Science (English Medium) इयत्ता ११ - CBSE Question Bank Solutions

In ∆ABC, prove  the following: 

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

In ∆ABC, prove the following: 

\[\left( c^2 - a^2 + b^2 \right) \tan A = \left( a^2 - b^2 + c^2 \right) \tan B = \left( b^2 - c^2 + a^2 \right) \tan C\] 

In ∆ABC, prove the following:

\[\frac{c - b \cos A}{b - c \cos A} = \frac{\cos B}{\cos C}\] 

In ∆ABC, prove that  \[a \left( \cos B + \cos C - 1 \right) + b \left( \cos C + \cos A - 1 \right) + c\left( \cos A + \cos B - 1 \right) = 0\]

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.