Given are 4 statements related to the chemical properties of Glucose.

Identify the two incorrect statements from the following.

A. It reacts with $$\mathrm{Br}_2$$ (aq) to form Saccharic acid.

B. Reacts with Acetic anhydride to form Glucose tetraacetate.

C. Reacts with Hydroxylamine to give Glucose oxime.

D. It reacts with ammoniacal $$\mathrm{AgNO}_3$$ to form ammonium salt of Gluconic acid with deposition of silver.

At $$700 \mathrm{~K}$$, the Equilibrium constant value for the formation of $$\mathrm{HI}$$ from $$\mathrm{H}_2$$ and $$\mathrm{I}_2$$ is 49.0 . 0.7 mole of $$\mathrm{HI}(\mathrm{g})$$ is present at equilibrium. What will be the concentrations of $$\mathrm{H}_2$$ and $$\mathrm{I}_2$$ gases if we initially started with $$\mathrm{HI}(\mathrm{g})$$ and allowed the reaction to reach equilibrium at the same temperature?

A compound having molecular formula $$\mathrm{C}_4 \mathrm{H}_{11} \mathrm{N}$$, reacts with $$\mathrm{CHCl}_3$$ in alcoholic $$\mathrm{KOH}$$, on heating, to form a compound with a foul smell. Identify the optically active isomer of the compound which also shows the above reaction.

A Hydrocarbon [A] (molecular formula $$\mathrm{C}_3 \mathrm{H}_6$$) on reaction with $$\mathrm{Br}_2 / \mathrm{CCl}_4$$ gave [B]. When [B] is heated with 2 moles of alcoholic $$\mathrm{KOH}$$, it gave compound [C]. 3 moles of Compound [C] when passed through red hot Iron tube forms [D]. Identify [D].

Identify the end-product [D] formed when solution salicylate undergoes the following series of reactions

For a given reaction, ,$$\mathrm{X}(\mathrm{g})+\mathrm{Y}(\mathrm{g} \rightarrow \mathrm{Z}(\mathrm{g})$$, the order of reaction with respect to $$\mathrm{X}$$ and $$\mathrm{Y}$$ are $$\mathrm{m}$$ and $$\mathrm{n}$$ respectively. If the concentration of $$\mathrm{X}$$ is tripled and that of $$\mathrm{Y}$$ is decreased to one third, what is the ratio between the new rate to the original rate of the reaction?

Study the graph between partial pressure and mole fraction of some gases and arrange the gases P, Q, R and S dissolved in H$$_2$$O, in the decreasing order of their K_H values.

Identify the final product formed when Toluene undergoes a series of reactions with reagents given in the order:

(i) $$\mathrm{Cl}_2$$ / Sunlight

(ii) $$\mathrm{H}_2 \mathrm{O} / 373 \mathrm{~K}$$

(iii) Acetophenone / $$\mathrm{OH^-}$$ at $$293 \mathrm{~K}$$

4 statements are given below. Identify the incorrect statement

A. Phenol has lower $$\mathrm{pK}_{\mathrm{a}}$$ value than $$\mathrm{p}$$-cresol

B. 2-Chlorophenol is more acidic than phenol

C. Ortho and para nitrophenols can be separated by steam distillation since $$\mathrm{p}$$-Nitrophenol is more steam volatile than o-Nitrophenol

D. Phenol on reaction with $$\frac{\mathrm{Cr}_2 \mathrm{O}_7^{2-}}{\mathrm{H}^{+}}$$ yields a conjugated diketone

One of the reactions A, B, C, D given below yields a product which will not answer Hinsberg's test when reacted with Benzene sulphonyl chloride. Identify the reaction.

Identify the correct statement from the following.

Identify the compound which is non-aromatic in nature.

