What is the $$\mathrm{pH}$$ of solution containing $$4.62 \times 10^{-4} \mathrm{M} \mathrm{H}^{+}$$ ions?

Which of the following is vinylic alcohol?

Which among the following is benzylic halide?

Which among the following is a simple ketone?

Identify glycosidic linkage present in maltose.

Which statement from following about nano-material is NOT correct?

Which from following species is NOT a monodentate ligand?

Two moles of an ideal gas expand freely and isothermally from $$5 \mathrm{~dm}^3$$ to $$50 \mathrm{~dm}^3$$. What is the value of $$\Delta H$$ ?

What is IUPAC name of following compound?

What is the molecular formula of an alkane if it exhibits three structural isomers?

What is the bond order in N$$_2^+$$?

Identify alkaline earth metal from following.

What is the molality of solution of a non-volatile solute having boiling point elevation $$7.15 \mathrm{~K}$$ and molal elevation constant $$2.75 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$$ ?

Which from following statements about rate constant is NOT true?

Identify '$$B$$' in the following conversions

$$\mathrm{CH}_3 \mathrm{Br} \xrightarrow{\mathrm{KCN}} A \xrightarrow{\mathrm{Na} / \mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}} B$$

Find the radius of metal atom in bcc unit cell having edge length $$450 \mathrm{~pm}$$.

Which from following statements is true for the molecule

Which among the following salt solution in water is acidic in nature?

Which among the following is intensive and extensive properties respectively?

Which of the following compounds has lowest boiling point?

Which of the following statements is NOT true about polymorphism?

Which from the following expression represents molar conductivity of an electrolyte $$\mathrm{A}_2 B_3$$ type?

Which of the following electromagnetic radiations possesses lowest energy?

Which among the following is an example of branched chain polymer?

For the reaction,

$$\mathrm{CH}_3 \mathrm{Br}(a q)+\mathrm{OH}^{-}(a q) \longrightarrow \mathrm{CH}_3 \mathrm{OH}(a q)+\mathrm{Br}^{-}(a q),$$

The rate law is rate $$=k\left[\mathrm{CH}_3 \mathrm{Br}\right]\left[\mathrm{OH}^{-}\right]$$. What is change in rate of reaction if concentration of both reactants is doubled?

For the reaction, $$2 A+2 B \longrightarrow 2 C+D$$, the rate law is expressed as rate $$=k[A]^2[B]$$. Calculate the rate constant if rate of reaction is $$0.24 \mathrm{~mol} \mathrm{~dm}^{-3} \mathrm{~s}^{-1}$$.

[[$$A$$]$$=0.5 \mathrm{M}$$ and $$[B]=0.2 \mathrm{M}$$]

What is the molar conductivity of $$0.005 \mathrm{~M} \mathrm{~NaI}$$ solution if it's conductivity is $$6.065 \times 10^{-4} \Omega^{-1} \mathrm{~cm}^{-1} \text { ? }$$

Which of the following is not a disaccharide?

Which from following is a correct decreasing order of ionisation enthalpy for different elements?

Which among the following salts exhibits inverse relation between it's solubility and temperature?

Which from following series of elements is correctly arranged according to their decreasing order of ionisation enthalpy $$\left(\mathrm{IE}_1\right)$$ ?

Identify the product formed by the action of $$\mathrm{H}_2(\mathrm{~g})$$ with $$\mathrm{CO}(\mathrm{g})$$ in presence of $$\mathrm{Ni}$$ ?

Calculate $$E^{\circ}$$ cell for following.

$$\mathrm{Zn}(s)\left|\mathrm{Zn}^{++}(1 \mathrm{M}) \| \mathrm{Pb}^{++}(1 \mathrm{M})\right| \mathrm{Pb}(s)$$ if $$E_{\mathrm{Zn}}^{\circ}=-0.763 \mathrm{~V}$$ and $$E_{\mathrm{Pb}}^{\circ}=-0.126 \mathrm{~V}$$

Which from following elements exhibits usual tendency to undergo reduction?

According to reaction,

$$\mathrm{Mg}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{MgCl}_2(a q)+\mathrm{H}_2(g) \uparrow .$$

Calculate the mass of $$\mathrm{Mg}$$ required to liberate $$4.48 \mathrm{~dm}^3 \mathrm{~H}_2$$ at STP?

