VITEEE 2021

The general solution of the linear differential equation \(\frac{d y}{d x}+\sec x \cdot y=\tan x\left(0 \leq x \leq \frac{\pi}{2}\right)\) is

On solving the differential equation \(x^2 y d x-\left(x^3+y^3\right) d y=0\), the value of \(\log y\) is

The value of \(\frac{d}{d x}\left(x^n \log _a x e^x\right)\) is

If \(y=1+\frac{x}{1 !}+\frac{x^2}{2 !}+\frac{x^3}{3 !} \ldots\), then \(\frac{d y}{d x}\) is equal to

The focal distance of the point \((x, y)\) from the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b\) is

The eccentricity of the ellipse \(25 x^2+9 y^2-150 x-90 y+225=0\) is

The quotient of the identity function by the reciprocal function is given by

Which of the following is not a function?

Evaluate \(\int \frac{3 x-2}{(x+3)(x+1)^2} d x\).

Using the principal values, the value of \(\sin ^{-1}\left\{\sin \frac{5 \pi}{6}\right\}+\tan ^{-1}\left\{\tan \frac{\pi}{6}\right\}\) is equal to

Find the value of \(\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{3}{5}\right)\).

If $$f(x)=\left\{\begin{array}{cc}\frac{(1-\cos 4 x)}{x^2}, & \text { if } x < 0 \\ a, & \text { if } x=0, \\ \frac{\sqrt{x}}{\sqrt{(16+\sqrt{x})}-4}, & \text { if } x > 0\end{array}\right.$$ then \(f(x)\) is continuous at \(x=0\), for $$a$...

The value of \(\lim _\limits{n \rightarrow \infty}\left\{\frac{1+2+3+\ldots+n}{n+2}-\frac{n}{2}\right\}\) is

\(\lim _\limits{x \rightarrow 0}\left\{\tan \left(\frac{\pi}{4}+x\right)\right\}^{1 / x}\) is equal to

The negation of \(\sim s \vee(\sim r \wedge s)\) is equivalent to

If $$A^{-1}=\left[\begin{array}{rr}5 & -2 \\ -7 & 3\end{array}\right]$$ and $$B^{-1}=\frac{1}{2}\left[\begin{array}{rr}9 & -7 \\ -8 & 6\end{array}\right]$$, then \((A B)^{-1}\) is equal to

If $$A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$$, then \(\left(A-A^{\prime}\right)\) is equal to (where, \(A^{\prime}\) is transpose of matrix \(A\) )

The locus of a point which moves in a plane such that its distance from a fixed point in the plane is always equal to its distance from a fixed straight line in the same plane represents

Five persons entered the lift cabin on the ground floor of an eight floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first, then the probability of all 5 per...

A bag contains 50 tickets numbered \(1,2,3, ..., 50\) of which five are drawn at random and arranged in ascending order of magnitude \(\left(x_1 < x_2 < x_3 < x_4< x_5\right)\), then the probability that $x_3=30$ is

If \(S=\{(a, b): b=|a-1|, a \in Z\) and \(|a|<3\}\), where \(Z\) denotes the set of integers. Then, the range set of \(S\) is

The cartesian product \(A \times A\) has 9 elements among which are found \((-1,0)\) and \((0,1)\), then set \(A\) is equal to

The equation of a straight line upon which the length of the perpendicular from the origin is 5 and slope of this perpendicular is 3/4 is

The equation of a straight line which cuts off intercept on \(X\)-axis which is twice that on \(Y\)-axis and is at a unit distance from origin is given by

If \((3,4,-1)\) and \((-1,2,3)\) be end points of the diameter of a sphere, then the radius of the sphere is

The following lines are
$$\begin{aligned}
\mathbf{r} & =(\hat{\mathbf{i}}+\hat{\mathbf{j}})+\lambda^{\prime}(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}), \\
\text { and } \quad \mathbf{r} & =(\hat{\mathbf{i}}+\hat{\mathbf{j}})+\mu...

