Engineer & Medical previous year questions (PYQ)

Let $P_1$, $P_2$, $P_3$ be the altitudes of a triangle ABC from the vertices A, B, C respectively. If $\triangle$ denotes the area of the triangle and s is the semi-perimeter of the triangle, then $\dfrac{\cos A}{P_1} + \dfrac{\cos B}{P_2} + \dfrac{\cos C}{P_3} =$

Water flows through a horizontal pipe at a speed 'V'. Internal diameter of the pipe is 'd'. If the water is emerging at a speed '$V_1$' then the diameter of the nozzle is

${ }^{\text { } }$ If $\alpha=1+\sum_{r=1}^6(-3)^{r-1} \quad{ }^{12} \mathrm{C}_{2 r-1}$, then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x-\sqrt{3} y+1=0$ is _________.

If the equation $\mathrm{a}(\mathrm{b}-\mathrm{c}) \mathrm{x}^2+\mathrm{b}(\mathrm{c}-\mathrm{a}) \mathrm{x}+\mathrm{c}(\mathrm{a}-\mathrm{b})=0$ has equal roots, where $\mathrm{a}+\mathrm{c}=15$ and $\mathrm{b}=\frac{36}{5}$, then $a^2+c^2$ is equal to

For the reaction, $\mathrm{A}+3 \mathrm{~B} \longrightarrow 2 \mathrm{C}$ rate of consumption of A is $1.4 \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{sec}^{-1}$. Calculate rate of formation of C ?

lectronic configuration of \(\mathrm{Cu}(\mathrm{II})\) is \(3 \mathrm{~d}^9\) whereas that of \(\mathrm{Cu}(\mathrm{I})\) is \(3 \mathrm{~d}^{10}\). Which of the following is correct?

An electron of charge ' $\mathrm{e}$ ' is moving round the nucleus of a hydrogen atom in a circular orbit of radius ' $r$ '. The coulomb force $\overrightarrow{\mathrm{F}}$ between the two is $\left(\right.$ here $\left.\mathrm{K}=\frac{1}{4 \pi \varepsilon_0}\right)$

In $\triangle A B C$, if $(a-b)(s-c)=(b-c)(s-a)$, then $r_1, r_2, r_3$ are in

A student needs to prepare a buffer solution of propanoic acid and its sodium salt with $\mathrm{pH}4$. The ratio of $\frac{\left[{\mathrm{CH}}_3{\mathrm{CH}}_2{\mathrm{COO}}^{-}\right]}{\left[{\mathrm{CH}}_3{\mathrm{CH}}_2\mathrm{COOH}\right]}$ required to make buffer is Given : ${\mathrm{K}}_{\mathrm{a}}\left({\mathrm{CH}}_3{\mathrm{CH}}_2\mathrm{COOH}\right)=1.3\times {10}^{-5}$

In a series $LR$ circuit $X_L=R$ and power factor of the circuit is $P_1$. When capacitor with capacitance $C$ such that $X_L=X_C$ is put in series, the power factor becomes $P_2$. The ratio $\frac{P_1}{P_2}$ is

The mole fraction of a solute in one molal ethanolic solution would be

If $4$ times the area of a $∆ABC$ is $c^2-(a-b{)}^2,$ then $\sin C$

The value of \(\left(\sin 210^{\circ}\right)\left(\sin 585^{\circ}\right)\) is

Which of the following chemical intermediates are the products of light reaction in photosynthesis?

The temporary hardness of water is due to:

Which of the following oxides can act both as oxidising agent as well as reducing agent?

In a meter bridge, as shown in the figure, it is given that resistance $Y=12.5 \Omega$ and that the balance is obtained at a distance $39.5 \mathrm{~cm}$ from end $A$ (by jockey J). After interchanging the resistances $X$ and $Y$, a new balance point is found at a distance $l_2$ from end $A$. What are the values of $X$ and $l_2$ ?

In a $\triangle A B C$, if $a=5, b=6, c=7$, then the length of the median drawn from $B$ is

The locus of the point $P$ such that the area of the $\triangle \mathrm{PAB}$ is 7 , where $\mathrm{A}(4,5)$ and $\mathrm{B}(-2,3)$ are given points, is

Purification of NaCl by passage of hydrogen chloride through brine is based on

A ray of light is incident along a line which meets another line $7x-y+1=0$ at the point $\left(0,1\right)$. The ray is then reflected from this point along the line $y+2x=1$. Then the equation of the line of incidence of the ray of light is :

A series LCR circuit is built with a variable resistor. The resonant frequency of the circuit is $\omega_0$. The circuit response ( $I_m-R$ profile, where, $I_m$ is peak current and $R$ is resistance) is recorded for three different values of frequencies $(\omega)$ of input signal, and these are depicted schematically in the accompanying figures, I, II and III. The figures respectively correspond to

If ${\displaystyle \sum _{n=1}^5}\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}=\frac{k}{3},$ then $k$ is equal to:

We have a galvanometer of resistance $25 \Omega$. It is shunted by $2.5 \Omega$ wire. The part of the total current that flows through the galvanometer is given as

The correct statement about the magnetic properties of [Fe(CN)₆]^3- and [FeF₆]^3- is : (Z = 26).

