MHT CET 2023 14th May Morning Shift

Let the curve be represented by \(x=2(\cos t+t \sin t), y=2(\sin t-t \cos t)\). Then normal at any point '\(t\)' of the curve is at a distance of ______ units from the origin.

Let \(\mathrm{B} \equiv(0,3)\) and \(\mathrm{C} \equiv(4,0)\). The point \(\mathrm{A}\) is moving on the line \(y=2 x\) at the rate of 2 units/second. The area of \(\triangle \mathrm{ABC}\) is increasing at the rate of

The maximum value of the function \(f(x)=3 x^3-18 x^2+27 x-40\) on the set \(\mathrm{S}=\left\{x \in \mathrm{R} / x^2+30 \leq 11 x\right\}\) is

The area bounded by the curves \(y=(x-1)^2, y=(x+1)^2\) and \(y=\frac{1}{4}\) is

If the line \(x-2 y=\mathrm{m}(\mathrm{m} \in \mathrm{Z})\) intersects the circle \(x^2+y^2=2 x+4 y\) at two distinct points, then the number of possible values of $m$ are

If \(a>0\) and \(z=\frac{(1+i)^2}{a-i}, i=\sqrt{-1}\), has magnitude \(\frac{2}{\sqrt{5}}\), then \(\bar{z}\) is

If \(\mathrm{f}^{\prime}(x)=\tan ^{-1}(\sec x+\tan x),-\frac{\pi}{2} < x < \frac{\pi}{2}\) and \(f(0)=0\), then \(\mathrm{f}(1)\) is

The integral \(\int_\limits{\frac{\pi}{6}}^{\frac{\pi}{3}} \sec ^{\frac{2}{3}} x \operatorname{cosec}^{\frac{4}{3}} x d x\) is equal to

General solution of the differential equation \(\cos x(1+\cos y) \mathrm{d} x-\sin y(1+\sin x) \mathrm{d} y=0\) is

The money invested in a company is compounded continuously. If ₹ 200 invested today becomes ₹ 400 in 6 years, then at the end of 33 years it will become ₹

The differential equation of \(y=\mathrm{e}^x(\mathrm{a} \cos x+\mathrm{b} \sin x)\) is

Let \(f: R \rightarrow R\) be a function such that \(\mathrm{f}(x)=x^3+x^2 \mathrm{f}^{\prime}(1)+x \mathrm{f}^{\prime \prime}(2)+6, x \in \mathrm{R}\), then \(\mathrm{f}(2)\) equals

\(\text { If } y=\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2, \text { then }\left(1-x^2\right) y_2-x y_1=\)

If \(y=[(x+1)(2 x+1)(3 x+1) \ldots \ldots(\mathrm{n} x+1)]^n\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=0\) is

The range of the function \(\mathrm{f}(x)=\frac{x^2}{x^2+1}\) is

The function \(\mathrm{f}(x)=[x] \cdot \cos \left(\frac{2 x-1}{2}\right) \pi\), where \([\cdot]\) denotes the greatest integer function, is discontinuous at

\(\int \frac{\sin 2 x\left(1-\frac{3}{2} \cos x\right)}{e^{\sin ^2 x+\cos ^3 x}} d x=\)

If \(\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^2 \theta} d \theta=A \log _e|f(\theta)|+c\) (where \(c\) is a constant of integration), then \(\frac{f(\theta)}{A}\) can be

\(\int \frac{\sin x+\sin ^3 x}{\cos 2 x} d x=A \cos x+B \log \mathrm{f}(x)+c\)
(where \(\mathrm{c}\) is a constant of integration). Then values of \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{f}(x)\) are

If \(\int \frac{x^3 \mathrm{~d} x}{\sqrt{1+x^2}}=\mathrm{a}\left(1+x^2\right) \sqrt{1+x^2}+\mathrm{b} \sqrt{1+x^2}+\mathrm{c}\)
(where \(\mathrm{c}\) is a constant of integration), then the value of \(3 \mathrm{ab}\) is

Let \(f(x)=\int \frac{x^2-3 x+2}{x^4+1} \mathrm{~d} x\), then function decreases in the interval

If \(\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi\), then the value of \(x^{2025}+x^{2026}+x^{2027}\) is

\(\lim _\limits{x \rightarrow \infty} x^3\left\{\sqrt{x^2+\sqrt{1+x^4}}-x \sqrt{2}\right\}=\)

The solution set of the inequalities \(4 x+3 y \leq 60, y \geq 2 x, x \geq 3, x, y \geq 0\) is represented by region

The negation of the statement
"The number is an odd number if and only if it is divisible by 3."

The statement \([(p \rightarrow q) \wedge \sim q] \rightarrow r\) is tautology, when \(r\) is equivalent to

Let $$A=\left[\begin{array}{cc}2 & -1 \\ 0 & 2\end{array}\right].$$ If \(B=I-{ }^3 C_1(\operatorname{adj} A)+{ }^3 C_2(\operatorname{adj} A)^2-{ }^3 C_3(\operatorname{adj} A)^3\), then the sum of all elements of the matrix B is

A linguistic club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this group including the selection of a leader (from among these 4 members) for the team. If the team has to include at most one boy, the number of...

