Differentiation of a Determinant

Q.

If $3\sin \left(xy\right)+4\cos \left(xy\right)=5$, then $\frac{dy}{dx}=$

1. $\frac{-y}{x}$

2. $\frac{y}{x}$

3. $\frac{x}{y}$

4. none of these

Q.

Solve: $2(\sin x-\cos 2x)-\sin 2x(1+2\sin x)+2\cos x=0$

Q. If $\left(\begin{array}{cc}{\lambda }^3+2\lambda & \lambda +1\\ {\lambda }^2+1& 2+\lambda \end{array}\right|=c{\lambda }^4+d{\lambda }^3+e{\lambda }^2+g\lambda +f$, the value of ‘f ’ is
यदि $\left(\begin{array}{cc}{\lambda }^3+2\lambda & \lambda +1\\ {\lambda }^2+1& 2+\lambda \end{array}\right|=c{\lambda }^4+d{\lambda }^3+e{\lambda }^2+g\lambda +f$ है, तब ‘f’ का मान है

1. –1
2. –3
3. 1
4. 3

Q. Let $f\left(x\right)=\left|\begin{array}{ccc}cosx& sinx& cosx\\ cox2x& sin2x& 2cos2x\\ cos3x& sin3x& 3cos3x\end{array}\right|$, then $f^{\prime }\left(\frac{\pi }{2}\right)$ is equal to

1. 8
2. 6
3. 4
4. 2

Q.

$\mathrm{If}f,g\mathrm{and}h\mathrm{are}\mathrm{differentiable}\mathrm{functions}\mathrm{of}x\mathrm{and}△\left(x\right)=\left(\begin{array}{ccc}f& g& h\\ \left(xf\right)\text{'}& \left(xg\right)\text{'}& \left(xh\right)\text{'}\\ \left(x^{2}f\right)\text{'}\text{'}& \left(x^{2}g\right)\text{'}\text{'}& \left(x^{2}h\right)\text{'}\text{'}\end{array}\right|,\mathrm{then}△\text{'}\left(x\right)\mathrm{is}$

1. $\left(\begin{array}{ccc}f& g& h\\ f\text{'}& g\text{'}& h\text{'}\\ (x^{3}f\text{'}\text{'})\text{'}& (x^{3}h\text{'}\text{'})\text{'}& (x^{3}h\text{'}\text{'})\text{'}\end{array}\right|$

2. $\left(\begin{array}{ccc}f& g& h\\ f\text{'}& g\text{'}& h\text{'}\\ (x^{3}f\text{'}\text{'})& (x^{3}h\text{'}\text{'})& (x^{3}h\text{'}\text{'})\end{array}\right|$

3. $\left(\begin{array}{ccc}f& g& h\\ f\text{'}& g\text{'}& h\text{'}\\ (x^{3}f\text{'}\text{'})\text{'}\text{'}& (x^{3}h\text{'}\text{'})\text{'}\text{'}& (x^{3}h\text{'}\text{'})\text{'}\text{'}\end{array}\right|$

4. $0$

Q. If $\Delta \left(x\right)=\left|\begin{array}{ccc}{x}^n& sinx& cosx\\ n!& sin\frac{n\pi }{2}& cos\frac{n\pi }{2}\\ a& a^2& a^3\end{array}\right|$, then the value of $\frac{d^n}{d{x}^n}\left[\Delta \right(x\left)\right]atx=0$ is

1. -1
2. \N
3. 1
4. Dependent of a

Q. Let $\left|\begin{array}{ccc}x& 2& x\\ x^2& x& 6\\ x& x& 6\end{array}\right|=A{x}^4+B{x}^3+C{x}^2+Dx+E$, then the value of 5A + 4B + 3C + 2D + E is equal to

1. -17
2. -14
3. -11
4. -8