1
KCET 2025
MCQ (Single Correct Answer)
+1
-0

If $y=\frac{\cos x}{1+\sin x}$, then

(a) $\frac{d y}{d x}=\frac{-1}{1+\sin x}$

(b) $\frac{d y}{d x}=\frac{1}{1+\sin x}$

(c) $\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{1}{2} \sec ^2\left(\frac{\pi}{4}-\frac{\mathrm{x}}{2}\right)$

(d) $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2} \sec ^2\left(\frac{\pi}{4}-\frac{\mathrm{x}}{2}\right)$

A
Only $b$ is correct
B
Only a is correct
C
Both a and c are correct
D
Both b and d are correct
2
KCET 2025
MCQ (Single Correct Answer)
+1
-0

Match the following:

In the following, $[\mathrm{x}]$ denotes the greatest integer less than or equal to x .

Column - I Column - II
(a) x | x | x | x | x|x| (i) continuous in (-1, 1)
(b) | x | | x | sqrt(|x|) (ii) differentiable in (-1, 1)
(c) x + [ x ] x + [ x ] x+[x] (iii) strictly increasing in (-1, 1)
(d) | x 1 | + | x + 1 | | x 1 | + | x + 1 | |x-1|+|x+1| (iv) not differentiable at, at least one point in (-1, 1)
A
$\mathrm{a}-\mathrm{i}, \mathrm{b}-\mathrm{ii}, \mathrm{c}-\mathrm{iv}, \mathrm{d}-\mathrm{iii}$
B
$a-i v, b-i i i, c-i, d-i i$
C
a - ii, b - iv, c - iii, d - i
D
a - iii, b - ii, c - iv, d - i
3
KCET 2025
MCQ (Single Correct Answer)
+1
-0

The function $f(x)=\left\{\begin{array}{ll}e^x+a x & , x<0 \\ b(x-1)^2 & , x \geq 0\end{array}\right.$ is differentiable at $x=0$. Then

A
$a=1, b=1$
B
$a=3, b=1$
C
$a=-3, b=1$
D
$a=3, b=-1$
4
KCET 2025
MCQ (Single Correct Answer)
+1
-0

$$ \text { A function } f(x)=\left\{\begin{array}{cl} \frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0 \end{array}\right. $$

A
continuous at $\mathrm{x}=0$
B
not continuous at $x=0$
C
differentiable at $x=0$
D
differentiable at $\mathrm{x}=0$, but not continuous at $\mathrm{x}=0$
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