(Molar mass of $$\mathrm{Mg}=24 \mathrm{~g} \mathrm{~mol}^{-1}$$)

A closed container contains mixture of non-reacting gases $$A$$ and $$B$$. Partial pressure of $$A$$ and $$B$$ are 4.5 bar and 5.5 bar respectively. Find mole fractions of $$A$$ and $$B$$ respectively?

Which from following compounds is obtained when toluene is treated with $$\mathrm{CrO}_2 \mathrm{Cl}_2$$ in presence of $$\mathrm{CS}_2$$ followed by acid hydrolysis?

Which of the following is NOT prepared by the action of Grignard's reagent on methanal?

What is the number of elements present in each series of transition element?

Which of the following is Wolf-Kishner reduction?

Calculate the wave number of photon emitted during transition from the orbit of $$n=3$$ to $$n=2$$ in hydrogen atom $$\left(R_H=109677 \mathrm{~cm}^{-1}\right)$$.

What is the molar concentration of acetic acid if value of it's, dissociation constant is $$1.8 \times 10^{-5}$$ and degree of dissociation is 0.02 ?

An ideal gas absorbs $$210 \mathrm{~J}$$ of heat and undergoes expansion from $$3 \mathrm{~L}$$ to $$6 \mathrm{~L}$$ against a constant external pressure of $$10^5 \mathrm{~Pa}$$. What is the value of $$\Delta U$$ ?

What is the molecular formula of $$p$$-toluidine?

If $$0.01 \mathrm{~m}$$ aqueous solution of an electrolyte freezes at $$-0.056 \mathrm{~K}$$. Calculate van't Hoff factor for an electrolyte (cryoscopic constant of water $$=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$$ )

Which element from following does NOT belong to chalcogen family?

What is IUPAC name of following compound?

Identify the monomers used to prepare novolac.

Calculate the volume of unit cell if an element having molar mass $$180 \mathrm{~g} \mathrm{~mol}^{-1}$$ forms fcc unit cell. $$\left[\rho \cdot N_A=120 \times 10^{21} \mathrm{~g} \mathrm{~cm}^{-3} \mathrm{~mol}^{-1}\right]$$

Identify homoleptic complex from following.

If $$x d y=y(d x+y d y), y(1)=1, y(x)>0$$, then $$y(-3)$$ is

Slope of the tangent to the curve $$y=2 e^x \sin \left(\frac{\pi}{4}-\frac{x}{2}\right) \cos \left(\frac{\pi}{4}-\frac{x}{2}\right)$$, where $$0 \leq x \leq 2 \pi$$ is minimum at $$x=$$

If the vectors $$p \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+q \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$\hat{\mathbf{i}}+\hat{\mathbf{j}}+r \hat{\mathbf{k}}(p \neq q \neq r \neq 1)$$ are coplanar, then the value of $$p q r-(p+q+r)$$ is

A tetrahedron has vertices at $$P(2,1,3), Q(-1,1,2), R(1,2,1)$$ and $$O(0,0,0)$$, then angle between the faces $$O P Q$$ and $$P Q R$$ is

Two dice are rolled. If both dice have six faces numbered $$1,2,3,5,7,11$$, then the probability that the sum of the numbers on upper most face is prime, is

The domain of the definition of the function $$y(x)$$ is given by the equation $$2^x+2^y=2$$ is

If $$\mathbf{a}=\frac{1}{\sqrt{10}}(3 \hat{\mathbf{i}}+\hat{\mathbf{k}}), \mathbf{b}=\frac{1}{7}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})$$, then the value of $$(2 \mathbf{a}-\mathbf{b}) \cdot[(\mathbf{a} \times \mathbf{b}) \times(\mathbf{a}+2 \mathbf{b})]$$ is

$$y=\frac{\sqrt[3]{1+3 x} \sqrt[4]{1+4 x} \sqrt[5]{1+5 x}}{\sqrt[7]{1+7 x} \sqrt[8]{1+8 x}} \text {. Then, } \frac{d y}{d x} \text { at } x=0$$ is

Five students are selected from $$n$$ students such that the ratio of number of ways in which 2 particular students are selected to the number of ways 2 particular students not selected is $$2: 3$$. Then, the value of $$n$$ is

