On the basis of crystal field theory, electronic configuration of a low spin $$\mathrm{d}^4$$ complex is:

In the reaction,

$$2 \mathrm{~S}_2 \mathrm{O}_3^{2-}+\mathrm{I}_2 \rightarrow \mathrm{S}_4 \mathrm{O}_6^{2-}+2 \mathrm{I}^{-}$$

An aqueous solution of glucose boils at $$100.01^{\circ} \mathrm{C}$$. The number of glucose molecules in a solution containing $$100 \mathrm{~g}$$ of water is _________ [$$\mathrm{K}_{\mathrm{b}}$$ for water is $$0.5 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$$]

The molar conductivity of the complex $$\mathrm{CoCl}_3 \cdot 4 \mathrm{NH}_3 \cdot 2 \mathrm{H}_2 \mathrm{O}$$ is found to be the same as that of a $$1: 3$$ electrolyte. The structural formula of the compound is :

$$ \text { From the following pair of compounds, identify the incorrect pair in terms of the covalent character. } $$

Arrange the following compounds in the decreasing order of reactivity towards electrophilic substitution reaction.

(I) Chlorobenzene

(II) Nitrobenzene

(III) Benzene

(IV) Isopropylbenzene

With reference to vitamins and its deficiency diseases, the correct statements are:

(i) Cheilosis is due to the deficiency of Riboflavin

(ii) The deficiency of Vitamin B6 causes pernicious anaemia

(iii) Osteomalacia is due to the deficiency of Vitamin E

(iv) The deficiency of ascorbic acid causes scurvy

(v) The deficiency of Vitamin A causes xerophthalmia

Predict the final major product 'S' of the following reaction

$$\mathrm{S}_8$$ on heating at a temperature above $$1000 \mathrm{~K}$$, changes to $$\mathrm{S}_2$$. When 1 mole of $$\mathrm{S}_8$$ is heated above $$1000 \mathrm{~K}$$, the pressure falls by $$32 \%$$ at equilibrium. The equilibrium constant for the conversion is:

Which of the following compounds will show geometrical isomerism?

The major product '$$\mathrm{X}$$' of the following reaction is

The system that forms minimum boiling azeotrope is :

Among the following peptides, identify the pair where the name and its structure are correctly matched.

The temperature $$(\mathrm{T})$$ and rate constant $$(\mathrm{k})$$ for a first order reaction $$\mathrm{R} \rightarrow \mathrm{P}$$, was found to follow the equation $$\log \mathrm{k}=-(2000) \frac{1}{\mathrm{~T}}+8.0$$. The pre-exponential factor '$$\mathrm{A}$$' and activation energy $$\mathrm{E}_{\mathrm{a}}$$, respectively are: [Given: $$\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$$]

Names of some organic compounds are given. Which one is not in IUPAC system?

When $$3.92 \mathrm{~g}$$ of Mohr salt is dissolved in $$100 \mathrm{~mL}$$ of water, and titrated against $$\mathrm{KMnO}_4$$ solution, $$20 \mathrm{~mL}$$ of this solution required $$18 \mathrm{~mL}$$ of $$\mathrm{KMnO}_4$$ for complete oxidation. The strength of $$\mathrm{KMnO}_4$$ is: [Molar mass of Mohr salt is $$392 \mathrm{~g} \mathrm{~mol}^{-1}$$ and $$\mathrm{KMnO}_4$$ is $$158 \mathrm{~g} \mathrm{~mol}^{-1}$$]

Among the following, which is not according to the property indicated against it?

