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[10×2] The binary operation is defined as . Find . If and is a symmetric matrix, show that . Solve…
[10×2]
(i)[2.0]
The binary operation $* : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is defined as $a * b = 2a + b$.
Find $(2 * 3) * 4$.
(ii)[2.0]
If $A = \begin{pmatrix} 5 & a \\ b & 0 \end{pmatrix}$ and $A$ is a symmetric matrix, show that $a = b$.
Given: $A = \begin{pmatrix} 5 & a \\ b & 0 \end{pmatrix}$ and $A$ is a symmetric matrix
To show: $a = b$
(iii)[2.0]
Solve: $3\tan^{-1}x + \cot^{-1}x = \pi$
(iv)[2.0]
Without expanding at any stage, find the value of:
$\begin{vmatrix} a & b & c \\ a+2x & b+2y & c+2z \\ x & y & z \end{vmatrix}$
(v)[2.0]
Find the value of constant $k$ so that the function $f(x)$ defined as:
$f(x) = \begin{cases} \frac{x^2 - 2x - 3}{x + 1}, & x \ne -1 \\ k, & x = -1 \end{cases}$
is continuous at $x = -1$.
(vi)[2.0]
Find the approximate change in the volume $V$ of a cube of side $x$ metres caused by decreasing the side by $1\%$.
(vii)[2.0]
Evaluate: $\int \frac{x^3 + 5x^2 + 4x + 1}{x^2} \, dx$.
(viii)[2.0]
Find the differential equation of the family of concentric circles $x^2 + y^2 = a^2$.
(ix)[2.0]
If $A$ and $B$ are events such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(A \cap B) = \frac{1}{4}$, then find:
(a) $P(A/B)$
(b) $P(B/A)$
(x)[2.0]
In a race, the probabilities of $A$ and $B$ winning the race are $\frac{1}{3}$ and $\frac{1}{6}$ respectively. Find the probability of neither of them winning the race.
Answer
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From ISC 2018 Mathematics Paper 1, question 1.