1.Algebra Booster

Quadratic Equations and Expressions 2.31
COMPREHENSIVE LINK PASSAGES
Passage I: 1. (b) 2. (c) 3. (a)
Passage II: 1. (c) 2. (c) 3. (c) 4. (d)
Passage III: 1. (c) 2. (b) 3. (c) 4. (c)
Passage IV: 1. (b) 2. (c) 3. (c)
Passage V: 1. (d) 2. (b) 3. (c)
MATCH MATRIX
1. A Æ (P, T); B Æ (Q, S); C Æ (R)
2. A Æ (R, S, T); B Æ (P, Q, R); C Æ (R, S, T)
3. A Æ (P, Q) ; B Æ (S, T) ; C((P, R)
4. A Æ (P, Q, R, S, T); B Æ (P, Q); C Æ (R)
5. (B)
6. (C)
7. (D)
8. (A)
ASSERTION AND REASON
1. (b) 2. (d) 3. (c) 4. (b) 5. (a)
6. (d) 7. (b) 8. (b) 9. (a) 10. (b)
HINTS AND SOLUTIONS
LEVEL I
2. Let x 2 2 2 2 ...... to
x= 2 + x
x 2 = 2 + x
x 2 – x – 2 = 0
(x – 2)(x + 1) = 0
x = 2, –1
since square always provide us non negative values, so
x = 2
3. Given x = 2 + 2 2/3 + 2 1/3
x – 2 = 2 2/3 + 2 1/3
(x – 2) 3 = (2 2/3 + 2 1/3 ) 3
x 3 – 6x 2 + 12x – 8
= (2 2/3 ) 3 + (2 1/3 ) 3 + 3.2 2/3 2 1/3 (2 2/3 + 2 1/3 )
= 4 + 2 + 3.2(x – 2)
= 6 + 6x – 12
= 6x – 6
x 3 – 6x 2 + 6x = 8 – 6 = 3
4. Divide by x 2 , we get
Ê 1 ˆ Ê 1ˆ
+ - + + 2=
0
2
Áx
Ë 2 ˜
x ¯
Áx
Ë
˜
x¯
2
Ê 1ˆ Ê 1ˆ
Áx+ - 2- x+ + 2=
0
Ë
˜
x¯ Á
Ë
˜
x¯
2
Ê 1ˆ Ê 1ˆ
Áx+ - x+ = 0
Ë
˜ Á ˜
x¯ Ë x¯
Ê 1ˆ
Put Áx+ = t
Ë
˜
x¯
Then t 2 – t = 0
t = 0 and t = 1
1 1
x + 0 and x 1
x
= + x
=
x 2 + 1 = 0 and x 2 – x + 1 = 0
1±
i 3
x=± iand
x=
2
Hence, the solutions are
Ï 1±
i 3¸
̱
i,
˝
Ó 2 ˛
5. (x + 1)(x + 2)(x + 3)(x + 4) = 120
{(x + 1)(x + 4)}{(x + 2)(x + 3)} = 120
{(x 2 + 5x + 4)}{(x 2 + 5x + 6)} = 120
Put x 2 + 5x = a
(a + 4)(a + 6) = 120
a 2 + 10a – 96 = 0
(a + 16)(a – 6) = 0
a = 6, –16
x 2 + 5x = 6, –16
x 2 + 5x – 6 = 0, x 2 + 5x + 16 – 0
(x + 6)(x – 1) = 0, x 2 + 5x + 16 – 0
- 5±
i 39
x = 1, – 6,
2
Hence, the solutions are
Ï - 5±
i 39¸
Ì1, – 6, ˝
Ó 2 ˛
6. It is true for all real values of x.
So it is an identity in x
Thus the number of real solution is infinite.
1 1
7. x - = 2 -
2 2
x -4 x -4
No real values of x satisfying the given equation
Thus, x = f
8. (x – 1) 2 + (x – 1) 2 + (x – 3) 2 = 0
3x 2 – 12x + 14 = 0
Clearly, D < 0
Number of real solutions is zero
x -ab x -bc x -ca
9. + + = a+ b+
c
a + b b+ c c+
a