Arrange the following in the order of increasing reactivity towards $$\mathrm{SN}_{\mathrm{2}}$$ reactions.

$$\begin{array}{ll} \left(\mathrm{CH}_3\right)_3 \mathrm{CCH}_2 \mathrm{Br}(\mathrm{P}) ; & \\\mathrm{CH}_3\left(\mathrm{CH}_2\right)_3 \mathrm{Br}(\mathrm{Q}) ; &\\ \mathrm{CH}_3 \mathrm{CH}_2-\mathrm{CH}\left(\mathrm{CH}_3\right)-\mathrm{CH}_2 \mathrm{Br}(\mathrm{R}) ; & \\\left(\mathrm{CH}_3\right)_2 \mathrm{CHCH}_2 \mathrm{CH}_2 \mathrm{Br}(\mathrm{S}) \end{array}$$

When 0.1 mole of $$\mathrm{MnO}_4{ }^{2-}$$ is oxidised, the quantity of electricity required to completely oxidise $$\mathrm{MnO}_4{ }^{2-}$$ to $$\mathrm{MnO}_4^{-}$$ is

Choose the incorrect statement from the following.

The correct sequence of filling of electrons in $$n=6$$ is:

The metallic ions that have almost same spin only magnetic moment are :

(i) $$\mathrm{Co}^{2+}$$

(ii) $$\mathrm{Mn}^{2+}$$

(iii) $$\mathrm{Cr}^{2+}$$

(iv) $$\mathrm{Cr}^{3+}$$

In a set of reactions, bromoethane yielded a final product 'S'.

The compound 'S' should be

Which of the following tests may be used to distinguish between phenol and cyclohexanol?

Which of the following compounds undergo Aldol condensation followed by dehydration to give 3-phenylprop-2-en-1-al?

The $$\Delta \mathrm{H}_{(\mathrm{f})}^{\mathrm{o}}$$ of $$\mathrm{NO}_2(\mathrm{~g})$$ and $$\mathrm{N}_2 \mathrm{O}_4(\mathrm{~g})$$ are 16.0 and $$4.0 \mathrm{k} \mathrm{cal} \mathrm{mol}^{-1}$$ respectively. The heat of dimerisation of $$\mathrm{NO}_2$$ in $$\mathrm{k}$$ cal is :

The bond dissociation enthalpy of the species in their correct order is:

Identify the incorrect statement among the following.

At Constant volume, the heat required to raise the temperature of $$4.48 \mathrm{~L}$$ of an ideal gas at STP by $$15^{\circ} \mathrm{C}$$ is 12.0 calories. The $$\mathrm{C_p}$$ of the gas is _____________ $$(\mathrm{R}=2 \mathrm{~Cal} \mathrm{~kg}^{-1} \mathrm{~mol}^{-1})$$

$$ 1 \mathrm{~cm}^3 \text { of } 0.01 \mathrm{~M} \mathrm{~HCl} \text { is added to } 1 \mathrm{~L} \text { of } \mathrm{NaCl} \text { solution. The } \mathrm{pH} \text { of the resulting solution is: } $$

An organic compound with molecular formula $$\mathrm{C}_3 \mathrm{H}_5 \mathrm{N}$$ on hydrolysis gives compound '$$\mathrm{X}$$' which on treatment with $$\mathrm{Cl}_2 / \mathrm{P}$$ gives compound '$$\mathrm{Y}$$'. Compound '$$\mathrm{Y}$$' on reaction with $$\mathrm{NH}_3$$ gives '$$\mathrm{Z}$$'. The compound '$$\mathrm{Z}$$' is :

Which of the following carbanions is the least stable?

The correct order of increasing melting point is

A solution of $$\mathrm{KCl}(\mathrm{M}=74.5 \mathrm{~g} \mathrm{~mol}^{-1})$$ containing $$1.9 \mathrm{~g}$$ per $$100 \mathrm{~mL}$$ of $$\mathrm{KCl}$$ is isotonic with a solution of urea $$(\mathrm{M}=60.0 \mathrm{~g} \mathrm{~mol}^{-1}$$) containing $$3 \mathrm{~g}$$ per $$100 \mathrm{~mL}$$ of urea. The degree of dissociation of $$\mathrm{KCl}$$ is: [Assume both the solutions are kept at same temperature]

$$\mathrm{Cu}^{+}$$ undergoes disproportionation, according to the equation,

$$ 2 \mathrm{Cu}^{+} \rightleftharpoons \mathrm{Cu}^{2+}+\mathrm{Cu} $$

The $$\mathrm{E}^{\circ}$$ value for the reaction is:

$$[\mathrm{E}_{\mathrm{Cu}^{2+} / \mathrm{Cu}}^{\mathrm{o}}=0.34 \mathrm{~V} \text { and } \mathrm{E}_{\mathrm{Cu}^{2+} / \mathrm{Cu}^{+}}^{\mathrm{O}}=0.15 \mathrm{~V}]$$

Arrange the following amines in the decreasing order of their solubilities in water.

