1
KCET 2021
+1
-0

For constant $$a, \frac{d}{d x}\left(x^x+x^a+a^x+a^a\right)$$ is

A
$$x^x(1+\log x)+a x^{a-1}$$
B
$$x^x(1+\log x)+a x^{a-1}+a^x \log a$$
C
$$x^x(1+\log x)+a^a(1+\log x)$$
D
$$x^x(1+\log x)+a^a(1+\log a)+a x^{a-1}$$
2
KCET 2021
+1
-0

Consider the following statements

Statement 1 : If $$y=\log _{10} x+\log _e x$$, then $$\frac{d y}{d x}=\frac{\log _{10} e}{x}+\frac{1}{x}$$

Statement 2 : If $$\frac{d}{d x}\left(\log _{10} x\right)=\frac{\log x}{\log 10}$$ and $$\frac{d}{d x}\left(\log _e x\right)=\frac{\log x}{\log e}$$

A
Statement 1 is true, Statement 2 is false.
B
Statement 1 is false, statement 2 is true.
C
Both statements 1 and 2 are true.
D
Both statements 1 and 2 are false.
3
KCET 2021
+1
-0

If the parametric equation of curve is given by $$x=\cos \theta+\log \tan \frac{\theta}{2}$$ and $$y=\sin \theta$$, then the points for which $$\frac{d y}{d x}=0$$ are given by

A
$$\theta=\frac{n \pi}{2}, n \in Z$$
B
$$\theta=(2 n+1) \frac{\pi}{2}, n \in Z$$
C
$$\theta=(2 n+1) \pi, n \in Z$$
D
$$\theta=n \pi, n \in z$$
4
KCET 2021
+1
-0

If $$y=(x-1)^2(x-2)^3(x-3)^5$$, then $$\frac{d y}{d x}$$ at $$x=4$$ is equal to

A
108
B
54
C
36
D
516
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NEET