VITEEE 2022

If a man running around a race-course notes that the sum of the distances of two flag-posts from him is always \(10 \mathrm{~m}\) and the distance between the flag-posts is \(8 \mathrm{~m}\), then the area of the path he encloses in square ...

The domain of
\(f(x)=\sqrt{\log \frac{1}{4}\left(\frac{5 x-x^2}{4}\right)}+{ }^{10} C_x\) is

Through a fixed point \(P(\alpha, \beta), a\), variable line is drawn to cut the coordinate axes at \(A\) and \(B\). The locus of the mid-point of \(A B\) is

The integral \(\int \frac{d x}{x^2\left(x^4+1\right)^{3 / 4}}\) equals

If \(\tan ^{-1} y=\tan ^{-1} x+\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)\), where \(|x|< \frac{1}{\sqrt{3}}\) then value of \(y\) is

If \(\cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha\) where \(-1 \leq x \leq 1, -2 \leq y \leq 2, x \leq \frac{y}{2}\), then for all \(x, y, 4 x^2-4 x y \cos \alpha+y^2\) is equal to

The value of \(f(x) = \mathop {\lim }\limits_{x \to 2} {{{x^3} - 3{x^2} + 4} \over {{x^4} - 7x - 2}}\)

The value of
\(\lim _\limits{x \rightarrow \infty}\left[\frac{p^{1 / x}+q^{1 / x}+r^{1 / x}+s^{1 / x}}{4}\right]^{3 x}, p, q, r, s>0 \text {, }\) is

The value of \(\lim _\limits{t \rightarrow \infty} \frac{\ln \left(\frac{3}{2} t\right)}{t^2}\)

If $$f(x)=\left\{\begin{array}{cc}(\sin x+\cos x)^{\operatorname{cosec} x} & ,-\frac{\pi}{2}< x<0 \\ a & ,x=0 \\ \frac{e^{1 / x}+e^{2 / x}+e^{3 / x}}{a e^{-2+\frac{1}{x}}+b e^{-1+\frac{3}{x}}} & , 0< x<\frac{\pi}{2}\end{array}\right.$$ is c...

For all values of \(\lambda\), rank of matrix
$$A=\left[\begin{array}{ccc} { }^h C_0 & { }^4 C_3 & { }^5 C_4 \\ \lambda & 8 & 8 \lambda-6 \\ 1+\lambda^2 & 8 \lambda+4 & 2 \lambda+21 \end{array}\right]$$

If the tangent at \(P\) on \(y^2=4 a x\) meets the tangent at the vertex in \(Q\) and \(S\) is the focus of the parabola, then \(\angle S Q P\) is equal to

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three of them are red is

There are 50 intermediate stations on a railway line from one terminus to another. The number of ways a train can stop at 3 of these intermediate stations if no two of these stopping stations are to be consecutive, are

Two dices are rolled. If both dices have six faces numbered \(1,2,3,5,7\) and \(11\) then the probability that the sum of the number on the top faces is less than or equal to 8 is

The value of \(\frac{4}{1 !}+\frac{11}{2 !}+\frac{22}{3 !}+\frac{37}{4 !}+\frac{56}{5 !}+\ldots \infty\) is

If the real numbers \(x, y, z, t\) be in GP then the value of \((x^2+y^2+z^2)(y^2+z^2+t^2)\) is euqal to

A ray of light is sent along the line \(x-2 y+5=0\). Upon reaching the line \(3 x-2 y+7=0\), the ray is reflected from it. The equation of the line containing the reflected ray, is

A force of magnitude \(\sqrt{6}\) acting along, the line joining points \(A(2,-1,1)\) and \(B(3,1,2)\) displaces a particle from \(A\) to \(B\). The work done by the force is

If a variable plane cuts the coordinate axes in \(A, B, C\) and is at constant distance $p$ from the origin, then the locus of the centroid of the tetrahedron \(A B C\) is equal to

The image of the point \((2,-3,4)\) with respect to the plane \(4 x+2 y-4 z+3=0\), is

The roots of the equation \(\cos x+\sqrt{3} \sin x=2 \cos 2 x\), are

If \(0< a<5,0< b<5\) and \(\frac{x^2+5}{2}=x-2[\cos (a+b x)]\) is satisfied for atleast one real \(x\), then the least value of \(\frac{a+b}{\pi}\) is equal to

\(\cos (x+y), \cos x, \cos (x-y)\) are in HP, then \(\cos x \sec \frac{y}{2}\) is

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

A nuclide at rest emits an $\alpha$-particle. In this process

The half-life period of a radioactive element \(x\) is same as the mean life time of another radioactive element \(y\). Initially, both of them have the same number of atoms. Then,

A parallel plate capacitor with plate area \(A\) and separation between the plates \(d\) is charged by a constant current \(i\). Consider a plane surface of area \(A / 2\) parallel to the plates and drawn simultaneously between the plates. ...

