Download App

INVERSE TRIGNOMETRIC FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINVERSE TRIGNOMETRIC FUNCTIONS3 Marks
Question
Using properties of determinants, prove the following:
$\begin{vmatrix}\text{a}+\text{b}+\text{c} & -\text{c} & -\text{b} \\-\text{c} & \text{a}+\text{b}+\text{c} & -\text{a}\\-\text{b} & -\text{a} & \text{a}+\text{b}+\text{c} \end{vmatrix}=2(\text{a}+\text{b})(\text{b}+\text{c})(\text{c}+\text{a}).$
✓

Answer

$\begin{vmatrix}\text{a}+\text{b}+\text{c} & -\text{c} & -\text{b} \\-\text{c} & \text{a}+\text{b}+\text{c} & -\text{a}\\-\text{b} & -\text{a} & \text{a}+\text{b}+\text{c} \end{vmatrix}$
Applying $\text{R}_1\rightarrow\text{R}_1+\text{R}_2+\text{R}_3,$ we get
$\begin{vmatrix}\text{a} & \text{b} & \text{c} \\-\text{c} & \text{a}+\text{b}+\text{c} & -\text{a}\\-\text{b} & -\text{a} & \text{a}+\text{b}+\text{c} \end{vmatrix}$
Applying $\text{R}_3\rightarrow\text{R}_3-\text{R}_1,+\text{R}_2\rightarrow\text{R}_2-\text{R}_1$
$\begin{vmatrix}\text{a} & \text{b} & \text{c} \\-(\text{a}+\text{c}) & (\text{a}+\text{c}) & -(\text{a}+\text{c})\\-(\text{a}+\text{b}) & -(\text{a}+\text{b}) & (\text{a}+\text{b}) \end{vmatrix}$
Taking (a + c) and (a + b) common from $R_2$ and $R_3$ respectively.
$(\text{a}+\text{b})(\text{a}+\text{c})\begin{vmatrix}\text{a} & \text{b} & \text{c} \\-1 & 1 & -1\\-1 & -1 & 1 \end{vmatrix}$
expanding along $R_3$
$(\text{a}+\text{c})(\text{a}+\text{b})[\text{a}(1-1)-\text{b}(-1-1)+\text{c}(1+1)]$
$(\text{a}+\text{c})(\text{a}+\text{b})[-2\text{b}-2\text{c}]$
$=2(\text{a}+\text{b})(\text{b}+\text{c})(\text{c}+\text{a})$

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 "3 Marks Question" from INVERSE TRIGNOMETRIC FUNCTIONS→INVERSE TRIGNOMETRIC FUNCTIONS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Write the value of b which $\text{f(x)}=\begin{cases}5\text{x}-4,&0<\text{x}\leq1\\4\text{x}^2+3\text{bx},&1<\text{x}<2\end{cases}$ is continuous at x = 1.
→
Show that function$
f(x)=\left\{\begin{array}{ll}
3-x, & \text { if } x<1 \\
2, & \text { if } x=1 \\
1+x, & \text { if } x>1
\end{array}\right.
$
is continuous at $x=1$.
→
Evaluate the following integrals:
$\int\text{x}^2\cos2\text{x dx}$
→
Write the angle between the lines 2x = 3y = -z and 6x = -y = -4z.
→
For each of the differential equations in find the general solution:
$\text{x}^5\frac{\text{dy}}{\text{dx}}=-\text{y}^5$
→
Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.
→
Solve the following differential equations:$\text{ye}^{\frac{\text{x}}{\text{y}}}\text{dx}=(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}^2)\text{dy, y}\neq0$
→
Give an example of a map:
  1. Which is one-one but not onto.
  2. Which is not one-one but onto.
  3. Which is neither one-one nor onto.
→
A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.
→
Find gof and fog when f : R → R and g : R → R are defined by:
f(x) = x and g(x) = |x|
→