Given that the freezing point of benzene is $$5.48^{\circ} \mathrm{C}$$ and its $$\mathrm{K}_{\mathrm{f}}$$ value is $$5.12{ }^{\circ} \mathrm{C} / \mathrm{m}$$. What would be the freezing point of a solution of $$20 \mathrm{~g}$$ of propane in $$400 \mathrm{~g}$$ of benzene?

Identify the final product formed when Benzamide undergoes the following reactions:

Identify the correct statement regarding corrosion of iron rod left exposed to atmosphere.

Two statements, Assertion and Reason are given below. Choose the correct option.

Assertion: n-propyl tert-butyl ether can be readily prepared in the laboratory by Williamson's synthesis.

Reason: The reaction occurs by $$\mathrm{S}_{\mathrm{N}} 1$$ attack of Primary alkoxide on Tert-alkyl halide to give good yield of the product, n-propyl tert-butyl ether.

Two statements, Assertion and Reason are given. Choose the correct option from the following.

Assertion: In the secondary structure of RNA, double helix structure is formed.

Reason: In the double structure of RNA, two nucleic acid chains are wound about each other and held together by hydrogen bonds between pairs of bases

An inorganic compound $$\mathrm{W}$$ undergoes the following reactions:

$$ \begin{gathered} W+\frac{\mathrm{Na}_2 \mathrm{CO}_3}{\mathrm{O}_2 / \text { heat }} \rightarrow X+\frac{\mathrm{H}^{+}}{} \rightarrow Y_{(s)} \\ Y(a q)+\mathrm{KCl}(\mathrm{aq}) \rightarrow Z_{(S)} \end{gathered} $$

Z appears in the form of orange crystals and is used as an oxidising agent in acid medium. Identify the compound W.

Given are the names of 4 compounds. Two of these compounds will not undergo Cannizzaro's reaction. Identify the two.

A. 2-Chlorobutanal

B. 2,2-Dimethylpropanal

C. Benzaldehyde

D. 2-Phenyl ethanol

Identify the product $$[\mathrm{C}]$$ formed at the end of the reaction below

1,1,2,2- Tetrabromopropane + 2 $$\mathrm{Zn}_{(\mathrm{S})}$$ / Ethanol $$\rightarrow$$ [B]

$$[\mathrm{B}]$$ + 2 moles of $$\mathrm{HBr} \rightarrow[\mathrm{C}]$$

What is/are the product/s formed when Benzaldehyde and Ethanal react in presence of dil. $$\mathrm{NaOH}$$ followed by heating the intermediate product formed?

In the estimation of element $$\mathrm{X}$$ in an organic compound, $$0.8 \mathrm{~g}$$ of the compound containing $$\mathrm{X}$$ was heated with fuming $$\mathrm{HNO}_3$$ and the cooled product was treated with barium chloride. The mass of barium sulphate precipitated was $$1.2 \mathrm{~g}$$. What is the percentage of element $$\mathrm{X}$$ and what is the formula of the violet-coloured compound formed when Lassaigne's extract of the organic compound is treated with sodium nitroprusside?

Identify the pair of molecules in which one of them is a molecule with an odd electron and the other has an expanded octet.

Arrange the following compounds in the increasing order of their reactivity when each of them is reacted with chloroethane / anhydrous AlCl$$_3$$.

What mass of Silver chloride, in grams, gets precipitated when $$150 \mathrm{~ml}$$ of $$32 \%$$ solution of Silver nitrate is reacted with $$150 \mathrm{ml}$$ of $$11 \%$$ Sodium chloride solution? Atomic mass in $$\mathrm{g} / \mathrm{mol}: \mathrm{Ag}=108, \mathrm{Na}=23, \mathrm{Cl}=35.5, \mathrm{~N}=14, \mathrm{O}=16$$

Which of the following 2 compounds exhibit both Geometrical and Structural isomerism?