If $$\int \frac{\log \left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}} d t=\frac{1}{2}[g(t)]^2+c$$, (where $$c$$ is a constant of integration), then $$g(2)$$ is

If $$A=\left[\begin{array}{cc}1 & \tan x \\ -\tan x & 1\end{array}\right]$$, then $$A^T \cdot A^{-1}=$$

$$\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$$ are three vectors. For a vector $$\mathbf{r}$$ with $$\mathbf{r} \times \mathbf{a}=\mathbf{b}$$ and $$\mathbf{r} \cdot \mathbf{c}=3,|\mathbf{r}|$$ is

Let $$P Q R$$ be a right angled isosceles triangle, right angled at $$Q(2,1)$$. If the equation of the line $$P R$$ is $$2 x+y=3$$, then the combined equation representing the pair of lines $$P Q$$ and $$Q R$$ is

If a curve $$y=a \sqrt{x}+b x$$ passes through the point $$(1,2)$$ and the area bounded by the curve, line $$x=4$$ and $$X$$-axis is 8 sq units, then

A plane is parallel to two lines whose direction ratios are $$2,0,-2$$ and $$-2,2,0$$ and it contains the point $$(2,2,2)$$. If it cuts coordinate axes at $$A, B, C$$, then the volume of the tetrahedron $$O A B C$$ (in cubic units) is

If $$q$$ is false and $$p \wedge q \leftrightarrow r$$ is true, then ............ is a tautology.

Negation of contrapositive of statement pattern $$(p \vee \sim q) \rightarrow(p \wedge \sim q)$$ is

The value of $$\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right)$$ is

Variance of first $$2 n$$ natural numbers is

$$\int \frac{x-3}{(x-1)^3} e^x d x=$$

If $$\mathbf{a}, \mathbf{b}, \mathbf{c}$$ are non-coplanar unit vectors such that $$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}$$, then the angle between $$\mathbf{a}$$ and $$\mathbf{b}$$ is

$$\int \frac{2+\cos \frac{x}{2}}{x+\sin \frac{x}{2}} d x=$$

In $$\triangle A B C$$, with usual notations, if $$\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$$, then the value of $$\cos A+\cos B+\cos C$$ is

If slope of the tangent to the curve $$x y+a x+b y=0$$ at the point $$(1,1)$$ on it is 2, then the value of $$3 a+b$$ is

If $$(3 x+2)-(5 y-3) i$$ and $$(6 x+3)+(2 y-4) i$$ are conjugates of each other, then the value of $$\frac{x-y}{x+y}$$ is (where $$\left.i=\sqrt{-1}, x, y \in R\right)$$

If $$\left(\tan ^{-1} x\right)^2+\left(\cot ^{-1} x\right)^2=\frac{5 \pi^2}{8}$$, then the value of $$x$$ is

The solution of $$(1+x y) y d x+(1-x y) x d y=0$$ is

The integral $$\int_\limits{\pi / 6}^{\pi / 3} \sec ^{\frac{2}{3}} x \operatorname{cosec}^{\frac{4}{3}} x d x$$ is equal to

$$\lim _\limits{x \rightarrow 0} \frac{\sqrt{1+x \sin x}-\sqrt{\cos x}}{\tan ^2 \frac{x}{2}}=$$

$$A(1,-3), B(4,3)$$ are two points on the curve $$y=x-\frac{4}{x}$$. The points on the curve, the tangents at which are parallel to the chord $$A B$$, are

Let $$f: R \rightarrow R$$ be a function such that $$f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in R \text {, }$$ then $$f(2)$$ equals

A radioactive substance, with initial mass $$m_0$$, has a half-life of $$h$$ days. Then, its initial decay rate is given by

The abscissae of two points $$A$$ and $$B$$ are the roots of the equation $$x^2+2 a x-b^2=0$$ and their ordinates are roots of the equation $$y^2+2 p y-q^2=0$$. Then, the equation of the circle with $$A B$$ as diameter is given by

The incentre of the $$\triangle A B C$$, whose vertices are $$A(0,2,1), B(-2,0,0)$$ and $$C(-2,0,2)$$, is

The acute angle between the line joining the points $$(2,1,-3),(-3,1,7)$$ and a line parallel to $$\frac{x-1}{3}=\frac{y}{4}=\frac{z+3}{5}$$ through the point $$(-1,0,4)$$ is

