Cognizant Algebra Quiz-1

x³+y³=(x+y)(x²-xy +y²)

or, 9=3(x²-xy+y²)

or, x²-xy +y²=3

or, x²+y²=3+xy ....(1)

Again (x+y)²=3²=9

or x²+y²=9-2xy........(2)

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

or, xy =2

Now x⁴+y⁴=(x²+y²)²-2.x².y²

=(3+xy)²-2(xy)²

=(3+2)²-2.(2²)=25-8=17(Ans)

 

Answer: (A)

Given, x +1/2x = 2

Multiplying through by 2x,

2x² + 1 = 4x …………………………………………………………………………………….(1)

Now,

8x³ + 1/x³ = (2x)³ +(1/x)³

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

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

= [(2x² + 1)/x] [4x² - 2 + 1/x²]

= [(2x² + 1)/x] [(4x⁴ - 2x² + 1)/x²]

= [(2x² + 1)/x] [(2x²)² + 2. 2x².1 + 1² - 6x²)/x²]

= [(2x² + 1)/x] [(2x² + 1)² - 6x²)/x²]

Replacing 2x² + 1 by 4x from (1),

= (4x/x) [(4x)² - 6x²)/x²] = 4(16x² - 6x²)/x²

= 4 (10x²/x²) = 4 . 10 = 40

Hence, the value of 8x³ + 1/x³ = 40

Answer: (C)
Explanation: for solving this particular question you will have to use hit and trial from the options given.

Case 1:-

taking x = 4, therefore
4 - 6(4) - (4)²
= 4 - 24 - 16 = -36

Case 2:-

taking x = 6, therefore
4 - 6(6) - (6)²
= 4 - 36 - 36 = - 68

Similarly taking x = 7 will also give a negative term but considering x = -3
4 - 6(-3) - (-3)²
= 4 + 18 - 9 = 13

Therefore -3 will be the correct option.

Answer: (D)


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 are the length of the hypotenuse.

In equation5^√x + 12^√x = 13^√x,the value of √x should be 2.

.’. √x=2

By squaring both sides, x = 4

Answer:

E = \frac{1}{\sqrt2 + 1} - \sqrt2 + 1 = \frac{1 - (\sqrt2 + 1)^{2}}{\sqrt2 + 1}

So E = \frac{1-(2+2\sqrt2 + 1)}{\sqrt2 + 1} = \frac{-2 - 2\sqrt2}{\sqrt2 + 1} = -2

answer is -2

Explanation:

E = (b + ¹⁄_b)(b² - 1 + ¹⁄_b²)

as b² + ¹⁄_b² = 1 therefore

E = (b + ¹⁄_b)(1 - 1)
E = (b + ¹⁄_b)(0)
E = 0

Answer: 

x(1/3)+y(1/3)-z(1/3)=0
=> x/3+y/3=z/3
=>(x+y)/3=z/3
=>x+y = z  [Multiply 3 on both sides]
=>(x+y-z) = 0
=>(x+y-z)³=0

Now, 3xyz.(x + y – z)³ = 3xyz.(0) = 0

So,the answer is 0.

Answer: (A)
Explanation: Let the number be x. Then,
4/15 of 5x/7 = 4/9 of 2x/5 + 8
4/15 * 5x/7 = 4/9 * 2x/5 + 8
4x/21 = 8x/45 + 8
x/21 = 2x/45 + 2
x/21 = (2x + 90)/45
45x = 21(2x + 90)
45x = 42x + 90*21
3x = 90*21
x = 30*21 = 630.

half of x = 315

Answer: (B)
Explanation:
We have,\frac{2x}{1+\frac{1}{1+ \frac{x}{1 - x}}} = 1
\frac{2x}{1 + \frac{1}{\frac{1}{1-x}}} = 1
\frac{2x}{1 + \frac{1 -x}{1}} = 1
\frac{2x}{1+1-x} = 1
2x = 2 - x
3x = 2
x = \frac{2}{3}

Answer: (B)
Explanation: We have
(1-1/3)(1-1/4)(1-1/5)…(1-1/100)
= (2/3)(3/4)(4/5)…(99/100)
= 2/100 = 1/50.