Consider the following reaction

$$(\mathrm{CH}_3)_2 \mathrm{CHCH}_2 \mathrm{Br}+(\mathrm{CH}_3)_3 \mathrm{CCH}_2 \mathrm{Br} \xrightarrow[\Delta]{\mathrm{Na} \text { /dry ether }} \mathrm{X}+\mathrm{Y}+\mathrm{Z}$$

Identify the product which is not formed?

Arrange the following compounds in the decreasing order of their acidic strength.

(I) m-cresol

(II) Phenol

(III) m-aminophenol

(IV) m-methoxyphenol

The ratio of the first three radii of Bohr's atom is:

Which of the following statement is not true about glucose?

The type of hybridisation and the spin only magnetic moment for the complex $$[\mathrm{Ni}(\mathrm{CO})_4]$$ are respectively:

Consider the following statements in respect of lanthanides, which of the statements are incorrect?

(i) $$\mathrm{La}(\mathrm{OH})_3$$ is least basic among the hydroxides of lanthanides

(ii) The lanthanide ions $$\mathrm{Yb}^{2+}, \mathrm{Lu}^{3+}$$ and $$\mathrm{Ce}^{4+}$$ are diamagnetic in nature.

(iii) $$\mathrm{Ce}^{4+}$$ can act as an oxidising agent

(iv) $$\operatorname{Ln}$$ (III) compounds are generally colourless

(v) Ionic radii of $$\mathrm{Ce}^{3+}$$ is greater than $$\mathrm{Yb}^{3+}$$

Which of the following is not aromatic?

Among the following compounds, the most acidic is :

Given below a first order reaction in the gas phase

$$\mathrm{A}(\mathrm{g}) \rightarrow \mathrm{B}(\mathrm{g})+\mathrm{C}(\mathrm{g})$$

If the initial pressure of the system is $$\mathrm{P}_{\mathrm{i}}$$ and the total pressure at $$\mathrm{t}$$ seconds is $$\mathrm{P}_{\mathrm{t}}$$, the rate constant $$\mathrm{k}$$ for the reaction is:

Which of the following reactions is an example for incomplete combustion?

The $$\mathrm{E}^{\circ}$$ values of $$\mathrm{A} . \mathrm{B}$$ and $$\mathrm{C}$$ are given. Which element/(s) is/(are) good for coating the surface of iron to prevent corrosion? Given: $$[\mathrm{E}_{\mathrm{Fe}^{2+} / \mathrm{Fe}}^0=-0.44 \mathrm{~V} ; \quad \mathrm{E}_{\mathrm{A}^{2+} / \mathrm{A}}^0=-2.37 \mathrm{~V} ; \quad \mathrm{E}_{\mathrm{B}^{2+} / \mathrm{B}}^0=-0.15 \mathrm{~V} ; \quad \mathrm{E}_{\mathrm{C}^{2+} / \mathrm{C}}^0=+0.34 \mathrm{~V}]$$

4-methyl benzamide (A) on reaction with ethanolic solution of $$\mathrm{KOH}$$ and bromine gives another compound (B). The compound (B) on treatment with benzoyl chloride gives compound (C). Identify the correct structure of compound (C) from the following.

Which among the following compounds has the highest freezing point of its 1 molal aqueous solution?

For the reaction, $$\mathrm{A}+3 \mathrm{~B} \rightarrow 2 \mathrm{C}+\mathrm{D}$$, the concentration of $$\mathrm{A}$$ changes from 0.0150 to 0.0125 in 1 minute. The rate of formation of $$\mathrm{C}$$ in $$\mathrm{mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$$ is:

The major product obtained in the following reaction is :

In the following question a statement of Assertion (A) followed by a statement of Reason (R) is given. Choose the correct option out of the choices given below.

