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Calculus and Optimization Practice Questions for GATE DA

GATE DA Calculus and Optimization: 2 units and 5 chapters, weightage from 13 previous year questions across 3 papers, a study order by exam weight and 511 pra

A question from this chapter

Question 1

Level 1: Warm-up

Match the double summation term a_{i, j} in List I with the most appropriate evaluation method in List II. List I: a_{i, j} = 2^{- i} 3^{- j} a_{i, j} = \frac{1}{i + j} List II: P. Evaluate inner sum first, then outer sum. Q. Split into product of two independent single sums.

Question 2

Level 1: Warm-up

Let h (x) = (3 - x)^{2} . The function h (x) is a composition of an outer function and an inner function. Using the chain rule, what is the maximum value of h^{'} (x) on the closed interval [0, 2] ?

Question 3

Level 1: Warm-up

How many of the following functions have a derivative that is a rational function (a ratio of two polynomials)? 1. f (x) = x^{- 2} 2. g (x) = x^{1/2} 3. h (x) = x^{- 3/2} 4. k (x) = x^{3}

Question 4

Level 1: Warm-up

Let f (x) = e^{x} - 2 x, which has a critical point at x = ln 2 . Using the Second Derivative Test, what does the test determine about x = ln 2 ?

Question 5

Level 1: Warm-up

Consider the following assertion and reason: Assertion (A): The function f (x, y) = x^{2} - y^{2} has a local minimum at (0, 0) . Reason (R): The Hessian matrix at (0, 0) is (20 0 - 2), which has eigenvalues 2 and - 2 . Which of the following is correct?

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Calculus and Optimization Practice Questions for GATE DA

GATE DA Calculus and Optimization: 2 units and 5 chapters, weightage from 13 previous year questions across 3 papers, a study order by exam weight and 511 practice questions.

About Calculus and Optimization Practice Questions

511 practice questions for Calculus and Optimization in GATE DA, sorted chapter by chapter and graded from basic to exam level, each with a full solution.

GATE DA Calculus and Optimization Unit-wise Weightage from Past Papers

We counted every GATE DA Calculus and Optimization previous year question in our bank (13 questions from 3 papers) and grouped them by unit.

Unit	Chapters	PYQs	Share of section	Avg per paper
Calculus	4	12	92%	4
Optimization	1	1	8%	0.3

Suggested Calculus and Optimization Study Order for GATE DA

Calculus: 92% of past Calculus and Optimization questions, about 4 per paper.
Optimization: 8% of past Calculus and Optimization questions, about 0.3 per paper.

Start where the marks are. Units at the top of this list have appeared most often in past GATE DA papers.

Units in GATE DA Calculus and Optimization

All Calculus and Optimization chapters

One Solved Question from Each Calculus and Optimization Chapter

Question 1 · Sequences, Series and Limits MCQ

Match the double summation term $a_{i, j}$ in List I with the most appropriate evaluation method in List II.

List I:

$a_{i, j} = 2^{- i} 3^{- j}$
$a_{i, j} = \frac{1}{i + j}$

List II:

P. Evaluate inner sum first, then outer sum.

Q. Split into product of two independent single sums.

A.
1-P, 2-Q
B.
1-Q, 2-P
C.
1-Q, 2-Q
D.
1-P, 2-P

Correct Answer:

Step-by-Step Solution

Key idea: Separable terms can be split into a product of single sums; non-separable terms must be evaluated iteratively.

Step 1: Analyze term 1: $a_{i, j} = 2^{- i} 3^{- j}$ . This is separable into $f (i) g (j)$ where $f (i) = 2^{- i}$ and $g (j) = 3^{- j}$ . Thus, it can be split into the product of two independent single sums (Method Q).

Step 2: Analyze term 2: $a_{i, j} = \frac{1}{i + j}$ . The variables $i$ and $j$ are coupled in the denominator. It cannot be factored into $f (i) g (j)$ . Thus, it must be evaluated by computing the inner sum first, then the outer sum (Method P).

Step 3: Match 1 with Q, and 2 with P.

Answer: 1-Q, 2-P.

Question 2 · Continuity and Differentiability MCQ

Let $h (x) = (3 - x)^{2}$ . The function $h (x)$ is a composition of an outer function and an inner function.

Using the chain rule, what is the maximum value of $h^{'} (x)$ on the closed interval $[0, 2]$ ?

A.
-2
B.
-6
C.
0
D.
2

Correct Answer:

Step-by-Step Solution

Key idea: The chain rule states that for $h (x) = f (g (x))$ , $h^{'} (x) = f^{'} (g (x)) \cdot g^{'} (x)$ .