A capacitor is filled with two dielectrics of the same dimensions but of dielectric constants 2 and 3 as shown in Fig.(a) and then in Fig.(b). Then ratio of the capacitor in the two arrangements is

A projectile is launched vertically from the earth with speed \(v_1\) hits a satellite at the height \(h\) moving with speed \(v_2\). If both have the same mass \(m\), then what is the common velocity if they move together after the collisi...

A train has to negotiate a curve of radius \(400 \mathrm{~m}\). By how much should the outer rail be raised with respect to the inner rail for a speed of \(48 \mathrm{~km} / \mathrm{h}\) ? The distance between the rails is \(1 \mathrm{~m}\)...

A cone filled with water is revolved in a vertical circle of radius \(4 \mathrm{~m}\) and the water does not fall down. What must be the maximum period of revolution?

A TV tower has a height of \(75 \mathrm{~m}\). What is the maximum distance upto which this TV transmission can be received? (Take, radius of the earth \(=6.4 \times 10^6 \mathrm{~m}\))

In which of the following photoelectric effect is not observed?

Light of wavelength \(\lambda\) strikes a photoelectric surface and electrons are ejected with kinetic energy \(K\). If \(K\) is to be increased to exactly twice its original value, the wavelength must be changed to \(\lambda^{\prime}\), su...

A wire of cross-section \(A\) is stretched horizontally between two clamps located \(2 l\) metre apart. A weight \(w \mathrm{~kg}\) is suspended from the mid-point of the wire. If the mid-point sags vertically through a distance \(x<1\), th...

A disc of \(10.0 \mathrm{~cm}\) diameter is rotated at speed of \(3600 \mathrm{~rpm}\) inside a long solenoid of 1200 turns per metre on the axis of the solenoid and perpendicular to the plane of the disc. When a current of $$1.5 \mathrm{~A...

A skylab or mass \(m \mathrm{~kg}\) is first launched from the surface of the earth in a circular orbit of radius \(2 R\) (from the centre of the earth) and then it is shifted from this circular orbit to another circular orbit of radius $$3...

An electrically heated coil is immersed in a calorimeter containing \(360 \mathrm{~g}\) of water at \(10^{\circ} \mathrm{C}\). The coil consumes energy at the rate of \(90 \mathrm{~W}\). The water equivalent of calorimeter and coil is $$40 ...

If 2 moles of an ideal monoatomic gas at temperature \(T_0\) is mixed with 4 moles of another ideal monoatomic gas at temperature \(2 T_0\), then the temperature of the mixture is

If two metallic plates of equal thickness and thermal conductivities \(K_1\) and \(K_2\) are put together face to face and a common plate is constructed, then the equivalent thermal conductivity of this plate will be

From Wien's displacement law, \(\lambda_m T=\) constant \(=0.00289 \mathrm{~m}-\mathrm{K}\)
Radiation from moon givens \(\lambda_m=4700 \mathop A\limits^o\) and another wavelength of \(14 \times 10^{-6} \mathrm{~m}\). Out of the following w...

If the ratio of specific heat of a gas at constant pressure to that at constant volume is \(\gamma\), the change in internal energy of the given mass of gas, when the volume changes from \(V\) to \(2 V\) at constant pressure \(p\) is

The system is pushed by a force \(F\) as shown in figure. All surfaces are smooth except between \(B\) and \(C\). Friction coefficient between \(B\) and \(C\) is \(\mu\). Minimum value of \(F\) to prevent block \(B\) from downward slipping ...

A bar magnet suspended by a horse's hair lies in the magnetic meridian where there is no twist in the hair, on turning the upper end of the hair through \(150^{\circ}\), the magnet is deflected through \(30^{\circ}\) from the meridian. Then...

Which of the following is deflected by the magnetic field?

In a cathode ray tube, a potential difference of 3000 volts is maintained between the deflector plates whose separation is \(2 \mathrm{~cm}\). A magnetic field of \(2.5 \times 10^{-3} \mathrm{~Wb} / \mathrm{m}^2\) at right angles to electri...

A stream of electrons is projected horizontally to the right. A straight conductor carrying, a current is supported parallel to the electron stream and above it. If the current in the conductor is from left to right, what will be the effect...

A coil of 100 turns and area \(2 \times 10^{-2} \mathrm{~m}^2\) is pivoted about a vertical diameter in a uniform magnetic field and carries a current of \(5 \mathrm{~A}\). When the coil is held with its plane in North-South direction, it e...

A plane glass mirror of thickness \(3 \mathrm{~cm}\) of material of \(\mu=\frac{3}{2}\) is silvered on the back surface. When a point object is placed \(9 \mathrm{~cm}\) from the front surface of the mirror, then the position of the brighte...