The major product 'P' for the following sequence of reactions is :

The magnetic behaviour of $$\mathrm{Li_2O,Na_2O_2}$$ and $$\mathrm{KO_2}$$, respectively, are :

Compound that will give positive Lassaigne's test for both nitrogen and halogen is :

Identify the correct order for the given property for following compounds.

Choose the correct answer from the option given below :

The bond dissociation energy is highest for

Number of cyclic tripeptides formed with 2 amino acids A and B is :

The shortest wavelength of hydrogen atom in Lyman series is $$\lambda$$. The longest wavelength is Balmer series of He$$^+$$ is

Match List I with List II

Choose the correct answer from the options given below :

The increasing order of $$\mathrm{pK_a}$$ for the following phenols is

(A) 2, 4 - Dinitrophenol

(B) 4 - Nitrophenol

(C) 2, 4, 5 - Trimethylphenol

(D) Phenol

(E) 3-Chlorophenol

Choose the correct answer from the option given below :

Chiral complex from the following is :

Here en = ethylene diamine

The standard electrode potential $$\mathrm{(M^{3+}/M^{2+})}$$ for V, Cr, Mn & Co are $$-$$0.26 V, $$-$$0.41 V, + 1.57 V and + 1.97 V, respectively. The metal ions which can liberate $$\mathrm{H_2}$$ from a dilute acid are :

Following chromatogram was developed by adsorption of compound 'A' on a 6 cm TLC glass plate. Retardation factor of the compound 'A' is _________ $$\times 10^{-1}$$.

Millimoles of calcium hydroxide required to produce 100 mL of the aqueous solution of pH 12 is $$x\times10^{-1}$$. The value of $$x$$ is ___________ (Nearest integer).

Assume complete dissociation.

For certain chemical reaction $$X\to Y$$, the rate of formation of product is plotted against the time as shown in the figure. The number of $$\mathrm{\underline {correct} }$$ statement/s from the following is ___________.

(A) Over all order of this reaction is one.

(B) Order of this reaction can't be determined.

(C) In region I and III, the reaction is of first and zero order respectively.

(D) In region-II, the reaction is of first order.

(E) In region-II, the order of reaction is in the range of 0.1 to 0.9.

The number of molecules or ions from the following, which do not have odd number of electrons are _________.

(A) NO$$_2$$

(B) ICl$$_4^ - $$

(C) BrF$$_3$$

(D) ClO$$_2$$

(E) NO$$_2^ + $$

(F) NO

Following figure shows dependence of molar conductance of two electrolytes on concentration. $$\Lambda \mathop m\limits^o $$ is the limiting molar conductivity.

The number of $$\mathrm{\underline {incorrect} }$$ statement(s) from the following is ___________

(A) $$\Lambda \mathop m\limits^o $$ for electrolyte A is obtained by extrapolation

(B) For electrolyte B, $$\Lambda \mathop m\limits $$ vs $$\sqrt c$$ graph is a straight line with intercept equal to $$\Lambda \mathop m\limits^o $$

(C) At infinite dilution, the value of degree of dissociation approaches zero for electrolyte B.

(D) $$\Lambda \mathop m\limits^o $$ for any electrolyte A and B can be calculated using $$\lambda^\circ$$ for individual ions

Solid Lead nitrate is dissolved in 1 litre of water. The solution was found to boil at 100.15$$^\circ$$C. When 0.2 mol of NaCl is added to the resulting solution, it was observed that the solution froze at $$-0.8^\circ$$ C. The solubility product of PbCl$$_2$$ formed is __________ $$\times$$ 10$$^{-6}$$ at 298 K. (Nearest integer)

Given : $$\mathrm{K_b=0.5}$$ K kg mol$$^{-1}$$ and $$\mathrm{K_f=1.8}$$ K kg mol$$^{-1}$$. Assume molality to the equal to molarity in all cases.

