KCET 2025

The function $f(x)=\tan x-x$

The area bounded by the curve $y=\sin \left(\frac{x}{3}\right)$, $x$ axis, the lines $x=0$ and $x=3 \pi$ is

The area of the region bounded by the curve $y=x^2$ and the line $y=16$ is

If the number of terms in the binomial expansion of $(2 \mathrm{x}+3)^{3 \mathrm{n}}$ is 22 , then the value of n is

If $Z_1$ and $Z_2$ are two non-zero complex numbers, then which of the following is not true?

The value of $\int_{-1}^1 \sin ^5 \mathrm{x} \cos ^4 \mathrm{xdx}$ is

\(\text { The value of } \int_0^{2 \pi} \sqrt{1+\sin \left(\frac{x}{2}\right)} d x \text { is }\)

$\int_0^1 \log \left(\frac{1}{x}-1\right) d x$ is

General solution of the differential equation $\frac{d y}{d x}+y \tan x=\sec x$ is

If ' $a$ ' and ' $b$ ' are the order and degree respectively of the differentiable equation. $\left(\frac{d^2 y}{d x^2}\right)^2+\left(\frac{d y}{d x}\right)^3+x^4=0$, then $a-b=$ $\qquad$

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{\ma...

If $\mathrm{y}=\mathrm{a} \sin ^3 \mathrm{t}, \mathrm{x}=\mathrm{a} \cos ^3 \mathrm{t}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{t}=\frac{3 \pi}{4}$ is

The derivative of $\sin \mathrm{x}$ with respect to $\log \mathrm{x}$ is

The length of the latus rectum of $x^2+3 y^2=12$ is

Domain of the function $f$, given by $f(x)=\frac{1}{\sqrt{(x-2)(x-5)}}$ is

If $f(x)=\sin \left[\pi^2\right] x-\sin \left[-\pi^2\right] x$, where $[x]=$ greatest integer $\leq x$, then which of the following is not true?

Let the functions " f " and " g " be $\mathrm{f}:\left[0, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$ given by $\mathrm{f}(\mathrm{x})=\sin \mathrm{x}$ and $\mathrm{g}:\left[0, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$ given by $g(x)=\c...

The value of $\int \frac{\mathrm{dx}}{(\mathrm{x}+1)(\mathrm{x}+2)}$ is

$\int \frac{\mathrm{dx}}{\mathrm{x}^2\left(\mathrm{x}^4+1\right)^{3 / 4}}$ equals

\(\sec ^2\left(\tan ^{-1} 2\right)+\operatorname{cosec}^2\left(\cot ^{-1} 3\right)=\)

$2 \cos ^{-1} x=\sin ^{-1}\left(2 x \sqrt{1-x^2}\right)$ is valid for all values of ' $x$ ' satisfying

$\lim _{x \rightarrow 1} \frac{x^4-\sqrt{x}}{\sqrt{x}-1}$ is

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

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

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



Column - I
Column - II


(a)


x


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x

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x


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x

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x|x|
(i)
continuo...

Consider the following statements:
Statement (I): In a LPP, the objective function is always linear.
Statement (II): Ina LPP, the linear inequalities on variables are called constraints. Which of the following is correct?

The maximum value of $\mathrm{z}=3 \mathrm{x}+4 \mathrm{y}$, subject to the constraints $\mathrm{x}+\mathrm{y} \leq 40, \mathrm{x}+2 \mathrm{y} \leq 60$ and $\mathrm{x}, \mathrm{y} \geq 0$ is

If $A$ is a square matrix such that $A^2=A$, then $(I-A)^3$ is

If $A$ and $B$ are two matrices such that $A B$ is an identity matrix and the order of matrix $B$ is $3 \times 4$, then the order of matrix $A$ is

Which of the following statements is not correct?

If a matrix $A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$ satisfies $A^6=k A^{\prime}$, then the value of $k$ is

If $A=\left[\begin{array}{ll}k & 2 \\ 2 & k\end{array}\right]$ and $\left|A^3\right|=125$, then the value of $k$ is

If $A$ is a square matrix satisfying the equation $A^2-5 A+7 I=0$, where $I$ is the $I$ dentity matrix and 0 is null matrix of same order, then $A^{-1}=$

If $A$ is a square matrix of order $3 \times 3, \operatorname{det} A=3$, then the value of $\operatorname{det}\left(3 A^{-1}\right)$ is

If $B=\left[\begin{array}{ll}1 & 3 \\ 1 & \alpha\end{array}\right]$ be the adjoint of a matrix $A$ and $|A|=2$, then the value of $\alpha$ is

The number of four digit even number that can be formed using the digits $0,1,2$ and 3 without repetition is

\(\text { The number of diagonals that can be drawn in an octagon is }\)

A random experiment has five outcomes $\mathrm{w}_1, \mathrm{w}_2, \mathrm{w}_3, \mathrm{w}_4$ and $\mathrm{w}_5$. The probabilities of the occurrence of the outcomes $w_1, w_2, w_3, w_4$ and $w_5$ are respectively $\frac{1}{6}, a, b$ and $...

A die has two face each with number ' 1 ', three faces each with number ' 2 ' and one face with number ' 3 '. If the die is rolled once, then $\mathrm{P}(1$ or 3$)$ is

Consider the following statements.
Statement (I): If E and F are two independent events, then $E^{\prime}$ and $F^{\prime}$ are also independent.
Statement (II): Two mutually exclusive events with non-zero probabilities of occurrence cannot...

If A and B are two non-mutually exclusive events such that $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\mathrm{P}(\mathrm{B} \mid \mathrm{A})$, then

If $A$ and $B$ are two events such that $A \subset B$ and $P(B) \neq 0$, then which of the following is correct?

Meera visits only one of the two temples A and B in her locality. Probability that she visits temple A is $\frac{2}{5}$. If she visits temple $A, \frac{1}{3}$ is the probability that she meets her friend, whereas it is $\frac{2}{7}$ if she ...

If $4^{\text {th }}, 10^{\text {th }}$ and $16^{\text {th }}$ terms of a G.P. are $x, y$ and $z$ respectively, then

If $\mathrm{A}=\left\{\mathrm{x}: \mathrm{x}\right.$ is an integer and $\left.\mathrm{x}^2-9=0\right\}$
$B=\{x: x$ is a natural number and $2 \leq x<5\}$
$\mathrm{C}=\{\mathrm{x}: \mathrm{x}$ is a prime number $\leq 4\}$
Then $(B-C) \cup A$...

$A$ and $B$ are two sets having 3 and 6 elements respectively. Consider the following statements.
Statement (I): Minimum number of elements in AUB is 3
Statement (II): Maximum number of elements in AB is 3 Which of the following is correct?

\(\text { Let } A=\{a, b, c\} \text {, then the number of equivalence relations on A containing }(b, c) \text { is }\)

Consider the following statements :
Statement(I) : The set of all solutions of the linear inequalities $3 \mathrm{x}+8<17$ and $2 \mathrm{x}+8 \geq 12$ are $\mathrm{x}<3$ and $x \geq 2$ respectively.
Statement(II) : The common set of soluti...

The mean deviation about the mean for the date $4,7,8,9,10,12,13,17$ is

The system of equations $4 x+6 y=5$ and $8 x+12 y=10$ has