Machine Learning GATE Practice Set
Mixed GATE DA style questions on regression, classification, k-NN, naive Bayes, SVM, decision trees, evaluation, neural networks, clustering and PCA, with answers and explanations.
These questions mix every topic in this track, in the three GATE formats: MCQ (one correct option), MSQ (one or more correct options) and NAT (type a number). Each answer is recomputed by a program when the site is built. Try each question before opening the explanation.
Regression
[Linear regression] A least-squares line ŷ = w₀ + w₁x is fitted to data with x̄ = 5, ȳ = 20, Sxy = 60 and Sxx = 15. What is the predicted value at x = 8?
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Answer: 32w₁ = 60/15 = 4 and w₀ = 20 − 4 × 5 = 0, so ŷ(8) = 0 + 4 × 8 = 32.
[Ridge regression] For one centred feature with Sxy = 30 and Sxx = 10, which value of λ makes the ridge slope exactly half of the least-squares slope?
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Answer: 10The least-squares slope is 30/10 = 3. Ridge gives 30/(10 + λ) = 1.5 when 10 + λ = 20, so λ = 10.
[R²] For a simple linear regression, the correlation between x and y is r = −0.8. What is R²?
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Answer: 0.64For simple linear regression, R² = r² = 0.64. The sign of r only gives the direction of the slope.
Classification
[Logistic regression] A logistic regression model gives log-odds z = −2 + 0.5x₁ + x₂. What is P(y = 1) for x₁ = 2, x₂ = 1? Give the answer to three decimal places.
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Answer: 0.5z = −2 + 1 + 1 = 0, and σ(0) = 0.5. The point lies on the decision boundary.
[k-NN] Training points: (0, 0) class P, (1, 0) class P, (0, 1) class Q, (2, 1) class Q, (3, 1) class Q. Using 3-NN with Manhattan distance, what is the predicted class for (1, 1)?
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Answer: BManhattan distances from (1, 1): (0, 0) → 2, (1, 0) → 1, (0, 1) → 1, (2, 1) → 1, (3, 1) → 2. The three nearest are (1, 0) P, (0, 1) Q and (2, 1) Q, all at distance 1, so the vote is 2 Q to 1 P and the prediction is Q.
[Naive Bayes] In a spam filter, P(spam) = 0.4. The word "offer" appears in 50% of spam and 5% of non-spam emails, and the word "meeting" appears in 10% of spam and 40% of non-spam emails. Assuming the words are conditionally independent given the class, what is P(spam | both words appear)? Give the answer to three decimal places.
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Answer: 0.625Spam score = 0.4 × 0.5 × 0.1 = 0.02. Non-spam score = 0.6 × 0.05 × 0.4 = 0.012. P(spam | words) = 0.02 / (0.02 + 0.012) = 0.02 / 0.032 = 0.625.
[SVM] The maximum-margin hyperplane of a linear SVM is 2x₁ + x₂ − 4 = 0, scaled so that support vectors satisfy |2x₁ + x₂ − 4| = 1. Which of these training points can be a support vector?
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Answer: BThe values of 2x₁ + x₂ − 4 are 2, −1, −4 and 2. Only (1, 1) gives ±1, so only it lies on the margin.
Trees and evaluation
[Decision trees] A node with 8 positive and 8 negative samples is split into a left child with 6 positive and 2 negative and a right child with 2 positive and 6 negative. What is the decrease in Gini impurity?
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Answer: 0.125Gini(root) = 0.5. Each child has proportions 0.75 and 0.25, so Gini = 1 − 0.5625 − 0.0625 = 0.375. Weighted child Gini = 0.375, so the decrease is 0.5 − 0.375 = 0.125.
[Evaluation] A test for a condition has recall 0.9 and the classifier made 45 true positives and 15 false positives. How many false negatives did it make?
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Answer: 5Recall = TP / (TP + FN) = 0.9, so TP + FN = 45 / 0.9 = 50, and FN = 5. The false positives do not affect recall.
[Bias and variance] Which of the following typically increase bias? (Select all that apply.)
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Answer: A, B, DA shallower tree, more neighbours and a smaller C all make the model simpler or smoother, which raises bias. Adding higher-degree features makes the model more flexible, lowering bias and raising variance.
Neural networks and unsupervised learning
[Neural networks] A network has 784 inputs, one hidden layer of 128 neurons and 10 outputs, fully connected with biases. How many parameters does it have?
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Answer: 101770(784 × 128 + 128) + (128 × 10 + 10) = 100480 + 1290 = 101770.
[k-means] k-means with k = 2 on the points 0, 2, 4, 10, 12 starts with centroids 0 and 12. What is the WCSS after convergence?
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Answer: 10The first assignment gives {0, 2, 4} and {10, 12} (4 is closer to 0 than to 12), with centroids 2 and 11. Nothing changes after that. WCSS = (4 + 0 + 4) + (1 + 1) = 10.
[PCA] A covariance matrix has eigenvalues 8, 4, 2, 1 and 1. What fraction of the variance is lost when the data is projected onto the first two principal components? Give the answer to three decimal places.
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Answer: 0.25The total variance is 16, and the first two components keep 12 of it. The lost fraction is (2 + 1 + 1) / 16 = 0.25.