[FeF₆]^3- is paramagnetic.

In allene $\left(\mathrm{C}_3 \mathrm{H}_4\right)$, the type(s) of hybridization of the carbon atoms is (are) :

Given sum of the first $n$ terms of an A.P. is $2 n+$ $3 n^2$. Another A.P. is formed with the same first term and double of the common difference, the sum of $n$ terms of the new A.P. is :

Which of the following statements/relationships is not correct in thermodynamic changes ?

At a certain place a magnet makes 30 oscillations per minute. At another place where the magnetic field is double, its time period will be

The probability distribution for the speed $(v)$ of molecules for an ideal gas at temperature $T$, is given by
$f(v)=4 \pi v^2\left(\frac{m}{2 \pi k_B T}\right)^{3 / 2} \exp \left(-\frac{m v^2}{2 k_B T}\right)$
where $m$ is the mass of a molecule and $k_{\mathrm{B}}$ is the Boltzmann constant. Among the following, the correct statement is

Under which one of the following conditions do real gases approach the ideal gas behaviour?

In plants when one sperm fertilizes the egg and the other fertilizes the polar bodies, the process is referred to as double fertilization. This results in the formation of

Water from a tap emerges vertically downwards with initial velocity $4 \mathrm{~ms}^{-1}$. The cross-sectional area of the tap is $A$. The flow is steady and pressure is constant throughout the stream of water. The distance $h$ vertically below the tap, where the cross-sectional area of the stream becomes $\left(\frac{2}{3}\right) A$, is $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$

Which of the following pairs of $2,3$-butanediol are enantiomeric?

If $3 \cos x \neq 2 \sin x$, then the general solution of $\sin ^2 x-\cos 2 x=2-\sin 2 x$ is

In any $\triangle A B C, a(b \cos C-c \cos B)$ equals

If $\alpha+\beta+\gamma=2 \theta$, then $\cos \theta+\cos (\theta-\alpha)$ $+\cos (\theta-\beta)+\cos (\theta-\gamma)$ is equal to

One junction of a certain thermoelectric couple is at a fixed temperature $T_r$ and the other junction is at temperature $T$. The thermo-electromotive force for this is expressed by $E=k\left(T-T_r\right)\left[T_0-\frac{1}{2}\left(T+T_r\right)\right]$. At temperature $T=\frac{1}{2} T_0$, the thermoelectric power is

Extraction for zinc from zinc blende is achieved by

A man walks a distance of 3 units from the origin towards the North-East $\left(\mathrm{N} 45^{\circ} \mathrm{E}\right)$ direction. From there, he walks a distance of 4 units towards the North-West ( $\mathrm{N} 45^{\circ} \mathrm{W}$ ) direction to reach a point $P$. Then, the position of $P$ in the Argand plane is

Which of the products is formed when acetone is reacted with barium hydroxide solution?

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

A tower, of $x$ metres high, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant $y$ metres from the foot of the tower. Then, the length of the flagstaff (in metres), is

Heating mixture of $\mathrm{Cu}_2 \mathrm{O}$ and $\mathrm{Cu}_2 \mathrm{~S}$ will give

$A$ metallic ring of radius $r$ and cross-sectional area $A$ is fitted into a wooden circular disc of radius $R$ $(R \gt r)$. If the Young's modulus of the material of the ring is $Y$,the force with which the metal ring expands is

The magnetic induction and the intensity of the magnetic field inside an iron core of an electromagnet are $1 \ Wb \ m^{-2}$ and $150 \ A \ m^{-1}$,respectively. The relative permeability of iron is $\left(\mu_0 = 4 \pi \times 10^{-7} \ H \ m^{-1}\right)$.

$A$ tower subtends angles $\alpha, 2 \alpha$ and $3 \alpha$ respectively at points $A, B$ and $C$,all lying on a horizontal line through the foot of the tower. Then $\frac{A B}{B C}$ is equal to:

$A$ ray of light is incident on the hypotenuse of a right-angled prism after travelling parallel to the base inside the prism. If $\mu$ is the refractive index of the material of the prism,the maximum value of the base angle for which light is totally reflected from the hypotenuse is

A body is projected up with a velocity equal to $\frac{3}{4}$ th of the escape velocity from the surface of the earth. The height it reaches is (Radius of the earth $=R$ )

A wave $y=a \sin (\omega t-k x)$ on a string meets with another wave producing a node at $x=0$. Then the equation of the unknown wave is

If $\operatorname{cosec} \theta=\frac{p+q}{p-q}$, then $\cot \left(\frac{\pi}{4}+\frac{\theta}{4}\right)$ is equal to

The coefficient of $x^4$ in the expansion of $\frac{(1-3 x)^2}{(1-2 x)}$ is equal to

In the special mode of reproduction observed in lemons, the embryos develop directly from the

When a diploid female plant is crossed with a tetraploid male, the ploidy of endosperm cells in the resulting seed is