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### Exams > Cat > Quantitaitve Aptitude

## ALGEBRA MCQs

Total Questions : 240
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**1**of 24 pages**Answer: Option D. ->**1 positive and 1 negative root

:

D

Given question isflexible. So, we can assume the roots and can make an other quardatic equation.

Let α = 1 and β = 2, then the expression becomes x

^{2}– 3x + 2 = 0. So p = -3, q = 2, r = 17, s = 16. The new expression is x

^{2}– 8x – 9 = 0. 1 root is positive and other is negative.

**Answer: Option C. ->**12 or -1

:

C

Option (c)

As ab+c=bc+a=ca+b=a+b+cb+c+c+a+a+b

=a+b+c2(a+b+c)=r=12 (Assuming a + b + c ≠ 0)

If a + b + c = 0

ab+c = aa+b+c−a (by adding and subtracting a in the denominator) =a0−a=a−a=r=−1

(similarly bc+a=ca+b= -1)

Hence r can take only 12 or -1 as values. Choice (c)

**Answer: Option D. ->**t3-1

:

D

Option (d)Assume t= 2The question changes to[(5-√3i)(5+√3i)]/4 [25-(√3i)

^{2}]/4, since (a-b)(a+b)=a

^{2}-b

^{2}(25+3)/4, since i

^{2}=-1=7Only answer option (d) gives 7 on substitution of t=2

**Answer: Option A. ->**aDn-bDn-1

:

A

Option (a)Best way to proceed is by assumption of values.Assume a quadratic equation. Let’s take x

^{2}+5x-6=0. Here the roots are p=1 and q=-6Thus, a = sum of roots = -5 and b= product of roots = -6D

_{1}= p

^{1}+q

^{1}= -5D

_{0}= 2D

_{2}= 37Assume n=1We need to find D

_{n+1}= D

_{2}= 37Look in the answer options for 37Option a is the only one which gives (-5)(-5)-(-6)(2) = 37

**Answer: Option C. ->**1

:

C

WXYZ = WXY * XZ => WXY (10) + Z = WXY (10x) + WXYZ.

Comparing we get => X = 1 and Z = 0

**Answer: Option C. ->**1

:

C

Option (c)d

^{3}+d

^{2}+ed+1 is divisible by (d-1). As 1 is a root, Put d=1 in the equation and equate it to 0, to get the value of e1+1+e+1=0 e=-3Similarly put d=3 in d

^{3}-4d

^{2}+fd-3 to get the value of f 27-36+3f-3=0 3f=12 f=4e+f= -3+4=1

**Answer: Option C. ->**2,7

:

C

Option c (x+1)(x+8)+10 can be expanded as x

^{2}+9x+8+10=0 x

^{2}+9x+18=0 Sum of roots p+q= -9 Product of roots pq = 18(x+p)(x+q)-4=0 can be written as x

^{2}+ (p+q)x+pq-4=0 x

^{2}-9x+18-4=0x

^{2}-9x+14=0Roots are 7 and 2

**Answer: Option C. ->**q2-p2

:

C

Shortcut: AssumptionOption (c)Product of roots=1 is the only constraint.Assume values that satisfy the constraint and substitute, take a= -1 and b= -1, then p= 2Assume c= - 1/2 and d= - 2; q= 5/2 Question is = (a-c)(b-c)(a+d)(b+d)=9/4Only option (c) satisfies this => q

^{2}–p

^{2}= 25/4 – 4 =9/4

**Answer: Option D. ->**q2-p2

:

D

Use answer options to arrive at the answer at the earliest

Since the highest power of x is even (12) , it will always be greater than its negative counterpart (9), hence, irrespective of the value of x, the net value will be positive. Answer is option (d)