$$\begin{aligned} & \mathrm{A}=\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_4 \mathrm{Cl}_2\right] \mathrm{NO}_2 \\ & \mathrm{~B}=\left[\mathrm{Co}\left(\mathrm{NH}_3\right) \mathrm{Br}\right] \mathrm{SO}_4 \\ & \mathrm{C}=\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_3\left(\mathrm{NO}_2\right)_3\right] \\ & \mathrm{D}=\left[\mathrm{Cr}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right] \mathrm{Cl}_3 \end{aligned}$$

What is the wave number (Units $$\mathrm{cm}^{-1}$$) of the longest wave length transition in the Balmer series of Hydrogen spectrum? $$Z=1 \text { for } H$$

Given below are 2 statements: Assertion and Reason. Choose the correct option.

Assertion: When Molar conductivity for a strong electrolyte is plotted versus $$\sqrt{C}(\mathrm{~mol} / \mathrm{L})^{1 / 2}$$, a straight line is obtained with intercept equal to Molar conductivity at infinite dilution for the electrolyte and Slope equal to $$-\mathrm{A}$$. All electrolytes of a given type have the same $$\mathrm{A}$$ value.

Reason: At infinite dilution, strong electrolytes of the same type will have different number of ions due to incomplete dissociation.

Based on Valence Bond Theory, match the complexes listed in Column I with the number of unpaired electrons on the central metal ion, given in Column II

What is the quantity of charge, in Faraday units, required for the reduction of 3.5 moles of $$\mathrm{Cr}_2 \mathrm{O} 7^{2-}$$ in acid medium?

A dry cell consists of a moist paste of $$\mathrm{NH}_4 \mathrm{Cl}$$ and $$\mathrm{ZnCl}_2$$ contained in a $$\mathrm{Zn}$$ casing which encloses a Carbon rod surrounded by black $$\mathrm{MnO}_2$$ paste. What is the role of $$\mathrm{ZnCl}_2$$ in it?

Compounds $$\mathrm{A}$$ and $$\mathrm{B}$$, having the same molecular formula $$(\mathrm{C}_4 \mathrm{H}_8 \mathrm{O})$$, react separately with $$\mathrm{CH}_3 \mathrm{MgBr}$$, followed by reaction with dil. $$\mathrm{HCl}$$ to form compounds $$\mathrm{X}$$ and $$\mathrm{Y}$$ respectively. Compound $$\mathrm{Y}$$ undergoes acidic dehydration in presence of Conc. $$\mathrm{H}_2 \mathrm{SO}_4$$ much more readily than $$\mathrm{X}$$. Compound $$\mathrm{Y}$$ also reacts with Lucas reagent, much more readily than $$\mathrm{X}$$, with appearance of turbidity. Identify $$\mathrm{X}$$ and $$\mathrm{Y}$$.

Match the names of reactions given in Column I with the appropriate reactions given in Column II.

With reference to Pauling's Electronegativity scale, which one of the following options shows the correct order of electronegativity values of elements?

What volume of $$0.2 \mathrm{~M}$$ Acetic acid is to be added to $$100 \mathrm{ml}$$ of $$0.4 \mathrm{M}$$ Sodium acetate so that a Buffer solution of $$\mathrm{pH}$$ equal to 4.94 is obtained? $$(\mathrm{pK}_{\mathrm{a}}$$ of $$\mathrm{CH}_3 \mathrm{COOH}=4.76)$$

Arrange the compounds $$\mathrm{A}, \mathrm{B}, \mathrm{C}$$ and $$\mathrm{D}$$ in the increasing order of their reactivity towards $$\mathrm{S_N}1$$ reaction.

$$ \begin{aligned} & \mathrm{A}=\mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{Cl} \\ & \mathrm{B}=\mathrm{C}_6 \mathrm{H}_5 \mathrm{Cl} \\ & \mathrm{C}=\mathrm{CH}_2=\mathrm{CH}-\mathrm{Cl} \\ & \mathrm{D}=\mathrm{CH}_3-\mathrm{CH}_2-\mathrm{Cl} \end{aligned}$$

For a reaction $$5 X+Y \rightarrow 3 Z$$, the rate of formation of $$Z$$ is $$2.4 \times 10^{-5} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$$ Calculate the average rate of disappearance of $$\mathrm{X}$$.

