Engineer & Medical previous year questions (PYQ)

A block of mass M is lying on a horizontal frictionless surface. One end of the uniform rope of half the mass of the block is fixed to the block which is pulled in the horizontal direction by applying a force F at the other end. The tension in the middle of the rope will be

The third overtone of a closed pipe of length '$L_c$' has the same frequency as the third overtone of the open pipe of length '$L_0$'. Both the pipes have same diameters. The ratio $L_c : L_0$ is equal to

For a reaction $\mathrm{A} \rightarrow$ Product, rate constant is $6.93 \times 10^{-3}$ hour $^{-1}$. What is order of reaction?

Let the domain of the function
$f(x)=\cos ^{-1}\left(\frac{4 x+5}{3 x-7}\right)$ be $[\alpha, \beta]$ and the domain of $\mathrm{g}(\mathrm{x})=\log _2\left(2-6 \log _{27}(2 \mathrm{x}+5)\right)$ be $(\gamma, \delta)$.
Then $|7(\alpha+\beta)+4(\gamma+\delta)|$ is equal to ________

In series LCR resonant circuit; the capacitance is changed from C to 3 C . To obtain the same resonant frequency, the inductance should be changed from $L$ to

Average power associated with an ideal inductor and ideal capacitor over a complete cycle of a.c. is respectively

About black body radiation, which of the following is the wrong statement?

In a parallel plate capacitor with air between the plates, the distance ' $\mathrm{d}$ ' between the plates is changed and the space is filled with dielectric constant 8 . The capacity of the capacitor is increased 16 times, the distance between the plates is

The complete combustion of $0.492 \mathrm{g}$ of an organic compound containing '$\mathrm{C}$', '$\mathrm{H}$' and '$\mathrm{O}$' gives $0.793 \mathrm{g}$ of ${\mathrm{CO}}_2$ and $0.442 \mathrm{g}$ of ${\mathrm{H}}_2\mathrm{O}$. The percentage of oxygen composition in the organic compound is____.(nearest integer)

Given two statements below :

Statement I : In ${\mathrm{Cl}}_2$ molecule the covalent radius is double of the atomic radius of chlorine.

Statement II : Radius of anionic species is always greater than their parent atomic radius.

Choose the most appropriate answer from options given below 

By using the non-zero digits, the number of 5 digit numbers that can be formed so that each number has largest digit in its middle place and the digits in the number are different is

If sum of the first $21$ terms of the series ${\log }_{9^{1/2}}x+{\log }_{9^{1/3}}x+{\log }_{9^{1/4}}x+\dots ..$ where $x>0$ is $504,$ then $x$ is equal to

Hydrogen peroxide, in the pure state, is :

If $\left|\sin x-\cos ^2 x\right| \geq\left|3-3 \sin x+\sin ^2 x\right|+4|\sin x-1|$, then $x=$

If \(f: R \rightarrow R\) is defined by \(f(x)= \begin{cases}|[x-5]|, & \text { for } x < 5 \\ {[|x-5|],} & \text { for } x \geq 5\end{cases}\) Then, \((f \circ f)\left(-\frac{7}{2}\right)=\) (here, \([x]\) is the greatest integer not exceeding \(x\))

When an inductor \(L\) and a resistor \(R\) in series are connected across a \(12 \mathrm{~V}, 50 \mathrm{~Hz}\) supply, a current of 0.5 A flows in the circuit. The current differs in the phase from applied voltage by \(\frac{\pi}{3}\) radian. Then the value of \(R\) is

What are $A, B$ and $C$ in the following reaction? Arene diazonium fluoroborate $$\stackrel{\mathrm{NaNO}_2}{\Delta} \stackrel{\mathrm{Cu}}{\longrightarrow} A+B+C$$

If $k \in R$, then roots of $(x-2)(x-3)=k^2$ are always

If the lengths of two sides of a triangle are the roots of the equation $x^2-2 \sqrt{3} x+2=0$ and the angle between these sides is $\frac{\pi}{3}$, then the perimeter of the triangle is

