Mbzuai Entry Exam Sample Questions !!top!! May 2026

Given vectors ( u = (1, 2, -1) ), ( v = (0, 1, 3) ). Compute the projection of ( u ) onto ( v ).

If ( A ) and ( B ) are square invertible matrices, then ( (A + B)^-1 = A^-1 + B^-1 ). Explain briefly. Section 2: Calculus & Optimization (25%) Question 4: Find the gradient ( \nabla f(x,y) ) of ( f(x,y) = \ln(1 + e^xy) ). Then compute the directional derivative at ( (1,0) ) in the direction of ( (1,1) ). mbzuai entry exam sample questions

For ( f(x) = \frac12 |Ax - b|^2 ), derive the closed-form solution for ( x ) that minimizes ( f ). Section 3: Probability & Statistics (20%) Question 7: You have two dice: one fair, one loaded (shows 6 with probability ( 1/3 ), others each ( 2/15 )). You pick a die at random (50% chance each), roll it once, and get a 6. What is the probability it was the loaded die? Given vectors ( u = (1, 2, -1) ), ( v = (0, 1, 3) )