Engaa help plsss urgent!!

Someone please help me with this integration question.
It’s question 49 from Engaa 2018 section 1.

What are you stuck with. You could evaluate 1 integral and try and spot/verify the answer, or evaluate them both and simplify.

I evaluated the first integral and tried spotting the ans but couldn’t get to the right ans.
Now I evaluated both and simplified as you said and I got it. Ty

Id probably have done the first way and quickly whizzed through the options and verified. Really only the last 2 were candidates and the 5/2 -> 7 on the denominator so thats the one. But Im sure the "proper" way to do it is the second way.

If you can could you please help me with ECAA 2019 section 1 question 23.
I multiplied everything by x and got a quadratic. Then simplified it and got option C as my answer. But F is the correct ans.

If you can could you please help me with ECAA 2019 section 1 question 23.
I multiplied everything by x and got a quadratic. Then simplified it and got option C as my answer. But F is the correct ans.

if its economics remember that when you multiply through by x it flips the inequality when x<0. note when x=-large, the lhs is negative so not a soln. Id have treated it as an equality to get the critical points (and vertical asymptote), then either sketch or sub a couple of values in to work out the right regions.

Sorry for not posting a pic. I’ve tried posting pics before but for some reason I can’t. And Yh it’s the econ one.
but how do you know when x < 0, or do you just do it with both cases. x>0 and x<0. I still don’t know how to go about solving it tbh.

If you multiply throgh by x, then yes you have to consider both cases. Then when you solve for the values, you have to ensure they satisfy the appropriate domain. For this one, its trivially true when x>0 as all the terms on the left are >0 and therefore their sum is > 0. So you only need to consider when x<0. So multiplying through by x you have
quadratic < 0
and as the quadratic has a positive x^2 term its a "u", so finding the roots then the inequality will be satisfied in the interval between the roots. So the interval between the roots and x>0.

Note as soon as you make the observation its the interval between the roots and x>0, pretty much the only option is even without finding the roots (though its not hard).

omg Tysm got it now.
I’m doing these maths question to prepare for TMUA on Wednesday. Do you have my last min tips? I’ve done all past papers but I still don’t feel very confident

Based on these questions, Id put a bit of practice into is a bit of problem solving/answer spotting. So
* sketch graphs (here you could sketch the lhs as its x + 10 + 9/x, so roughly linear (large x), vertical asymptote at x=0, ...
* sub answers into the question to "guess" / validate your working, as indicated above
* ...
Also maybe use the mat mutliple choice for a bit of extra practice for tmua / watch their livestream videos / ... But the reality is youve not a lot of time, so good luck anyway.

if its economics remember that when you multiply through by x it flips the inequality when x<0. note when x=-large, the lhs is negative so not a soln. Id have treated it as an equality to get the critical points (and vertical asymptote), then either sketch or sub a couple of values in to work out the right regions.