Download App

Question Bank [2022] — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]3 Marks
Question
Using properties of scalar triple product, prove that $\left[\begin{array}{llll}\overline{ a }+\overline{ b } & \overline{ b }+\overline{ c } & \overline{ c }+\overline{ a }\end{array}\right]=2\left[\begin{array}{lll}\overline{ a } & \overline{ b } & \overline{ c }\end{array}\right]$.
✓

Answer

$\text { L.H.S }=\left[\begin{array}{lll}\overline{ a }+\overline{ b } & \overline{ b }+\overline{ c } & \overline{ c }+\overline{ a }\right]$
$=(\overline{ a }+\overline{ b }) \cdot[(\overline{ b }+\overline{ c }) \times(\overline{ c }+\overline{ a })]$
$=(\overline{ a }+\overline{ b }) \cdot[\overline{ b } \times \overline{ c }+\overline{ b } \times \overline{ a }+\overline{ c } \times \overline{ c }+\overline{ c } \times \overline{ a }]$
$=(\overline{ a }+\overline{ b }) \cdot[\overline{ b } \times \overline{ c }+\overline{ b } \times \overline{ a }+\overline{ c } \times \overline{ a }] \quad \ldots[\because \overline{ c } \times \overline{ c }=\overline{0}]$
$=\overline{ a } \cdot[(\overline{ b } \times \overline{ c })+(\overline{ b } \times \overline{ a })+(\overline{ c } \times \overline{ a })]+\overline{ b } \cdot[(\overline{ b } \times \overline{ c })+(\overline{ b } \times \overline{ a })+(\overline{ c } \times \overline{ a })]$
$=\overline{ a } \cdot(\overline{ b } \times \overline{ c })+\overline{ a } \cdot(\overline{ b } \times \overline{ a })+\overline{ a } \cdot(\overline{ c } \times \overline{ a })+\overline{ b } \cdot(\overline{ b } \times \overline{ c })+\overline{ b }(\overline{ b } \times \overline{ a })+\overline{ b }(\overline{ c } \times \overline{ a })$
$=[\overline{ a } \overline{ b } \overline{ c }]+[\overline{ a } \overline{ b } \overline{ a }]+[\overline{ a } \overline{ c } \overline{ a }]+[\overline{ b } \overline{ b } \overline{ c }]+[\overline{ b } \overline{ b } \overline{ a }]+[\overline{ b } \overline{ c } \overline{ a }]$
$=[\overline{ a } \overline{ b } \overline{ c }]+0+0+0+0+[\overline{ a } \overline{ b } \overline{ c }]$
$=2[\overline{ a } \overline{ b } \overline{ c }]$
$=R \cdot H \cdot S$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(3 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the vector equation of the line passing through the point A(1, 2, -1) and parallel to the line 5x - 25 = 14 - 7y = 35z.
→
Classify the following functions as injection, surjection or bijection:
$f : R \rightarrow R,$ defined by $f(x) = \sin x$
→
Let $g(x)$ be the inverse of an invertible function $f(x)$ which is derivable at $x = 3$. If $f(3) = 3$ and $f(3) = 9$, write the value of $g'(9)$.
→
Determine whether $\text{f}(\text{x})=\frac{-\pi}{2}+\sin\text{x}$ is a increasing or decreasing on $\Big(\frac{-\pi}{3},\frac{\pi}{3}\Big).$
→
Evaluate the following integrals:
$\int\frac{1}{\sqrt{1-\text{x}^2}(\sin^{-1}\text{x})^2}\text{dx}$
→
Show that lines $\bar{r}=(\hat{i}+4 \hat{j})+\lambda(\hat{i}+2 \hat{j}+3 \hat{k})$ and

$\bar{r}=(3 \hat{j}-\hat{k})+\mu(2 \hat{i}+3 \hat{j}+4 \hat{k})$

→
Prove that : $\int_0^{2 a} f(x) d x=\int_0^a f(x) d x+\int_0^a f(2 a-x) d x$
→
Check the commutativity and associativity of the following binary operations:
'*' on Q defined by a * b = a + ab for all a, b ∈ Q.
→
Show that the lines $\frac{\text{x}-5}{7}=\frac{\text{y}+2}{-5}=\frac{\text{z}}{1}$ and $\frac{\text{x}}{1}=\frac{\text{y}}{2}=\frac{\text{z}}{3}$ are perpendicular to each other.
→
Evaluate: $\int_{-1}^1|5 x-3| d x$
→