List of top Questions asked in GATE MA

Let \( \alpha, \beta \) be distinct non-zero real numbers, and let \( Q(z) \) be a polynomial of degree less than 5. If the function \[ f(z) = \frac{\alpha^6 \sin \beta z - \beta^6 (e^{2az} - Q(z))}{z^6} \] satisfies Morera's theorem in \( \mathbb{C} \setminus \{0\} \), then the value of \( \frac{\alpha}{4\beta} \) is equal to (in integer).
A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?

A square paper, shown in figure (I), is folded along the dotted lines as shown in figures (II) and (III). Then a few cuts are made as shown in figure (IV). Which one of the following patterns will be obtained when the paper is unfolded?

A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled thrice. What is the probability of rolling ‘6’ exactly once?

In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and the line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT? 
Note: The figure shown is representative


 

“I put the brown paper in my pocket along with the chalks, and possibly other things. I suppose every one must have reflected how primeval and how poetical are the things that one carries in one’s pocket: the pocket-knife, for instance the type of all human tools, the infant of the sword. Once I planned to write a book of poems entirely about the things in my pocket. But I found it would be too long: and the age of the great epics is past.” (From G.K. Chesterton’s “A Piece of Chalk”) 
Based only on the information provided in the above passage, which one of the following statements is true?
 

According to the map shown in the figure, which one of the following statements is correct?
Note: The figure shown is representative.


 

Consider the relationships among P, Q, R, S, and T: 
• P is the brother of Q. 
• S is the daughter of Q. 
• T is the sister of S. 
• R is the mother of Q. 
The following statements are made based on the relationships given above.
(1) R is the grandmother of S. 
(2) P is the uncle of S and T. 
(3) R has only one son. 
(4) Q has only one daughter. 
Which one of the following options is correct?

The average marks obtained by a class in an examination were calculated as 30.8. However, while checking the marks entered, the teacher found that the marks of one student were entered incorrectly as 24 instead of 42. After correcting the marks, the average becomes 31.4. How many students does the class have?
The CEO’s decision to downsize the workforce was considered myopic because it sacrificed long-term stability to accommodate short-term gains.
Select the most appropriate option that can replace the word “myopic” without changing the meaning of the sentence.

Ravi had _________ younger brother who taught at _________ university. He was widely regarded as _________ honorable man. 
Select the option with the correct sequence of articles to fill in the blanks.

Let \( (\mathbb{R}^2, d_1) \) and \( (\mathbb{R}^2, d_2) \) be two metric spaces with \[ d_1\left( (x_1, x_2), (y_1, y_2) \right) = |x_1 - y_1| + |x_2 - y_2| \] \[ {and} \quad d_2\left( (x_1, x_2), (y_1, y_2) \right) = \frac{d_1\left( (x_1, x_2), (y_1, y_2) \right)}{1 + d_1\left( (x_1, x_2), (y_1, y_2) \right)}. \] If the open ball centered at \( (0,0) \) with radius \( \frac{1}{7} \) in \( (\mathbb{R}^2, d_1) \) is equal to the open ball centered at \( (0,0) \) with radius \( \frac{1}{\alpha} \) in \( (\mathbb{R}^2, d_2) \), then the value of \( \alpha \) is (in integer).
Let \( G \) be a group with identity element \( e \), and let \( g, h \in G \) be such that the following hold: \[ g \neq e, \quad g^2 = e, \quad h \neq e, \quad h^2 \neq e, \quad {and} \quad ghg^{-1} = h^2. \] Then, the least positive integer \( n \) for which \( h^n = e \) is (in integer).

For \( X = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \), consider the quadratic form:

\[ Q(X) = 2x_1^2 + 2x_2^2 + 3x_3^2 + 4x_1x_2 + 2x_1x_3 + 2x_2x_3. \] Let \( M \) be the symmetric matrix associated with the quadratic form \( Q(X) \) with respect to the standard basis of \( \mathbb{R}^3 \). 
Let \( Y = (y_1, y_2, y_3)^T \in \mathbb{R}^3 \) be a non-zero vector, and let

\[ a_n = \frac{Y^T(M + I_3)^{n+1}Y}{Y^T(M + I_3)^n Y}, \quad n = 1, 2, 3, \dots \] Then, the value of \( \lim_{n \to \infty} a_n \) is equal to (in integer).

