If in a , + =6, then = - Vidyadip

MCQ

If in $\text{a}\triangle\text{ABC},\tan\text{B}+\tan\text{C}=6,$ then $\cot\text{A}\cot\text{B}\cot\text{C}=$

A
$6$
B
$1$
✓
$\frac16$
D
None of these

✓

Answer

Correct option: C.

$\frac16$

In $\triangle\text{ABC},$
$\text{A+B+C}=\pi$
We know that $\tan(\text{A+B+C)}=\frac{\tan\text{A}+\tan\text{B}+\tan\text{C}-\tan\text{A}\tan\text{B}\tan\text{C}}{1-\tan\text{A}\tan\text{B}-\tan\text{B}\tan\text{C}-\tan\text{C}\tan\text{A}}$ and $\tan\pi=0.$
$\therefore\tan\text{A}+\tan\text{B}+\tan\text{C}-\tan\text{A}\tan\text{B}\tan\text{C}=0$
$\tan\text{A}+\tan\text{B}+\tan\text{C}=\tan\text{A}\tan\text{B}\tan\text{C}$
If $\tan\text{A}+\tan\text{B}+\tan\text{C}=6,\tan\text{A}\tan\text{B}\tan\text{C}=6$
$\Rightarrow\frac{1}{\tan\text{A}\tan\text{B}\tan\text{C}}=\frac16$
$\Rightarrow\cot\text{A}\cot\text{B}\cot\text{C}=\frac16$

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 Arithmetic Progressions→Arithmetic Progressions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Find the mean deviation about the mean for the data

Height in cm	$95-105$	$105-115$	$115-125$	$125-135$	$135-145$	$145-155$
Number of boys	$9$	$13$	$26$	$30$	$12$	$10$

$\text { If } u =\{1,2,3,4,5,6,7,8,9,10,12,24\}$
$A =\{ x : x \text { is prime and } x \leq 10\}$
$B =\{ x : x \text { is a factor of } 24\}$
Verify the following result
$i. A - B = A \cap B^{\prime}$
$ii. (A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$
$iii. (A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$

Find the sum of n terms of the geometric progression $1, -a, a^2, -a^3 ... n$ terms $($if a $\ne -1).$

There are $10$ lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.
$[$Hint: Required number $= 2^{10} – 1].$

Which term of the sequence $12+8\text{i},\ 6\text{i},\ 10+4\text{i},\ ...$ is $(a)$ real $(b)$ purely imaginary$?$

If $\cos\text{P}=\frac{1}{7}$ then $\cos\text{Q}=\frac{13}{14},$ where $P$ and $Q$ both are acute angles. Then, the value of $P - Q$ is:

Match each item given under the column $C_1$ to its correct answer given under the column $C_2.$
There are $3$ books on Mathematics$, 4$ on Physics and $5$ on English. How many different collections can be made such that each collection consists of :

	$C_1$		$C_2$
$(a)$	One book of each subject.	$(i)$	$3968$
$(b)$	At least one book of each subject.	$(ii)$	$60$
$(c)$	At least one book of English.	$(iii)$	$3255$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{\sqrt{2}-\cos\text{x}-\sin\text{x}}{(4\text{x}-\pi)^2}$

From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them, will join or none of them will join. In how many ways can the excursion party be chosen?

Show that the statement
$p:$ “If $x$ is a real number such that $x^3 + 4x = 0,$ then $x$ is $0”$ is true by

direct method,
method of contradiction,
method of contrapositive