edited by
1,433 views
3 votes
3 votes

If $a,b,c$ and $d$ satisfy the equations

  • $a+7b+3c+5d =16$
  • $8a+4b+6c+2d = -16$
  • $2a+6b+4c+8d = 16$
  • $5a+3b+7c+d= -16$

Then $(a+d)(b+c)$ equals

  1. $-4$
  2. $0$
  3. $16$
  4. $-16$
edited by

4 Answers

11 votes
11 votes
$\color{maroon}{a+7b+3c+5d=16}$   

$\color{maroon}{2a+6b+4c+8d=16}$  

Or, $(a+a)+(7b-b)+(3c+c)+ (5d+3d) = 16$

Or, $(a+7b+3c+5d)+(a-b+c+3d) = 16$

Or, $a-b+c+3d = 0 \qquad\qquad ∵\big[\color{blue}{ a+7b+3c+5d=16}\big]$ ------------- 1)

$\color{maroon}{8a+4b+6c+2d = -16}$

$\color{maroon}{5a+3b+7c+d = -16}$

Or, $(5a+3a)+(3b+b)+(7c-c)+(d+d) = -16$

Or, $(5a+3b+7c+d)+(3a+b-c+d) = -16$

Or, $3a+b-c+d = 0 \qquad \qquad \big[∵\color{blue}{5a+3b+7c+d = -16}\big]$ ------------- 2)

Adding 1) & 2)

$3a+b-c+d=0\\a-b+c+3d=0$

Or, $4a +4d = 0$

Or, $a+d = 0$

Now, we've to find $\color{maroon}{(a+d)(b+c) = ?}$

$(a+d)(b+c) = (0)\times(b+c) \qquad \qquad ∵ \big[\color{blue}{a+d=0}\big]$

∴$\color{red}{(a+d)(b+c) = 0}$

$\color{green}{\text{Hence, option }} \color{orange}{ B)} \color{green}{ \text{ is the right answer}}$.
edited by
6 votes
6 votes
$a+7b+3c+5d=16 ..............eq.1$

$8a+4b+6c+2d=−16.................eq.2$

$2a+6b+4c+8d=16...................eq.3$

$5a+3b+7c+d=−16....................eq.4$

add $eq.2$ and $eq.3$, we get,

$10a + 10b + 10c + 10d = 0$

$a + b + c + d = 0.....................eq.5$

add $eq.1$ and $eq.4$, we get,

$6a + 10b + 10c + 6d = 0$

$6(a+d) + 10(b+c) = 0$

on putting the value of $(a+d)$ from eq.5, we get

$-6(b+c) + 10(b+c) = 0$

$b+c = 0$

$a+d = 0$ is also 0

So, $(a+d)*(b+c) = 0 $
0 votes
0 votes
a+7b+3c+5d=16....(1)

8a+4b+6c+2d=-16......(2)

2a+6b+4c+8d=16......(3)

5a+3b+7c+d=-16......(4)

add (1) and (4) and add(2) and (3)

u will get

a+d=-(b+c)

and a+d=-10/6(b+c)

subtract both u will get (b+c)=0

so (a+d)(b+c)=0

Related questions

0 votes
0 votes
1 answer
1
jjayantamahata asked Mar 30, 2018
514 views
Let $$f(x,y) = \begin{cases} \dfrac{x^2y}{x^4+y^2}, & \text{ if } (x,y) \neq (0,0) \\ 0 & \text{ if } (x,y) = (0,0)\end{cases}$$Then $\displaystyle{\lim_{(x,y)...
0 votes
0 votes
1 answer
2
15 votes
15 votes
3 answers
4
Shreya Roy asked Apr 5, 2017
2,307 views
Consider all possible trees with $n$ nodes. Let $k$ be the number of nodes with degree greater than $1$ in a given tree. What is the maximum possible value of $k$?