Algebra Quiz-1

p³+q³=(p+q)(p² -pq + q²)

or, 9=3(p² - pq + q²)

or, p² - pq + q² = 3

or, p² + q² = 3 + pq ....(1)

Again (p+q)² = 3² = 9

or p² + q² = 9 - 2pq........(2)

From (1) &(2), 3+pq =9-2pq

or, pq =2

Now p⁴ + q⁴ = (p² + q²)² - 2.p².q²

=(3+pq)² - 2(pq)²

=(3+2)² - 2.(2²) = 25 - 8 = 17(Ans)

 

Given, e +1/2e = 2

Multiplying through by 2e,

2e² + 1 = 4e …………………………………………………………………………………….(1)

Now,

8e³ + 1/e³ = (2e)³ +(1/e)³

(Using the identity a³ + b³ = (a+b) (a² - ab + b²) for a=2x and b=1/x)

= [(2e + 1/e)] [(2e)² - 2e.1/e + (1/e)²]

= [(2e² + 1)/e] [4e² - 2 + 1/e²]

= [(2e² + 1)/e] [(4e⁴ - 2e² + 1)/e²]

= [(2e² + 1)/e] [(2e²)² + 2. 2e².1 + 1² - 6e²)/e²]

= [(2e² + 1)/e] [(2e² + 1)² - 6e²)/e²]

Replacing 2e² + 1 by ex from (1),

= (4e/e) [(4e)² - 6e²)/e²] = 4(16e² - 6e²)/e²

= 4 (10e²/e²) = 4 . 10 = 40

Hence, the value of 8e³ + 1/e³ = 40

Explanation:

Put the value on the equation:4−6e−e2

If e=4

Then 4-6*4-4² = 4-24-16 = -36

If e=6

Then 4-6*6-6² = 4-36-36 = -68

If e=-3

Then 4-6*(-3)-(-3)² = 4+18-9 = 13

If e=7

Then 4-6*7-7² = 4-42-49 = -87

Hence, Maximum value of e = -3.

Here we are seeing a Pythagorean Triplet, where 5 units and 12 units can be the respective lengths of the sides of a triangle and 13 units is the length of hypotenuse.

In the equation 5√f+12√f=13√f, t

he value of √x should be
2.5√f + 12√f = 13√f, the value of √f should be 2.

Here we go,

.’. √f=2√f=2

By squaring both sides,

f = 4

OR

Put the value on the equation:

If f=4

Then 5² + 12² = 25 + 144 = 169

Hence, f=4.

Answer:

Explanation:

E = (f + ¹⁄_f)(f² - 1 + ¹⁄_f²)

E = (f + ¹⁄_f) *0

Answer: 

p(1/3)+pq(1/3)-r(1/3)=0
p/3+q/3-r/3=0
(p+q-r)/3=0
Multiply 3/3 on both sides,
(p+q-r)/3×3/3=0×3/3
3(p+q-r)/9=0
3(p+q-r)=0

So,the answer is 0.

Explanation: Let the number be x. Then,
7/23 of 11x/9 = 7/21 of 3x/8 + 12
7/23 * 11x/9 = 7/21 * 3x/8 + 12
77x/207 = 21x/168 + 12
x[(77/207) - (21/168)] = 12

x[ (12936-4347)/34776 ] = 12

x [ 8589/34776 ] = 12

x(0.25) = 12

x = 48

Explanation:

(2h/(1+(1/(1+(h/(1-h))))) = 1

(2h/(1+1/((1-h+h)/(1-h)))) = 1

(2h/(1+1/(1-h)) = 1

(2h / 1+1-h) = 1

(2h/2-h) = 1

2h = 2-h

3h=2

h=⅔

Explanation: We have
(1-1/7)(1-1/8)(1-1/9)…(1-1/100)
= (6/7)(7/8)(8/9)…(99/100)
= 6/100 = 3/50.