In the redox reaction between $$\mathrm{Cr}_2 \mathrm{O}_7^{2-} / \mathrm{H}^{+}$$ and sulphite ion, what is the number of moles of electrons involved in producing 3.0 moles of the oxidised product?

Which of the following two molecular species are Diamagnetic in nature?

$$[\mathrm{A}]=\mathrm{O}_2^{-} \quad[\mathrm{B}] .=\mathrm{N}_2^{+} \quad[\mathrm{C}]=\mathrm{C}_2 \quad[\mathrm{D}]=\mathrm{O}_2{ }^{2-}$$

A solute $$\mathrm{X}$$ is found to exist as a dimer in water. A 4 molal solution of $$\mathrm{X}$$ shows a boiling point of $$101.04^{\circ} \mathrm{C}$$. What is the percentage association of $$\mathrm{X}$$ ? ($$\mathrm{K}_{\mathrm{b}}$$ for water $$=0.52 \mathrm{~K} / \mathrm{m}$$).

The Lanthanoid ion which would form coloured compounds is -------------.

Atomic numbers: $$\mathrm{Yb}=70, \quad \mathrm{Lu}=71, \mathrm{Pr}=59, \quad \mathrm{La}=57$$

Given below are 4 statements. Two of these are correct statements. Identify them.

A. $$\mathrm{Co}^{2+}$$ is easily oxidised to $$\mathrm{Co}^{3+}$$ in the presence of a strong ligand like $$\mathrm{CN}^{-}$$

B. $$[\mathrm{Fe}(\mathrm{CN})_6]^{4-}$$ is an octahedral complex ion which is paramagnetic in nature.

C. Removal of $$\mathrm{H}_2 \mathrm{O}$$ molecules from $$[\mathrm{Ti}(\mathrm{H}_2 \mathrm{O})_6] \mathrm{Cl}_3$$ on strong heating converts it to a colourless compound.

D. Crystal Field splitting in Octahedral and Tetrahedral complexes is given by the equation $$\Delta_0=4 / 9 \Delta_t$$

The Enthalpy of combustion of $$\mathrm{C}_6 \mathrm{H}_5 \mathrm{COOH}(\mathrm{s})$$ at $$25^{\circ} \mathrm{c}$$ and 1.0 atm pressure is $$-2546 \mathrm{~kJ} / \mathrm{mol}$$. What is the Internal energy change for this reaction?

Sulphuric acid used in Lead Storage battery has a concentration of $$4.5 \mathrm{~M}$$ and a density of $$1.28 \mathrm{~g} / \mathrm{~ml}$$. The molality of the acid is __________.

Given :

$$ \begin{gathered} \Delta \mathrm{H}^0 \mathrm{f}_{\text {of }} \mathrm{CO}_2(\mathrm{~g})=-393.5 \mathrm{~kJ} / \mathrm{mol} \\ \Delta \mathrm{H}^0{ }_{\mathrm{f}} \text { of } \mathrm{H}_2 \mathrm{O}(\mathrm{l})=-286 \mathrm{~kJ} / \mathrm{mol} \\ \Delta \mathrm{H}^0{ }_{\mathrm{f}} \text { of } \mathrm{C}_3 \mathrm{H}_6(\mathrm{~g})=+20.6 \mathrm{~kJ} / \mathrm{mol} \end{gathered} $$

$$\Delta \mathrm{H}^0$$ isomerisation of Cyclopropane to Propene $$=-33 \mathrm{~kJ} / \mathrm{mol}$$

What is the standard enthalpy of combustion of Cyclopropane?

