1
KCET 2025
MCQ (Single Correct Answer)
+1
-0
$\lim _{x \rightarrow 1} \frac{x^4-\sqrt{x}}{\sqrt{x}-1}$ is
A
0
B
7
C
Does not exist
D
$\frac{1}{2}$
2
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
3
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
4
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
$a=3, b=-1$
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