MHT CET 2021 22th September Evening Shift

If \(f(x)=x^2+a x+b\) has minima at \(x=3\) whose value is 5 , then the values of \(a\) and \(b\) are respectively.

The function \(f(x)=\frac{\lambda \sin x+6 \cos x}{2 \sin x+3 \cos x}\) is increasing, if

The slant height of a right circular cone is \(3 \mathrm{~cm}\). The height of the cone for maximum volume is

The area bounded by the parabola \(y^2=x\) and the line \(x+y=2\) in the first quadrant is

Two circles centred at \((2,3)\) and \((4,5)\) intersects each other. If their radii are equal, then the equation of the common chord is

If \(\omega\) is the complex cube root of unity, then \(\left(3+5 \omega+3 \omega^2\right)^2+\left(3+3 \omega+5 \omega^2\right)^2=\)

If \(\int_\limits0^a \sqrt{\frac{a-x}{x}} d x=\frac{k}{2}\), then \(k=\)

\(\int_\limits{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{\operatorname{cosec} x \cdot \cot x}{1+\operatorname{cosec}^2 x} d x=\)

\(\int_\limits0^2|2 x-3| d x=\)

The degree of the differential equation whose solution is \(y^2=8 a(x+a)\), is

The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is

A spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is 3 mm. and 1 hour later has been reduced to 2 mm, then the expression of radius r of the raindrop at any time t is (where 0 \(\le\) t < 3...

The particular solution of the differential equation \(\frac{d y}{d x}=\frac{y+1}{x^2-x}\), when \(x=2\) and \(y=1\) is

The general solution of \(\frac{d y}{d x}=\frac{x+y}{x-y}\) is

If \(y=\tan ^{-1} \sqrt{\frac{1+\cos x}{1-\cos x}}\), then \(\frac{d y}{d x}=\)

If \(x^y \cdot y^x=16\), then $\frac{d y}{d x}$ at $(2,2)$$ is

If \(f(x)=[8 x]-3\), where \([x]\) is greatest integer function of \(x\), then \(f(\pi)=\) (where \(\pi=3,14\))

\(\int \cos ^{-1} x d x=\)

\(\int \frac{\tan ^4 \sqrt{x} \cdot \sec ^2 \sqrt{x}}{\sqrt{x}} d x=\)

\(\int \frac{1}{\cos x+\sqrt{3} \sin x} d x=\)

If \(2 \tan ^{-1}(\cos x)=\tan ^{-1}(2 \operatorname{cosec} x)\), then the value of \(x\) is

\(\lim _\limits{x \rightarrow 2}(x-1)^{ \frac{1}{3 x-6}}=\)

Let
$$\begin{aligned} f(x) & =x+a \sqrt{2} \sin x & & , 0 \leq x<\frac{\pi}{4} \\ & =2 x \cot x+b & & \frac{\pi}{4} \leq x<\frac{\pi}{2} \\ & =a \cos 2 x-b \sin x & & \frac{\pi}{2} \leq x \leq \pi \end{aligned}$$
If $$\mathrm{f}(\mathrm{x})...

The common region of the solution of the inequations \(x+y \geq 5, y \leq 4, x \geq 2, x, y \geq 0\) is

Given \(\mathrm{p}\) : A man is a judge, \(\mathrm{q}\) : A man is honest
If \(\mathrm{S} 1\) : If a man is a judge, then he is honest
S2 : If a man is a judge, then he is not honest
S3 : A man is not a judge or he is honest Then
S4 : A man...

The statement pattern \((p \wedge q) \wedge[(p \wedge q) \vee(\sim p \wedge q)]\) is equivalent to

Let \(a: \sim(p \wedge \sim r) \vee(\sim q \vee s)\) and \(b:(p \vee s) \leftrightarrow(q \wedge r)\).
If the truth values of \(p\) and \(q\) are true and that of \(r\) and \(s\) are false, then the truth values of \(a\) and \(b\) are respe...

