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

In triangle ABC, prove the following: 

\[a \left( \cos B \cos C + \cos A \right) = b \left( \cos C \cos A + \cos B \right) = c \left( \cos A \cos B + \cos C \right)\]

In ∆ABC, prove that \[a \left( \cos C - \cos B \right) = 2 \left( b - c \right) \cos^2 \frac{A}{2} .\] 

In ∆ABC, prove that if θ be any angle, then b cosθ = c cos (A − θ) + a cos (C + θ). 

In ∆ABC, if sin² A + sin² B = sin² C. show that the triangle is right-angled. 

In ∆ABC, if a², b² and c² are in A.P., prove that cot A, cot B and cot C are also in A.P. 

The upper part of a tree broken by the wind makes an angle of 30° with the ground and the distance from the root to the point where the top of the tree touches the ground is 15 m. Using sine rule, find the height of the tree. 

At the foot of a mountain, the elevation of it summit is 45°; after ascending 1000 m towards the mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the mountain. 

A person observes the angle of elevation of the peak of a hill from a station to be α. He walks c metres along a slope inclined at an angle β and finds the angle of elevation of the peak of the hill to be ϒ. Show that the height of the peak above the ground is \[\frac{c \sin \alpha \sin \left( \gamma - \beta \right)}{\left( \sin \gamma - \alpha \right)}\] 

If the sides a, b and c of ∆ABC are in H.P., prove that \[\sin^2 \frac{A}{2}, \sin^2 \frac{B}{2} \text{ and } \sin^2 \frac{C}{2}\]

In \[∆ ABC, if a = 5, b = 6 a\text{ and } C = 60°\]  show that its area is \[\frac{15\sqrt{3}}{2} sq\].units. 

In \[∆ ABC, if a = \sqrt{2}, b = \sqrt{3} \text{ and } c = \sqrt{5}\] show that its area is \[\frac{1}{2}\sqrt{6} sq .\] units.

The sides of a triangle are a = 4, b = 6 and c = 8. Show that \[8 \cos A + 16 \cos B + 4 \cos C = 17\]

In ∆ ABC, if a = 18, b = 24 and c = 30, find cos A, cos B and cos C. 

In ∆ABC, prove the following: \[b \left( c \cos A - a \cos C \right) = c^2 - a^2\]

In ∆ABC, prove the following: \[c \left( a \cos B - b \cos A \right) = a^2 - b^2\]

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  \[a \left( \cos B + \cos C - 1 \right) + b \left( \cos C + \cos A - 1 \right) + c\left( \cos A + \cos B - 1 \right) = 0\]