KCET 2019

The sides of an equilateral triangle are increasing at the rate of \(4 \mathrm{~cm} / \mathrm{sec}\). The rate at which its area is increasing, when the side is \(14 \mathrm{~cm}\)

The interval in which the function \(f(x)=x^3-6 x^2+9 x+10\) is increasing in

The area of the region above \(X\)-axis included between the parabola \(y^2=x\) and the circle \(x^2+y^2=2 x\) in square units is

The area of the region bounded by \(Y\)-axis, \(y=\cos x\) and \(y=\sin x, 0 \leq x \leq \frac{\pi}{2}\) is

The number of terms in the expansion of \(\left(x^2+y^2\right)^{25}-\left(x^2-y^2\right)^{25}\) after simplification is

\(\int_\limits{-3}^3 \cot ^{-1} x d x=\)

\(\int_\limits0^2\left[x^2\right] d x=\)

\(\int_\limits0^1 \sqrt{\frac{1+x}{1-x}} d x=\)

The integrating factor of the differential equation \(\left(2 x+3 y^2\right) d y=y d x(y>0)\) is

The equation of the curve passing through the point \((1,1)\) such that the slope of the tangent at any point \((x, y)\) is equal to the product of its co-ordinates is

The order of the differential equation \(y=C_1 e^{C_2+x}+C_3 e^{C_4+x}\) is

If \([x]\) represents the greatest integer function and \(f(x)=x-[x]-\cos x\), then \(f^{\prime}\left(\frac{\pi}{2}\right)=\)

If \(x=a \sec ^2 \theta\) & \(y=a \tan ^2 \theta\), then \(\frac{d^2 y}{d x^2}=\)

\(\sqrt[3]{y} \sqrt{x}=\sqrt[6]{(x+y)^5}\), then \(\frac{d y}{d x}=\)

The eccentricity of the ellipse \(9 x^2+25 y^2=225\) is

\(f: R \rightarrow R\) and \(g:[0, \infty) \rightarrow R\) is defined by \(f(x)=x^2\) and \(g(x)=\sqrt{x}\). Which one of the following is not true?

If \(|3 x-5| \leq 2\) then

The value of \(\sqrt{24.99}\) is

The domain of the function \(f: R \rightarrow R\) defined by \(f(x)=\sqrt{x^2-7 x+12}\) is

\(\int x^3 \sin 3 x d x=\)

\(\int \frac{1}{\sqrt{x}+x \sqrt{x}} d x=\)

$$\begin{aligned} & \int \frac{2 x-1}{(x-1)(x+2)(x-3)} d x \\ & \quad=A \log |x-1|+B \log |x+2|+C \log |x-3|+K \end{aligned}$$
Then \(A, B, C\) are respectively

If \(f(x)=\sin ^{-1}\left(\frac{2^{x+1}}{1+4^x}\right)\) then \(f^{\prime}(0)=\)

\(\cos \left[2 \sin ^{-1} \frac{3}{4}+\cos ^{-1} \frac{3}{4}\right]=\)

If \(a+\frac{\pi}{2}<2 \tan ^{-1} x+3 \cot ^{-1} x< b\) then '\(a\)' and '\(b\)' are respectively.

If $$f(x)=\left\{\begin{array}{cl}\frac{\sin 3 x}{e^{2 x}-1} ; & x \neq 0 \\ k-2 ; & x=0\end{array}\right.$$ is continuous at \(x=0\), then \(k=\)

\(\sum_\limits{r=1}^n(2 r-1)=x\) then, \(\lim _\limits{n \rightarrow \infty}\left[\frac{1^3}{x^2}+\frac{2^3}{x^2}+\frac{3^3}{x^2}+\ldots+\frac{n^3}{x^2}\right]=\)

Rolle's theorem is not applicable in which one of the following cases?

The shaded region in the figure is the solution set of the inequations.

The negative of the statement "All continuous functions are differentiable."

The inverse of the matrix $$\left[\begin{array}{ccc}2 & 5 & 0 \\ 0 & 1 & 1 \\ -1 & 0 & 3\end{array}\right]$$ is

If \(P\) and \(Q\) are symmetric matrices of the same order then \(P Q-Q P\) is

If $$3 A+4 B^{\prime}=\left[\begin{array}{ccc}7 & -10 & 17 \\ 0 & 6 & 31\end{array}\right]$$ and $$2 B+3 A^{\prime}\left[\begin{array}{cc}-1 & 18 \\ 4 & 0 \\ -5 & -7\end{array}\right]$$ then \(B=\)

If $$A=\left[\begin{array}{ll}1 & 3 \\ 4 & 2\end{array}\right], B=\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]$$, Then \(\left|A B B^{\prime}\right|=\)

If the value of a third order determinant is 16, then the value of the determinant formed by replacing each of its elements by its cofactor is

The constant term in the expansion of $$\left|\begin{array}{ccc}3 x+1 & 2 x-1 & x+2 \\ 5 x-1 & 3 x+2 & x+1 \\ 7 x-2 & 3 x+1 & 4 x-1\end{array}\right|$$ is

The number of 4 digit numbers without repetition that can be formed using the digits \(1,2,3,4,5,6,7\) in which each number has two odd digits and two even digits is

If \(P(n): 2^n< n !\) Then the smallest positive integer for which \(P(n)\) is true, is

Two letters are chosen from the letters of the word 'EQUATIONS'. The probability that one is vowel and the other is consonant is

A random variable '\(X\)' has the following probability distribution

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If \(A\) and \(B\) are two events of a sample space \(S\) such that \(P(A)=0.2, P(B)=0.6\) and \(P(A \mid B)=0.5\) then \(P\left(A^{\prime} \mid B\right)=\)

If '\(X\)' has a binomial distribution with parameters \(n=6, p\) and \(P(X=2)=12\), \(P(X=3)=5\) then \(P=\)

A man speaks truth 2 out of 3 times. He picks one of the natural numbers in the set \(S=\{1,2,3,4,5,6,7\}\) and reports that it is even. The probability that is actually even is

If \(\alpha\) and \(\beta\) are roots of the equation \(x^2=x+1=0\), then \(\alpha^2+\beta^2\) is

The third term of a GP is 9. The product of its first five terms is

If \(A=\{x \mid x \in N, x \leq 5\},B=\left\{x \mid x \in Z, x^2-5 x+6=0\right\}\), then the number of onto functions from \(A\) to \(B\) is

On the set of positive rational, a binary operation * is defined by \(a * b=\frac{2 a b}{5}\). If \(2 * x=3^{-1}\), then \(x=\)

If \(U\) is the universal set with 100 elements; \(A\) and \(B\) are two set such that \(n(A)=50, n(B)=60, n(A \cap B)=20\) then \(n\left(A^{\prime} \cap B^{\prime}\right)=\)

Mean and standard deviation of 100 items are 50 and 4 , respectively. The sum of all squares of the items is

A line cuts off equal intercepts on the co-ordinate axes. The angle made by this line with the positive direction of \(X\)-axis is