answer: 0

480 Views

10.

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

• 1

D.

1

The system of equations x – cy – bz = 0, cx – y + az = 0 and bx + ay – z = 0 have non-trivial solution if 391 Views