MHT CET 2023 13th May Evening Shift

Slope of the tangent to the curve \(y=2 e^x \sin \left(\frac{\pi}{4}-\frac{x}{2}\right) \cos \left(\frac{\pi}{4}-\frac{x}{2}\right)\), where \(0 \leq x \leq 2 \pi\) is minimum at \(x=\)

If slope of the tangent to the curve \(x y+a x+b y=0\) at the point \((1,1)\) on it is 2, then the value of \(3 a+b\) is

\(A(1,-3), B(4,3)\) are two points on the curve \(y=x-\frac{4}{x}\). The points on the curve, the tangents at which are parallel to the chord \(A B\), are

Water is running in a hemispherical bowl of radius \(180 \mathrm{~cm}\) at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is \(120 \mathrm{~cm}\) ? (1 decimeter $$=1...

If Rolle's theorem holds for the function \(f(x)=x^3+b x^2+a x+5\) on \([1,3]\) with \(c=2+\frac{1}{\sqrt{3}}\), then the values of \(a\) and \(b\) respectively are

If a curve \(y=a \sqrt{x}+b x\) passes through the point \((1,2)\) and the area bounded by the curve, line \(x=4\) and \(X\)-axis is 8 sq units, then

The abscissae of two points \(A\) and \(B\) are the roots of the equation \(x^2+2 a x-b^2=0\) and their ordinates are roots of the equation \(y^2+2 p y-q^2=0\). Then, the equation of the circle with \(A B\) as diameter is given by

If \((3 x+2)-(5 y-3) i\) and \((6 x+3)+(2 y-4) i\) are conjugates of each other, then the value of \(\frac{x-y}{x+y}\) is (where \(\left.i=\sqrt{-1}, x, y \in R\right)\)

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

If \(x d y=y(d x+y d y), y(1)=1, y(x)>0\), then \(y(-3)\) is

The solution of \((1+x y) y d x+(1-x y) x d y=0\) is

A radioactive substance, with initial mass \(m_0\), has a half-life of \(h\) days. Then, its initial decay rate is given by

\(y=\frac{\sqrt[3]{1+3 x} \sqrt[4]{1+4 x} \sqrt[5]{1+5 x}}{\sqrt[7]{1+7 x} \sqrt[8]{1+8 x}} \text {. Then, } \frac{d y}{d x} \text { at } x=0\) is

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

If \(x=\log _e\left(\frac{\cos \frac{y}{2}-\sin \frac{y}{2}}{\cos \frac{y}{2}+\sin \frac{y}{2}}\right), \tan \frac{y}{2}=\sqrt{\frac{1-t}{1+t}}\) Then, \(\left(y_1\right)_{t=1 / 2}\) has the value

The domain of the definition of the function \(y(x)\) is given by the equation \(2^x+2^y=2\) is

If \(\int \frac{\log \left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}} d t=\frac{1}{2}[g(t)]^2+c\), (where \(c\) is a constant of integration), then \(g(2)\) is

\(\int \frac{x-3}{(x-1)^3} e^x d x=\)

\(\int \frac{2+\cos \frac{x}{2}}{x+\sin \frac{x}{2}} d x=\)

If \(I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x\) and
\(J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x\), then for any arbitrary constant \(C\), than the value of \(J-I\) equals

The value of \(\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right)\) is

If \(\left(\tan ^{-1} x\right)^2+\left(\cot ^{-1} x\right)^2=\frac{5 \pi^2}{8}\), then the value of \(x\) is

The principal value of \(\sin ^{-1}(\sin (3 \pi / 4))\) is

\(\lim _\limits{x \rightarrow 0} \frac{\sqrt{1+x \sin x}-\sqrt{\cos x}}{\tan ^2 \frac{x}{2}}=\)

If \(f(x)\) is continuous on its domain \([-2,2]\), where
$$f(x)=\left\{\begin{array}{cc}
\frac{\sin a x}{x}+3 & , \text { for }-2 \leq x<0 \\
2 x+7 & , \text { for } 0 \leq x \leq 1 \\
\sqrt{x^2+8}-b & , \text { for } 1< x \leq 2
\end{arra...

