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Sum of the first six terms of an arithmetic sequence is $9$. Sum of the first twelve terms is $90$. Find the sum of the thirteenth and ...

Solve for matrices $A$ and $B$: $2A + B = \begin{bmatrix} 3 & -4 \\ 2 & 7 \end{bmatrix}$ and $A - 2B = \begin{bmatrix} 4 ...

The eighth term of an A.P is half its second term and the eleventh term exceeds one third of its fourth term by $1$. Find the $15^{th}$...

If $\;$ $\text{cosec} \theta - \sin \theta = a^3$, $\;$ $\sec \theta - \cos \theta = b^3$, then prove that $\;$ $a^2 \; b^2 \left(a^2...

Prove the identity: $\dfrac{\sin \left(2 \alpha + \beta\right)}{\sin \alpha} - 2 \cos \left(\alpha + \beta\right) = \dfrac{\sin \beta...

Prove the identity: $\sec \left(\dfrac{\pi}{4} + \alpha\right) \sec \left(\dfrac{\pi}{4} - \alpha\right) = 2 \sec 2 \alpha$ LH...

Two cyclists left the same point simultaneously and traveled in the same direction. The speed of the first was $15$ km/h and that of th...