Science (English Medium) Class 11 - CBSE Question Bank Solutions

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

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.