The shaded area in the given figure is a solution set for some system of inequations. The maximum value of the function \(z=10 x+25 y\) subject to the linear constraints given by the system is

If \(q\) is false and \(p \wedge q \leftrightarrow r\) is true, then ............ is a tautology.

Negation of contrapositive of statement pattern \((p \vee \sim q) \rightarrow(p \wedge \sim q)\) is

If $$A=\left[\begin{array}{cc}1 & \tan x \\ -\tan x & 1\end{array}\right]$$, then \(A^T \cdot A^{-1}=\)

Five students are selected from \(n\) students such that the ratio of number of ways in which 2 particular students are selected to the number of ways 2 particular students not selected is \(2: 3\). Then, the value of \(n\) is

Two dice are rolled. If both dice have six faces numbered \(1,2,3,5,7,11\), then the probability that the sum of the numbers on upper most face is prime, is

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

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In \(\triangle A B C\), with usual notations, if \(\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}\), then the value of \(\cos A+\cos B+\cos C\) is

Variance of first \(2 n\) natural numbers is

If the sum of the mean and the variance of a binomial distribution for 5 trials is 1.8 , then the value of \(p\) is

The c.d.f. \(F(x)\) associated with p.d.f. \(f(x)\)
$$f(x)=\left\{\begin{array}{cl}12 x^2(1-x), & \text { if } 0< x <1 \\ 0 ; & \text { otherwise }\end{array}\right.$$ is

Let \(P Q R\) be a right angled isosceles triangle, right angled at \(Q(2,1)\). If the equation of the line \(P R\) is \(2 x+y=3\), then the combined equation representing the pair of lines \(P Q\) and \(Q R\) is

\(P S\) is the median of the triangle with vertices at \(P(2,2), Q(6,-1)\) and \(R(7,3)\), then the intercepts on the coordinate axes of the line passing through point \((1,-1)\) and parallel to PS are respectively

A tetrahedron has vertices at \(P(2,1,3), Q(-1,1,2), R(1,2,1)\) and \(O(0,0,0)\), then angle between the faces \(O P Q\) and \(P Q R\) is

A plane is parallel to two lines whose direction ratios are \(2,0,-2\) and \(-2,2,0\) and it contains the point \((2,2,2)\). If it cuts coordinate axes at \(A, B, C\), then the volume of the tetrahedron \(O A B C\) (in cubic units) is

The incentre of the \(\triangle A B C\), whose vertices are \(A(0,2,1), B(-2,0,0)\) and \(C(-2,0,2)\), is

The acute angle between the line joining the points \((2,1,-3),(-3,1,7)\) and a line parallel to \(\frac{x-1}{3}=\frac{y}{4}=\frac{z+3}{5}\) through the point \((-1,0,4)\) is

The foot of the perpendicular from the point \((1,2,3)\) on the line \(\mathbf{r}=(6 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})\) has the coordinates

The distance of the point \((1,6,2)\) from the point of intersection of the line \(\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}\) and the plane \(x-y+z=16\) is

The solution of \(\sin x+\sin 5 x=\sin 3 x\) in \((0, \pi / 2)\) are

If \((1+\sqrt{1+x}) \tan x=1+\sqrt{1-x}\), then \(\sin 4 x\) is

If the vectors \(p \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+q \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+r \hat{\mathbf{k}}(p \neq q \neq r \neq 1)\) are coplanar, then the value ...

If \(\mathbf{a}=\frac{1}{\sqrt{10}}(3 \hat{\mathbf{i}}+\hat{\mathbf{k}}), \mathbf{b}=\frac{1}{7}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})\), then the value of $$(2 \mathbf{a}-\mathbf{b}) \cdot[(\mathbf{a} \times \mathbf{b})...

\(\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) are three vectors. For a vector $$\mathb...

If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are non-coplanar unit vectors such that \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}\), then the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is