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Direction: Study the following questions carefully and choose the right answer.
1
If x = 2 then the value of x3 + 27x2 + 243x + 631 is:
» Explain it
C
Given equation,

f(x) = x3 + 27x2 + 243x + 631

⇒ x(x2 + 27x+ 243) + 631

Now, put the value of x = 2

⇒ 2(22 + 27 × 2 + 243) + 631

⇒ 2 (4 + 54 + 243) + 631

⇒ 2(301) + 631 = 602 + 631 = 1233.

Hence, option C is correct.

2
If x  1  then (x + 1) equals to :
2 + 1

» Explain it
D
On multiplying the numerator and denominator by the conjugate of the existing denominator, we get,

x =  1  × 
2  – 1
2  + 1
2  – 1

⇒ x =
2 – 1
  ⇒  x = 
2  – 1
2 – 1

Then (x + 1) = 
2  – 1 + 1 = 
2

Hence, option D is correct.

3
if x +  1  = 99, find the value of  100x  
x 2x+ 102x + 2

» Explain it
C
x +  1  = 99
x

∴   100x  =  100x
2x2 + 102x + 2 2x2 + 2 + 102x

On dividing by x,

=  
 
100x
 = 
 
100  
2x +  2  + 102
x
2 ( x +  1 )  + 102
x

=   100  =  100  =  1
2 × 99 + 102 300 3

Hence, option C is correct.

4
The expression x4 – 2x2 + k will be a perfect square when the value of k is
» Explain it
B
Method I:

(a – b)2 = a2 –2ab + b2

x4 – 2x2 + k = (x2)2 – 2.x2.1 + k 

⇒   (12)2 – 2.(1)2.1 + k = 0

⇒  1 – 2 + k  = 0

⇒ – 1 + k = 0

For a perfect square, 

 k = 1.

Method II:

Let's assume x2 = m, therefore the given eq. will be:

m2 – 2m + k which is a quadratic equation (ax2 + bx + c).

Now we know that a quadratic eqn. is a perfect square if its discriminant (b2 – 4ac) is equal to zero.

In the eq. a = 1, b = – 2, c = k

∴    (– 2)2 – 4 (1).k = 0

–4k = –4

∴ k = 1

Hence, option B is correct.

5
If x +  1  = 2, find the value of 8x3 +  1 .
2x x3

» Explain it
C
x +  1  = 2
2x

Multiplying both sides by 2

⇒  2x +  2  = 4
2x

⇒ 2x +  1  = 4
x

On Cubing both sides, we get

⇒   ( 2x +  1 ) 3  =  (4)3
x  

⇒  8x3 +  1  + 3 × 2x ×  1 ( 2x +  1 )  = 64
x3 x x

⇒   8x3 +  1  + 6 × 4 = 64
x3

⇒  8x3 +  1  = 64 – 24 = 40.
x3

Hence, option C is corrrect.