A current of 3.0A is passed through 750 ml of 0.45 M solution of CuSO$$_4$$ for 2 hours with a current efficiency of 90%. If the volume of the solution is assumed to remain constant, what would be the final molarity of CuSO$$_4$$ solution?

Match the Vitamins given in Column I with the diseases caused by their deficiency as given in Column II.

Which one of the following is the correct order of reagents to be used to convent [A] to [X] ?

$$ \text { Reagents: } \mathrm{PCC}, \mathrm{PCl}_5, \mathrm{LiAlH}_4, \frac{\mathrm{KCN}}{\mathrm{H}_3 \mathrm{O}^{+}} $$

Arrange the following redox couples in the increasing order of their reducing strength:

$$\begin{array}{ll} {[\mathrm{A}]=\mathrm{Cu} / \mathrm{Cu}^{2+}} & \mathrm{E}^0=-0.34 \mathrm{~V} \\ {[\mathrm{~B}]=\mathrm{Ag} / \mathrm{Ag}^{+}} & \mathrm{E}^0=-0.8 \mathrm{~V} \\ {[\mathrm{C}]=\mathrm{Ca} / \mathrm{Ca}^{2+}} & \mathrm{E}^0=+2.87 \mathrm{~V} \\ {[\mathrm{D}]=\mathrm{Cr} / \mathrm{Cr}^{3+}} & \mathrm{E}^0=+0.74 \mathrm{~V} \end{array}$$

In the presence of a catalyst at a given temperature of $$27^{\circ} \mathrm{C}$$, the Activation energy of a specific reaction is reduced by $$100 \mathrm{~J} / \mathrm{mol}$$. What is the ratio between the rate constants for the catalysed $$(\mathrm{k}_2)$$ and uncatalysed $$(\mathrm{k}_1)$$ reactions?

5.0 moles of an Ideal gas at 3.0 atm pressure and $$27^{\circ} \mathrm{C}$$ is compressed isothermally to half its volume by application of an external pressure of $$3.5 \mathrm{~atm}$$. What is the amount of work done (in joules) on the gas? Given: $$1 \mathrm{~L} \mathrm{~atm}=101.3 \mathrm{~J}: \mathrm{R}=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~K}{ }^{-1} \mathrm{~mol}^{-1}$$

Identify the products C, D and F formed in the following sets of reactions.

Given below are 4 reactions. Two of these reactions will give product which is an equimolar mixture of the d and 1 forms. Identify these 2 reactions.

[A] 2- Methylpropene + $$\mathrm{HI} \rightarrow$$ ---------

[B] But-1-ene + $$\mathrm{HBr} \rightarrow$$ -----------

[C] 3-Methylbut-1-ene + $$\mathrm{HI} \rightarrow$$ -----------

[D] 3- Phenylpropene + $$\mathrm{HBr}$$ (Peroxide) $$\rightarrow$$ -----------

$$200 \mathrm{ml}$$ of an aqueous solution contains $$3.6 \mathrm{~g}$$ of Glucose and $$1.2 \mathrm{~g}$$ of Urea maintained at a temperature equal to $$27^{\circ} \mathrm{C}$$. What is the Osmotic pressure of the solution in atmosphere units?

$$\mathrm{R}=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~K}{ }^{-1} \mathrm{~mol}^{-1}$$ : Molecular Formula: Glucose is $$\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6$$ and of Urea is $$\mathrm{NH}_2 \mathrm{CONH}_2$$

The following data was recorded for the decomposition of XY compound at 750K

What is the order of reaction with respect to decomposition of XY?

