In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the…
Mathematics202515 marksShort answer
In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
(i)[1.0]
A matrix which is both symmetric and skew symmetric matrix is a / an: [Understanding]
(a)triangular matrix
(b)identity matrix
(c)diagonal matrix
(d)null matrix
(ii)[1.0]
The value of $\int a^x \cdot e^x \\, dx$ equals [Recall]
(a)$(a^x \cdot \log_e a)e^x + c$
(b)$\frac{a^x \cdot e^x}{\log_e(ae)} + c$
(c)$\frac{a^x \cdot e^x}{\log_{ae} e} + c$
(d)$\log_e(ae)(ae)^x + c$
(iii)[1.0]
The trigonometric equation $\tan^{-1} x = 3\tan^{-1} a$ has solution for [Application]
(a)$|a| \le \frac{1}{\sqrt{3}}$
(b)$|a| > \frac{1}{\sqrt{3}}$
(c)$|a| < \frac{1}{\sqrt{3}}$
(d)all real value of a.
(iv)[1.0]
Assertion: Degree of the differential equation: $a\left(\frac{dy}{dx}\right)^2 + b\frac{dx}{dy} = c$, is 3
Reason: If each term involving derivatives of a differential equation is a polynomial (or can be expressed as polynomial) then highest exponent of the highest order derivative is called the degree of the differential equation. [Analysis]
Which of the following is correct?
(a)Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
(b)Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.
(c)Assertion is true and Reason is false.
(d)Assertion is false and Reason is true.
(v)[1.0]
Five numbers $x_1, x_2, x_3, x_4, x_5$ are randomly selected from the numbers $1, 2, 3, \dots, 18$ and are arranged in the increasing order such that $x_1 < x_2 < x_3 < x_4 < x_5$. What is the probability that $x_2 = 7$ and $x_4 = 11$? [Application]
(a)$\frac{26}{51}$
(b)$\frac{3}{104}$
(c)$\frac{1}{68}$
(d)$\frac{1}{34}$
(vi)[1.0]
If $\begin{vmatrix} a & b & c \\\\ m & n & p \\\\ x & y & z \end{vmatrix} = k$, then what is the value of $\begin{vmatrix} 6a & 2b & 2c \\\\ 3m & n & p \\\\ 3x & y & z \end{vmatrix}$? [Understanding]
(a)$\frac{k}{6}$
(b)$2k$
(c)$3k$
(d)$6k$
(vii)[1.0]
Consider the graph $y = x^{1/3}$
Statement 1: The above graph is continuous at $x = 0$
Statement 2: The above graph is differentiable at $x = 0$ [Application]
Which of the following is correct?
(a)Statement 1 is true, and Statement 2 is false.
(b)Statement 2 is true, and Statement 1 is false.
(c)Both the statements are true.
(d)Both the statements are false.
(viii)[1.0]
The value of $\frac{dy}{dx}$ if $y = |x-1| + |x-4|$ at $x = 3$ is
(a)$-2$
(b)$0$
(c)$2$
(d)$4$
(ix)[1.0]
Statement 1: The intersection of two equivalence relations is always an equivalence relation.
Statement 2: The Union of two equivalence relations is always an equivalence relation.
Which one of the following is correct? [Understanding]
(a)Statement 1 implies Statement 2.
(b)Statement 2 implies Statement 1.
(c)Statement 1 is true only if Statement 2 is true.
(d)Statement 1 and 2 are independent of each other.
(x)[1.0]
Assertion: Matrix ‘A’ is not invertible.
Reason: Determinant A = 0
In a third order matrix $a_{ij}$ denotes the element of the $i^{\text{th}}$ row and the $j^{\text{th}}$ column.
$A = [a_{ij}] = \begin{cases} 0, & \text{for } i = j \\\\ 1, & \text{for } i > j \\\\ -1, & \text{for } i < j \end{cases}$
Which of the following is correct? [Application]
(a)Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
(b)Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.
(c)Assertion is true and Reason is false.
(d)Assertion is false and Reason is true.
(xi)[1.0]
Given two events A and B such that $P(A/B) = 0.25$ and $P(A \cap B) = 0.12$.
The value $P(A \cap B')$ is: [Understanding]
(a)$0 \cdot 36$
(b)$0 \cdot 48$
(c)$0 \cdot 88$
(d)$0 \cdot 036$
(xii)[1.0]
The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of $C^2$? [Analysis]
(xiii)[1.0]
If a relation R on the set $\{a, b, c\}$ defined by $R = \\{(b,b)\\}$, then classify the relation. [Understanding]
(xiv)[1.0]
The given function $f: \mathbb{R} \to \mathbb{R}$ is not ‘onto’ function. Give reason. [Understanding]
(xv)[1.0]
There are three machines and 2 of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. What is the probability that only two tests are needed to identify the faulty machines? [Application]