Excess of $\mathrm{NaOH} \left(\mathrm{aq}\right)$ was added to $100 \mathrm{mL}$ of ${\mathrm{FeCl}}_3\mathrm{ }\left(\mathrm{aq}\right)$ resulting into $2.14 \mathrm{g}$ of $\mathrm{Fe}{\left(\mathrm{OH}\right)}_3$. The molarity of ${\mathrm{FeCl}}_3\left(\mathrm{aq}\right)$ is:

(Given the molar mass of$\mathrm{Fe}=56 \mathrm{g}$${\mathrm{mol}}^{-1}$ and molar mass of $\mathrm{Cl}=35.5\mathrm{ }\mathrm{g}\mathrm{ }{\mathrm{mol}}^{-1}$)

Alternating current of peak value $\left(\frac{2}{\pi }\right)$ ampere flows through the primary coil of the transformer. The coefficient of mutual inductance between primary and secondary coil is 1 henry. The peak emf induced in secondary coil is (Frequency of AC = 50 Hz)

The domain of the function $f(x)=\sqrt{\log _{0.5} x !}$ is

Which one of the following is applicable for an adiabatic expansion of an ideal gas?

2.76 g of silver carbonate on being strongly heated yeild a residue weighing

Let $f\left(n\right)=\left[\frac{1}{3}+\frac{3n}{100}\right]n$, where $\left[n\right]$ denotes the greatest integer less than or equal to $n$. Then $\sum _{n=1}^{56}f\left(n\right)$ is equal to

In the diagram shown, the difference in the two tubes of the manometer is $5 \mathrm{cm}$, the cross-section of the tube at $A$ and $B$ is $6 {\mathrm{mm}}^2$ and $10 {\mathrm{mm}}^2$ respectively. The rate at which water flows through the tube is $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

Let $a=\operatorname{Im}\left(\frac{1+z^2}{2 i z}\right)$, where $z$ is any non-zero complex number. The set $\mathrm{A}=\{a:|z|=1$ and $z \neq \pm 1\}$ is equal to:

The magnetic moment of the complex anion $\left[\mathrm{Cr}(\mathrm{NO})\left(\mathrm{NH}_3\right)(\mathrm{CN})_4\right]^{2-}$ is :

A thick-walled hollow sphere has outside radius $R_0$. It rolls down an incline without slipping and its speed at the bottom is $v_0$. Now the incline is waxed, so that it is practically frictionless and the sphere is observed to slide down (without any rolling). Its speed at the bottom is observed to be $5 v_0 / 4$. The radius of gyration of the hollow sphere about an axis through its centre is

A uniform rope of mass $0.1 \mathrm{~kg}$ and length $2.45 \mathrm{~m}$ hangs from a rigid support. The time taken by the transverse wave formed in the rope to travel through the full length of the rope is (Assume $g=9.8 \mathrm{~m} / \mathrm{s}^2$ )

What is the entropy change in $\mathrm{JK}^{-1}$ during the melting of $27.3 \mathrm{~g}$ of ice at $0^{\circ} \mathrm{C}$ ? (Latent heat of fusion of ice $=330 \mathrm{Jg}^{-1}$ )

The angle between the asymptotes of the hyperbola $x^2-3 y^2=3$ is

$A$ bimetallic strip is formed out of two identical strips,one of copper and the other of brass. The coefficients of linear expansion of the two metals are $\alpha_C$ and $\alpha_B$. On heating,the temperature of the strip increases by $\Delta T$ and the strip bends to form an arc of radius $R$. Then $R$ is proportional to

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-6x^2+11x-6=0$ and if $a=\alpha^2+\beta^2+\gamma^2$,$b=\alpha\beta+\beta\gamma+\gamma\alpha$ and $c=(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)$,then the correct inequality among the following is

Let $A$ and $B$ be two symmetric matrices of same order. Then, the matrix $A B-B A$ is

Four light sources produce the following four waves : (i) $y_1=a \sin \left(\omega t+\phi_1\right)$ (ii) $y_2=a \sin 2 \omega t$ (iii) $y_3=d^{\prime} \sin \left(\omega t+\phi_2\right)$ (iv) $y_4=d^{\prime} \sin (3 \omega t+\phi)$ Superposition of which two waves give rise to interference?