Two cards are drawn successively with replacement from well shuffled pack of 52 cards, then the probability distribution of number of queens is

For an initial screening of an entrance exam, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is \(\frac{4}{5}\), then the probability, that he is unable to solve less than two probl...

$$\text { If } f(x)= \begin{cases}3\left(1-2 x^2\right) & ; 0< x < 1 \\ 0 & ; \text { otherwise }\end{cases}$$ is a probability density function of \(\mathrm{X}\), then \(\mathrm{P}\left(\frac{1}{4} < x < \frac{1}{3}\right)\) is

Three critics review a book. For the three critics the odds in favour of the book are \(2: 5, 3: 4\) and \(4: 3\) respectively. The probability that the majority is in favour of the book, is given by

In \(\triangle \mathrm{ABC}\), with usual notations, \(2 \mathrm{ac} \sin \left(\frac{1}{2}(\mathrm{~A}-\mathrm{B}+\mathrm{C})\right)\) is equal to

If the angles \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\) of a triangle are in an Arithmetic Progression and if \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) denote the lengths of the sides opposite to A, B and C respectively, then the v...

The equation \(x^3+x-1=0\) has

The variance of 20 observations is 5. If each observation is multiplied by 2, then variance of resulting observations is

\(\mathrm{p}\) is the length of perpendicular from the origin to the line whose intercepts on the axes are a and \(\mathrm{b}\) respectively, then \(\frac{1}{\mathrm{a}^2}+\frac{1}{\mathrm{~b}^2}\) equals

The perpendiculars are drawn to lines \(L_1\) and \(L_2\) from the origin making an angle \(\frac{\pi}{4}\) and \(\frac{3 \pi}{4}\) respectively with positive direction of \(\mathrm{X}\)-axis. If both the lines are at unit distance from the...

If \(\triangle \mathrm{ABC}\) is right angled at \(\mathrm{A}\), where \(A \equiv(4,2, x), \mathrm{B} \equiv(3,1,8)\) and \(C \equiv(2,-1,2)\), then the value of \(x\) is

The angle between the lines, whose direction cosines \(l, \mathrm{~m}, \mathrm{n}\) satisfy the equations \(l+\mathrm{m}+\mathrm{n}=0\) and \(2 l^2+2 \mathrm{~m}^2-\mathrm{n}^2=0\), is

Equation of the plane passing through \((1,-1,2)\) and perpendicular to the planes \(x+2 y-2 z=4\) and \(3 x+2 y+z=6\) is

A line with positive direction cosines passes through the point \(\mathrm{P}(2,-1,2)\) and makes equal angles with the co-ordinate axes. The line meets the plane \(2 x+y+z=9\) at point \(\mathrm{Q}\). The length of the line segment \(P Q\) ...

If the shortest distance between the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda}\) and \(\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}\) is \(\frac{1}{\sqrt{3}}\), then sum of possible values of \(\lambda\) is

Consider the lines \(\mathrm{L}_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{\mathrm{z}+1}{2}\)
\(\mathrm{L}_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{\mathrm{z}-3}{3}\), then the unit vector perpendicular to both \(\mathrm{L}_1\) and \(\mathrm{L}_2\) i...

The principal solutions of the equation \(\sec x+\tan x=2 \cos x\) are

If \(\sin (\theta-\alpha), \sin \theta\) and \(\sin (\theta+\alpha)\) are in H.P., then the value of \(\cos 2 \theta\) is

If \(\bar{a}, \bar{b}, \bar{c}\) are three vectors such that \(|\bar{a}+\bar{b}+\bar{c}|=1, \overline{\mathrm{c}}=\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}})\) and $$|\overline{\mathrm{a}}|=\frac{1}{\sqrt{3}},|\overline{\mat...

Let \(\bar{a}, \bar{b}, \bar{c}\) be three vectors such that \(|\bar{a}|=\sqrt{3}, |\bar{b}|=5, \bar{b} \cdot \bar{c}=10\) and the angle between \(\bar{b}\) and \(\bar{c}\) is \(\frac{\pi}{3}\). If \(\bar{a}\) is perpendicular to the vector...

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are unit vectors and \(\theta\) is angle between \(\overline{\mathrm{a}}\) and \(\bar{c}\) and \(\bar{a}+2 \bar{b}+2 \bar{c}=\overline{0}\), then $$|\bar{a} \times \...

If \(\bar{a}, \bar{b}, \bar{c}\) are three vectors with magnitudes \(\sqrt{3}\), 1, 2 respectively, such that \(\bar{a} \times(\bar{a} \times \bar{c})+3 \bar{b}=\overline{0}\), if \(\theta\) is the angle between \(\bar{a}\) and \(\bar{c}\),...