# Tutor profile: Mohammed A.

## Questions

### Subject: Trigonometry

If $$ sin(2x - 5) = cos(5x+25), $$ find $$ x $$.

The rule says: $$Sin A = cos(90 - A)$$, therefore, we can rewrite the the equation in question as follow: $$ sin(2x - 5) = cos(5x+25) $$ $$ sin(2x - 5) = sin(90-(5x+25)) $$ Consequently, if the two $$sines$$ are equal, the angles must be equal. Therefore: $$(2x - 5) = (90-(5x+25))$$ $$2x-5=90-5x-25$$ $$2x-5=65-5x$$ $$7x=70$$ $$x=10$$

### Subject: Calculus

Prove that the sequence U(n) = (2n - 3)/(n+2) is: (a) monotone; (b) bounded;

(a) Let us find the difference U(n+1) - U(n): $$U(n+1) - U(n) = \frac {2(n+1) -3)} {(n+1+2)} - \frac {(2n-3)}{(n+2)}$$ $$= \frac {2n^2 + 4n - n -2 - 2n^2 - 6n + 3n +9}{(n + 3)(n + 2)}$$ $$= \frac {7}{(n + 3)(n + 2)}$$ This difference remains positive for all n, that says U(n+1) > U(n) for all n, hence, the sequence is monotone increasing. (b) Let's find U(1): $$U(1)=\frac{2-3}{1+2}= - \frac{1}{3}$$ From (a), the sequence increases and consequently $$U(n) \geqslant \frac{-1}{3}$$ for every n. Let us now check whether the sequence is bounded from above by evaluating the difference: $$A - U(n) = A- \frac{2n-3}{n+2}= \frac{An+2A-2n+3}{n+2} = \frac{n(A-2)+2A+3}{n+2}$$ For A=2 this difference is positive, that is U(n)<2 for all n.

### Subject: Algebra

Solve the following system by substitution: First equation: -3x + y= -9 Second equation: 5x + 4y = 32

Notice that both equations are not solved for 'x' and 'y'. As a result, the first thing to do is to solve the equation for either 'x' or 'y'. Step 1: Solve one of the equations for one of the variables (e.g., for 'y'). First equation: -3x + y = -9 -3x + y + 3x = -9 + 3x (Adding 3x to each side) y = -9 + 3x Step 2: Let's substitute the resulted equation from "step 1" into the other equation, and solve for 'x'. Second equation: 5x + 4y = 32 5x + 4(-9 + 3x) = 32 (Substitute -9 + 3x for y) 5x - 36 + 12x = 32 17x - 36 = 32 17x = 68 (Divide each side by 17) x=4 Step 3: Substitute x = 4 into one of the original equations, and solve for y. First equation: -3x + y= -9 -3(4) + y =-9 (Substitute 4 for x) -12 + y = -9 y = 3 (Add 12 to each side) Hence, our solution is (4,3)