Assertion (A): van't Hoff factor for a solution of benzoic acid in benzene is less than one.

Reason(R): Benzoic acid undergoes dissociation in benzene and its calculated molecular mass using colligative properties is lower than its actual molecular mass.

In the following question a statement of Assertion (A) followed by a statement of Reason (R) is given. Choose the correct option out of the choices given below.

Assertion (A): A molecule of $$\mathrm{SF}_4$$ is see-saw shaped, while that of $$\mathrm{CIF}_3$$ is T-shaped.

Reason(R): SF$$_4$$ has two lone pair of electrons. But CIF$$_3$$ has one pair of electrons.

The rate of appearance of bromine is related to the disappearance of bromide ion in the equation given below is:

$$\mathrm{BrO}_3^{-} \text {(aq) }+5 \mathrm{Br}^{-} \text {(aq) }+6 \mathrm{H}^{+} \rightarrow 3 \mathrm{Br}_2(\mathrm{l})+3 \mathrm{H}_2 \mathrm{O}(\mathrm{l})$$

When 1 mole of $$\mathrm{O}_2$$ and 1 mole of ammonia are made to react in the reaction,

$$4 \mathrm{NH}_3(\mathrm{~g})+5 \mathrm{O}_2(\mathrm{~g}) \rightarrow 4 \mathrm{NO}(\mathrm{g})+6 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$$

The limiting molar conductivity of $$\mathrm{NH}_4 \mathrm{OH}$$ is $$238 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$$. At $$25^{\circ} \mathrm{C}$$, molar conductance of $$0.1 \mathrm{M}$$ aqueous solution of ammonium hydroxide is $$9.54 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$$. The degree of ionisation of $$\mathrm{NH}_4 \mathrm{OH}$$ at the same concentration and temperature is:

The half-life for a zero order reaction is

Identify the oxidation reaction in which acidified $$\mathrm{KMnO}_4$$ is required

The correct option for free expansion of an ideal gas under adiabatic condition is:

Which of the following compound cannot be prepared by Williamson's synthesis?

P and Q are considering to apply for a job. The probability that P applies for the job is $$\frac{1}{4}$$. The probability that $$\mathrm{P}$$ applies for the job given that $$\mathrm{Q}$$ applies for the job is $$\frac{1}{2}$$, and the probability that Q applies for the job given that P applies for the job is $$\frac{1}{3}$$. Then the probability that $$\mathrm{P}$$ does not apply for the job given that $$\mathrm{Q}$$ does not apply for the job is

$$ \text { The value of } \frac{i^{1004}+i^{1006}+i^{1008}+i^{1010}+i^{1012}}{i^{510}+i^{508}+i^{506}+i^{504}+i^{502}} \text { is } $$

There are some baskets. The chances of picking a loaded basket and choosing a red coloured one is 0.2 . For every 100 tries to pick one basket, 60 times a basket is either loaded or red in colour. What is the probability of choosing an empty basket plus choosing not a red coloured one.

The sum of first three terms of a geometric progression is 16 and the sum of next three terms is 128 . The sum to $$\mathrm{n}$$ terms of the geometric progression is

$$ \text { The general solution of the differential equation } \frac{d y}{d x}=\frac{x y}{x^2+y^2} \text { is } $$

The points on the ellipse $$16 x^2+9 y^2=400$$ at which the ordinate decreases at the same rate at which the abscissa increases are

$$ \text { If } a \mathcal{N}=\{a x: x \in \mathcal{N}\} \text {, then } 3 \mathcal{N} \cap 7 \mathcal{N} \text { is } $$

$$ \int \frac{f^{\prime}(x)}{f(x) \log (f(x))} d x \text { is equal to } $$

$$ \text { The maximum value of } Z=3 x+4 y \text { for the given constraints } x+2 y \leq 76,2 x+y \leq 104, x \geq 0, y \geq 0 \text { is } $$

