Discrete Mathematics Previous Year Questions

Explanation

legend() adds a legend box that labels each line or marker on the plot, using the label= values given when plotting.

Explanation

Huffman merges: 0.1 + 0.15 = 0.25, then 0.2 + 0.25 = 0.45, then 0.25 + 0.3 = 0.55, then 1. The code lengths are A 2, D 2, E 2, B 3, C 3, totalling 2 + 2 + 2 + 3 + 3 = 12 bits.

Explanation

Only 90% of the code was tested, and the success probability is 0.9. So reliability is at most 0.9 × 0.9 = 0.81.

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x²/(1+x²) is 0 at x = 0 and approaches 1 as x grows, but never reaches 1 because x² < 1 + x². So the range is [0, 1).

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An embedded system is electronics and software built into a product for a dedicated function, such as a washing machine or microwave controller, rather than a general-purpose computer.

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Curriculum is shaped by learners' needs and abilities and by social aspects. Gender roles should not determine curriculum, which must be gender-neutral.

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The range is the set of second elements (outputs) of the ordered pairs: {8, 2, −4}.

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Emitter-Coupled Logic (ECL) is the fastest logic family, because its transistors never saturate, but it uses a lot of power.

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Declaring the formal (reference) parameters as const lets the function read the caller's data without being able to modify it.

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DCTL stands for Direct-Coupled Transistor Logic, an early logic family in which transistors are connected directly without coupling resistors.

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A POS expression is a product of maxterms (OR terms), and an SOP expression is a sum of minterms (AND terms).

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In f(x, y), x and y are spatial coordinates, and the amplitude of f at any point is the intensity (grey level) of the image there.

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Every row of the second matrix is 2 times the corresponding row of the first. Multiplying all three rows of a 3×3 matrix by 2 multiplies the determinant by 2³ = 8, so the new determinant is 8 × (−p) = −8p.

Explanation

The chance that all four shots miss is 0.4 × 0.5 × 0.6 × 0.8 = 0.096. So P(at least one hit) = 1 − 0.096 = 0.904.

Explanation

A dot-matrix printer forms characters by pins striking an inked ribbon against the paper, so it is an impact printer. Inkjet, laser and thermal printers are non-impact printers.

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Path costs: V1→V2→V3→V5 = 130, V1→V4→V3→V5 = 120, V1→V4→V5 = 130 and V1→V5 = 120. The minimum is 120.

Explanation

By Bayes' theorem, P(M | red) = (½ × 5/9) / (½ × 3/5 + ½ × 5/9) = (5/9) / (52/45) = 25/52.

Explanation

With two points, Lagrange's formula gives f(x) = 3(x−4)/(2−4) + 5(x−2)/(4−2) = −1.5x + 6 + 2.5x − 5 = x + 1. Check: f(2) = 3 and f(4) = 5.

Explanation

The probabilities must add to 1: 9K + 10K² = 1, so 10K² + 9K − 1 = 0, i.e. (10K − 1)(K + 1) = 0. Since K > 0, K = 1/10.

Explanation

With replacement the draws are independent: P = (8/18) × (8/18) = (4/9)² = 16/81.

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Ctrl + T applies a hanging indent in MS Word (Ctrl + Shift + T removes it). Undo is Ctrl + Z, redo is Ctrl + Y and justify is Ctrl + J.

Explanation

For a single trial (Bernoulli), variance = p × q = 0.7 × 0.3 = 0.21.

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P(defective | red) = P(red and defective) / P(red) = 0.04 / 0.20 = 0.2.

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By Bayes' theorem, P(P | standard) = (0.6 × 0.8) / (0.6 × 0.8 + 0.4 × 0.9) = 0.48 / 0.84 = 4/7.

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y = x² is increasing on [1, 6], so its minimum is at x = 1: y = 1.

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A dot-matrix printer forms characters by pins striking an inked ribbon against paper, so it is an impact printer. Inkjet, laser and thermal printers are non-impact.

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Multiplexers can implement any logic function directly, which simplifies design, reduces the IC package count and removes the need to minimise the expression. All three are advantages.

