Engineering Mat...

Let X be an exponential random variable with rate parameter λ . Then find the variance of X.

What is the determinant of A?

\(A = \;\left| {\begin{array}{*{20}{c}} 3&1&1&1&1\\ 1&3&1&1&1\\ 1&1&3&1&1\\ 1&1&1&3&1\\ 1&1&1&1&3 \end{array}} \right|\)

Consider the given set of equations:

x + y = 1

y + z = 1

x + z = 1

Which one of the following does NOT equal \(\left| {\begin{array}{*{20}{c}}1&x&{{x^2}}\\1&y&{{y^2}}\\1&z&{{z^2}}\end{array}} \right|\)?

Consider a matrix Mn×n, the two eigenvalues of the matrix are 12 and 6 + √-1). If n is equal to three then what is the determinant of M?

The rank of the matrix, \(M = \left[ {\begin{array}{*{20}{c}} 0&1&1\\ 1&0&1\\ 1&1&0 \end{array}} \right]\), is _________.

For what values of a and b, the system of equation x + 2y + z = 6, x + 4y + 3z = 10, x + 4y + az = b has no solution

If probability density function of a random variable x is

f(x) = x² for -1 ≤ x ≤ 1, and

= 0 for any other value of x

Then, the percentage probability P\(\left( { - \frac{1}{3}\; \le x\; \le \;\frac{1}{3}} \right)\) is

Consider the function f(x) = \(\mathop {{\rm{lim}}}\limits_{x \to 0} \frac{{{4x^2}}}{{\left( {\cos 8x - cos4x} \right)}}\;\). What is the value of 18× f(x)?

Let \(A = \left( {\begin{array}{*{20}{c}} 1&{ - 1}\\ 0&1 \end{array}} \right)\) then which of the following statements is/are true?

Consider two events X and Y. X and Y are independent events and both are equally likely events. If the probability of having X or Y is \(\frac{3}{4}\) then what is the probability of Y?

NOTE:
Answer up to 2 decimal places

In the LU decomposition of the matrix \(\left[ {\begin{array}{*{20}{c}} 2&2\\ 4&9 \end{array}} \right]\).The diagonal elements of U are both 1 and U₁₂ is p, and the elements of L are x, 0, z and y in row major order. Which of the following is/are true?

f(x) and g(x) are two functions differentiable in [0,1] such that f(0) = 2; g(0) = 0; f(1) = 6; and g(1) = 2. Then there must exist a constant c in ?

Find the eigen vector corresponding to Largest eigen value of matrix A = \(\left[ {\begin{array}{*{20}{c}} 1&{ - 1}&1\\ 1&{ - 1}&1\\ { - 1}&1&1 \end{array}} \right]\)

Consider two candidates A and B are considering applying for a job. The probability that A applies for the job is \(\frac{3}{5}\), the probability that A applies for the job given that B applies for the job is \(\frac{4}{5}\), and the probability that B applies for the job given that A applies for the job is \(\frac{2}{3}\). What is the probability that A does not apply for the job given that B does not apply for the job?

NOTE:

Answer up to 2 decimal places

Consider a differentiable function f(x) on the set of real numbers such that f(−1) = 0 and |f′(x)| ≤ 2. Given these conditions, which one of the following inequalities is necessarily true for all x ∈ [−2 , 2] ?

Let M and N two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. 

Which of the below-given statements is/are FALSE?

A function f (x) is defined as \(g\left( t \right) = \left\{ {\begin{array}{*{20}{c}} {{e^x},}&{x < 1}\\ {\ln x + a{x^2}+bx,}&{x \ge 1} \end{array}} \right.\), where x ϵ R. Which one of the following statements is TRUE?

Find the point at which absolute minima values lies for f(x) where x ϵ [4, 8]:

\(f(x) = x^3 -18x^2 + 105x -10\)

The integral of \(\mathop \smallint \limits_0^{{\bf{\pi }}/2} sin2x{\rm{log}}(\tan x)dx\) is given by -

Let \(A = \left[ {\begin{array}{*{20}{c}} 1&0&{ - 1}\\ { - 1}&2&0\\ 0&0&{ - 2} \end{array}} \right]\) and B = A³ – A² – 4A + 5I, where I is the 3 × 3 identity matrix. The determinant of B is _______ (up to 1 decimal place).

Consider a 2 × 2 matrix \(M = \left[ {\begin{array}{*{20}{c}}{{v_1}}&{{v_2}}\end{array}} \right]\), where, v₁ and v₂ are the column vectors. Suppose \({M^{ - 1}} = \left[ {\begin{array}{*{20}{c}}{u_1^T}\\{u_2^T}\end{array}} \right]\), where u^T₁ and u^T₂ are the row vectors. Consider the following statements.

Statement: u^T₁v₁ = 1 and u^T₂v₂ = 1

Statement: u^T₁v₂ = 0 and u^T₂v₁ = 0

For the linear equation

a + 3b – 2c = -1

5b + 3c = -8

a – 2b – 5c = 7