Direction: Study the following questions carefully and choose the right answer.
1
The quantity which must be added to (1 – x) (1 + x2) to obtain x3 is:
» Explain it
C
f(x) = (1 – x)(1 + x2) = 1 + x2 – x – x3

So,  2x3 – x2 + x – 1  is added to  1 + x2 – x – x to obtain x3.

Hence, option C is correct.

2
If the expression x3 + 3x2 + 4x + k has a factor x + 5, then what is the value of k?
» Explain it
B
Here, x + 5 is a factor.

So, x + 5 = 0 ⇒ x = –5

Now, x3 + 3x2 + 4x + k = (–5)3 + 3 × (–5)2 + 4 × (–5) + k

⇒ –125 + 75 – 20 + k = 0

Now, –70 + k = 0

So, k = 70.

Hence, option B is correct.

3
What is   (x2 + y2)(x – y) – (x –y)3  equal to?
x2y – xy2
» Explain it
B
  (x2 + y2)(x – y) – (x –y)3  
x2y – xy2

(x3 + xy2 – x2y – y3 – (x3 – y3 – 3x2y + 3xy2)  
x2y – xy2

(x3 + xy2 – x2y – y3 – x3 + y3 + 3x2y – 3xy2)  
x2y – xy2

2x2y – 2xy2  =  2(x2y – 2xy2)  = 2
x2y – xy2 x2y – xy2

Hence, option B is correct.

4
Consider the following statements

I. x + 3 is the factor of x3 + 2x2 + 3x + 8.

II. x – 2 is the factor of x3 + 2x2 + 3x + 8.


Which of the statements given above is/are correct?
» Explain it
D
Put x = –3 in equation x3 + 2x2 + 3x + 8

= (–3)3 + 2(–3)2 + 3(–3) + 8

= –10 ≠ 0

So, (x + 3) is not the factor of x3 + 2x2 + 3x + 8

Similarly, put x = 2 in above equation

= (2)3 + 2(2)2 + 3(2) + 8

= 30 ≠ 0

So, (x – 2) is also not the factor of x3 + 2x2 + 3x + 8.

Hence, option D is correct.

5
If  ( x2 +  1 )  =  17 , then what is ( x3 –   1 )  equal to?  
x2 4 x3
» Explain it
B
( x2 +  1 )  =  17
x2 4

⇒  x2 +  1  + 2 – 2 =  17   ⇒   ( x –  1 ) 2  + 2 =  17
x2 4 x   4

⇒   ( x –  1 ) 2  =  17  – 2   ⇒   ( x –  1 ) 2  =  9
x   4 x   4

⇒   ( x –  1 )  =  3
x 2

On cubing both side, we get
 
⇒   ( x –  1 ) 3  =  ( 3 ) 3
x   2  

⇒   x3 –  1  – 3 ×  1 .x  ( x –  1 )  =  27  
x3 x x 8

⇒   x3 –  1  =  27  + 3 ×  ( 3 )
x3 8 2

⇒  x3 –  1  =  27  +  9
x3 8 2

⇒    ( x3 –  1 )  =  63  
x3 8

Hence, option B is correct.