Water decomposes at 2300 K

$$\mathrm{H_2O(g)\to H_2(g)+\frac{1}{2}O_2(g)}$$

The percent of water decomposing at 2300 K and 1 bar is ___________ (Nearest integer).

Equilibrium constant for the reaction is $$2\times10^{-3}$$ at 2300 K.

Consider the following reaction approaching equilibrium at 27$$^\circ$$C and 1 atm pressure

$$\mathrm{A+B}$$ $$\mathrel{\mathop{\kern0pt\rightleftharpoons} \limits_{{k_r} = {{10}^2}}^{{k_f} = {{10}^3}}} $$ $$\mathrm{C+D}$$

The standard Gibb's energy change $$\mathrm{(\Delta_r G^\theta)}$$ at 27$$^\circ$$C is ($$-$$) ___________ kJ mol$$^{-1}$$ (Nearest integer).

(Given : $$\mathrm{R=8.3~J~K^{-1}~mol^{-1}}$$ and $$\mathrm{\ln 10=2.3}$$)

The sum of bridging carbonyls in $$\mathrm{W(CO)_6}$$ and $$\mathrm{Mn_2(CO)_{10}}$$ is ____________.

17 mg of a hydrocarbon (M.F. $$\mathrm{C_{10}H_{16}}$$) takes up 8.40 mL of the H$$_2$$ gas measured at 0$$^\circ$$C and 760 mm of Hg. Ozonolysis of the same hydrocarbon yields

The number of double bond/s present in the hydrocarbon is ___________.

Let $$\lambda \ne 0$$ be a real number. Let $$\alpha,\beta$$ be the roots of the equation $$14{x^2} - 31x + 3\lambda = 0$$ and $$\alpha,\gamma$$ be the roots of the equation $$35{x^2} - 53x + 4\lambda = 0$$. Then $${{3\alpha } \over \beta }$$ and $${{4\alpha } \over \gamma }$$ are the roots of the equation

Let $$B$$ and $$C$$ be the two points on the line $$y+x=0$$ such that $$B$$ and $$C$$ are symmetric with respect to the origin. Suppose $$A$$ is a point on $$y-2 x=2$$ such that $$\triangle A B C$$ is an equilateral triangle. Then, the area of the $$\triangle A B C$$ is :

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $$\mu$$ and $$\sigma^2$$ represent mean and variance of X, respectively, then $$10(\mu^2+\sigma^2)$$ is equal to :

Let $$f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )$$ and $$S = \left\{ {\theta \in [0,\pi ]:f'(\theta ) = - {{\sqrt 3 } \over 2}} \right\}$$. If $$4\beta = \sum\limits_{\theta \in S} \theta $$, then $$f(\beta )$$ is equal to

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is :

Let the tangents at the points $$A(4,-11)$$ and $$B(8,-5)$$ on the circle $$x^{2}+y^{2}-3 x+10 y-15=0$$, intersect at the point $$C$$. Then the radius of the circle, whose centre is $$C$$ and the line joining $$A$$ and $$B$$ is its tangent, is equal to :

Let $$f(x) = x + {a \over {{\pi ^2} - 4}}\sin x + {b \over {{\pi ^2} - 4}}\cos x,x \in R$$ be a function which

satisfies $$f(x) = x + \int\limits_0^{\pi /2} {\sin (x + y)f(y)dy} $$. then $$(a+b)$$ is equal to

Let $$\alpha$$ and $$\beta$$ be real numbers. Consider a 3 $$\times$$ 3 matrix A such that $$A^2=3A+\alpha I$$. If $$A^4=21A+\beta I$$, then

A light ray emits from the origin making an angle 30$$^\circ$$ with the positive $$x$$-axis. After getting reflected by the line $$x+y=1$$, if this ray intersects $$x$$-axis at Q, then the abscissa of Q is :

For two non-zero complex numbers $$z_{1}$$ and $$z_{2}$$, if $$\operatorname{Re}\left(z_{1} z_{2}\right)=0$$ and $$\operatorname{Re}\left(z_{1}+z_{2}\right)=0$$, then which of the following are possible?

