List of top Questions asked in GATE EE, GATE MA

Consider the problem of maximizing \[ z = \begin{pmatrix} x_1 & x_2 & x_3 \end{pmatrix} \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 1 & 2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \] subject to \[ \begin{pmatrix} x_1 & x_2 & x_3 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = 1, \] where \( \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \in \mathbb{R}^3 \).

Then the maximum value of \( z \) is ______. (Answer in integer)

Four people P, Q, R, and S, of different ages, make the following observations.
P - I am younger than S.
Q - I am neither the youngest nor the oldest.
R - P is older than me.
Based on these observations, the youngest person is ______.

Let \( M_3(\mathbb{R}) \) be the vector space of all \( 3 \times 3 \) real matrices over \( \mathbb{R} \) under usual matrix addition and scalar multiplication. Let \( T_3(\mathbb{R}) \) be the set of all \( 3 \times 3 \) real upper triangular matrices. Which one of the following is TRUE?

In a population, patients who have high cholesterol also have high blood-pressure (BP). Some patients with high BP also have diabetes. There are no patients who have both high cholesterol and diabetes. Furthermore,
1. the total number of patients with at least one of these conditions is 75,
2. the number of patients with high cholesterol is 10,
3. the number of patients with high BP is 45, and
4. the number of patients with only high BP and no other conditions is 20.
Then the number of patients who have both diabetes and high BP is _______

A coin with heads facing up is shown as H and a coin with tails facing up is shown as T. Six coins are placed in the Starting Arrangement, as shown in the figure below. A "step" is defined as interchanging a pair of adjacent coins without flipping them.

Starting Arrangement: H H H T T T
Final Arrangement: T T T H H H
The minimum number of steps needed to go from the Starting Arrangement to the Final Arrangement, as shown in the figure, is ________.

Let \( P_3(\mathbb{R}) \) be the vector space of all polynomials of degree at most three with real coefficients under usual polynomial addition and scalar multiplication. Let \( T: P_3(\mathbb{R}) \to \mathbb{R}^2 \) be the linear transformation defined as \[ T(p) = \left(p(1), p'(1)\right) \] for all \( p \in P_3(\mathbb{R}) \), where \( p' \) is the derivative of \( p \).

Then the nullity of \( T \) is equal to ______. (Answer in integer)

To continue the sequence of tiles shown, the tile indicated by the question mark should be:

(The tiles in the sequence contain, in order, 0 dots, 1 dot, 1 dot, 2 dots, 3 dots, and 5 dots, followed by a tile marked with a question mark.)

Let \( f: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined as
\[ f(x, y) = (e^x \cos y,\ e^x \sin y) \]
Which one of the following is TRUE?

Circles \(C_1\), \(C_2\), and \(C_3\), with centers \(O_1\), \(O_2\), and \(O_3\), and radii \(r_1\), \(r_2\), and \(r_3\), respectively, touch each other as shown in the following figure.

Given \(r_1 = 2\) cm, \(r_2 = 1\) cm and the angle \(\angle O_1 O_3 O_2\) is \(90^{\circ}\), \(r_3 =\) _____ cm.

Let \(T: \mathbb{R}^3 \to \mathbb{R}^3\) be the linear transformation which reflects every vector in \(\mathbb{R}^3\) through a two-dimensional subspace of \(\mathbb{R}^3\). Let \(P \in \mathbb{R}^{3 \times 3}\) be the matrix representation of \(T\) using the basis

\[ \left\{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right\}. \]

Then the value of \(2 \times \text{trace}(P) - 3 \times \text{determinant}(P)\) is equal to ______. (answer in integer)

Let \( I \) be the integral defined as follows:
\[ I = \int_0^1 \int_0^{\sqrt{y}} dx\, dy \;+\; \int_1^2 \int_{\sqrt{y-1}}^{1} dx\, dy \]
If the order of integration is changed, then which one of the following is the correct expression for \( I \)?

Suresh said, "I did it yesterday." Which one of the following options is the correct form of this sentence in indirect speech?