A random variable $$X$$ has the probability distribution

For the events $$E=\{X$$ is a prime number $$\}$$ and $$F=\{x<5\}, P(E U F)$$ is

The shaded area in the given figure is a solution set for some system of inequations. The maximum value of the function $$z=10 x+25 y$$ subject to the linear constraints given by the system is

The foot of the perpendicular from the point $$(1,2,3)$$ on the line $$\mathbf{r}=(6 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})$$ has the coordinates

Water is running in a hemispherical bowl of radius $$180 \mathrm{~cm}$$ at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is $$120 \mathrm{~cm}$$ ? (1 decimeter $$=10 \mathrm{~m}$$)

The solution of $$\sin x+\sin 5 x=\sin 3 x$$ in $$(0, \pi / 2)$$ are

If $$(1+\sqrt{1+x}) \tan x=1+\sqrt{1-x}$$, then $$\sin 4 x$$ is

If the sum of the mean and the variance of a binomial distribution for 5 trials is 1.8 , then the value of $$p$$ is

If $$I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x$$ and $$J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x$$, then for any arbitrary constant $$C$$, than the value of $$J-I$$ equals

The principal value of $$\sin ^{-1}(\sin (3 \pi / 4))$$ is

If $$x=\log _e\left(\frac{\cos \frac{y}{2}-\sin \frac{y}{2}}{\cos \frac{y}{2}+\sin \frac{y}{2}}\right), \tan \frac{y}{2}=\sqrt{\frac{1-t}{1+t}}$$ Then, $$\left(y_1\right)_{t=1 / 2}$$ has the value

The c.d.f. $$F(x)$$ associated with p.d.f. $$f(x)$$

$$f(x)=\left\{\begin{array}{cl}12 x^2(1-x), & \text { if } 0< x <1 \\ 0 ; & \text { otherwise }\end{array}\right.$$ is

If $$f(x)$$ is continuous on its domain $$[-2,2]$$, where

$$f(x)=\left\{\begin{array}{cc} \frac{\sin a x}{x}+3 & , \text { for }-2 \leq x<0 \\ 2 x+7 & , \text { for } 0 \leq x \leq 1 \\ \sqrt{x^2+8}-b & , \text { for } 1< x \leq 2 \end{array}\right.$$ $$\text { then the value of } 2 a+3 b \text { is }$$

$$P S$$ is the median of the triangle with vertices at $$P(2,2), Q(6,-1)$$ and $$R(7,3)$$, then the intercepts on the coordinate axes of the line passing through point $$(1,-1)$$ and parallel to PS are respectively

If Rolle's theorem holds for the function $$f(x)=x^3+b x^2+a x+5$$ on $$[1,3]$$ with $$c=2+\frac{1}{\sqrt{3}}$$, then the values of $$a$$ and $$b$$ respectively are

The distance of the point $$(1,6,2)$$ from the point of intersection of the line $$\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$$ and the plane $$x-y+z=16$$ is

A sample of oxygen gas and a sample of hydrogen gas both have the same mass, same volume and the same pressure. The ratio of their absolute temperature is

If two planets have their radii in the ratio $$x: y$$ and densities in the ratio $$m: n$$, then the acceleration due to gravity on them are in the ratio

An excited hydrogen atom emits a photon of wavelength $$\lambda$$ in returning to ground state. The quantum number $$n$$ of the excited state is ($$R=$$ Rydberg's constant)

There is hole of area $$A$$ at the bottom of a cylindrical vessel. Water is filled to a height $$h$$ and water flows out in $$t$$ second. If water is filled to a height $$4 h$$, it will flow out in time (in second)

The number of turns in the primary and the secondary of a transformer are 1000 and 3000 , respectively. If $$80 \mathrm{~V} \mathrm{~AC}$$ is applied to the primary coil of the transformer, then the potential difference per turn of the secondary coil would be

The ratio of magnetic field at the centre of the current carrying circular loop and magnetic moment is $$X$$. When both the current and radius are doubled, then the ratio will be