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Logical cohesion groups elements that do logically similar operations, such as all error handling or all data input routines. B describes functional cohesion and C procedural cohesion.

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P(no one solves) = (2/3)(3/4)(4/5) = 2/5. So P(solved) = 1 − 2/5 = 3/5 = 0.6.

Explanation

Lagrange's formula with two points: f(3) = 4·(3−5)/(1−5) + 6·(3−1)/(5−1) = 4×0.5 + 6×0.5 = 5.

Explanation

P(defective) = 0.25×0.05 + 0.35×0.04 + 0.40×0.02 = 0.0125 + 0.014 + 0.008 = 0.0345. By Bayes' theorem, P(Q | defective) = 0.014/0.0345 ≈ 0.41.

Explanation

A matrix is usually called sparse when most of its elements are zero, that is, when more than half of the m × n elements are zero.

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The WWW is a system of linked hypertext documents and resources that are reached over the internet with HTTP through browsers.

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A scanner reads printed documents or photos and converts them into digital images.

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In a skew-symmetric matrix aᵢⱼ = −aⱼᵢ, so the diagonal is all zeros. Option B has a12 = 1, a21 = −1; a13 = 2, a31 = −2; a23 = −4, a32 = 4.

Explanation

Matrix × (1, 1, 0) = (3 − 1, 7 − 5, 6 − 6) = (2, 2, 0) = 2 × (1, 1, 0), so the eigenvalue is 2.

Explanation

2A = [2 8; 6 10] and 3B = [9 6; 6 3], so 2A − 3B = [2−9 8−6; 6−6 10−3] = [−7 2; 0 7].

Explanation

A Hermitian matrix equals its conjugate transpose, so aᵢⱼ = conj(aⱼᵢ). aᵢⱼ = aⱼᵢ is the rule for a symmetric matrix, and the minus forms are for skew types.

Explanation

Row 1 × columns: 1−1+4 = 4, 2+0+6 = 8, 1+3+2 = 6. Row 2: 3, 8, 9. Row 3: 2−3+6 = 5, 4+0+9 = 13, 2+9+3 = 14. This matches option B.

Explanation

The conjugate replaces every i by -i in each element: i becomes -i, 1+i becomes 1-i, 3-2i becomes 3+2i, 2-i becomes 2+i, -3i becomes 3i, 1-i becomes 1+i and 2i becomes -2i. Real entries 4 and 5 stay the same.

Explanation

Interchanging rows and columns gives the transpose, and det(Aᵀ) = det(A), so the determinant value is unchanged.

Explanation

A Hermitian matrix equals its conjugate transpose: diagonal entries are real and a_ji is the conjugate of a_ij. In A the diagonal is 1 and 5 and the off-diagonal entries 2+3i and 2-3i are conjugates.

Explanation

P(even) = P(H) x 4/9 + P(T) x 2/5 = (2/3)(4/9) + (1/3)(2/5) = 8/27 + 2/15 ≈ 0.296 + 0.133 = 0.43.

Explanation

C passes arguments by value, so to change the caller's variable you pass its address (a pointer) and modify it through *ptr inside the function.

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The cut tool removes the selected part of an image (placing it on the clipboard). Resolution, resizing and colour effects are done with other tools.

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Squaring gives row 3 = [2, 0, 1] and the middle entry (-1)² = 1. In general A^n has row 3 = [n, 0, 1] and middle entry (-1)^n, so A⁶ has row 3 = [6, 0, 1] and middle entry 1.

Explanation

For a 2x2 matrix the characteristic equation is λ² - (trace)λ + det = 0. Trace = 1 + 5 = 6 and det = 5 - 12 = -7, so λ² - 6λ - 7 = 0.

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A die has 3 even faces (2, 4, 6) out of 6, so the probability is 3/6 = 1/2.

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The Poisson distribution is discrete; it counts events in a fixed interval. Normal, log-normal and exponential distributions are continuous.

Explanation

Newton's divided difference formula works for unequally spaced xᵢ. Newton's forward/backward and Gauss formulas need equally spaced points.