This set of Aptitude Questions and Answers (MCQs) focuses on “Algebraic Variables – Set 2”.
1. If (x-1) is the HCF of Px2-Qx+R and Qx2-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 Px2-Qx+R and Qx2-Px+R.
P(12)-Q(1)+R = 0
R = Q-P …… (i)
Q(12)-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 2x2+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 = 2x2+6x+4 …… (i)
A – B = 4x+8 …… (ii)
On solving (i) and (ii), we get,
A = x2+5x+6 = (x+2)(x+3) and B = x2+x-2 = (x+2)(x-1).
Therefore, HCF is (x+2).
3. The sum and difference of two expressions are 2x2+x-12 and 5x+20. Find the LCM.
a) (x2-4)(x+1)
b) (x2-16)(x-1)
c) (x2-16)(x+1)
d) (x-4)(x+1)
View Answer
Explanation: Let A and B be the two expressions.
A + B = 2x2+x-12 …… (i)
A – B = 5x+20 …… (ii)
On solving (i) and (ii), we get,
A = x2+5x+4 = (x+1)(x+4) and B = x2-16 = (x+4)(x-4).
Therefore, LCM is (x2-16)(x+1).
4. Find the HCF of (x-1)6 and (x6+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) = (x6+1) = (x2+1)(x4+1+x2).
There is no common factor between them. Hence, HCF is 1.
5. If (z-2) is the HCF of (z2-4) and pz2-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 (z2-4) and pz2-q(z-3).
p(22)-q(2-3) = 0.
i.e., 4p = -q.
6. What is the value of k for which the HCF of x2+kx+6 and x2+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+x2+12, B = x2+2x-24 and C = x2+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+x2+12 = (x+2)(x+6), B = x2+2x-24 = (x-4)(x+6) and C = x2+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 p2-p-6 and p6-93 is 1.
II. HCF of x+y and x10-y10 is x+y.
III. HCF of x+y and x10+y10 is x+y.
a) Only I
b) Only II
c) I and III
d) II and III
View Answer
Explanation: p2-p-6 = (p-3)(p+2) and p6-93 = p6-36 = (p+3)(p-3)(p4+9p+81). Hence, HCF is p-3.
We know that x+y and x-y are factors of x10-y10. Therefore, HCF of x+y and x10-y10 is x+y.
We know that x+y and x-y are not the factors of x10+y10. Therefore, HCF of x+y and x10+y10 is 1.
Therefore, only statement II is true.
9. Find the HCF of x2-y2, x6-y6 and x10+y10.
a) x-y
b) 1
c) x+y
d) x2-y2
View Answer
Explanation: x2-y2 = (x+y(x-y) and x6-y6 = (x+y)(x-y)(x4+y4+x2y2).
We know that x+y and x-y are not the factors of x10+y10.
Therefore, HCF is 1.
10. What is the value of k, if HCF of x2-x-2 and 2x2-kx-4 is (x-k)?
a) 3
b) 2
c) 4
d) -2
View Answer
Explanation: x-k is HCF of x2-x-2 and x2+kx-8.
2k2-k2-4 = 0
So, k = ±2.
x2-x-2 = (x-2)(x+1).
Therefore k = 2, so that the HCF is x-2.
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