## Elementary Algebra

#### Elementary Algebra

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

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. 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 - 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. 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. 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