If the line $$\frac{1-x}{-3}=y=\frac{z+2}{2}$$ is perpendicular to the line $$\frac{3 x-1}{2 b}=3-y=\frac{z-1}{a}$$, then find the value of $$3 a+3 b$$

The probability of inviting three friends on 5 consecutive days, exactly one friend a day and no friend is invited on more than two days is

$$ \lim _\limits{x \rightarrow 0} \frac{a^x-b^x}{c^x-d^x}= $$

$$ \text { If } f(x)=\frac{a \sin x+b \cos x}{c \sin x+d \cos x} \text { is decreasing for all } x \text {, then } $$

$$ \int_\limits0^{\frac{\pi}{2}} \frac{\cos x}{1+\cos x+\sin x} d x= $$

The inequality $$4 x-3 \geq \frac{10 x-1}{3}$$ represents which of the following interval when $$x \in R$$

If the distance between the foci and the distance between the two directrixes are in the ratio $$3: 2$$ for a hyperbola $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$, then a : b is

$$ \text { If } \frac{x}{\cos \theta}=\frac{y}{\cos \left(\theta+\frac{2 \pi}{3}\right)}=\frac{z}{\cos \left(\theta-\frac{2 \pi}{3}\right)} \text { then } x+y+z \text { is equal to } $$

$$ \frac{\cos 9^{\circ}+\sin 9^{\circ}}{\cos 9^{\circ}-\sin 9^{\circ}}= $$

$$ \text { The function } y=\tan x-x \text { is } $$

If the sum of the coefficients of the first three terms in the expansion of $$\left(x-\frac{a}{x^2}\right)^{12}, x \neq 0$$ is 559. Find the value of '$$a$$' if '$$a$$' belongs to positive integers

Evaluate :

$$ \operatorname{cosec}^{-1}\left(-\frac{2 \sqrt{3}}{3}\right)+\tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right)+\sec ^{-1} 2+\cos ^{-1}\left(-\frac{1}{2}\right)-\sin ^{-1}\left(\frac{\sqrt{2}}{2}\right)$$

$$ \text { Integrating factor of the differential equation } \frac{d y}{d x}+y=\frac{x^3+y}{x} \text { is } $$

Let $$\mathrm{ABC}$$ be a triangle with equations of its sides $$\mathrm{AB}, \mathrm{BC}$$. $$\mathrm{CA}$$ respectively are $$x-2=0, y-5=0$$ and $$5 x+2 y-10=0$$. Then the orthocentre of triangle lies on the line

$$ \text { If } y=\sin ^{-1}(\sqrt{\sin x}) \text {, then } \frac{d y}{d x} \text { equals } $$

Let a, b, c be three vector such that $$a \neq 0$$ and $$\vec{a} \times \vec{b}=2 \vec{a} \times \vec{c},|a|=|c|=1,|b|=4$$ and $$|\vec{b} \times \vec{c}|=\sqrt{15}$$. If $$\vec{b}-2 \vec{c}=\lambda \vec{a}$$ then $$\lambda$$ equals to

$$\text { Which of the following function is injective? }$$

$$ \int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{\cos x-\cos ^3 x} d x \text { is equal to } $$

$$ \int \log x(\log x+2) d x \text { equals to } $$

$$ \text { If } A=\left[\begin{array}{cc} 1 & -2 \\ 4 & 5 \end{array}\right] \text { and } f(t)=t^2-3 t+7 \text { then } f(A)+\left[\begin{array}{cc} 3 & 6 \\ -12 & -9 \end{array}\right] \text { is } $$

A and B are two independent events. The probability of their simultaneous occurrence is $$\frac{1}{8}$$ and the probability that neither of them occurs is $$\frac{3}{8}$$. Then their individual probabilities are

A determinant of the second order is made with elements 0 and 1 . What is the probability that the determinant made is non-negative?

$$ \text { If } x^2+y^2=t+\frac{1}{t} \text { and } x^4+y^4=t^2+\frac{1}{t^2} \text { then } \frac{d y}{d x}= $$

