GATE CSE 2010 | Question: 8

Question

$P$ is a $16$-bit signed integer. The $2$'s complement representation of $P$ is $(F87B)_{16}$. The $2$'s complement representation of $8\times P$ is

• A. $(C3D8)_{16}$
• B. $(187B)_{16}$
• C. $(F878)_{16}$
• D. $(987B)_{16}$

Comments

Is there any overflow

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just Left shift is needed 

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option (A) is correct

first convt. into binary and then add 000 zero's at last then again convt. into hexa decimal.

Answer 1

Multiplication can be directly carried in $2$’s complement form.

$\textsf{F87B} = 1111 1000 0111 1011$ can be left shifted $3$ times to give $8P = 1100 0011 1101 1000 = \textsf{C3D8}.$

Alternate Method:

As MSB in $\textsf{(F87B)}$ is $1$, $P$ is a negative number.

So, $P = -1 \times 2$'s complement of $\textsf{(F87B)}$

$ P = -1 \times (0785) = -1 \times (0000\; 0111 \;1000 \;0101)$

$8 \ast P = -1 \times  (0011 \;1100 \;0010 \;1000)\;\;\; (P$ in binary left shifted $3$ times$)$

In $2$'s complement representation , this equals, $1100 0011 1101 1000 = \textsf{C3D8}$

Correct Answer: $A$

Comments

while performing Left shift Trailing bits will be filled with zeros, on right shift leading bits will filled with zeros.

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on right shift leading bits will filled with zeros – this happens only when number is unsigned

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"Multiplication can be directly carried in 2’s complement form."

Valid only for numbers multiplied by the powers of 2. i.e; 2,4,8,16, and so on.
Like, here the given 2's complement number is being multiplied by 2^3.

Correct me if i am wrong.

Answer 2

$P=(F87B)_{16}=(1111|1000|0111|1011)_{2}:$ 2's compliment representation

What is this number ?

$(0000|0111|1000|0100)_{2}+1=(0000|0111|1000|0101)_{2}=1925$

$So, P\ is =-1925$

$-1925\times 8=-15400$

Find 2's compliment representation of $-15400$

$+15400=(0011|1100|0010|1000)_{2}$

$(1100|0011|1101|0111)_{2}+1=(1100|0011|1101|1000)_{2}=(C3D8)_{16}$

 

Correct Answer is (A)

Answer 3

Other way to see.

Answer - A

Answer 4

left shift = x2 

so we will have to do 3 left shifts 
convert the number to binary and then left shift 3 times , that will be your ans