Theory of Equations Test

Result

Theory of Equations Test - 1

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Question 1
1 / -0

If b > a, then the equation (x– a) (x– b) – 1 = 0 has

A
both roots in (b, ∞)

B
both roots in [a, b]

C
one root in (- ∞, a) and other in (b, ∞).

D
both roots in [– ∞, a]

Solution

Your answer is incorrect.

Let, f (x) = (x - a) (x - b) - 1

⇒f (a) = -1 and f (b) = -1.

Also, The coefficient x² = 1 > 0

Hence a and b both lie between the roots of the equation f (x) = 0.

∴ The equation ( x - a ) ( x - b) - 1 = 0 has one root

In (- ∞, a) and other in ( b, ∞ ) [ ∵ b > a ]
Question 2
1 / -0

If 4ac > b² and a + c > b for real numbers a, b and c, then which of the following is true ?

A
a + b + c >0

B
a > 0

C
4a +c > 2b

D
c > 0

Solution

Let f (x) = ax² + bx + c

its discriminant D = b² - 4ac < 0

and, f (- 1) = a - b + c > 0

So, f (x) > 0 for all x ∈ R

f (0) > 0

⇒ a + b+ c > 0

f (- 2) > 0

⇒ 4a - 2b + c > 0
Question 3
1 / -0

If x + y + z = 5 and xy + yz + zx = 3 then find the greatest value of 3x.

A
17

B
22

C
13

D
16

Solution

Y + z = 5 - x and (y + z) x + yz = 3

yz = 3 - x (5 - x) = 3 - 5x + x²

Now, (y - z)² ≥ 0

⇒ ( 5 - x² ) - 4 (3 - 5x + x²) ≥ 0

3x² - 10x - 13 ≤ 0

( x + 1) (3x - 13) ≤ 0

⇒ - 1 ≤ x ≤ 13/3

⇒ - 3 ≤ 3x ≤ 13