$A$ body is thrown vertically upwards and reaches its maximum height in $t$ seconds. The total time from the time of projection to reach a point at half of its maximum height while returning (in seconds) is:

Statement (S) : Using Huygen's eye-piece measurements can be taken but are not correct. Reason (R) : The cross wires, scale and final image are not magnified proportionately because the image of the object is magnified by two lenses, whereas the cross wire scale is magnified by one lens only. Identify the correct one of the following

Let $f(x)=\frac{x^2-6 x+5}{x^2-5 x+6}$.

If $f: R \rightarrow R$ and is defined by $f(x)=\frac{1}{2-\cos 3 x}$ for each $x \in R$, then the range of $f$ is

A mass of $\mathrm{M} \mathrm{kg}$ is suspended by a weightless string. The horizontal force that is required to displace it until the string makes an angle of $45^{\circ}$ with the initial vertical direction is

For natural numbers $m, n$ if $(1-y)^m(1+y)^n=1+a_1 y+a_2 y^2+\ldots$, and $a_1=a_2=10$ then $(\mathrm{m}, \mathrm{n})$ is

A uniform electric field and a uniform magnetic field are acting along the same direction in a certain region. If an electron is projected along the direction of the fields with a certain velocity then

The function $f: C \rightarrow C$ defined by $f(x)=\frac{a x+d}{c x+d}$ for $x \in C$ where $b d \neq 0$ reduces to a constant function, if

A particle of mass $1 \mathrm{~kg}$ is projected with an initial velocity $10 \mathrm{~ms}^{-1}$ at an angle of projection $45^{\circ}$ with the horizontal. The average torque acting on the projectile, between the time at which it is projected and the time at which it strikes the ground, about the point of projection in newton-metre is

At a given instant of time,two particles have position vectors $4 \hat{i} + 4 \hat{j} + 57 \hat{k} \text{ m}$ and $2 \hat{i} + 2 \hat{j} + 5 \hat{k} \text{ m}$ respectively. If the velocity of the first particle is $0.4 \hat{i} \text{ ms}^{-1}$,what is the velocity of the second particle in $\text{ms}^{-1}$ if they collide after $10 \text{ s}$?

Three unequal resistors in parallel are equivalent to a resistance $1 \mathrm{ohm}$. If two of them are in the ratio $1: 2$ and if no resistance value is fractional, then the largest of the three resistances in ohms is

If the circle $x^2+y^2+6x-2y+k=0$ bisects the circumference of the circle $x^2+y^2+2x-6y-15=0$,then $k$ is equal to :

Which one of the following statements is correct with respect to basic character?

Which one of the following statements is not correct ?

The products of lengths of perpendiculars from any point of hyperbola $x^2-y^2=8$ to its asymptotes, is

The velocities of three particles of masses $20 \mathrm{~g}$, $30 \mathrm{~g}$ and $50 \mathrm{~g}$ are $10 \hat{\mathbf{i}}, 10 \hat{\mathbf{j}}$ and $10 \hat{\mathbf{k}}$, respectively. The velocity of the centre of mass of the three particles is

If $\pi <\theta <\frac{3\pi }{2}$ and $\cos \theta =-\frac{3}{5}$, then $\mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{\theta }{4}\right)$ is equal to

Let $f\left(\theta \right)=\frac{1}{1+{\left(\tan \theta \right)}^{2021}},$ then the value of $\sum _{\theta =1°}^{89°}f\left(\theta \right)$ is equal to