Quadratic Equations Test

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Quadratic Equations Test - 4

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Question 1
3 / -1

Find the maximum value of the expression

A
15/2

B
1

C
4/15

D
1/3

Solution

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.
Question 2
3 / -1

Find the minimum value of the expression (p +1/p); p > 0.

A
1

B
0

C
2

D
Depends upon the value of p

Solution

The minimum value of (p + 1/p) is at p = 1. The value is 2.
Question 3
3 / -1

The expression a² + ab + b² is _________for a < 0, b < 0

A
≠ 0

B
<0

C
> 0

D
= 0

Solution

For a, b negative the given expression will always be positive since, a², b² and ab are all positive.
Question 4
3 / -1

Find the roots of the quadratic equation: x2 + 2x - 15 = 0?

A
-5, 3

B
3, 5

C
-3, 5

D
-3, -5

Solution

x² + 5x - 3x - 15 = 0
x(x + 5) - 3(x + 5) = 0
(x - 3)(x + 5) = 0
=> x = 3 or x = -5.
Question 5
3 / -1

A
ab = cd

B
ad = bc

C
ad = √bc

D
ab = √cd

Solution

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.
Question 6
3 / -1

Two numbers a and b are such that the quadratic equation ax² + 3x + 2b = 0 has - 6 as the sum and the product of the roots. Find a + b.

A
2

B
-1

C
1

D
-2

Solution
Question 7
3 / -1

Find the value of the expression

A

B

C

D

Solution

Solving quadratically, we have option (b) as the root of this equation.
Question 8
3 / -1

If the roots of the equation (a² + b²) x² - 2b(a + c) x + (b² + c²) = 0 are equal then a, b, c, are in

A
AP

B
GP

C
HP

D
Cannot be said

Solution

Solve by assuming values of a, b, and c in AP, GP and HP to check which satisfies the condition.
Question 9
3 / -1

If x² + ax + b leaves the same remainder 5 when divided by x - 1 or x + 1 then the values of a and b are respectively

A
0 and 4

B
3and0

C
0 and 3

D
4and0

Solution
Question 10
3 / -1

Read the data given below and it solve the questions based on.
If α and β? are roots of the equation x² + x - 7 = 0 then.
Find α² + β²

A
10

B
15

C
5

D
18

Solution
Question 11
3 / -1

For what values o f c in the equation 2x² - (c³ + 8c - l)x + c² - 4c = 0 the roots of the equation would be opposite in signs?

A
c € (0 ,4 )

B
c € ( - 4 , 0 )

C
c € (0 ,3 )

D
c € ( - 4 , 4 )

Solution

For the roots to be opposite in sign, the product should be negative.
(c² - 4x)/2 < 0 ⇒ 0 < c < 4 .
Question 12
3 / -1

The expression x² + kx + 9 becomes positive for what values of k (given that x is real)?

A
k < 6

B
k > 6

C
|K|<6

D
|k|> 6

Solution

The roots should be imaginary for the expression to be positive i.e.
k² - 36 < 0
thus - 6 < k < 6 or |k| < 6.
Question 13
3 / -1

In the Maths Olympiad of 2020 at Animal Planet, two representatives from the donkey’s side, while solving a quadratic equation, committed the following mistakes: (i) One of them made a mistake in the constant term and got the roots as 5 and 9. (ii) Another one committed an error in the coefficient of x and he got the roots as 12 and 4.But in the meantime, they realised that they are wrong and they managed to get it right jointly. Find the quadratic equation.

A
x² + 4x + 14 = 0

B
2x² + lx - 24 = 0

C
x² - 14x + 48 = 0

D
3x² - 17x + 52 = 0

Solution

From (i) we have sum o f roots = 1 4 and from (ii) we have product of roots = 48.
Option (c) is correct.
Question 14
3 / -1

For what value of b and c would the equation x² + bx + c = 0 have roots equal to b and c.

A
(0,0)

B
(1,-2)

C
(1,2)

D
Both (a) and (b)

Solution

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 .
Question 15
3 / -1

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

Solution

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