The lines $$\vec{r}=(2 \hat{\jmath}-3 \hat{k})+\lambda(\hat{\imath}+2 \hat{\jmath}+3 \hat{k})$$ and $$\vec{r}=(2 \hat{\imath}+6 \hat{\jmath}+3 \hat{k})+\mu(2 \hat{\imath}+3 \hat{\jmath}+4 \hat{k})$$ are

$$ \text { If } \alpha=\tan ^{-1}\left(\tan \frac{5 \pi}{4}\right) \text { and } \beta=\tan ^{-1}\left(-\tan \frac{2 \pi}{3}\right) \text { then } $$

A line makes the same angle $$\theta$$ with each of the $$x$$ and $$z$$-axes. If the angle $$\beta$$, which it makes with the $$y$$-axis is such that $$\sin ^2 \beta=3 \sin ^2 \theta$$, then $$\cos ^2 \theta$$ equals

$$ \text { The number of points of discontinuity of the rational function } f(x)=\frac{x^2-3 x+2}{4 x-x^3} $$

$$ \text { The value of } \lim _\limits{x \rightarrow 0} \frac{\sin (a+x)-\sin (a-x)}{x} \text { is } $$

If $$f(x)=\log x+b x^2+a x, x \neq 0$$ has extreme values (or turning points) at $$x=-1$$ and $$x=2$$ then the values of $$\mathrm{a}$$ and $$\mathrm{b}$$ are

$$ \left|\begin{array}{ccc} \cos (\alpha+\beta) & -\sin (\alpha+\beta) & \cos 2 \beta \\ \sin \alpha & \cos \alpha & \sin \beta \\ -\cos \alpha & \sin \alpha & \cos \beta \end{array}\right| $$ is independent of

The area of the region (in sq units) bounded by the curve $$ y=\sqrt{16-x^2} \text { and } x \text {-axis is } $$

$$ \text { Let } \mathrm{A} \text { and } \mathrm{B} \text { be two sets then } A-(A \cap B) \text { is equal to } $$

The sum of four numbers in a geometric progression is 60 , and the arithmetic mean of the first and the last number is 18 . Then the numbers are

The number of four digit numbers strictly greater than 4321 formed using the digits $$0,1,2,3,4,5$$ with repetition of digit is

Let 'P' be the mean deviation of the first five odd natural numbers about their mean and 'Q' be the mean deviation of the first five prime numbers about their mean. The $$Q-P=$$

A relation $$R$$ is defined from $$\{2,3,4\}$$ to $$\{3,6,7,10\}$$. If $$x R y \Leftrightarrow x$$ and $$y$$ are co prime numbers. Then range of $$R$$ is

$$ \text { The general solution of } \frac{d y}{d x}=\sin ^{-1} x \text { is } $$

For what value of $$\mathrm{a}$$ and $$\mathrm{b}$$ the intercepts cut off on the co-ordinate axes by the line $$a x-b y+8=0$$ are equal in length but opposite in signs to those cut off by the line $$2 x-3 y+6=0$$ on the axes

$$ \text { If A }(\operatorname{adj} A)=5 I \text {, where I is the identity matrix of order } 3 \text {, then }|\operatorname{adj} A|= $$

A student has 3 library cards and 8 books of his interest in the library. Out of these 8 books he does not want to borrow Chemistry part 2 unless he can borrow Chemistry part 1 also. In how many ways can he choose the three books to be borrowed?

The dimensions of the largest rectangle of side $$x$$ and $$y$$ that can be inscribed in the right angled triangle of sides $$\mathrm{a}$$ and $$\mathrm{b}$$ is

$$ \text { The angle between } \hat{\imath}-\hat{\jmath} ~\&~ \hat{\jmath}-\hat{k} \text { is } $$

