This set of Aptitude Questions and Answers (MCQs) focuses on “Algebraic Variables – Set 2”.

1. If (x-1) is the HCF of Px^{2}-Qx+R and Qx^{2}-Px+R, then what is the value of R?

a) 1

b) A

c) B

d) 0

View Answer

Explanation: (x-1) is the HCF of Px

^{2}-Qx+R and Qx

^{2}-Px+R.

P(1

^{2})-Q(1)+R = 0

R = Q-P …… (i)

Q(1

^{2})-P(1)+R = 0

R = P-Q …… (ii)

Solving (i) and (ii), we get,

R = 0.

2. The sum and difference of two expressions are 2x^{2}+6x+4 and 4x+8. Find the HCF.

a) x-2

b) x+2

c) x-1

d) x+3

View Answer

Explanation: Let A and B be the two expressions.

A + B = 2x

^{2}+6x+4 …… (i)

A – B = 4x+8 …… (ii)

On solving (i) and (ii), we get,

A = x

^{2}+5x+6 = (x+2)(x+3) and B = x

^{2}+x-2 = (x+2)(x-1).

Therefore, HCF is (x+2).

3. The sum and difference of two expressions are 2x^{2}+x-12 and 5x+20. Find the LCM.

a) (x^{2}-4)(x+1)

b) (x^{2}-16)(x-1)

c) (x^{2}-16)(x+1)

d) (x-4)(x+1)

View Answer

Explanation: Let A and B be the two expressions.

A + B = 2x

^{2}+x-12 …… (i)

A – B = 5x+20 …… (ii)

On solving (i) and (ii), we get,

A = x

^{2}+5x+4 = (x+1)(x+4) and B = x

^{2}-16 = (x+4)(x-4).

Therefore, LCM is (x

^{2}-16)(x+1).

4. Find the HCF of (x-1)^{6} and (x^{6}+1).

a) x-1

b) (x-1)^{2}

c) 1

d) x+1

View Answer

Explanation: Let p(x) = (x-1)

^{6}and q(x) = (x

^{6}+1) = (x

^{2}+1)(x

^{4}+1+x

^{2}).

There is no common factor between them. Hence, HCF is 1.

5. If (z-2) is the HCF of (z^{2}-4) and pz^{2}-q(z-3), then which of the following is true?

a) 4p = -q

b) p = 2q

c) q = -2p

d) p = q

View Answer

Explanation: (z-2) is the HCF of (z

^{2}-4) and pz

^{2}-q(z-3).

p(2

^{2})-q(2-3) = 0.

i.e., 4p = -q.

6. What is the value of k for which the HCF of x^{2}+kx+6 and x^{2}+x-2 is (x+2)?

a) 5

b) 6

c) -5

d) -4

View Answer

Explanation: (x+2) is the HCF.

(-2)

^{2}+k(-2)+6 = 0.

Therefore, k = 5.

7. If A = 8x+x^{2}+12, B = x^{2}+2x-24 and C = x^{2}+15x+54. Then which of the following is true?

I. Their LCM is (x+6)(x-4)(x+2)(x+9).

II. Their HCF is (x+6)(x-2).

a) Only I

b) Only II

c) Both I & II

d) Neither I nor II

View Answer

Explanation: A = 8x+x

^{2}+12 = (x+2)(x+6), B = x

^{2}+2x-24 = (x-4)(x+6) and C = x

^{2}+15x+54 = (x+6)(x+9).

HCF = (x+6) and LCM = (x+6)(x-4)(x+2)(x+9).

Hence, only statement I is true.

8. Which of the following is/are true?

I. HCF of p^{2}-p-6 and p^{6}-9^{3} is 1.

II. HCF of x+y and x^{10}-y^{10} is x+y.

III. HCF of x+y and x^{10}+y^{10} is x+y.

a) Only I

b) Only II

c) I and III

d) II and III

View Answer

Explanation: p

^{2}-p-6 = (p-3)(p+2) and p

^{6}-9

^{3}= p

^{6}-3

^{6}= (p+3)(p-3)(p

^{4}+9p+81). Hence, HCF is p-3.

We know that x+y and x-y are factors of x

^{10}-y

^{10}. Therefore, HCF of x+y and x

^{10}-y

^{10}is x+y.

We know that x+y and x-y are not the factors of x

^{10}+y

^{10}. Therefore, HCF of x+y and x

^{10}+y

^{10}is 1.

Therefore, only statement II is true.

9. Find the HCF of x^{2}-y^{2}, x^{6}-y^{6} and x^{10}+y^{10}.

a) x-y

b) 1

c) x+y

d) x^{2}-y^{2}

View Answer

Explanation: x

^{2}-y

^{2}= (x+y(x-y) and x

^{6}-y

^{6}= (x+y)(x-y)(x

^{4}+y

^{4}+x

^{2}y

^{2}).

We know that x+y and x-y are not the factors of x

^{10}+y

^{10}.

Therefore, HCF is 1.

10. What is the value of k, if HCF of x^{2}-x-2 and 2x^{2}-kx-4 is (x-k)?

a) 3

b) 2

c) 4

d) -2

View Answer

Explanation: x-k is HCF of x

^{2}-x-2 and x

^{2}+kx-8.

2k

^{2}-k

^{2}-4 = 0

So, k = ±2.

x

^{2}-x-2 = (x-2)(x+1).

Therefore k = 2, so that the HCF is x-2.

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