VITEEE 2023

If \(x d y / d x=x^2+y-2, y(1)=1\), then \(y(2)\) is equal to

The solution of differential equation \(y y^{\prime}=x\left(\frac{y^2}{x^2}+\frac{f\left(y^2 / x^2\right)}{f^{\prime}\left(y^2 / x^2\right)}\right)\) is

If \(\left(\sin ^{-1} x\right)^2-\left(\cos ^{-1} x\right)^2=a \pi^2\), then the range of \(a\) is

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

\(\tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{8}\right)\)
equals to

The equation \(3 \cos ^{-1} x-\pi x-\pi / 2=0\) has

For \(x \in R, f(x)=|\log 2-\sin x|\) and \(g(x)=f(f(x))\), then

If matrix $$A=\left[\begin{array}{ccc}0 & 2 b & -2 \\ 3 & 1 & 3 \\ 3 a & 3 & -1\end{array}\right]$$ is given to be symmetric, then the value of \(a b\) is

The determinant of the matrix $$\left[\begin{array}{ccc}1 & 4 & 8 \\ 1 & 9 & 27 \\ 1 & 16 & 64\end{array}\right]$$ is

Suppose, $$A=\left[\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right]$$ is an adjoint of the matrix $$\left[\begin{array}{rrr}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right]$$. The value of $$\fr...

If the 4th term in the expansion of \(\left(p x+\frac{1}{x}\right)^n, n \in N\) is \(\frac{5}{2}\) and three normals to the parabola \(y^2=x\) are drawn through a point \((q, 0)\), then

Out of 9 consonants and 4 vowels, how many words of 4 consonants and 3 vowels can be formed?

The sum of all the numbers of four different digits that can be made using the digits 0, 1, 2 and 3 is

Let \(A\) and \(B\) be two independent events such that the odds in favour of \(A\) and \(B\) are \(1: 1\) and \(3: 2\), respectively. Then, the probability that only one of the two occurs is

Let \(a_n\) be a sequence of numbers which is defined by relation \(a_1=2, \frac{a_n}{a_{n+1}}=3^{-n}\), then \(\log _2\left(a_{50}\right)\) is equal to (take \(\log _2 3=1.6\) )

The value of \(\frac{1}{2}\left(\frac{1}{5}\right)^2+\frac{2}{3}\left(\frac{1}{5}\right)^3+\frac{3}{4}\left(\frac{1}{5}\right)^4+\ldots . . \infty\) is

If \(R\) is a relation from a finite set A having \(m\) elements to finite set \(B\) having \(n\) elements, then the number of relation from \(A\) to \(B\) is

The image of the point \((2,3,7)\) in the plane \(2 x+5 y-3 z-19=0\), is

The distance between the point \((7,2,4)\) and the plane determined by the points \((2,5,-3),(-2,-3,5)\) and \((5,3,-3)\) is

The plane is perpendicular to the planes \(x-y+2 z-4=0\) and \(2 x-2 y+z=0\) and passes through \((1,-2,1)\) is

If \(\theta_1, \theta_2\) and \(\theta_3\) are the angles made by a line with the positive direction of \(X, Y\) and \(Z\)-axes, then \(\cos 2 \theta_1+\cos 2 \theta_2+\cos 2 \theta_3\) is equal to

The minium value of \(\left[2-\cos \theta+\sin ^2 \theta\right]\) is

If \(\sin A, \sin B\) and \(\cos A\) are in GP, then the roots of \(x^2+2 x \cot B+1=0\) are always

A unit vector perpendicular to both the vectors \(\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+\hat{\mathbf{k}}\) is

Let \(a, b\) and \(c\) be three unit vectors such that \(a \times(b \times c)=\frac{\sqrt{3}}{2}(b+c)\). If \(b\) is not parallel to \(c\), then the angle between \(a\) and \(b\) is

The activity of a radioactive sample is measured as No counts per minute at \(t=0\) and \(\mathrm{N}_0 / \mathrm{e}\) counts per minute at \(t=6 \mathrm{~min}\). The time (in minutes) at which the activity reduces to half its value is.

The wavelength of \(k_\alpha\)-line characteristic \(X\)-rays emitted by an element is \(0.32 \mathop A\limits^o\). The wavelength of the \(k_\beta\)-line emitted by the same element will be

A radioactive element \(X\) convents into another stable element \(Y\). Half-life of \(X\) is 2 hrs. Initially only \(X\) is present. After time \(t\), the ratio of atoms of \(X\) and \(Y\) is found to be \(1: 4\), then \(t\) in hours is

A parallel plate capacitor with air between the plates has a capacitance of \(15 \mathrm{~pF}\). the separation between the plates is \(d\). The space between the plates is now filled with two dielectrics constant \(k_1=3\) and thickness $$...