If $$A=\left[\begin{array}{ccc}1 & 2 & 1 \\ -1 & 1 & 3\end{array}\right]$$ and $$B=\left[\begin{array}{cc}1 & 2 \\ -3 & 1 \\ 0 & 2\end{array}\right]$$, then \((A B)^{-1}\)

If $$A=\left[\begin{array}{cc}5 a & -b \\ 3 & 2\end{array}\right]$$ and \(A\) adj \(A=A A^T\), then \(5 a+b=\)

For an invertible matrix \(A\), if $$A(\operatorname{adj} A)=\left[\begin{array}{cc}20 & 0 \\ 0 & 20\end{array}\right]$$, then \(|A|=\)

The numbers can be formed using the digits \(1,2,3,4,3,2,1\) so that odd digits always occupy odd places in __________ ways.

If the mean and variance of a binomial distribution are 4 and 2 respectively, then probability of getting 2 heads is

The distribution function \(F(X)\) of discrete random variable \(X\) is given by

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First bag contains 3 red and 5 black balls and second bag contains 6 red and 4 black balls. A ball is drawn from each bag. The probability that one ball is red and the other is black, is

A fair coin is tossed 4 times. If \(X\) is a random variable which indicates number of heads, then \(\mathrm{P}[\mathrm{X}<3]=\)

In \(\Delta ABC\), with usual notations \(\mathrm{\frac{b\sin B-c\sin C}{\sin(B-C)}}=\)

With usual notations, in any \(\triangle A B C\), if \(a\cos B=b \cos A\), then the triangle is

Following data shows the information about marks obtained in Physics, Chemistry, Mathematics and Biology by 100 students in a class. Then subject shows the highest variability in marks

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If lines represented by the equation \(\mathrm{px}^2-\mathrm{qy^{2 }}=0\) are distinct, then

If the sum of slopes of lines represented by \(\mathrm{ax^2+8xy+5y^2=0}\) is twice their product, then a =

If the line joining two points \(\mathrm{A}(2,0)\) and \(\mathrm{B}(3,1)\) is rotated about \(\mathrm{A}\) in anticlockwise direction through an angle of \(15^{\circ}\), then the equation of the line in new position is

If the points \(P(4,5, x), Q(3, y, 4)\) and \(R(5,8,0)\) are collinear, then the value of \(x+y\) is

The d.r.s. of the normal to the plane passing through the origin and the line of intersection of the planes \(x+2 y+3 z=4\) and \(4 x+3 y+2 z=1\) are

The line \(\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}\) lies in the plane \(x+3 y-\alpha z+\beta=0\), then value of \(\alpha \beta\) is

A line drawn from a point \(A(-2,-2,3)\) and parallel to the line \(\frac{x}{-2}=\frac{y}{2}=\frac{z}{-1}\) meets the \(\mathrm{YOZ}\) plane in point \(\mathrm{P}\), then the co-ordinates of the point \(\mathrm{P}\) are

If \(\mathrm{G}(4,3,3)\) is the centroid of the triangle \(\mathrm{ABC}\) whose vertices are \(\mathrm{A}(\mathrm{a}, 3,1), \mathrm{B}(4,5, \mathrm{~b})\) and \(C(6, c, 5)\), then the values of \(a, b, c\) are

The area of triangle with vertices \((1,2,0),(1,0, a)\) and \((0,3,1)\) is \(\sqrt{6}\) sq. units, then the values of '\(a\)' are

If \(2 \cos \theta=x+\frac{1}{x}\), then \(2 \cos 3 \theta=\)

If the vectors \(2 \hat{i}-\hat{j}-\hat{k} ; \hat{i}+2 \hat{j}-3 \hat{k}\) and \(3 \hat{i}+\lambda \hat{j}+5 \hat{k}\) are coplanar, then the value of \(\lambda\) is

The vector equation of the line whose Cartesian equations are \(y=2\) and \(4 x-3 z+5=0\) is