Light of wavelength $$5000 \mathop A\limits^o$$ is incident normally on a slit. The first minimum of the diffraction pattern is observed to lie at a distance of $$5 \mathrm{~mm}$$ from the central maximum on a screen placed at a distance of $$2 \mathrm{~m}$$ from the slit. The width of the slit is

When two tuning forks are sounded together, 5 beats per second are heard. One of the forks is in unison with $$0.97 \mathrm{~m}$$ length of sonometer wire and the other is in unison with $$0.96 \mathrm{~m}$$ length of the same wire. The frequencies of the two tuning forks are

The work done in rotating a dipole placed parallel to the electric field through $$180^{\circ}$$ is W. So, the work done in rotating it through $$60^{\circ}$$ is $$\left(\cos 0^{\circ}=1, \cos 60^{\circ}=\frac{1}{2}, \cos 180^{\circ}=-1\right)$$

When an electron is excited from its 4 th orbit to 5 th stationary orbit, the change in the angular momentum of electron is approximately.

(Planck's constant $$=h=6.63 \times 10^{-34} \mathrm{~J}-\mathrm{s}$$ )

The Boolean expression for the following combination is

Under the influence of force $$F_1$$ the body oscillates with a period $$T_1$$ and due to another force $$F_2$$ body oscillates with period $$T_2$$. If both forces acts simultaneously, then the resultant period is (consider displacement is same in all three cases)

The internal energy of a monoatomic ideal gas molecule is

A gas at pressure $$p_0$$ is contained in a vessel. If the masses of all the molecules are halved and their velocities are doubled, then the resulting pressure would be equal to

The equation of a progressive wave is $$Y=a \sin 2 \pi\left(n t-\frac{x}{5}\right)$$. The ratio of maximum particle velocity to wave velocity is

For an adiabatic process, which one of the following is wrong statement?

A mine is located at depth $$R / 3$$ below earth's surface. The acceleration due to gravity at that depth in mine is ($$R=$$ radius of earth, $$g=$$ acceleration due to gravity)

A $$10 \mathrm{~m}$$ long wire of resistance $$20 \Omega$$ is connected in series with a battery of emf $$3 \mathrm{~V}$$ (negligible internal resistance) and a resistance of $$10 \Omega$$. The potential gradient along the wire is

A group of lamps having total power rating of $$1000 \mathrm{~W}$$ is supplied by an AC voltage of $$E=200 \sin \left(310 t+60^{\circ}\right)$$, the rms value of current flowing through the circuit is

At a particular angular frequency, the reactance of capacitor and that of inductor is same. If the angular frequency is doubled, the ratio of the reactance of the capacitor to that of the inductor will be

A metal disc of radius $$R$$ rotates with an angular velocity $$\omega$$ about an axis perpendicular to its plane passing through its centre in a magnetic field of induction $$B$$ acting perpendicular to the plane of the disc. The magnitude of induced emf between the rim and axis of the disc is

A mass $$M$$ is suspended from a light spring. An additional mass $$M_1$$ added extends the spring further by a distance $$x$$. Now, the combined mass will oscillate on the spring with period $$T=$$

The path difference between two identical light waves at a point $$Q$$ on the screen is $$3 \mu \mathrm{m}$$. If wavelength of the waves is $$5000 \mathop A\limits^o$$, then at point $$Q$$ there is

If the maximum efficiency of a full wave rectifier is $$x \%$$ and that of half-wave rectifier is $$y \%$$, then the relation between $$x$$ and $$y$$ is

Two bodies $$A$$ and $$B$$ start from the same point at the same instant and move along a straight line. body $$A$$ moves with uniform acceleration $$a$$ and body $$B$$ moves with uniform velocity $$v$$. They meet after time $$t$$. The value of $$t$$ is

A small steel ball is dropped from a height of $$1.5 \mathrm{~m}$$ into a glycerine jar. The ball reaches the bottom of the jar $$1.5 \mathrm{~s}$$ after it was dropped. If the retardation is $$2.66 \mathrm{~m} / \mathrm{s}^2$$, the height of the glycerine in the jar is about (acceleration due to gravity $$g=9.8 \mathrm{~m} / \mathrm{s}^2$$ )

In a parallel plate capacitor with air between the plates, the distance $$d$$ 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

A ray of light passes through an equilateral prism such that the angle of incidence $$(i)$$ is equal to angle of emergence $$(e)$$. The angle of emergence is equal to $$\left(\frac{3}{4}\right)$$th the angle of prism. The angle of deviation is

