@AHSAN  Your problems with integration are related to the fact that for some values ​​of the parameter  k  the denominator of the expression  Pre  is 0 and the function is not defined (its graph goes to infinity). Therefore, we can first find such values ​​of the parameter  k  and use the corresponding assumption on  k  when integrating:

restart; 
Pre := -(6*(x-1))*(k-1)*x/(((x-1)*k-x)^2*(k+1)); 
sol := solve(denom(Pre) = 0, x); 
solve({sol[1] >= 0, sol[1] <= 1}, k); 
int(Pre, x = 0 .. 1)  assuming k > 0;

@AHSAN  I don't understand why you integrate the expression  Pre ? What does it have to do with your problem  "Need to plot maximum relation by considering Pre_max alog vertical axis and k along x axis or horizontal axis" ?

@mmcdara 

`(Probability(Total)=50)` = (Probability(Total <=50)+Probability(Total >=50)) - 1;

@nm  The answer is not necessarily the greatest common divisor of the numbers 45 and 63. The answer can be any common divisor of this numbers.

@Aixleft math 

restart;
solutions := solve({v1^2 = 2*g*(h-h1), (1/2)*g*t^2 = h2, v1*t+(1/2)*g*t^2 = h1}, {h, t, v1}, explicit): 
selected_solutions := select(p->op(1,eval(t,p))<>-1, [solutions])[]; 

@Raafat  Maybe you want these points to be connected by line segments:

S:=seq(P(1,2,n), n=0..10):
A:=plots:-pointplot([S][1..4], color=red, symbol=solidcircle, symbolsize=15):
B:=plot([S][1..4], linestyle=3, color=blue):
plots:-display(A,B, view=[0..110,0..16]);

@GFY  Maple 2018.2 solves this system without any problems. Try to specify the list of unknown functions in the code:

Download question928_new.mw

@Saha In fact, you are plotting the same graph for one value of the parameter Nc. To plot graphs for different values, all calculations must be done again (solving the system, etc.). This can be easily done using a for-loop:

Download MSNL_new1.mw

@Alfred_F  What did you find difficult in my solution? It is based on direct calculation of triangle areas using coordinates of vertices of arbitrary parallelogram. Of course, you can solve it in another way, for example using commands of geometry package:

restart;
local D: 
with(geometry):
assume(a>0, c>0):
point(A,0,0): point(B,b,c): point(C,a+b,c): point(D,a,0): 
OnSegment(M,D,C,2/3): OnSegment(N,B,M,1): 
area_AND:=area(triangle(t,[A,N,D])):
area_BCM:=area(triangle(t1,[B,C,M])):
area_BCM/area_AND; 

@Ronan If you solve it in your head, it is easier to consider similar triangles  AFE and BCF  with a similarity coefficient of  1/2 . It follows that   BF = 2*FE . Also  S(BCE) = 1/2*S(ABCD) . So  S(EFC) = (1/2)*(1/3)*S(ABCD) = (1/6)*24 = 4  

@Carl Love Thanks for this. I didn't know this command before. Of course Iterator:-TopologicalSorts is the shortest way to generate permutations with restrictions.

@Susana30  You've already been shown one way. Here's another way, where we replace this region with a polygon for coloring:

restart;
f:=y->y^2-3: g:=y->y-y^2:
L1:=seq([g(y),y], y=-1..3/2, 0.1):
L2:=seq([f(y),y], y=3/2..-1, -0.1):
plots:-display(plot([L1], color=red, thickness=3),plot([L2], color=blue, thickness=3), plottools:-polygon([L1,L2], color="LightGreen"), scaling=constrained);

@minhthien2016  Why do you think this angle is 3*Pi/4?

Here is another solution to your problem if S(0,0,sqrt(2)/2). This solution uses the symmetry of this polyhedron about the ASC plane:

restart;
local D, O:
with(Student:-MultivariateCalculus):
A, B, C, D, S :=  [0,0,0], [1,0,0], [1,1,0], [0,1,0], [0,0,sqrt(2)/2]:
L:=Line(C,S);
O:=Projection(B, L);
v1:=convert(B-O,Vector);
v2:=convert(D-O,Vector);
Angle(v1,v2); # The answer

@Carl Love  Through the points M, M1, M2 (from my answer) we draw a plane intersecting the line of intersection of our half-planes at point O. Consider the quadrilateral OM1MM2. Obviously, the sum of the angles M1MM2 and M1OM2 is equal to Pi (we are looking for the angle M1OM2).

@JAMET  Insert  the line  s1:=polygon([A,M,N,P], color = "Pink", transparency = 0.7);  and add  s1  to the arguments of the  display  command.