The angle between the lines \(\frac{x-5}{-3}=\frac{y+3}{-4}=\frac{z-7}{0}, \frac{x}{1}=\frac{y-1}{-2}=\frac{z-6}{2}\) is

The position vector of a point \(R\) which divides the line joining \(P(6,3,-2)\) and \(Q(3,1,-4)\) in the ratio \(2 : 1\) externally is

If \(\theta=\frac{\pi}{2^n+1}\), then the value of \(2^n \cos \theta \cos 2 \theta \cos 2^2 \theta \ldots \cos 2^{n-1} \theta\) is

The value of \(\cos \left(\frac{3 \pi}{2}+x\right) \cos (2 \pi+x)\left\{\cot \left(\frac{3 \pi}{2}-x\right)+\cot (2 \pi+x)\right\}\) is

The alternating current in a circuit is given by \(I=50 \sin 314 t\). The peak value and frequency of the current are

If the alternating current \(I=I_1 \cos \omega t+ I_2 \sin \omega t\) then the rms current is given by

A \(50 \mathrm{~Hz}\) \(\mathrm{AC}\) signal is applied in a circuit of inductance of \((1 / \pi) \mathrm{H}\) and resistance \(2100 \Omega\). The impedance offered by the circuit is

The Rutherford scattering experiment proves that an atom consists of

Polonium has a half-life of 140 days. If we take \(20 \mathrm{~g}\) of polonium initially then the amount of it that remains after 280 days is

According to Bohr model of hydrogen atom, only those orbits are permissible which satisfy the condition

A parallel plate capacitor of capacitance \(5 \mu \mathrm{F}\) is charged to \(120 \mathrm{~V}\) and then connected to another uncharged capacitor. If the potential falls to \(40 \mathrm{~V}\), and capacitance of the second capacitor is

Two identical capacitors are first connected in series and then in parallel. The ratio of equivalent capacitance is

The demodulator or detector circuit consists of a

A silver wire of radius \(0.1 \mathrm{~cm}\) carries a current of \(2 \mathrm{~A}\). If the charge density in silver is \(5.86 \times 10^{28} \mathrm{~m}^{-3}\), the drift velocity is

24 cells of emf \(1.5 \mathrm{~V}\) each having internal resistance of \(1 \mathrm{~ohm}\) are connected to an external resistance of \(1.5 \mathrm{~ohms}\). To get maximum current

An electron revolves in a circle at the rate of \(10^{19}\) rounds per second. The equivalent current is \(\left(e=1.6 \times 10^{-19} \mathrm{C}\right)\)

The temperature coefficient of the resistance of a wire is 0.00125 per \({ }^{\circ} \mathrm{C}\). At \(300 \mathrm{~K}\) its resistance is \(1 \Omega\). The resistance of wire will be \(2 \Omega\) at

Einstein's photoelectric equation is

The Young's modulus of a rope of \(10 \mathrm{~m}\) length and having diameter of \(2 \mathrm{~cm}\) is \(200 \times 10^{11} \mathrm{~dyne} / \mathrm{cm}^2\). If the elongation produced in the rope is \(1 \mathrm{~cm}\), the force applied o...

The induced emf in a coil of \(10 \mathrm{H}\) inductance in which current varies from \(9 \mathrm{~A}\) to \(4 \mathrm{~A}\) in \(0.2 \mathrm{~s}\) is

A conductor of length \(5 \mathrm{~cm}\) is moved parallel to itself with a speed of \(2 \mathrm{~m} / \mathrm{s}\), perpendicular to a uniform magnetic field of \(10^{-3} \mathrm{~Wb} / \mathrm{m}^3\). The induced e.m.f. generated is

Which component of electromagnetic spectrum have maximum wavelength?

The net electric force on a charge of \(+3 \mu \mathrm{C}\) at the mid-point on the line joining two charges of magnitude \(+2 \mu \mathrm{C}\) and \(-2 \mu \mathrm{C}\) separated by the distance of \(6 \mathrm{~mm}\), is

An electric charge does not have which of the following properties?