If $$(x-a)^2+(y-b)^2=c^2$$, where $$\mathrm{a}, \mathrm{b}, \mathrm{c}$$ are some constants, $$c>0$$ then $$\frac{\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2 y}{d x^2}}$$ is independent of

$$ \text { A square matrix } P \text { satisfies } P^2=I-P \text { where } I \text { is identity matrix. If } P^n=5 I-8 P \text {, then } n \text { is equal to } $$

$$ \text { If } 6^{\text {th }} \text { term of a geometric progression is }-\frac{1}{32} \text { and } 9^{\text {th }} \text { term is } \frac{1}{256} \text { then } r \text { is } $$

$$ \left(\cos \frac{\pi}{12}-\sin \frac{\pi}{12}\right)\left(\tan \frac{\pi}{12}+\cot \frac{\pi}{12}\right)= $$

$$ \int \frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}} d x \text { is equal to } $$

If $$A=\left[\begin{array}{lll}5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1\end{array}\right] \quad B^{-1}=\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right]$$ then $$(A B)^{-1}$$ is equal to

Degree of the differential equation $$\log \left(\frac{d y}{d x}\right)^{\frac{1}{2}}=5 x+4 y$$ is

$$ \text { The area of the region (in square units) bounded by the line } y+3=x ; x=1 \text { and } x=5 \text { is } $$

If an ellipse has an equation in the standard form and it passes through the points $$\left(\frac{5}{2}, \frac{\sqrt{6}}{4}\right)$$ and $$\left(-2, \frac{\sqrt{15}}{5}\right)$$ then the length of its latus rectum is

A transformer has 400 turns in its primary winding and 800 turns in its secondary winding. The primary voltage is $$20 \mathrm{~V}$$ and the load in the secondary is 4 ohm. The current in the primary, assuming it to be an ideal transformer, is

An object is placed at a distance of $$12 \mathrm{~cm}$$ from a convex lens on its principal axis and a virtual image of certain size is formed. If the object is moved $$4 \mathrm{~cm}$$ away from the lens, a real image of the same size as that of the virtual image is formed. The focal length of the lens in $$\mathrm{cm}$$ is

The graph of an object moving with speed v for a time t is shown below.

The graph that shows the distance s travelled by the same object for a time t is

An object of mass $$1 \mathrm{~kg}$$ is allowed to hang tangentially from the rim of the wheel of radius R. When released from the rest, the block falls vertically through $$4 \mathrm{~m}$$ height in 2 seconds. The moment of inertia is $$1 \mathrm{~kg} \mathrm{~m}^2$$. The radius of the wheel $$\mathrm{R}$$ is

A planet has double the mass of the earth and double the radius. The gravitational potential at the surface of the Earth is $$\mathrm{V}$$ and the magnitude of the gravitational field strength is $$\mathrm{g}$$. The gravitational potential and gravitational field strength on the surface of the planet are

A tuning fork of unknown frequency produces 4 beats with tuning fork of frequency $$310 \mathrm{~Hz}$$. It gives the same number of beats on filing. The initial frequency of a tuning fork is

A wire ' 1 ' $$\mathrm{cm}$$ long bent into a circular loop is placed perpendicular to the magnetic field of flux density '$$B^{\prime} W b \mathrm{~m}^{-2}$$. Within $$0.1 \mathrm{sec}$$, the loop is changed into a square of side '$$a$$' $$\mathrm{cm}$$ and flux density is doubled. The value of e.m.f. induced is

An iron piece of mass $$200 \mathrm{~g}$$ is kept inside a furnace for some time and then put in a calorimeter of water equivalent $$20 \mathrm{~g}$$ containing $$230 \mathrm{~g}$$ of water at $$20 \mathrm{C}$$. The steady state temperature attained by the mixture is $$60^{\circ}$$. The temperature of the furnace is (Specific heat capacity of iron is $$470 \mathrm{~J~kg}^{-1} \mathrm{C}^{-1}$$ )

A slit of width $$10 \times 10^{-7} \mathrm{~m}$$ is illuminated by light of wavelength $$500 \mathrm{~nm}$$. Angular position of the first minimum is

An ideal gas changes its state from $$\mathrm{A}$$ to $$\mathrm{C}$$ in two different paths $$\mathrm{ABC}$$ and $$\mathrm{AC}$$. The internal energy of the gas at state $$\mathrm{C}$$ is $$20 \mathrm{~J}$$ and at state $$\mathrm{B}$$ is $$10 \mathrm{~J}$$. Heat supplied to the gas to go from $$\mathrm{B} \rightarrow \mathrm{C}$$ is

