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In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the…
In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed.
(i)[1.0]
Which of the following is NOT an equivalence relation on $\mathbb{Z}$?
- (a)$aRb \iff a + b$ is an even integer
- (b)$aRb \iff a - b$ is an even integer
- (c)$aRb \iff a < b$
- (d)$aRb \iff a = b$
(ii)[1.0]
Let $A$ be the set of all $50$ cards numbered from $1$ to $50$. Let $f: A \to \mathbb{N}$ be a function defined by $f(x) =$ card number of the card ‘$x$’. Then the function ‘$f$’ is:
- (a)one to one but not onto.
- (b)onto but not one to one.
- (c)neither one to one nor onto.
- (d)one to one and onto.
(iii)[1.0]
The value of $\tan^{-1} \sqrt{3} - \sec^{-1}(-2)$ is equal to
- (a)$\frac{\pi}{3}$
- (b)$\frac{2\pi}{3}$
- (c)$-\frac{\pi}{3}$
- (d)$\frac{\pi}{4}$
(iv)[1.0]
If $A = \begin{pmatrix} 1 & b \\\\ 0 & 1 \end{pmatrix}$, then $A^n$ ($n \in \mathbb{N}$) is equal to
- (a)$\begin{pmatrix} 1 & nb \\\\ 0 & 1 \end{pmatrix}$
- (b)$\begin{pmatrix} 1 & b^n \\\\ 0 & 1 \end{pmatrix}$
- (c)$\begin{pmatrix} 1 & nb \\\\ 0 & 1 \end{pmatrix}$
- (d)$\begin{pmatrix} 1 & nb \\\\ 0 & 0 \end{pmatrix}$
(v)[1.0]
If $A$ is a $3 \times 3$ matrix such that $A(\text{adj } A) = \begin{pmatrix} 3 & 0 & 0 \\\\ 0 & 3 & 0 \\\\ 0 & 0 & 3 \end{pmatrix}$, then the value of $|A^T|$ will be
- (a)$27$
- (b)$9$
- (c)$3$
- (d)$-3$
(vi)[1.0]
If $\begin{vmatrix} a & b & c \\\\ x+a & y+2a & z+3a \\\\ 1 & 2 & 3 \end{vmatrix} = \begin{vmatrix} a & b & c \\\\ x & y & z \\\\ 1 & 2 & 3 \end{vmatrix} + |B|$, then the value of $|B|$ is equal to
- (a)$a + 2b + 3c$
- (b)$a - b + 2c$
- (c)$0$
- (d)$3a$
(vii)[1.0]
If $\sin^{-1} x + \sin^{-1} y = \frac{\pi}{2}$, then $\frac{dy}{dx}$ is equal to
- (a)$\frac{x}{y}$
- (b)$-\frac{x}{y}$
- (c)$\frac{y}{x}$
- (d)$-\frac{y}{x}$
(viii)[1.0]
If $f(x) = \frac{4-x^2}{4x-x^3}$ then the function is:
- (a)discontinuous at only one point.
- (b)discontinuous at exactly two points.
- (c)discontinuous at exactly three points.
- (d)discontinuous at exactly four points.
(ix)[1.0]
The degree of the differential equation $\frac{d^2y}{dx^2} + 3 \left(\frac{dy}{dx}\right)^2 = x^2 \log \left(\frac{d^2y}{dx^2}\right)$ is:
- (a)$2$
- (b)$1$
- (c)$3$
- (d)not defined
(x)[1.0]
If $A$ and $B$ are two events such that $P(A) > 0$ and $P(B) \neq 1$, then $P(\bar{A}/\bar{B})$ is
- (a)$1 - P(\bar{A}/B)$
- (b)$1 - P(A/B)$
- (c)$\frac{1 - P(A \cup B)}{P(\bar{B})}$
- (d)$\frac{P(\bar{A})}{P(B)}$
(xi)[1.0]
Write the smallest equivalence relation on the set $A = \\{a, b, c\\}$
(xii)[1.0]
If $\begin{pmatrix} 3 & 5 \\\\ 7 & 9 \end{pmatrix} = X + Y$, where $X$ is skew-symmetric matrix and $Y$ is symmetric matrix. Find $|X|$.
(xiii)[1.0]
If $\int \log 2x \\, dx = x \log 2x - k + c$ where $k$ is a function of $x$, then find $k$.
(xiv)[1.0]
$50$ tickets in a box are numbered $00, 01, 02, \dots, 49$. One ticket is drawn randomly from the box. Find the probability of the ticket having the product of its digits $7$, given that the sum of the digits is $8$?
(xv)[1.0]
A speaks truth in $60\\%$ of cases and B speaks truth in $90\\%$ of the cases. In what percentage of cases are they likely to contradict each other in stating the same fact?
Answer
No answer yet.
From ISC 2023 Specimen Mathematics Paper 1, question 1.