Step 1: Identify the inner and outer functions. Let $g (x) = 3 - x$ and $f (u) = u^{2}$ .

Step 2: Differentiate both. $g^{'} (x) = - 1$ and $f^{'} (u) = 2 u$ .

Step 3: Apply the chain rule. $h^{'} (x) = 2 (3 - x) \cdot (- 1) = 2 x - 6$ .

Step 4: Find the maximum of $h^{'} (x) = 2 x - 6$ on the interval $[0, 2]$ . Since $h^{'} (x)$ is a strictly increasing linear function, its maximum on a closed interval occurs at the right boundary, $x = 2$ .

Step 5: Evaluate at $x = 2$ . $h^{'} (2) = 2 (2) - 6 = - 2$ .

Answer: -2

Question 3 · Differentiation and Higher Order Derivatives NAT

How many of the following functions have a derivative that is a rational function (a ratio of two polynomials)?

f (x) = x^{- 2}

g (x) = x^{1/2}

h (x) = x^{- 3/2}

k (x) = x^{3}

Correct Answer:

2.00

Step-by-Step Solution

Key idea: A rational function must only have integer powers of $x$ in the numerator and denominator.

Step 1: Derivative of $f (x) = x^{- 2}$

$f^{'} (x) = - 2 x^{- 3} = \frac{- 2}{x ^{3}}$ . This is a ratio of polynomials. (Yes)

Step 2: Derivative of $g (x) = x^{1/2}$

$g^{'} (x) = \frac{1}{2} x^{- 1/2} = \frac{1}{2 x}$ . The denominator contains a square root, so it is not a polynomial. (No)

Step 3: Derivative of $h (x) = x^{- 3/2}$

$h^{'} (x) = - \frac{3}{2} x^{- 5/2} = \frac{- 3}{2 x ^{2} x}$ . Contains a square root. (No)

Step 4: Derivative of $k (x) = x^{3}$

$k^{'} (x) = 3 x^{2} = \frac{3 x ^{2}}{1}$ . This is a polynomial, which is a rational function. (Yes)

Count: Functions 1 and 4 have rational derivatives.

Answer: 2.00

Question 4 · Maxima, Minima and Applications of Derivatives MCQ

Let $f (x) = e^{x} - 2 x$ , which has a critical point at $x = ln 2$ . Using the Second Derivative Test, what does the test determine about $x = ln 2$ ?

A.
Local minimum
B.
Local maximum
C.
Inflection point
D.
Test fails

Correct Answer:

Step-by-Step Solution

Key idea: observation-type application of the second derivative test: sign of $f^{''}$ at the critical point classifies it.

Step 1: $f^{'} (x) = e^{x} - 2$ , and indeed $f^{'} (ln 2) = 2 - 2 = 0$ .

Step 2: $f^{''} (x) = e^{x}$ , so $f^{''} (ln 2) = e^{l n 2} = 2 > 0$ .

Step 3: Positive second derivative ⇒ concave up ⇒ local minimum by the test.

Note: the question asks only what the TEST determines; no boundary/global claim is needed.

Answer: A

Question 5 · Optimization: Local and Global Extrema MCQ

Consider the following assertion and reason: Assertion (A): The function

f (x, y) = x^{2} - y^{2}

has a local minimum at

(0, 0)

. Reason (R): The Hessian matrix at

(0, 0)

(20 0 - 2)

, which has eigenvalues

2

and

- 2

. Which of the following is correct?

A.
Both A and R are true, and R is the correct explanation of A
B.
Both A and R are true, but R is NOT the correct explanation of A
C.
A is true but R is false
D.
A is false but R is true

Correct Answer:

Step-by-Step Solution

Key idea: The second-order sufficient condition classifies stationary points based on the definiteness of the Hessian.

Step 1: Verify the stationary point.

$\nabla f = (2 x, - 2 y) = (0, 0) ⟹ (0, 0)$ is a stationary point.

Step 2: Compute the Hessian at $(0, 0)$ .

$H = (\frac{\partial ^{2} f}{\partial x ^{2}} \frac{\partial ^{2} f}{\partial y \partial x} \frac{\partial ^{2} f}{\partial x \partial y} \frac{\partial ^{2} f}{\partial y ^{2}}) = (20 0 - 2)$

The eigenvalues are $2$ and $- 2$ . So Reason (R) is TRUE.

Step 3: Classify using the second-order condition.

Since the eigenvalues have mixed signs ( $2 > 0$ and $- 2 < 0$ ), the Hessian is indefinite.

An indefinite Hessian at a stationary point indicates a saddle point, NOT a local minimum.

Therefore, Assertion (A) is FALSE.

Answer: A is false but R is true