Identify the final product $$[\mathrm{D}]$$ formed when Benzyl alcohol undergoes the following series of reactions

$$\mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{OH}+\mathrm{SOCl}_2 \rightarrow[\mathrm{A}]+\mathrm{KCN} \rightarrow[\mathrm{B}]+\mathrm{H}_3 \mathrm{O}^{+} \text {(partial hydrolysis) } \rightarrow[\mathrm{C}]+\mathrm{H}_3 \mathrm{O}^{+} \text {(complete hydrolysis) } \rightarrow[\mathrm{D}]$$

Choose the incorrect statement from the following:

A. Isoelectronic molecules/ions have the same bond order.

B. Dipole moment of $$\mathrm{NH}_3$$ is greater than that of $$\mathrm{NF}_3$$.

C. The Carbon in Methyl Carbocation is $$\mathrm{sp}^3$$ hybridised.

D. The stability of an ionic compound is measured in terms of its lattice enthalpy and not simply based on attaining Octet configuration.

Based on Crystal Field theory, match the Complex ions listed in Column I with the electronic configuration in the d orbitals of the central metal ion listed in Column II.

The Activation energy for the reaction $$A \rightarrow B+C$$, at a temperature $$\mathrm{TK}$$ was $$0.04606 \mathrm{~RT} \mathrm{~J} / \mathrm{mol}$$. What is the ratio of Arrhenius factor to the Rate constant for this reaction?

$$ \text { The rate of change of the volume of a sphere with respect to its surface area } \mathrm{S} \text { is } $$

While shuffling a pack of cards, 3 cards were accidently dropped, then find the probability that the missing cards belong to different suits?

The area of the triangle whose vertices are $$(-2, a)(2,-6)$$ and $$(5,4)$$ is 35 sq units then the value of '$$\mathrm{a}$$' is

$$ \text { If } 3 A+4 B^t=\left(\begin{array}{ccc} 7 & -10 & 17 \\ 0 & 6 & 31 \end{array}\right) \text { and } 2 B-3 A^t=\left(\begin{array}{cc} -1 & 18 \\ 4 & -6 \\ -5 & -7 \end{array}\right) \text { then }(5 B)^t= $$

The domain of the function $$y=\frac{1}{\log _{10}(3-x)}+\sqrt{x+7}$$ is

Consider an infinite geometric series with first term '$$a$$' and common ratio '$$r$$'. If the sum of infinite geometric series is 4 and the second term is $$\frac{3}{4}$$ then

Let $$\alpha$$ and $$\beta$$ be the distinct roots of $$a x^2+b x+c=0$$, then $$\lim _\limits{x \rightarrow \alpha} \frac{1-\cos \left(a x^2+b x+c\right)}{(x-\alpha)^2}$$ is equal to

If two positive numbers are in the ratio $$3+2 \sqrt{2}: 3-2 \sqrt{2}$$, then the ratio between their A.M (arithmetic mean) and G.M (geometric mean) is

$$ \text { The value of the integral } \int_\limits{\frac{1}{3}}^1 \frac{\left(x-x^3\right)^{\frac{1}{3}}}{x^4} d x \text { is } $$

$$ \text { The area of the region enclosed by the curve }\left\{(x, y): 4 x^2+25 y^2=100\right\} \text { is } $$

The mean of five observations is 4 and their variance is 5.2 . If three of these observations are 1, 2 and 6, then the other two observations are

The line joining two points $$A(2,0) B(3,1)$$ is rotated about $$A$$ in anticlockwise direction through an angle of $$15^{\circ}$$. If $$B$$ goes to $$C$$ in the new position, then the coordinates of $$C$$ is

$$ \text { The value of } \lim _\limits{x \rightarrow 1} \frac{x^{15}-1}{x^{10}-1}= $$

Let $$\mathrm{A}$$ and $${B}$$ be two events such that $$P(A / B)=\frac{1}{2}$$ and $$P(B / A)=\frac{1}{3}$$ and $$P(A \cap B)=\frac{1}{6}$$ then, which one of the following is not true?

$$ \text { If } \frac{\cos x}{\cos (x-2 y)}=\lambda \text { then } \tan (x-y) \tan y= $$

$$ \text { If } y=f(x), \quad p=\frac{d y}{d x} ; q=\frac{d^2 y}{d x^2} \text { then } \frac{d^2 x}{d y^2} \text { is equal to } $$

$$ \text { If } \hat{\imath}+\hat{\jmath}-\hat{k} \quad \&~ 2 \hat{\imath}-3 \hat{\jmath}+\hat{k} \text { are adjacent sides of a parallelogram, then length of its diagonals are } $$

Which of the following relations on the set of real numbers $$\mathrm{R}$$ is an equivalence relation?

