Exercise 5


Directions (Q. 1-4): In each of the following questions two equations I and II are given. You have to solve both the equations and give answer

(a) if a < b

(b) if a ≤ b

(c) if a ≥ b

(d) if a = b

(e) if a > b

Question.1
I. a2 – 5a + 6 = 0
II. b2 – 3b + 2 = 0
Ans.c

For eqn 1, the roots (a) will be 2,3. As –2 × –3 = 6 (ac) and (– 2) + (– 3) = – 5

(b). Similarly, for eqn II, the roots (b) will be 2, 1.

Question.2
I. 2a + 3b = 31
II. 3a = 2b + 1
Ans.a

2a + 3b = 31 ... (i)

3a – 2b = 1 ..(ii)

Multiply (i) by 2 and (ii) by 3 and then adding (i) and (ii), we get a = 6513 = 5. Putting the value of ‘a’ in any equation, we get b = 7.

Hence, b > a or a < b.

Question.3
I. 2a2 + 5a + 3 = 0
II. 2b2 – 5b + 3 = 0
Ans.a

a = –32 & –1; b = 32 & 1

Question.4
I. 4a2 = 1
II. 4b2 – 12b + 5 = 0
Ans.b

a = ±1/2; b = 1/2, 5/2

Directions (Qs. 5 - 9): In each of the following questions there are two equations. Solve them and give answer

(a) If P < Q

(b) If P > Q

(c) If P ≤ Q

(d) If P ≥ Q

(e) If P = Q

Question.5
I. 4P2 – 8P + 3 = 0
II. 2Q2 – 13Q + 15 = 0
Ans.c

I. 4P2 – 8P + 3 = 0

4P2 – 2P – 6P + 3= 0

2P(2P – 1) – 3(2P – 1) = 0

(2P – 3)(2P –1) = 0

⇒ P = 1/2, 3/2

II. 2Q2 – 13Q + 15 = 0

2Q2 – 10Q – 3Q + 15 = 0

2Q(Q – 5) – 3(Q – 5) = 0

⇒ (2Q – 3)(Q – 5) = 0

⇒ Q = 3/2 , 5

∴ Q ≥ P

Question.6
I. P2 + 3P – 4 = 0
II. 3Q2 – 10Q + 8 = 0
Ans.a

I. P2 + 3P – 4 = 0

P2 + 4P – P – 4 = 0

⇒ P(P + 4) – 1(P + 4) = 0

⇒ P = 1, – 4

II. 3Q2 – 10Q + 8 = 0

3Q2 – 6Q – 4Q + 8 = 0

3Q(Q – 2) – 4(Q – 2) = 0

(3Q – 4) (Q – 2) = 0

⇒ Q = 4/3, 2

∴ Q >P

Question.7
I. 3P2 – 10P + 7 = 0
II. 15Q2 – 22Q + 8 = 0
Ans.b

I. 3P2 – 10P + 7 = 0

3P2 – 3P – 7P + 7 = 0

3P(P – 1) – 7(P – 1) = 0

⇒ (3P – 7)(P – 1) = 0

⇒ P = 7/3, 1

II. 15Q2 – 22Q + 8 = 0

15Q2 – 10Q – 12Q + 8 = 0

5Q(3Q – 2) – 4(3Q – 2) = 0

(5Q – 4)(3Q – 2) = 0

⇒ Q = 45, 23

∴ P > Q

Question.8
I. 20P2 – 17P + 3 = 0
II. 20Q2 – 9Q + 1 = 0
Ans.b

I. 20P2 – 17P + 3 = 0

20P2 – 12P – 5P + 3 = 0

5P(4P – 1) – 3(4P – 1) = 0

⇒ P = 3/5, 1/4

II. 20Q2 – 9Q + 1 = 0

20Q2 – 4Q – 5Q + 1 = 0

4Q(5Q – 1) – 1 (5Q – 1) = 0

(4Q – 1)(5Q – 1) = 0

⇒ Q = 1/4, 1/5

∴ P ≥ Q

Question.9
I. 20P2 + 31P + 12 = 0
II. 21Q2 + 23Q + 6 = 0
Ans.b

I. 20P2 + 31P + 12 = 0

20P2 + 16P + 15P + 12 = 0

5P(4P + 3) + 4(4P + 3) = 0

∴ P = – 4/5, –3/4

II. 21Q2 + 23Q + 6 = 0

21Q2 + 14Q + 9Q + 6 = 0

7Q(3Q + 2) + 3(3Q + 2) = 0

(7Q + 3)(3Q + 2) = 0

⇒ Q = – 3/7, –2/3

∴ Q > P