**Question 1:**

Given that three roots of f(x) = x

^{4}+ax

^{2}+bx+c are 2, -3, and 5, what is the value of a+b+c?

[1] -79

[2] 79

[3] -80

[4] 80

We have to find out a + b + c

r = - 4

a + b + c = f(1) – 1 =

f(1) is 1 + a + b + c

So, we need to find out f(1) – 1

Let the 4^{th} root be r

Coefficient of x^{3} is - (Sum of the roots)

r = - 4

So, f(x) = (x – 2) (x + 3) (x + 4)(x – 5)

f(1) = (-1)×4×5×(-4) = 80a + b + c = f(1) – 1 =

**79. Option B****Question 2:**

If both a and b belong to the set (1, 2, 3, 4), then the number of equations of the form ax

^{2}+bx+1=0 having real roots is

[1] 10

[2] 7

[3] 6

[4] 12

For the equation to have real roots

b^{2} – 4a ≥ 0

b = 1, No equation exists

b = 2, a = 1. 1 equation exists

b = 3, a = 1 or 2. 2 equations exist

b = 4, a = 1 or 2 or 3 or 4. 4 equations exist

Total equations = 0 + 1 + 2 + 4 = **7. Option B**

**Question 3:**

Rakesh and Manish solve an equation. In solving Rakesh commits a mistake in constant term and finds the root 8 and 2. Manish commits a mistake in the coefficient of x and finds the roots -9 and -1. Find the correct roots.

[1] 9,1

[2] -9,1

[3] -8,-2

[4] None of these

Rakesh’s equation

Roots are

(x – 8)(x – 2) = 0

x^{2}– 10x + 16 = 0Manish’s equation

(x + 9)(x + 1) = 0

x^{2}+ 10x + 9 = 0Correct equation is x^{2} – 10x + 9 = 0

Roots are

**9, 1. Option A****Question 4:**

The number of quadratic equations which are unchanged by squaring their roots is

[1] 2

[2] 4

[3] 6

[4] None of these.

This would happen if and only if the roots and their squares are the same value.

Both roots are 1

One root is 0 and the other root is 1

The roots can be 0 or 1 or a combination of these.

So, valid equations will be formed when

Both roots are 0Both roots are 1

One root is 0 and the other root is 1

Number of equations = **3. Option D**

**Question 5:**

If the roots of px

^{2}+qx+2=0 are reciprocals of each other, then

[1] p = 0

[2] p = -2

[3] p= +2

[4] p = √2

If the roots are reciprocals of each other, product of the roots is 1

2/p = 1

p =

2/p = 1

p =

**2. Option C**- Algebra Practice Test 1
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