A number consists of three digits in geometric progression. The sum of the right hand and left hand digits exceeds twice the middle digit by 1 and the sum of left hand and middle digits is two third of the sum of the middle and right hand digits. Then the sum of digits of number is

$$ \text { If } y=\sqrt{\sin x+y} \text { then find } \frac{d y}{d x} \text { at } x=0, \quad y=1 $$

$$ \text { If } A=\left[\begin{array}{cc} 5 a & -b \\ 3 & 2 \end{array}\right] \text { and } A \operatorname{adj} A=A A^t \text {, then } 5 a+b \text { is equal to } $$

$$ \text { If } f(x)=\left\{\begin{array}{cc} x & , \quad 0 \leq x \leq 1 \\ 2 x-1 & , \quad x>1 \end{array}\right. \text { then } $$

The measure of the angle between the lines $$x=k+1, \quad y=2 k-1, \quad z=2 k+3, \quad k \in R \quad$$ and $$\quad \frac{x-1}{2}=\frac{y+1}{1}=\frac{z-1}{-2}$$ is

$$ \text { Evaluate: } \cot ^{-1}\left(-\frac{3}{\sqrt{3}}\right)-\sec ^{-1}\left(-\frac{2}{\sqrt{2}}\right)-\operatorname{cosec}^{-1}(-1)-\tan ^{-1}(1) $$

The shaded region in the Venn diagram represents

$$ \text { The solution set for the inequality } 13 x-5<15 x+4<7 x+12 ; x \in W \text { is } $$

The general solution of the differential equation $$x \frac{d y}{d x}=y+x \tan \left(\frac{y}{x}\right)$$ is

$$ \sqrt{2+\sqrt{2+\sqrt{2+2 \cos 8 \theta}}} \text { where } \theta \in\left[-\frac{\pi}{8}, \frac{\pi}{8}\right] \text { is equal to } $$

The turning point of the function $$y=\frac{a x-b}{(x-1)(x-4)}$$ at the point $$P(2,-1)$$ is

A coin is tossed until a head appears or until the coin has been tossed three times. Given that 'head' does not appear on the first toss, what is the probability that the coin is tossed thrice?

$$ \text { The value of } \int \frac{d x}{\sqrt{2 x-x^2}} \text { is } $$

The equation of the circle which touches the $$x$$-axis, passes through the point $$(1,1)$$ and whose centre lies on the line $$x+y=3$$ in the first quadrant is

If the matrix $A$ is such that $$A\left(\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right)=\left(\begin{array}{cc}-4 & 1 \\ 7 & 7\end{array}\right)$$ then $$A$$ is equal to

In the parabola $$y^2=4 a x$$ the length of the latus rectum is 6 units and there is a chord passing through its vertex and the negative end of the latus rectum. Then the equation of the chord is

The points on the $$x$$-axis whose perpendicular distance from the line $$\frac{x}{3}+\frac{y}{4}=1$$ is 4 units are

The side of a cube is equal to the diameter of a sphere. If the side and radius increase at the same rate then the ratio of the increase of their surface area is

Suppose we have three cards identical in form except that both sides of the first card are coloured red, both sides of the second are coloured black, and one side of the third card is coloured red and the other side is coloured black. The three cards are mixed and a card is picked randomly. If the upper side of the chosen card is coloured red, what is the probability that the other side is coloured black.

$$ \text { The function } f(x)=\tan ^{-1}(\sin x+\cos x) \text { is an increasing function in } $$

For an examination a candidate has to select 7 questions from three different groups $$\mathrm{A}, \mathrm{B}$$ and C. The three groups contain 4, 5 and 6 questions respectively. In how many different ways can a candidate make his selection if he has to select atleast 2 questions from each group?