A particle of mass \(10 \mathrm{~kg}\) moving eastwards with a speed \(10 \mathrm{~m} / \mathrm{s}\) collides with another particle of the same mass moving northward with the same speed \(10 \mathrm{~m} / \mathrm{s}\). The two particles coa...

What is the voltage gain in a common emitter amplifier, where input resistance is \(6 \Omega\) and load resistance \(48 \Omega, \beta=0.3\) ?

A transmitting antenna is kept on the surface of the Earth. The minimum height of receiving antenna required to receive the signal in line of sight at \(4 \mathrm{~km}\) distance from it is \(x \times 10^{-2} \mathrm{~m}\). The value of $$x...

A constant voltage is applied between the two ends of a uniform metallic wire, Some heat is developed in it. The heat developed is halved if

When \({ }_{92} \mathrm{U}^{235}\) undergoes fission, \(0.1 \%\) of its original mass is changed into energy. How much energy is released if \(5 \mathrm{Kg}\) of \({ }_{92} \mathrm{U}^{235}\) undergoes fission?

The de-Broglie wavelength of a proton \(\left(m=1.67 \times 10^{-27} \mathrm{~kg}\right)\) acclerated through a potential difference of \(2 \mathrm{~kV}\) is

The upper end of a wire of diameter \(24 \mathrm{~mm}\) and length \(1 \mathrm{~m}\) is damped and its other end is twisted through an angle is \(30^{\circ}\). The angle of shear is

The elastic limit of \(1 \mathrm{~kg}\) mass is \(3.5 \times 10^{10} \mathrm{~N} / \mathrm{m}^2\). Find the maximum load that can be applied to a brass wire of a \(1 \mathrm{~mm}\) diameter without exceeding the elastic unit.

A coil 10 turns and a resistance of \(40 \Omega\) is connected in series with B.G. of resistance \(30 \Omega\). The coil is placed with its plane perpendicular to the direction of a uniform magnetic field of induction $$10^{-2} \mathrm{~T}$...

Electric field in the region is given by \(\mathbf{E}=\left(M / x^4\right) \hat{\mathbf{i}}\), then the correct expression for the potential in the region is (assume potential at infinity is zero)

With increasing temperature, the angle of contact,

The gravitational field in a region is given by \(\mathbf{E}=5 \mathrm{~N} / \mathrm{kg} \hat{\mathbf{i}}+12 \mathrm{~N} / \mathrm{kg} \hat{\mathbf{j}}\). The change in the gravitational potential energy of a particle of mass $$1 \mathrm{~k...

A geostationary satellite revolves around the Earth in a circular orbit of radius \(4 R\). Here, $R$ is the radius of the Earth. Then, the time period of another satellite moving in a circular orbit of radius \(2 R\) is:

A copper sphere cools from \(82^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\) in 10 minutes and to \(42^{\circ} \mathrm{C}\) in the next \(10 \mathrm{~min}\). Calculate the temperature of the surrounding?

Two gases occupy two containers \(A\) and \(B\). The gas in \(A\) of volume \(0.20 \mathrm{~m}^3\), exerts a pressure of \(1.40 \mathrm{~MPa}\) and that in \(B\), of volume \(0.30 \mathrm{~m}^3\) exerts a pressure of \(0.7 \mathrm{~MPa}\). ...

0.5 mole of an ideal gas at constant temperature \(27^{\circ} \mathrm{C}\) kept inside a cylinder of length \(L\) and cross-section \(A\) closed by a massless piston. The cylinder is attached with a conducting rod of length L\(_1\) cross-se...

Three blocks of masses \(m_1, m_2\) and \(m_3\) are connected by massless strings, as shown, on a frictionless table. They are pulled with a force, \(T_3=40 \mathrm{~N}\). If \(m_1=10 \mathrm{~kg}, m_2=8 \mathrm{~kg}\) and $$m_3=2 \mathrm{~...

Susceptibility of ferromagnetic substance is

The susceptibility of a magnetism at \(300 \mathrm{~K}\) is \(1.5 \times 10^{-5}\). The temperature at which the susceptibility increases to \(4.5 \times 10^{-5}\) is

The position of a projectile launched from the origin at \(t=0\) is given \(\mathbf{r}=(40 \hat{\mathbf{i}}+50 \hat{\mathbf{j}}) \mathrm{m}\) at \(t=2 \mathrm{~s}\). If the projectile was launded at an angle \(\theta\) from the horizontal, ...

Two long parallel wires carry equal current \(i\) flowing in the same directions are at a distance \(4 d\) apart. The magnetic field \(B\) at a point \(P\) lying on the perpendicular line joining the wires and at a distance \(x\) from the m...