KCET 2021

The maximum slope of the curve \(y=-x^3+3 x^2+2 x-27\) is

A particle starts form rest and its angular displacement (in radians) is given by \(\theta=\frac{t^2}{20}+\frac{t}{5}\). If the angular velocity at the end of \(t=4\) is \(k\), then the value of \(5 k\) is

The cost and revenue functions of a product are given by \(c(x)=20 x+4000\) and \(R(x)=60 x+2000\) respectively, where \(\mathrm{x}\) is the number of items produced and sold. The value of \(x\) to earn profit is

The function \(f(x)=x^2-2 x\) is strictly decreasing in the interval

The area of the region bounded by \(y=-\sqrt{16-x^2}\) and \(X\)-axis is

If \(\left(\frac{1+i}{1-i}\right)^x=1\), then

The value of \(\int_0^{4042} \frac{\sqrt{x} d x}{\sqrt{x}+\sqrt{4042-x}}\) is equal to

If \(I_n=\int_0^{\frac{\pi}{4}} \tan ^n x d x\), where \(n\) is positive integer, then \(I_{10}+I_8\) is equal to

The number of solutions of \(\frac{d y}{d x}=\frac{y+1}{x-1}\), when \(y(\mathrm{l})=2\) is

Solution of differential equating \(x d y-y d x=0\) represents

If \(y=\left(\cos x^2\right)^2\), then \(\frac{d y}{d x}\) is equal to

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...

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

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

If \(a\) and \(b\) are fixed non-zero constants, then the derivative of \(\frac{a}{x^4}-\frac{b}{x^2}+\cos x\) is \(m a+n b-p\), where

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

If the area of the ellipse is \(\frac{x^2}{25}+\frac{y^2}{\lambda^2}=1\) is \(20 \pi\) sq units, then \(\lambda\) is

\(f: R \rightarrow R\) defined by \(f(x)\) is equal to $$\left\{\begin{array}{l}2 x, x> 3 \\ x^2, 1< x \leq 3, \text { then } f(-2)+f(3)+f(4) \text { is } \\ 3 x, x \leq 1\end{array}\right.$$

The function \(f(x)=\sqrt{3} \sin 2 x-\cos 2 x+4\) is one-one in the interval

Domain of the function
\(f(x)=\frac{1}{\sqrt{\left[x^2\right]-[x]-6}},\)
where \([x]\) is greatest integer \(\leq x\) is

Domain of \(f(x)=\frac{x}{1-|x|}\) is

Let \(A=\{x: x \in R, x\) is not a positive integer) Define \(f: A \rightarrow R\) as \(f(x)=\frac{2 x}{x-1}\), then \(f\) is

The value of \(\int \frac{x e^x d x}{(1+x)^2}\) is equal to

The value of \(\int e^x\left[\frac{1+\sin x}{1+\cos x}\right] d x\) is equal to

\(\int \frac{x^3 \sin \left(\tan ^{-1}\left(x^4\right)\right)}{1+x^8} d x\) is equal to

The value of \(\int \frac{x^2 d x}{\sqrt{x^6+a^6}}\) is equal to

\(\tan ^{-1}\left[\frac{1}{\sqrt{3}} \sin \frac{5 \pi}{2}\right] \sin ^{-1}\left[\cos \left(\sin ^{-} \frac{\sqrt{3}}{2}\right)\right]\) is equal to

\(\cos \left[\cot ^{-1}(-\sqrt{3})+\frac{\pi}{6}\right]\) is equal to

If $$f(x)=\left|\begin{array}{ccc}\cos x & 1 & 0 \\ 0 & 2 \cos x & 3 \\ 0 & 1 & 2 \cos x\end{array}\right|$$, then \(\lim _\limits{x \rightarrow \pi} f(x)\) is equal to

Consider the following statements
Statement 1 : \(\lim _\limits{x \rightarrow 1} \frac{a x^2+b x+c}{x^2+b x+a}\) is 1
(where \(a+b+c \neq 0\)).
Statement 2 : \(\lim _\limits{x \rightarrow -2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}\) is $$\frac...

At \(x=1\), the function
$$f(x)=\left\{\begin{array}{cc} x^3-1, & 1< x < \infty \\ x-1, & -\infty< x \leq 1 \end{array}\right. \text { is }$$

The shaded region is the solution set of the inequalities

If $$A=\left[\begin{array}{ccc}1 & -2 & 1 \\ 2 & 1 & 3\end{array}\right]$$
$$ B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 1\end{array}\right]$$, then \((A B)^{\prime}\) is equal to

Let \(M\) be \(2 \times 2\) symmetric matrix with integer entries, then \(M\) is invertible if

If \(A\) and \(B\) are invertible matrices then which of the following is not correct?

If \(A\) and \(B\) are matrices of order 3 and \(|A|=5,|B|=3\), then \(|3 A B|\) is

If \(x^3-2 x^2-9 x+18=0\) and $$A=\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & x & 6 \\ 7 & 8 & 9\end{array}\right|$$ then the maximum value of \(A\) is

If the parabola \(y=\alpha x^2-6 x+\beta\) passes through the point \((0,2)\) and has its tangent at \(x=\frac{3}{2}\) parallel to \(X\)-axis, then

\(A\) and \(B\) are non-singleton sets and \(n(A \times B)=35\). If \(B \subset A\), then \({ }^{n(A)} C_{n(B)}\) is equal to

A student has to answer 10 questions, choosing at least 4 from each of the parts \(A\) and \(B\). If there are 6 questions in part \(A\) and 7 in part \(B\), then the number of ways can the student choose 10 questions is

Given that, \(A\) and \(B\) are two events such that \(P(B)=\frac{3}{5}, P\left(\frac{A}{B}\right)=\frac{1}{2}\) and \(P(A \cup B)=\frac{4}{5}\), then \(P(A)\) is equal to

Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6 the probability of getting a sum as 3 is

If \(P(A)=0.59, P(B)=0.30\) and \(P(A \cap B)=0.21\) then \(P\left(A^{\prime} \cap B^{\prime}\right)\) is equal to

If \(A, B\) and \(C\) are three independent events such that \(P(A)=P(B)=P(C)=P\), then \(P\) (at least two of \(A, B\) and \(C\) occur) is equal to

A car manufacturing factory has two plants \(X\) and \(Y\). Plant \(X\) manufactures \(70 \%\) of cars and plant \(Y\) manufactures \(30 \%\) of cars. \(80 \%\) of cars at plant \(X\) and \(90 %\) of cars at plant \(Y\) are rated as standar...

If the middle term of the AP is 300, then the sum of its first 51 terms is

In a certain two \(65 \%\) families own cell phones, 15000 families own scooter and \(15 \%\) families own both. Taking into consideration that the families own at least one of the two, the total number of families in the town is

The standard deviation of the numbers \(31,32,33 \ldots 46,47\) is

The equation of straight line which passes through the point \(\left(a \cos ^3 \theta, a \sin ^3 \theta\right)\) and perpendicular to \(x \sec \theta+y \operatorname{cosec} \theta=a\) is

If a plane meets the coordinate axes at \(A, B\) and \(C\) in such a way that the centroid of \(\triangle A B C\) is at the point \((1,2,3)\), then the equation of the plane is