GATE|| Engineering Mathematics || Linear Algebra, Calculus and Probability and Statistics || Pyqs (2010 to 2025 )

[ NAT || GATE 2018 || 1 Marks]

In an examination, a student can choose the order in which two questions (Ques A and Ques B) must be attempted.

      - If the first question is answered wrong, the student gets zero marks. 

      - If the first question is answered correctly and the second question is not answered correctly, the student gets the marks only for the first question. 

      - If both the questions are answered correctly, the student gets the sum of the marks of the two questions. 

The following table shows the probability of correctly answering a question and the marks of the question respectively.

Assuming that the student always wants to maximize her expected marks in the examination, in which order should she attempt the questions and what is the expected marks for that order (assume that the questions are independent)?

 [ MCQ || GATE 2021 Set-2 || 2 Marks]

A bag has r red balls and b black balls. All balls are identical except for their colours. In a trial, a ball is randomly drawn from the bag, its colour is noted and the ball is placed back into the bag along with another ball of the same colour.

Note that the number of balls in the bag will increase by one after each trial.

A sequence of four such trials is conducted.

Which one of the following choices gives the probability of drawing a red ball in the fourth trial?

[ MCQ || GATE 2021 Set-2 || 2 Marks]

Consider the two statements.

S₁ : There exist random variables X and Y such that

      (E[(X – E(X)) (Y – E(Y))])²> Var[X] Var[Y]

S₂ : For all random variables X and Y,

      Cov[X,Y] = E[|X – E[X]||Y – E[Y]|]

Which one of the following choices is correct? [ MCQ || GATE 2021 Set-1 || 2 Marks]

Consider the following matrix.

The largest eigenvalue of the above matrix is __________.[ NAT || GATE 2021 Set - 1 || 2 Marks]

Suppose that P is a 4 × 5 matrix such that every solution of the equation Px = 0 is a scalar multiple of [2 5 4 3 1]^T. The rank of P is _______ . [ NAT || GATE 2021 Set-2 || 1 Marks]

Consider the functions

I. e^-x

II. x² - sin x

III. √(x³+1)

Which of the above functions is/are increasing everywhere in [0, 1]? [ MCQ || GATE 2020 || 1 Marks]

Let A and B be two n×n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.

I. rank(AB) = rank(A) × rank (B)
II. det(AB) = det(A) × det(B)
III. rank(A+B) ≤ rank(A) + rank(B)
IV. det(A+B) ≤ det(A) + det(B)

Which of the above statements are TRUE? [MCQ || GATE 2020 || 2 Marks]

1.

Let U = {1, 2, …,n} and A = {(x, X)∣ x∈X, X⊆U }. Consider the following two statements on |A|.

[Tex]\begin{aligned} &\text{I.} \quad\quad\quad |A| = n2^{n-1} \\ &\text{II.} \quad\quad |A| = \sum_{k=1}^n k\binom{n}{k} \end{aligned}[/Tex]

Which of the above statements is/are TRUE? [ MCQ || GATE 2019 || 2 Marks]