In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the…
Mathematics202615 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]
If A and B are square matrices of order 3, A is non-singular matrix and AB = O, then the matrix B is: (Understanding)
(a)unit matrix
(b)Scalar matrix
(c)non-singular matrix
(d)null matrix
(ii)[1.0]
If $m$ and $n$ are respectively the order and degree of the differential equation $\frac{d}{dx}\left[\left(\frac{dy}{dx}\right)^3\right] = 0$ then the value of $(m - n)$ is: (Recall)
(a)0
(b)1
(c)2
(d)3
(iii)[1.0]
The derivative of $xy = c^2$ with respect to $x$ is: (Understanding)
(a)$\frac{dy}{dx} = \frac{2c - y}{x}$
(b)$\frac{dy}{dx} = \frac{c^2}{x^2}$
(c)$\frac{dy}{dx} = -\frac{y}{x}$
(d)$\frac{dy}{dx} = \frac{y}{x}$
(iv)[1.0]
Assertion: Consider $\Delta = \begin{vmatrix} 2a & 2b & 2c \\ 2e & f & g \\ 2i & j & k \end{vmatrix}$.
The value of $\Delta = 4 \times \begin{vmatrix} a & b & c \\ e & f & g \\ i & j & k \end{vmatrix}$
Reason: If all elements of one row or one column of a determinant are multiplied by a scalar, $k$ then the value of the determinant is multiplied by $k$. (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 $A = \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{pmatrix}$, then $A^n$ equals to (Understanding)
(d)$\begin{pmatrix} na & 0 & 0 \\ 0 & na & 0 \\ 0 & 0 & a^n \end{pmatrix}$
(vii)[1.0]
Observe the following graphs (a), (b), (c), and (d), each representing different types of functions.
Statement 1: A function which is continuous at a point may not be differentiable at that point.
Statement 2: Graph (c) is an example of a function that is continuous but not differentiable at the origin. (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]
If $f(x) = kx^2 + 7x - 4$ and $f'(5) = 97$ then what is the value of $k$? (Understanding)
(a)$-4$
(b)$0$
(c)$4$
(d)$9$
(ix)[1.0]
Statement 1: If a relation $R$ on a set $A$ satisfies $R = R^{-1}$, then $R$ is symmetric.
Statement 2: For a relation $R$ to be symmetric, it is necessary that $R = R^{-1}$
Which one of the following is correct? (Understanding)
(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.
(x)[1.0]
Assertion: The equality $\tan(\cot^{-1} x) = \cot(\tan^{-1} x)$, is true for all $x \in R$
Reason: The identity $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$, is true for all $x \in R$
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 $P(A \cap B) = 0\cdot 12$.
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 $f: R \to R$ is many to one 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)
