Arts (English Medium) कक्षा ११ - CBSE Question Bank Solutions for Mathematics

a cos A + b cos B + c cos C = 2b sin A sin C 

In ∆ABC, prove the following: 

\[a^2 = \left( b + c \right)^2 - 4 bc \cos^2 \frac{A}{2}\]

In ∆ABC, prove the following:

\[4\left( bc \cos^2 \frac{A}{2} + ca \cos^2 \frac{B}{2} + ab \cos^2 \frac{C}{2} \right) = \left( a + b + c \right)^2\]

In ∆ABC, prove the following: 

\[\sin^3 A \cos \left( B - C \right) + \sin^3 B \cos \left( C - A \right) + \sin^3 C \cos \left( A - B \right) = 3 \sin A \sin B \sin C\]

In \[∆ ABC, \frac{b + c}{12} = \frac{c + a}{13} = \frac{a + b}{15}\]  Prove that \[\frac{\cos A}{2} = \frac{\cos B}{7} = \frac{\cos C}{11}\] 

In \[∆ ABC, if \angle B = 60°,\]  prove that \[\left( a + b + c \right) \left( a - b + c \right) = 3ca\]

If in \[∆ ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1\] prove that the triangle is right-angled. 

In \[∆ ABC \text{ if } \cos C = \frac{\sin A}{2 \sin B}\] prove that the triangle is isosceles.  

Two ships leave a port at the same time. One goes 24 km/hr in the direction N 38° E and other travels 32 km/hr in the direction S 52° E. Find the distance between the ships at the end of 3 hrs. 

Answer  the following questions in one word or one sentence or as per exact requirement of the question. 

Find the area of the triangle ∆ABC in which a = 1, b = 2 and \[\angle C = 60º\] 

Answer  the following questions in one word or one sentence or as per exact requirement of the question.In a ∆ABC, if b =\[\sqrt{3}\] and \[\angle A = 30°\]  find a. 

Answer  the following questions in one word or one sentence or as per exact requirement of the question. 

In a ∆ABC, if \[\cos A = \frac{\sin B}{2\sin C}\]  then show that c = a. 

Answer  the following questions in one word or one sentence or as per exact requirement of the question. 

In a ∆ABC, if b = 20, c = 21 and \[\sin A = \frac{3}{5}\] 

Answer  the following questions in one word or one sentence or as per exact requirement of the question.

In a ∆ABC, if sinA and sinB are the roots of the equation  \[c^2 x^2 - c\left( a + b \right)x + ab = 0\]  then find \[\angle C\]  

Answer the following questions in one word or one sentence or as per exact requirement of the question.  

In ∆ABC, if a = 8, b = 10, c = 12 and C = λA, find the value of λ. 

Answer the following questions in one word or one sentence or as per exact requirement of the question. 

If the sides of a triangle are proportional to 2, \[\sqrt{6}\] and \[\sqrt{3} - 1\] find the measure of its greatest angle. 

Answer the following questions in one word or one sentence or as per exact requirement of the question.  

If in a ∆ABC, \[\frac{\cos A}{a} = \frac{\cos B}{b} = \frac{\cos C}{c}\] then find the measures of angles A, B, C. 

Answer the following questions in one word or one sentence or as per exact requirement of the question. 

In any triangle ABC, find the value of \[a\sin\left( B - C \right) + b\sin\left( C - A \right) + c\sin\left( A - B \right)\ 

Answer the following questions in one word or one sentence or as per exact requirement of the question. 

In any ∆ABC, find the value of

\[\sum^{}_{}a\left( \text{ sin }B - \text{ sin }C \right)\]

Mark the correct alternative in each of the following:
In any ∆ABC, \[\sum^{}_{} a^2 \left( \sin B - \sin C \right)\] =