Consider the inner product space of all real-valued continuous functions defined on \( [-1, 1] \) with the inner product \[ \langle f, g \rangle = \int_{-1}^{1} f(x) g(x) \, dx. \] If \( p(x) = \alpha + \beta x^2 - 30x^4 \), where \( \alpha, \beta \in \mathbb{R} \), is orthogonal to all the polynomials having degree less than or equal to 3, with respect to this inner product, then \( \alpha + 5\beta \) is equal to (in integer).
Let \( E \subset F \) and \( F \subset K \) be field extensions which are not algebraic. Let \( \alpha \in K \) be algebraic over \( F \) and \( \alpha \notin F \). Let \( L \) be the subfield of \( K \) generated over \( E \) by the coefficients of the monic polynomial of minimal degree over \( F \) which has \( \alpha \) as a zero. Then, which of the following is/are TRUE?
All rings considered below are assumed to be associative and commutative with \( 1 \neq 0 \). Further, all ring homomorphisms map 1 to 1.

Consider the following statements about such a ring \( R \):

P1: \( R \) is isomorphic to the product of two rings \( R_1 \) and \( R_2 \).
P2: \( \exists r_1, r_2 \in R \) such that \( r_1^2 = r_1 \neq 0 \), \( r_2^2 = 0 \), and \( r_1 + r_2 = 1 \).
P3: \( R \) has ideals \( I_1, I_2 \subset R \) with \( R \neq I_1 \), \( (0) \neq I_2 \), and \( R = I_1 + I_2 \) and \( I_1 \cap I_2 = (0) \).
P4: \( \exists a, b \in R \) with \( a \neq 0 \), \( b \neq 0 \) such that \( ab = 0 \).

Then, which of the following is/are TRUE?
Let \( X \) be an uncountable set. Let the topology on \( X \) be defined by declaring a subset \( U \subset X \) to be open if \( X - U \) is either empty or finite or countable, and the empty set to be open. Then, which of the following is/are TRUE?
Let \( \mathbb{R}^1 \) and \( \mathbb{R}^2 \) be provided with the respective Euclidean topologies, and let \[ S^1 = \{ (x_1, x_2) \in \mathbb{R}^2 : x_1^2 + x_2^2 = 1 \} \] be assigned the subspace topology induced from \( \mathbb{R}^2 \). If \( f: S^1 \to \mathbb{R}^1 \) is a non-constant continuous function, then which of the following is/are TRUE?
For \( x \in (0, \pi) \), let \[ u_n(x) = \frac{\sin(nx)}{\sqrt{n}}, \quad n = 1, 2, 3, \dots \] Then, which of the following is TRUE?
Consider the sequence \( \{ f_n \} \) of continuous functions on \( [0, 1] \) defined by \[ f_1(x) = \frac{x}{2}, \quad f_{n+1}(x) = f_n(x) - \frac{1}{2} \left( (f_n(x))^2 - x \right), \quad n = 1, 2, 3, \dots \] Then, which of the following is/are TRUE?
Let \( \{x_k\}_{k=1}^\infty \) be an orthonormal set of vectors in a real Hilbert space \( X \) with inner product \( \langle \cdot, \cdot \rangle \). Let \( n \in \mathbb{N} \), and let \( Y \) be the linear span of \( \{ x_k \}_{k=1}^n \) over \( \mathbb{R} \). For \( x \in X \), let \[ S_n(x) = \sum_{k=1}^n \langle x, x_k \rangle x_k. \] Then, which of the following is/are TRUE?
Consider the function \( f : \mathbb{R}^2 \to \mathbb{R}^2 \) given by \[ f(x, y) = (e^{2\pi x} \cos 2\pi y, e^{2\pi x} \sin 2\pi y). \] Then, which of the following is/are TRUE?
Let \( u(x, t) \) be the solution of the initial-boundary value problem \[ \frac{\partial u}{\partial t} = 2 \frac{\partial^2 u}{\partial x^2}, \quad 0<x<1, \quad t>0, \] with the boundary conditions \[ u(0, t) = u(1, t) = 0, \quad u(x, 0) = 2x(1 - x). \] Then, which of the following is/are TRUE?
Let \( u(x, t) \) be the solution of the initial value problem \[ \frac{\partial u}{\partial t} + 3 \frac{\partial u}{\partial x} = u, \quad x \in \mathbb{R}, \quad t>0, \quad u(x, 0) = \cos x, \] and let \( v(x, t) \) be the solution of the initial value problem \[ \frac{\partial v}{\partial t} + 3 \frac{\partial v}{\partial x} = v^2, \quad x \in \mathbb{R}, \quad t>0, \quad v(x, 0) = \cos x. \] Then, which of the following is/are TRUE?