In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the…
Mathematics202415 marksShort answer
In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed.
(i)[1.0]
Let $L$ be a set of all straight lines in a plane. The relation $R$ on $L$ defined as 'perpendicular to' is:
(a)Symmetric and Transitive
(b)Transitive
(c)Symmetric
(d)Equivalence
(ii)[1.0]
The order and the degree of differential equation $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{\frac{1}{3}} = \frac{d^2y}{dx^2}$ are:
(a)$2$ and $\frac{3}{2}$
(b)$2$ and $3$
(c)$3$ and $4$
(d)$2$ and $1$
(iii)[1.0]
Let $A$ be a non-empty set.
Statement 1: Identity relation on $A$ is Reflexive.
Statement 2: Every Reflexive relation on $A$ is an Identity relation.
(a)Both the statements are true.
(b)Both the statements are false.
(c)Statement 1 is true and Statement 2 is false.
(d)Statement 1 is false and Statement 2 is true.
(iv)[1.0]
The graph of the function $f$ is shown below.
Of the following options, at what values of $x$ is the function $f$ NOT differentiable?
(a)At $x = 0$ and $x = 2$
(b)At $x = 1$ and $x = 3$
(c)At $x = -1$ and $x = 1$
(d)At $x = -1.5$ and $x = 1.5$
(v)[1.0]
The value of $\operatorname{cosec}\left(\sin^{-1}\left(-\frac{1}{2}\right)\right) - \sec\left(\cos^{-1}\left(-\frac{1}{2}\right)\right)$ is equal to:
(a)$-4$
(b)$0$
(c)$-1$
(d)$4$
(vi)[1.0]
The value of $\int_1^{\sqrt{3}} \frac{dx}{1 + x^2}$ is:
(a)$\frac{\pi}{2}$
(b)$\frac{2\pi}{3}$
(c)$\frac{\pi}{6}$
(d)$\frac{\pi}{12}$
(vii)[1.0]
Assertion: Let the matrices $A = \begin{bmatrix} -3 & 2 \\ -5 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -2 \\ 5 & -3 \end{bmatrix}$ be such that $A^{100}B = BA^{100}$.
Reason: $AB = BA$ implies $A^n B = BA^n$ for all positive integers $n$.
(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.
(viii)[1.0]
If $\int (\cot x - \operatorname{cosec}^2 x)e^x dx = e^x f(x) + c$ then $f(x)$ will be:
(a)$\cot x + \operatorname{cosec} x$
(b)$\cot^2 x$
(c)$\cot x$
(d)$\operatorname{cosec} x$
(ix)[1.0]
In which one of the following intervals is the function $f(x) = x^3 - 12x$ increasing?
(a)$(-2, 2)$
(b)$(-\infty, -2) \cup (2, \infty)$
(c)$(-2, \infty)$
(d)$(-\infty, 2)$
(x)[1.0]
If $A$ and $B$ are symmetric matrices of the same order, then $AB - BA$ is:
(a)Skew-symmetric matrix
(b)Symmetric matrix
(c)Diagonal matrix
(d)Identity matrix
(xi)[1.0]
Find the derivative of $y = \log x + \frac{1}{x}$ with respect to $x$.
(xii)[1.0]
Teena is practising for an upcoming Rifle Shooting tournament. The probability of her shooting the target in the $1^{\text{st}}$, $2^{\text{nd}}$, $3^{\text{rd}}$ and $4^{\text{th}}$ shots are $0.4$, $0.3$, $0.2$ and $0.1$ respectively. Find the probability of at least one shot of Teena hitting the target.
(xiii)[1.0]
Which one of the following graphs is a function of $x$?
Graph A
Graph B
(xiv)[1.0]
Evaluate: $\int_0^6 |x + 3| dx$
(xv)[1.0]
Given that $\frac{1}{y} + \frac{1}{x} = \frac{1}{12}$ and $y$ decreases at a rate of $1\text{ cm s}^{-1}$, find the rate of change of $x$ when $x = 5\text{ cm}$ and $y = 1\text{ cm}$.