A. $$\operatorname{Im}\left(z_{1}\right)>0$$ and $$\operatorname{Im}\left(z_{2}\right) > 0$$

B. $$\operatorname{Im}\left(z_{1}\right) < 0$$ and $$\operatorname{Im}\left(z_{2}\right) > 0$$

C. $$\operatorname{Im}\left(z_{1}\right) > 0$$ and $$\operatorname{Im}\left(z_{2}\right) < 0$$

D. $$\operatorname{Im}\left(z_{1}\right) < 0$$ and $$\operatorname{Im}\left(z_{2}\right) < 0$$

Choose the correct answer from the options given below :

Let $$y=f(x)$$ be the solution of the differential equation $$y(x+1)dx-x^2dy=0,y(1)=e$$. Then $$\mathop {\lim }\limits_{x \to {0^ + }} f(x)$$ is equal to

Let $$\Delta$$ be the area of the region $$\left\{ {(x,y) \in {R^2}:{x^2} + {y^2} \le 21,{y^2} \le 4x,x \ge 1} \right\}$$. Then $${1 \over 2}\left( {\Delta - 21{{\sin }^{ - 1}}{2 \over {\sqrt 7 }}} \right)$$ is equal to

The domain of $$f(x) = {{{{\log }_{(x + 1)}}(x - 2)} \over {{e^{2{{\log }_e}x}} - (2x + 3)}},x \in \mathbb{R}$$ is

Let $$f:R \to R$$ be a function such that $$f(x) = {{{x^2} + 2x + 1} \over {{x^2} + 1}}$$. Then

Let $$[x]$$ denote the greatest integer $$\le x$$. Consider the function $$f(x) = \max \left\{ {{x^2},1 + [x]} \right\}$$. Then the value of the integral $$\int\limits_0^2 {f(x)dx} $$ is

Let $$A=\left\{(x, y) \in \mathbb{R}^{2}: y \geq 0,2 x \leq y \leq \sqrt{4-(x-1)^{2}}\right\}$$ and

$$ B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: 0 \leq y \leq \min \left\{2 x, \sqrt{4-(x-1)^{2}}\right\}\right\} \text {. } $$.

Then the ratio of the area of A to the area of B is

Consider the following system of equations

$$\alpha x+2y+z=1$$

$$2\alpha x+3y+z=1$$

$$3x+\alpha y+2z=\beta$$

for some $$\alpha,\beta\in \mathbb{R}$$. Then which of the following is NOT correct.

If the vectors $$\overrightarrow a = \lambda \widehat i + \mu \widehat j + 4\widehat k$$, $$\overrightarrow b = - 2\widehat i + 4\widehat j - 2\widehat k$$ and $$\overrightarrow c = 2\widehat i + 3\widehat j + \widehat k$$ are coplanar and the projection of $$\overrightarrow a $$ on the vector $$\overrightarrow b $$ is $$\sqrt {54} $$ units, then the sum of all possible values of $$\lambda + \mu $$ is equal to :

Let $$x=2$$ be a root of the equation $$x^2+px+q=0$$ and $$f(x) = \left\{ {\matrix{ {{{1 - \cos ({x^2} - 4px + {q^2} + 8q + 16)} \over {{{(x - 2p)}^4}}},} & {x \ne 2p} \cr {0,} & {x = 2p} \cr } } \right.$$

Then $$\mathop {\lim }\limits_{x \to 2{p^ + }} [f(x)]$$, where $$\left[ . \right]$$ denotes greatest integer function, is

Let $$a_1,a_2,a_3,...$$ be a $$GP$$ of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then $$a_1a_9+a_2a_4a_9+a_5+a_7$$ is equal to __________.

