Various Topics Doubt, Please Solve Exam Near

1,3) put z = x+iy

2) use sinxcosx = (sinx+cosx)² -12
then you get a quadratic in sinx + cosx i.e √2sin(x+pi4) solve this normally by setting x+pi/4 = θ

4) n(s) = ⁴⁰C₅ i.e just selection of 5 cards since there is only 1 way to arrange each selection.

n(E) = ²³C₂ * ¹⁶C₂ i.e. selecting 2 cards each from the left and right of 24

=> 23C_{2 * 16C2} 40C₅

5) take general point on the curve as (3t³4,t)
now differentiate the curve to get slope of normal and write equation of normal since you have a point and slope. (0,1) should satisfy it from this we get ,
27t⁵ + 16t -16 =0
differentiate it to get , (27*5) t^4 +16 which is always positive hence the curve is increasing so it has only 1 root .
hence the answer is 1.

1. I was wondering if there was any method without going to co ordinate systems
which gives answer in |z-c|=a form of circle

by the way the co ordinate system worked

2. i got
Sin (∂) = (1/√2)-1
Now For what Value of Sin ∂ it satisfies ?
acc to calculator:
-17.03124846...

how am i supposed to calculate it or solve it on the exam hall ?

5.I understood your Answer BUT
Your general point 3t³4 and t Should satisfy the equation
of the curve
just like
Parabola , at² , 2at Satisfies y² = 4ax
SO if you ask my opinion , this Should be something else !

thanks for the rest of the answers !

2) you only need to find number of solutions not the solns. so just draw the graph of sinx and check at how many points the line y= 1/√2 -1 intersects it in the given domain.