Test: Quadratic Equations- 4 - CAT MCQ

For the given expression to be a maximum, the denominator should be minimized. (Since, the function in the denominator has imaginary roots and is always positive), x2 + 5x + 10 will be minimized at x = -2.5 and its minimum values at x = -2.5 is 3.75.
Hence, required answer = 1/3.75 =4/15.

GIVEN:

 f(x) = x² - 14x + 45

FORMULA USED:

Sum of roots = (-b/a) and Product of roots = c/a for f(x) = ax² + bx + c 

CALCULATION:

f(x) = x² - 14x + 45

⇒ a = 1, b = -14, c = 45

⇒ Sum of roots(P + Q) = (-b/a)

⇒ 14

⇒ Product of roots (PQ) = c/a

⇒ 45

⇒ (1/P + 1/Q) = (P + Q)/PQ

⇒ 14/45

∴  (1/P + 1/Q) = 14/45

For a, b negative the given expression will always be positive since, a², b² and ab are all positive.

Solve this by assuming each option to be true and then check whether the given expression has equal roots for the option under check.

Thus, if we check for option (b). ad = be.
We assume a = 6, d = 4 b = 12 c = 2 (any set o f values that satisfies ad = bc).
Then (a² + b²) x² - 2(ac + bc) x + (c² + d²) = 0

180 x²- 120 x +20 = 0.
We can see that this has equal roots. Thus, option (b) is a possible answer. The same way if we check for a, e and d we see that none of them gives us equal roots and can be rejected.

Solving quadratically, we have option (b) as the root of this equation.

Solve by assuming values of a, b, and c in AP, GP and HP to check which satisfies the condition.

For the roots to be opposite in sign, the product should be negative.

(c² - 4x)/2 < 0 ⇒ 0 < c < 4 .

The roots should be imaginary for the expression to be positive i.e.

k² - 36 < 0

thus - 6 < k < 6 or |k| < 6.

From (i) we have sum o f roots = 1 4 and from (ii) we have product of roots = 48.
Option (c) is correct.

Solve using options. It can be seen that at b = 0 and c = 0 the condition is satisfied. It is also satisfied at b = 1 and c = - 2 .

Solve using options, If the car’s speed is 48 kmph, the bus’s speed would be 32 kmph. The car would take 4 hours and the bus 6 hours.A journey between Mumbai and Pune (192 km apart) takes two hours less by a car than by a truck. Determine the average speed of the car if the average speed of the truck is 16 km/h less than the car. (a) 48 km/h (b) 64 km/h (c) 16 km/h (d) 24 km/h