Suppose $$f$$ is a function satisfying $$f(x + y) = f(x) + f(y)$$ for all $$x,y \in N$$ and $$f(1) = {1 \over 5}$$. If $$\sum\limits_{n = 1}^m {{{f(n)} \over {n(n + 1)(n + 2)}} = {1 \over {12}}} $$, then $$m$$ is equal to __________.

Let the coefficients of three consecutive terms in the binomial expansion of $$(1+2x)^n$$ be in the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of those three terms, is __________.

If the co-efficient of $$x^9$$ in $${\left( {\alpha {x^3} + {1 \over {\beta x}}} \right)^{11}}$$ and the co-efficient of $$x^{-9}$$ in $${\left( {\alpha x - {1 \over {\beta {x^3}}}} \right)^{11}}$$ are equal, then $$(\alpha\beta)^2$$ is equal to ___________.

If all the six digit numbers $$x_1\,x_2\,x_3\,x_4\,x_5\,x_6$$ with $$0< x_1 < x_2 < x_3 < x_4 < x_5 < x_6$$ are arranged in the increasing order, then the sum of the digits in the $$\mathrm{72^{th}}$$ number is _____________.

Let $$f:\mathbb{R}\to\mathbb{R}$$ be a differentiable function that satisfies the relation $$f(x+y)=f(x)+f(y)-1,\forall x,y\in\mathbb{R}$$. If $$f'(0)=2$$, then $$|f(-2)|$$ is equal to ___________.

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is ____________.

Let the co-ordinates of one vertex of $$\Delta ABC$$ be $$A(0,2,\alpha)$$ and the other two vertices lie on the line $${{x + \alpha } \over 5} = {{y - 1} \over 2} = {{z + 4} \over 3}$$. For $$\alpha \in \mathbb{Z}$$, if the area of $$\Delta ABC$$ is 21 sq. units and the line segment $$BC$$ has length $$2\sqrt{21}$$ units, then $$\alpha^2$$ is equal to ___________.

Two particles of equal mass '$$m$$' move in a circle of radius '$$r$$' under the action of their mutual gravitational attraction. The speed of each particle will be :

Find the mutual inductance in the arrangement, when a small circular loop of wire of radius '$$R$$' is placed inside a large square loop of wire of side $$L$$ $$(L \gg R)$$. The loops are coplanar and their centres coincide :

In a cuboid of dimension $$2 \mathrm{~L} \times 2 \mathrm{~L} \times \mathrm{L}$$, a charge $$q$$ is placed at the center of the surface '$$\mathrm{S}$$' having area of $$4 \mathrm{~L}^{2}$$. The flux through the opposite surface to '$$\mathrm{S}$$' is given by

Surface tension of a soap bubble is $$2.0 \times 10^{-2} \mathrm{Nm}^{-1}$$. Work done to increase the radius of soap bubble from $$3.5 \mathrm{~cm}$$ to $$7 \mathrm{~cm}$$ will be:

Take $$\left[\pi=\frac{22}{7}\right]$$

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: If $$d Q$$ and $$d W$$ represent the heat supplied to the system and the work done on the system respectively. Then according to the first law of thermodynamics $$d Q=d U-d W$$.

Reason R: First law of thermodynamics is based on law of conservation of energy.

In the light of the above statements, choose the correct answer from the options given below:

The magnitude of magnetic induction at mid point $$\mathrm{O}$$ due to current arrangement as shown in Fig will be

A stone is projected at angle $$30^{\circ}$$ to the horizontal. The ratio of kinetic energy of the stone at point of projection to its kinetic energy at the highest point of flight will be -

A block of mass $m$ slides down the plane inclined at angle $$30^{\circ}$$ with an acceleration $$\frac{g}{4}$$. The value of coefficient of kinetic friction will be:

Match List I with List II :

Choose the correct answer from the options given below:

A car is moving on a horizontal curved road with radius 50 m. The approximate maximum speed of car will be, if friction between tyres and road is 0.34. [take g = 10 ms$$^{-2}$$]

Which of the following are true?

