Minimization Of Boolean Expression in Digital Logic : Practice MCQ Questions & Answers with Explanations

$\begin{aligned} & \cos ^{-1} \mathrm{a}-\cos ^{-1} \mathrm{~b}=\cos ^{-1}\left(a b+\sqrt{1-\mathrm{a}^{2}} \cdot \sqrt{1-\mathrm{b}^{2}}\right) \\\ \therefore & \cos ^{-1} x-\cos ^{-1} \frac{y}{3} \\\ = & \cos ^{-1}\left(\frac{x y}{3}+\sqrt{1-x^{2}} \cdot \sqrt{1-\frac{y^{2}}{9}}\right)=a \\\ \therefore & \frac{x y}{3}+\frac{\sqrt{1-x^{2}} \cdot \sqrt{9-y^{2}}}{3}=\cos \alpha \\\ & x y+\sqrt{1-x^{2}} \cdot \sqrt{9-y^{2}}=3 \cos \alpha \\\ & x y-3 \cos \alpha=-\sqrt{1-x^{2}} \cdot \sqrt{9-y^{2}} \end{aligned}$ squaring on both sides, we get $x^{2} y^{2}-6 x y \cos \alpha+9 \cos ^{2} \alpha=\left(1-x^{2}\right)\left(9-y^{2}\right)$ $x^{2} y^{2}-6 x y \cos \alpha+9 \cos ^{2} \alpha=9-y^{2}-9 x^{2}+x^{2} y^{2}$ i.e., $9 x^{2}-6 x y \cos \alpha+y^{2}=9 \sin ^{2} \alpha$

(A + B')(C' + D). Given Boolean function: A'B + CD' + A'B + CD' First simplify the given expression: A'B + A'B = A'B (redundant term) CD' + CD' = CD' (redundant term) So, the simplified function = A'B + CD' Now, we are asked to find the complement of this function: Let F = A'B + CD' Then the complement is: F' = (A'B + CD')' Apply De Morgans Law: (A'B + CD')' = (A'B)' · (CD')' (A'B)' = A + B' (CD')' = C' + D Therefore, F' = (A + B')(C' + D) Hence, the correct answer is:Option 3) (A + B')(C' + D)

correct matching of Boolean Algebra Laws with their corresponding Axioms is as follows: Absorption Law: a + (a b) = a Bounded Law: a + 1 = 1 Identity Law: a + 0 = a Distributive Law: a (b + c) = (a b) + (a c) Based on this, the correct matching is: A - IV (Absorption Law - a + (a b) = a) B - I (Bounded Law - a + 1 = 1) C - II (Identity Law - a + 0 = a) D - III (Distributive Law - a (b + c) = (a b) + (a c)) Additional Information Understanding Boolean Algebra is essential in digital electronics and computer science for designing and analyzing digital circuits. Boolean Algebra allows for the simplification of logic expressions, making it easier to implement and optimize digital systems. These fundamental laws and axioms provide the foundation for more complex theorems and applications in Boolean logic and digital design.