The radii of curvature of both the surfaces of a convex lens of focal length $$f$$ and power $$P$$ are equal. One of the surfaces is made by plane grinding. The new focal length and focal power of the lens is

A spring balance is attached to the ceiling of a lift. A man hangs his bag on the spring and the spring balance reads $$49 \mathrm{~N}$$, when the lift is stationary. If the lift moves downward with an acceleration of $$5 \mathrm{~m} / \mathrm{s}^2$$, the reading of the spring balance will be $$\left(g=9.8 \mathrm{~m} / \mathrm{s}^2\right)$$

In a $$L$$-$$R$$ circuit the inductive reactance is equal to the resistance $R$ in the circuit. An emf $$E=E_0 \cos \omega t$$ is applied to the circuit. The power consumed in the circuit is

Four identical uniform solid spheres each of same mass $$M$$ and radius $$R$$ are placed touching each other as shown in figure with centres $$A, B, C, D. I_A, I_B, I_C, I_D$$ are the moment of inertia of these spheres respectively about an axis passing through centre and perpendicular to the plane, then

A transverse wave strike against a wall,

A circular current carrying coil has radius $$R$$. The magnetic induction at the centre of the coil is $$B_C$$. The magnetic induction of the coil at a distance $$\sqrt{3} R$$ from the centre along the axis is $$B_A$$. The ratio $$B_A: B_C$$ is

In the electric field due to a charge $$Q$$, a charge $$q$$ moves from point $$A$$ to $$B$$. The work done is ( $$\varepsilon_0=$$ permittivity of vacuum)

Of the two slits producing interference in Young's experiment, one is covered with glass so that light intensity passing is reduced to $$50 \%$$. Which of the following is correct?

Magnetic shielding is done by surrounding the instrument to be protected from magnetic field by

In the given figure, the equivalent capacitance between points A and B is

If work done in blowing a soap bubble of volume $$V$$ is $$W$$, then the work done in blowing the bubble of volume $$2 \mathrm{~V}$$ from same soap solution is

Which one of the following is based on convection?

A simple spring has length $$l$$ and force constant $$K$$. It is cut in to two springs of length $$l_1$$ and $$l_2$$ such that $$l_1=n l_2$$($$n$$ is an integer). The force constant of spring of length $$l_1$$ is

A closed pipe and an open pipe have their first overtone equal in frequency. Then, the lengths of these pipe are in the ratio

A conductor $$10 \mathrm{~cm}$$ long is moves with a speed $$1 \mathrm{~m} / \mathrm{s}$$ perpendicular to a field of strength $$1000 \mathrm{~A} / \mathrm{m}$$. The emf induced in the conductor is (Given : $$\mu_0=4 \pi \times 10^{-7} \mathrm{~Wb} / \mathrm{Am}$$ )

Dual nature of light is exhibited by

A carnot engine operates with source at $$227^{\circ} \mathrm{C}$$ and sink at $$27^{\circ} \mathrm{C}$$. If the source supplies $$50 \mathrm{~kJ}$$ of heat energy, the work done by the engine is

Doping of a semiconductor (with small impurity atoms) generally changes the resistivity as follows.

A stone is projected at angle $$\theta$$ with velocity $$u$$. If it executes nearly a circular motion at its maximum point for short time, then the radius of the circular path will be ( $$g=$$ acceleration due to gravity)

When radiations of wavelength $$\lambda$$ is incident on a metallic surface the stopping potential required is $$4.8 \mathrm{~V}$$. If same surface is illuminated with radiations of double the wavelength, then required stopping potential becomes $$1.6 \mathrm{~V}$$, then the value of threshold wavelength for the surface is

A thin uniform $$\operatorname{rod} A B$$ of mass $$m$$ and length $$l$$ is hinged at one end $$A$$ to the ground level. Initially the rod stands vertically and is allowed to fall freely to the ground in the vertical plane. The angular velocity of the rod when its end $$B$$ strikes the ground is ( $$g=$$ acceleration due to gravity)

Only $$4 \%$$ of the total current in the circuit passes through a galvanometer. If the resistance of the galvanometer is $$G$$, then the shunt resistance connected to the galvanometer is