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[15×1] In sub-parts (i) to (x) choose the correct options and in sub-parts (xi) to (xv), answer the…
[15×1] In sub-parts (i) to (x) choose the correct options and in sub-parts (xi) to (xv), answer the questions as instructed.
(i)
Let $R$ be the relation in the set $N$, given by $R = \{(x, y) : x = y + 3 \wedge y > 5\}$.
Choose the correct answer from the following:
- (a)$(7, 4) \in R$.
- (b)$(9, 6) \in R$.
- (c)$(4, 1) \in R$.
- (d)$(8, 5) \in R$.
(ii)
If $A = \{1, 2, 3\}$, $B = \{x, y\}$, then the number of functions from $A$ to $B$ is:
- (a)3
- (b)6
- (c)8
- (d)12
(iii)
Simplified value of $\sin\left[\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$ is:
- (a)$\frac{1}{2}$
- (b)$\frac{1}{\sqrt{2}}$
- (c)$\frac{\sqrt{3}}{2}$
- (d)$-\frac{\sqrt{3}}{2}$
(iv)
Which of the following statements is correct:
- (a)Determinant is number associated to a matrix.
- (b)Determinant is a square matrix.
- (c)Determinant is a number associated to a square matrix.
- (d)None of the above.
(v)
If $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, then $A^2$ is equal to:
- (a)$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
- (b)$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
- (c)$\begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}$
- (d)$\begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}$
(vi)
For what value of $k$ inverse does not exist for the matrix $\begin{bmatrix} 1 & 2 \\ k & 6 \end{bmatrix}$:
- (a)0
- (b)3
- (c)6
- (d)2
(vii)
Identify from the given options the slope of the normal to the curve $x^2 + 3y + y^2 = 5$ at $(1, 1)$:
- (a)$-\frac{2}{5}$
- (b)$\frac{5}{2}$
- (c)$\frac{2}{5}$
- (d)$-\frac{5}{2}$
(viii)
The function $f(x) = x^3 - 3x$ is strictly decreasing on:
- (a)$-1 < x < 1$
- (b)$-1 \le x \le 1$
- (c)$1 \le x < \infty$
- (d)$-\infty < x < -1$
(ix)
The function $f(x) = \sin x + \cos x$, $x \in (0, \pi/2)$; Maximum value of $f(x)$ is:
- (a)1
- (b)2
- (c)$\frac{1}{\sqrt{2}}$
- (d)$\sqrt{2}$
(x)
If $P(A) = \frac{4}{5}$ and $P(A \cap B) = \frac{7}{10}$, then the value of $P(B/A)$ is:
- (a)$\frac{1}{8}$
- (b)$\frac{4}{7}$
- (c)$\frac{5}{7}$
- (d)$\frac{7}{8}$
(xi)
Write total number of functions from set A to set B, where $A = \{1, 2, 3\}$, $B = \{p, q, r, s\}$.
(xii)
If A is square matrix of order 3, with $|A| = 4$, then write the value of $|-2A|$.
(xiii)
Find the sum of order and degree of the differential equation:
$\left(\frac{dy}{dx}\right)^5 + 3xy\left(\frac{d^3y}{dx^3}\right)^2 + y^2\left(\frac{d^2y}{dx^2}\right)^3 = 0$
(xiv)
$A$ and $B$ appeared for an interview for two vacancies. The probability of $A$’s selection is $\frac{4}{5}$ and $B$’s selection is $\frac{1}{3}$. Find the probability that none of them will be selected.
(xv)
If $A$ and $B$ are independent events such that $P(A) = 1/4$ and $P(B) = 2/3$. Find $P(A \cap B)$.
Answer
No answer yet.
From ISC 2021 Specimen Mathematics Paper 1, question 1.