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[10×2] Determine whether the binary operation on defined by is commutative. Also, find the value of…
[10×2]
(i)[2.0]
Determine whether the binary operation $*$ on $\mathbb{R}$ defined by $a * b = |a - b|$ is commutative. Also, find the value of $(-3) * 2$.
(ii)[2.0]
Prove that: $\tan^2(\sec^{-1} 2) + \cot^2(\operatorname{cosec}^{-1} 3) = 11$.
Given: $\tan^2(\sec^{-1} 2) + \cot^2(\operatorname{cosec}^{-1} 3)$
To show: $11$
(iii)[2.0]
Without expanding at any stage, find the value of the determinant:
$\Delta = \begin{vmatrix} 20 & a & b + c \\ 20 & b & a + c \\ 20 & c & a + b \end{vmatrix}$
(iv)[2.0]
If $\begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \begin{pmatrix} 1 & -3 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -4 & 6 \\ -9 & x \end{pmatrix}$, find $x$.
(v)[2.0]
Find $\frac{dy}{dx}$ if $x^3 + y^3 = 3axy$.
(vi)[2.0]
The edge of a variable cube is increasing at the rate of $10\text{ cm/sec}$. How fast is the volume of the cube increasing when the edge is $5\text{ cm}$ long?
(vii)[2.0]
Evaluate: $\int_{4}^{5} |x - 5| \, dx$.
(viii)[2.0]
Form a differential equation of the family of the curves $y^2 = 4ax$.
(ix)[2.0]
A bag contains 5 white, 7 red and 4 black balls. If four balls are drawn one by one with replacement, what is the probability that none is white?
(x)[2.0]
Let $A$ and $B$ be two events such that $P(A) = \frac{1}{2}$, $P(B) = p$ and $P(A \cup B) = \frac{3}{5}$, find ‘$p$’ if $A$ and $B$ are independent events.
Answer
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From ISC 2020 Mathematics Paper 1, question 1.