## Elementary Algebra

#### Elementary Algebra

1. The degree f polynomial p(x) = x3 + 1 + 2x = 6x + 1/x is ?
1. 2
2. 4
3. 3
4. 5

1. x3 + 1 + 2x =6x + 1/x
x3 + 1 + 2x = (6x2 + 1)/x
(x3 + 1 + 2x)x = 6x2 + 1
x4 - 4x2 - 6x2 -1 =0
x4 - 4x2 + x - 1 =0

##### Correct Option: B

x3 + 1 + 2x =6x + 1/x
x3 + 1 + 2x = (6x2 + 1)/x
(x3 + 1 + 2x)x = 6x2 + 1
x4 - 4x2 - 6x2 -1 =0
x4 - 4x2 + x - 1 =0
Degree of polynomial is highest exponent degree term i.e.,4.

1. If x = 1 + √2, then what is the value of x4 - 4x3 + 4x2 ?
1. -1
2. 0
3. 1
4. 2

1. x = 1 + √2
∴ x4 - 4x3 + 4x2 = x2(x2 - 4x + 4)
= x2(x - 2)2
= (1 + √2)2(1 + √2 - 2)2

##### Correct Option: C

x = 1 + √2
∴ x4 - 4x3 + 4x2 = x2(x2 - 4x + 4)
= x2(x - 2)2
= (1 + √2)2(1 + √2 - 2)2
=(√2 + 1)2 (√2 - 1)2
=[(√2)2 - (1)2]2
=(2 - 1)2 =1

1. What is the solution of the equations x - y = 0.9 and 11(x + y)-1 = 2 ?
1. x = 3.2, y = 2.3
2. x = 1, y = 0.1
3. x = 2, y = 1.1
4. x = 1.2, y = 0.3

1. x - y = 0.9 ...(i)
and 11(x + y)-1=2
⇒ 11/ (x + y) = 2
⇒ 2(x + y) =11

##### Correct Option: A

x - y = 0.9 ...(i)
and 11(x + y)-1=2
⇒ 11/ (x + y) = 2
⇒ 2(x + y) =11
⇒ x + y = 11/2 ...(ii)
On solving Eqs.(i) and (ii),we get
x = 3.2
and y = 2.3

1. The product of two alternate odd integers exceeds three times the smaller by 12. What is the larger integer ?
1. 9
2. 7
3. 3
4. 5

1. Let two alternate odd integers odd integers be (2x+1) and (2x+5).
Then according to the question,
(2x + 1) (2x + 5) = 3(2x + 1) + 12
⇒ (2x + 1)(2x + 5 - 3) = 12
⇒ 2x2 + 3x - 5 = 0

##### Correct Option: B

Let two alternate odd integers odd integers be (2x+1) and (2x+5).
Then according to the question,
(2x + 1) (2x + 5) = 3(2x + 1) + 12
⇒ (2x + 1) (2x + 5 - 3) = 12
⇒ 2x2 + 3x - 5 = 0
On solving this quadratic equation,we get
x = 1 and x = -5/2
x = -5/2 is not a integer ∴ x = 1
Then, larger integer = 2x + 5 = 2 x 1 + 5 = 7

1. The sum of two numbers is 24 and their product is 143. The sum of their squares is
1. 296
2. 295
3. 290
4. 228

1. Let the two numbers be P and Q.
According to given question,
Sum of two numbers = 24
∴ P + Q = 24
Product of two numbers = 143
and, PQ = 143
As we know the formula,
∴ P 2 + Q2 = (P + Q)2 – 2PQ

##### Correct Option: C

Let the two numbers be P and Q.
According to given question,
Sum of two numbers = 24
∴ P + Q = 24 ...................... (1)
Product of two numbers = 143
and, PQ = 143 .......................... (2)
As we know the formula,
∴ P 2 + Q2 = (P + Q)2 – 2PQ
Put the value from the equation (1) and (2), We will get
∴ P 2 + Q2 = (24)2 – 2 × 143
∴ P 2 + Q2 = 576 – 286 = 290