The letters of the word "COCHIN" are permuted and all the permutations are arranged in alphabetical order as in an English dictionary. The number of words that appear before the word "COCHIN" is

The co-ordinate of the foot of the perpendicular from $$P(1,8,4)$$ on the line joining $$R(0,-1,3)$$ and $$Q(2,-3,-1)$$ is

$$ \text { The general solution of the differential equation }(1+\tan y)(d x-d y)+2 x d y=0 \text { is } $$

If $$a$$ is a real number such that $$\int_\limits0^a x d x \leq a+4$$ then

Find the value of '$$b$$' such that the scalar product of the vector $$\hat{\imath}+\hat{\jmath}+\hat{k}$$ with the unit vector parallel to the sum of the vectors $$2 \hat{\imath}+4 \hat{\jmath}-5 \hat{k}$$ and $$b \hat{\imath}+2 \hat{\jmath}+3 \hat{k}$$ is unity

$$ \text { Value of } \cos 105^{\circ} \text { is } $$

The area bounded by the curve $$y=\cos x, x=0$$ and $$x=\pi$$ is

$$ \text { If } I_n=\int_\limits0^{\frac{\pi}{4}} \tan ^n x d x \text {, for } n \geq 2 \text {, then } I_n+I_{n-2}= $$

In the expansion $$\left(\frac{1}{x}+x \sin x\right)^{10}, \quad$$ the co - efficient of $$6^{\text {th }}$$ term is equal to $$7 \frac{7}{8}$$, then the principal value of $$x$$ is

$$ \text { If }(1-4 i)^3=a+i b \text { then the value of } \mathrm{a} \text { and } \mathrm{b} \text { is } $$

Two finite sets have '$$m$$' and '$$n$$' number of elements respectively. The total number of subsets of the first set is 112 more than the total number of subsets of the second set. Then the values of $$\mathrm{m}$$ and $$\mathrm{n}$$ are respectively.

The sum of the order and degree of the differential equation $$\left(\frac{d^2 y}{d x^2}\right)^5+\frac{4\left(\frac{d^2 y}{d x^2}\right)^3}{\left(\frac{d^3 y}{d x^3}\right)}+\frac{d^3 y}{d x^3}=x^2-1$$ is

$$ \int e^x\left[\frac{x^2+1}{(x+1)^2}\right] d x \quad \text { is equal to } $$

$$ \text { Evaluate: } \cos ^{-1}\left(\cos \frac{35 \pi}{18}\right)-\sin ^{-1}\left(\sin \frac{35 \pi}{18}\right) $$

The maximum value of $$P=500 x+400 y$$ for the given constraints $$x+y \leq 200, \quad x \geq 20, \quad y \geq 4 x, \quad y \geq 0$$ is

$$ \text { If } y=\sin ^{-1}\left(\frac{5 x+12 \sqrt{1-x^2}}{13}\right) \text { then } \frac{d y}{d x} \text { equals } $$

If the straight lines $$\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-t}$$ and $$\frac{x-1}{t}=\frac{y-4}{2}=\frac{z-5}{1}$$ are intersecting then $$t$$ can have

$$ \text { If } \frac{d y}{d x}=y+3>0 \text { and } y(0)=2 \text { then } y(\log 2) \text { is equal to } $$

What is the probability of a randomly chosen 2 digit number being divisible by 3 ?

If $$A=\left[\begin{array}{ccc}0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x\end{array}\right]$$ is a singular matrix then $$x$$ is equal to

What is the nature of the function $$f(x)=x^3-3 x^2+4 x$$ on real numbers?