Engineering Mathematics-III 2022 Question Paper — Government College…

Instructions

• Attempt five questions in all, selecting at least two questions from each section.
• Each question carries 20 marks.

Questions

Section-A

• Q1. (a) A random variable X has the following probability function : | $X=x$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | |---|---|---|---|---|---|---|---|---|---| | $P(x)$ | $k$ | $3k$ | $5k$ | $7k$ | $9k$ | $11k$ | $13k$ | $15k$ | $17k$ | (i) Find the value of $k$. (ii) Evaluate $P(X \ge 6)$, $P(3 < X < 5)$. (iii) Calculate the minimum value of $k$ such that $P(X \le 3) \ge 0.5$. (b) Find the variance of the number of divisors of an integer from 1 through 20 chosen at random. [20 marks]
• Q2. (a) Let X be a random variable having probability density function $$f(x) = \begin{cases} x, & 0 \le x \le 1 \\ 2 - x, & 1 \le x \le 2 \end{cases}$$ Find the mean and variance of X. (b) The first three moments of a distribution about the value 2 are 1, 6 and 40. Show that the first three moments about zero are 3, 14, 16. [20 marks]
• Q3. (a) Derive the mode of Binomial distribution. (b) Show that for $p = \frac{1}{2}$, the binomial distribution has a maximum probability at $x = \frac{n}{2}$, if $n$ is even and at $x = \frac{n + 1}{2}$, if $n$ is odd. [20 marks]
• Q4. (a) Define Poisson distribution and find its mean and variance. (b) If X is a Poisson variate with mean $\lambda$, show that $\frac{x - \lambda}{\sqrt{\lambda}}$ is a variate with mean zero and variance unity. Find the moment generating function (m.g.f) for this variate and show that it approaches $e^{\frac{t^2}{2!}}$ as $\lambda \to \infty$. [20 marks]

Section-B

• Q5. (a) Prove that a non-empty subset S of a vector space V over a field F is a subspace of V if and only if $\alpha x + \beta y \in S$ for all $\alpha, \beta \in F$ and $x, y \in S$. (b) Let V be a vector space over a field F. Then prove the following : (i) Any subset of V containing a zero element is linearly dependent. (ii) Every subset of a linearly independent set of vectors is linearly independent. [20 marks]
• Q6. (a) Prove that if S is a non-empty subset of a vector space V over a field F, then L(S), the linear span of S is the smallest subspace of V containing S. (b) Determine whether or not the set $S = \{(1, 1, 2), (1, 2, 5), (5, 3, 4)\}$ forms a basis for the vector space $V = \mathbf{R}^3$ over the field R. [20 marks]
• Q7. (a) Show that the mapping $T : \mathbf{R}^3 \to \mathbf{R}^2$ defined by $T(x, y, z) = (x - y, x + z), \forall (x, y, z) \in \mathbf{R}^3$ is a linear transformation. (b) Find kernel, nullity, range and rank of the linear transformation $T : \mathbf{R}^2 \to \mathbf{R}^3$ defined by $T(x, y) = (x, x + y, y)$. Also verify rank-nullity theorem. [20 marks]
• Q8. (a) Define kernel of a linear transformation. Prove that if U and V are two vector spaces over the same field F, and $T : U \to V$ is a linear transformation, then kernel of T is a subspace of U. (b) Find the matrix associated to the linear transformation $T : \mathbf{R}^3 \to \mathbf{R}^3$ defined by $T(x, y, z) = (9x + 4y + 5z, -4x - 3z, -6x - 4y - 2z)$ w.r.t the basis $B = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ of $\mathbf{R}^3$. [20 marks]

Related pages

• Engineering Mathematics-III syllabus
• All Engineering Mathematics-III papers from 2022