A. Speed of light in vacuum is dependent on the direction of propagation.

B. Speed of light in a medium is independent of the wavelength of light.

C. The speed of light is independent of the motion of the source.

D. The speed of light in a medium is independent of intensity.

Choose the correct answer from the options given below:

In a Young's double slit experiment, two slits are illuminated with a light of wavelength $$800 \mathrm{~nm}$$. The line joining $$A_{1} P$$ is perpendicular to $$A_{1} A_{2}$$ as shown in the figure. If the first minimum is detected at $$P$$, the value of slits separation 'a' will be:

The distance of screen from slits D = 5 cm

A single current carrying loop of wire carrying current I flowing in anticlockwise direction seen from +ve $$\mathrm{z}$$ direction and lying in $$x y$$ plane is shown in figure. The plot of $$\hat{j}$$ component of magnetic field (By) at a distance '$$a$$' (less than radius of the coil) and on $$y z$$ plane vs $$z$$ coordinate looks like

Which one of the following statement is not correct in the case of light emitting diodes?

A. It is a heavily doped p-n junction.

B. It emits light only when it is forward biased.

C. It emits light only when it is reverse biased.

D. The energy of the light emitted is equal to or slightly less than the energy gap of the semiconductor used.

Choose the correct answer from the options given below:

A bicycle tyre is filled with air having pressure of $$270 ~\mathrm{kPa}$$ at $$27^{\circ} \mathrm{C}$$. The approximate pressure of the air in the tyre when the temperature increases to $$36^{\circ} \mathrm{C}$$ is

The threshold wavelength for photoelectric emission from a material is 5500 $$\mathop A\limits^o $$. Photoelectrons will be emitted, when this material is illuminated with monochromatic radiation from a

A. 75 W infra-red lamp

B. 10 W infra-red lamp

C. 75 W ultra-violet lamp

D. 10 W ultra-violet lamp

Choose the correct answer from the options given below :

Ratio of thermal energy released in two resistors R and 3R connected in parallel in an electric circuit is :

Two simple harmonic waves having equal amplitudes of 8 cm and equal frequency of 10 Hz are moving along the same direction. The resultant amplitude is also 8 cm. The phase difference between the individual waves is _________ degree.

A solid sphere of mass 2 kg is making pure rolling on a horizontal surface with kinetic energy 2240 J. The velocity of centre of mass of the sphere will be _______ ms$$^{-1}$$.

In a metre bridge experiment the balance point is obtained if the gaps are closed by 2$$\Omega$$ and 3$$\Omega$$. A shunt of X $$\Omega$$ is added to 3$$\Omega$$ resistor to shift the balancing point by 22.5 cm. The value of X is ___________.

A point charge $$q_1=4q_0$$ is placed at origin. Another point charge $$q_2=-q_0$$ is placed at $$x=12$$ cm. Charge of proton is $$q_0$$. The proton is placed on $$x$$ axis so that the electrostatic force on the proton is zero. In this situation, the position of the proton from the origin is ___________ cm.

As shown in the figure, three identical polaroids P$$_1$$, P$$_2$$ and P$$_3$$ are placed one after another. The pass axis of P$$_2$$ and P$$_3$$ are inclined at angle of 60$$^\circ$$ and 90$$^\circ$$ with respect to axis of P$$_1$$. The source S has an intensity of 256 $$\frac{W}{m^2}$$. The intensity of light at point O is ____________ $$\frac{W}{m^2}$$.

A 0.4 kg mass takes 8s to reach ground when dropped from a certain height 'P' above surface of earth. The loss of potential energy in the last second of fall is __________ J.

(Take g = 10 m/s$$^2$$)

A certain elastic conducting material is stretched into a circular loop. It is placed with its plane perpendicular to a uniform magnetic field B = 0.8 T. When released the radius of the loop starts shrinking at a constant rate of 2 cms$$^{-1}$$. The induced emf in the loop at an instant when the radius of the loop is 10 cm will be __________ mV.