JEE Advance - Mathematics (2013 - Paper 2 Offline - No. 1)
$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$I$$
(P.)$$\,\,\,\,$$ Volume of parallelopiped determined by vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ is $$2.$$ Then the volume of the parallelepiped determined by vectors $$2\left( {\overrightarrow a \times \overrightarrow b } \right),3\left( {\overrightarrow b \times \overrightarrow c } \right)$$ and $$\left( {\overrightarrow c \times \overrightarrow a } \right)$$ is
(Q.)$$\,\,\,\,$$ Volume of parallelopiped determined by vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ is $$5.$$ Then the volume of the parallelepiped determined by vectors $$3\left( {\overrightarrow a + \overrightarrow b } \right),\left( {\overrightarrow b + \overrightarrow c } \right)$$ and $$2\left( {\overrightarrow c + \overrightarrow a } \right)$$ is
(R.)$$\,\,\,\,$$ Area of a triangle with adjacent sides determined by vectors $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$20.$$ Then the area of the triangle with adjacent sides determined by vectors $$\left( {2\overrightarrow a + 3\overrightarrow b } \right)$$ and $$\left( {\overrightarrow a - \overrightarrow b } \right)$$ is
(S.)$$\,\,\,\,$$ Area of a parallelogram with adjacent sides determined by vectors $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$30.$$ Then the area of the parallelogram with adjacent sides determined by vectors $$\left( {\overrightarrow a + \overrightarrow b } \right)$$ and $${\overrightarrow a }$$ is
$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$II$$
(1.)$$\,\,\,\,$$ $$100$$
(2.)$$\,\,\,\,$$ $$30$$
(3.)$$\,\,\,\,$$ $$24$$
(4.)$$\,\,\,\,$$ $$60$$
Explanation
(P) Given, the volume of parallelopiped formed by $$\vec{a}, \vec{b}, \vec{c}$$ is 2
$$\Rightarrow[\vec{a} \vec{b} \vec{c}]=2$$
Let V be the volume of parallelopiped formed by $$2(\vec{a} \times \vec{b}), 3(\vec{b} \times \vec{c})$$ and $$(\vec{c} \times \vec{a})$$
$$\begin{aligned} & \Rightarrow \mathrm{V}=\left[\begin{array}{lll} 2(\vec{a} \times \vec{b}) & 3(\vec{b} \times \vec{c}) & (\vec{c} \times \vec{a}) \end{array}\right] \\ & \Rightarrow \mathrm{V}=2 \times 3[(\vec{a} \times \vec{b})(\vec{b} \times \vec{c})(\vec{c} \times \vec{a})] \\ & \Rightarrow \mathrm{V}=6[\vec{a} \vec{b} \vec{c}]^2 \\ & \Rightarrow \mathrm{V}=6 \times 2^2 \\ & \Rightarrow \mathrm{V}=24 \\ \end{aligned}$$
Hence, P match with 3.
(Q) Given, the volume of parallelopiped formed by $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ is 5.
$$\Rightarrow[\vec{a} \vec{b} \vec{c}]=5 \quad \text{... (i)}$$
Let V be the volume of parallelopiped formed by $$3(\vec{a}+\vec{b}),(\vec{b}+\vec{c})$$ and $$2(\vec{c}+\vec{a})$$.
$$\Rightarrow \mathrm{V}=\left[\begin{array}{lll} 3(\vec{a}+\vec{b}) & (\vec{b}+\vec{c}) & 2(\vec{c}+\vec{a}) \end{array}\right]$$
$$\begin{aligned} & \Rightarrow \mathrm{V}=3.2[(\vec{a}+\vec{b})(\vec{b}+\vec{c})(\vec{c}+\vec{a})] \\ & \Rightarrow \mathrm{V}=6 \times 2\left[\begin{array}{lll} \vec{a} & \vec{b} & \vec{c} \end{array}\right] \\ & \Rightarrow \mathrm{V}=12 \times 5 \\ & \Rightarrow \mathrm{V}=60 \end{aligned}$$
Hence, Q match with 4.
(R) Given, the area of a triangle with adjacent sides $$\vec{a}$$ and $$\vec{b}$$ is 20 .
$$\begin{array}{ll} \Rightarrow & \frac{1}{2}|\vec{a} \times \vec{b}|=20 \\ \Rightarrow & |\vec{a} \times \vec{b}|=40 \quad \text{.... (i)} \end{array}$$
Let $$\Delta$$ be the area of a triangle with adjacent sides $$(2 \vec{a}+3 \vec{b})$$ and $$(\vec{a}-\vec{b})$$.
$$\begin{aligned} & \Rightarrow \quad \Delta=\frac{1}{2}|(2 \vec{a}+3 \vec{b}) \times(\vec{a}-\vec{b})| \\ & \Rightarrow \Delta=\frac{1}{2}|2 \vec{a} \times \vec{a}-2 \vec{a} \times \vec{b}+3 \vec{b} \times \vec{a}-3 \vec{b} \times \vec{b}| \\ & \Rightarrow \Delta=\frac{1}{2}|0-2 \vec{a} \times \vec{b}-3 \vec{a} \times \vec{b}-0| \\ & \Rightarrow \Delta=\frac{5}{2}|\vec{a} \times \vec{b}| \\ & \Rightarrow \Delta=\frac{5}{2} \times 40 \\ & \Rightarrow \quad \Delta=100 \end{aligned}$$
Hence, R match with 1.
(S) Given, the area of parallelogram with adjacent sides $$\vec{a}$$ and $$\vec{b}$$ is 30.
$$\Rightarrow|\vec{a} \times \vec{b}|=30 \quad \text{... (i)}$$
Let, $$\Delta^{\prime}$$ be the area of parallelogram with adjacent sides $$(\vec{a}+\vec{b})$$ and $$\vec{a}$$.
$$\begin{array}{ll} \Rightarrow & \Delta^{\prime}=|(\vec{a}+\vec{b}) \times \vec{a}| \\ \Rightarrow & \Delta^{\prime}=|\vec{a} \times \vec{a}+\vec{b} \times \vec{a}| \\ \Rightarrow & \Delta^{\prime}=|0-\vec{a} \times \vec{b}| \\ \Rightarrow & \Delta^{\prime}=|\vec{a} \times \vec{b}| \\ \Rightarrow & \Delta^{\prime}=30 \end{array}$$
Hence, S match with 2.
Hints :
(i) $$[\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}]=[\vec{a} \vec{b} \vec{c}]^2$$
(ii) $$\left[\begin{array}{lll}\vec{a}+\vec{b} & \vec{b}+\vec{c} & \vec{c}+\vec{a}\end{array}\right]=2\left[\begin{array}{ll}\vec{a} \vec{b} \vec{c}\end{array}\right]$$
(iii) $$\vec{A} \cdot(\vec{B} \times \vec{C})=[\vec{A} \vec{B} \vec{C}]$$
(iv) $$\vec{A} \times \vec{B}=-\vec{B} \times \vec{A}$$
(v) $$\vec{A} \times \vec{A}=\vec{B} \times \vec{B}=0$$
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