Let \( D = \{(x, y) \in \mathbb{R}^2 : 0 \leq x \leq 2,\ 0 \leq y \leq 2\} \) and let \( f(t) \) denote the smallest integer greater than or equal to \( t \). Then the value of the integral
\[ \iint_D f(x+y)\, dx\, dy \]
is ______.

Let \(P\) be a \(3 \times 3\) real symmetric positive definite matrix. Which of the following statements is/are TRUE?

Let \(f: \mathbb{R}^2 \to \mathbb{R}\) be a function defined as follows:
\[ f(x,y) = \begin{cases} \dfrac{x\sqrt{x^2+y^2}}{|x|} & \text{for } x \neq 0 \\ 0 & \text{for } x = 0 \end{cases} \]
Which of the following statements is/are TRUE?

Consider an art gallery whose walkways are shown as lines in the diagram. A black dot represents a junction of two walkways. A guard may be placed at a junction to watch over the walkways that join at that junction. The minimum number of guards needed to watch all the walkways is ________.

(The diagram shows five junctions joined in a closed loop shaped like a pentagon: a top junction connects down to an upper-left junction and an upper-right junction, the upper-left junction connects down to a lower-left junction, the upper-right junction connects down to a lower-right junction, and the lower-left and lower-right junctions connect to each other. This gives 5 junctions and 5 walkways in total.)

A paper shown in Panel I is folded along the dashed lines to construct a cube. The shaded regions shown in Panel I appear on the outer surface of the cube. Referring to the cubes shown in Panel II, which one of the options is correct?

(Panel I shows a flat cross-shaped net of six squares: a vertical strip of four squares one above another, with one extra square attached to the left of the second square in that strip and another extra square attached to its right, all joined along dashed fold lines. Some squares carry grey shaded regions and some are left plain white. Panel II shows two separate solid cubes, labelled (i) and (ii), each with grey shaded regions on some of their visible faces.)

Let \(P\) and \(Q\) be \(3 \times 3\) nonzero real matrices. Assume that there exists a \(3 \times 3\) real nonsingular matrix \(S\) such that \(S^{-1}PS\) and \(S^{-1}QS\) are both upper triangular. Which of the following statements is/are TRUE?

Let \[ \alpha = \lim_{n \to \infty} \int_0^{1} \frac{n^2 + (\sin e^x)^n}{7n^2 + x^8} \, dx. \] The value of \(14\alpha\) is equal to ______. (Answer in integer)

The 2nd of June is a Thursday in a certain year. Which day of the week is the 3rd of July in that year?

Exacerbate : Mitigate :: __________
Choose the option with the correct pair of words to fill the blank.

Let \( f: \mathbb{C} \to \mathbb{C} \) be defined by \( f(z) = |z|^2 - 5\bar{z} + 2 \).
Which one of the following is TRUE?

Let \( C \) denote the Cantor set and \( f: [0,1] \to \mathbb{R} \) be defined as follows:
\[ f(x) = \begin{cases} x^{2026} & \text{for } x \in C \\ \cos(\pi x) & \text{for } x \in \left[0, \dfrac{1}{2}\right] \setminus C \\ \sin(\pi x) & \text{for } x \in \left[\dfrac{1}{2}, 1\right] \setminus C \end{cases} \]
The value of the Lebesgue integral of \( f(x) \) over the interval \( [0,1] \) is equal to

Let \( \alpha = \iint_S \vec{F} \cdot \hat{n} \, dS \), where \( \vec{F} = (2x + 3z)\hat{i} + (xz - y)\hat{j} + (y^2 + 2z)\hat{k} \) and \( S \) is the sphere with centre at \( (3, -1, 2) \) and radius \( 9 \). Here \( \hat{n} \) is the unit normal drawn outward and \( \hat{i}, \hat{j}, \hat{k} \) are unit vectors.

Then the value of \( \dfrac{1}{36\pi}\alpha \) is equal to ______. (Answer in integer)

Let \[ J = \begin{pmatrix} 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{pmatrix}. \] Then the geometric multiplicity of the eigenvalue \(2\) of \(J\) is equal to ________. (Answer in integer)