If $f(x) = \sin x \cdot \sin 2x \cdot \sin 3x$ and $f''(x) = a(\sin bx) + c(\sin dx) + e(\sin kx)$,then the value of $(a+c+e) - (b+d+k)$ equals

Let $f(x)$ be a twice differentiable function such that $f''(x) = -f(x)$, $f'(x) = g(x)$, and $h(x) = \{f(x)\}^2 + \{g(x)\}^2$. If $h(5) = 11$, then $h(10)$ is equal to ...

If $y = (\sin^{-1} 2x)^2 + (\cos^{-1} 2x)^2$, then $(1 - 4x^2) y_2 - 4x y_1 = $

If $y^{2}=a x^{2}+b x+c$,where $a, b, c$ are constants,then $y^{3} \frac{d^{2} y}{d x^{2}}$ is equal to

If $y = a{e^x} + b{e^{-x}} + c$ where $a, b, c$ are parameters,then $y''' = $

Let $f(x)=ax^3+bx^2+cx+41$ be such that $f(1)=40, f'(1)=2$ and $f''(1)=4$. Then $a^2+b^2+c^2$ is equal to :