If pressure of an ideal gas is increased by keeping temperature constant the kinetic energy will

In Young's double slit experiment the ratio of phase difference between light waves reaching the third bright fringe and third dark fringe is

An electromagnetic wave of frequency $$3 \mathrm{~MHz}$$ passes from vacuum into a dielectric medium $$(\mu_r=1)$$ of relative permittivity 4.0. Then,

A group of devices having a total power rating of 500 watt is supplied by an $$\mathrm{AC}$$ voltage $$E=200 \sin \left(3.14 t+\frac{\pi}{4}\right)$$. Then the r.m.s. value of the circuit current is

$$ \text { In a p-n junction, the depletion layer of thickness } 1 \mu \mathrm{m} \text { has } 0.05 \mathrm{~V} \text { potential across it. The electric field in } N C^{-1} \text { is } $$

A long horizontal wire $$\mathrm{P}$$ carries current of $$50 \mathrm{~A}$$ from left to right. It is rigidly fixed. Another fine wire $$\mathrm{Q}$$ is placed directly above and parallel to $$\mathrm{P}$$. The mass of the wire is '$$\mathrm{m}$$' $$\mathrm{kg}$$ and carries a current of '$$\mathrm{I}$$' A. The direction of current in $$\mathrm{Q}$$ and position of wire $$\mathrm{Q}$$ from $$\mathrm{P}$$ so that the wire $$\mathrm{Q}$$ remains suspended are

A record player is spinning at an angular velocity of $$45 \mathrm{~rpm}$$ just before it is turned off. It then decelerates at a constant rate of $$0.8 \mathrm{~rad} \mathrm{~s}^{-1}$$. The angular displacement is

The resistance of a wire at room temperature $$20^{\circ} \mathrm{C}$$ is found to be $$10 \Omega$$. If resistance of the wire increases by $$10 \%$$, then the temperature of the wire will be (The temperature coefficient of the material of the wire is $$0.002 /{ }^{\circ} \mathrm{C}$$)

The potential difference between the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ of the arrangement shown in figure is

If units of mass, length and gravitational constant are chosen to fundamental units, the dimensions of time would be

An electric motor raises a mass of $$1.5 \mathrm{~kg}$$, a distance of $$1.128 \mathrm{~m}$$ in time of $$4.79 \mathrm{~s}$$. Calculate the power to an appropriate significant figures. (take $$g=9.81 \mathrm{~ms}^{-2}$$)

An electron starting from rest and moving with the velocity $$\mathrm{v}$$ through a potential difference $$\mathrm{V}$$ is shown by the graphs below. Identify the correct graph.

In an inductor of self-inductance $$2 \mathrm{~mH}$$, current changes with time (in sec) according to the relation, $$I=(3 t^2-3 t+8) A$$. The emf becomes zero at

A tiny ball of mass $$\mathrm{m}$$ and charge $$\mathrm{q}$$ is suspended from the fixed support using an insulating string of length $$1 \mathrm{~m}$$. The horizontal uniform electric field $$\mathrm{E}$$ is switched on. The angle made by the string with vertical when the ball is in equilibrium is $$45^{\circ}$$. The magnitude of uniform electric field is

To a fish under water, viewing obliquely, a fisherman standing on the bank of a lake looks

A cube bar slides thrice the time with friction than that without friction when it slides on an inclined plane of inclination $$45^{\circ}$$. The coefficient of friction between the block and the surface is

The magnetic susceptibility of an ideal diamagnetic substance is

Water flows from a tap with steady flow, through a cross sectional area of $$10^{-3} \mathrm{~m}^2$$ with a speed of $$0.5 \mathrm{~ms}^{-1}$$. Assume the pressure is constant throughout the stream of water. The cross sectional area of the stream $$0.19 \mathrm{~m}$$ below the tap is

The mobility of the charge carriers increases with

Two balls are thrown horizontally, one from the window of first floor which is $$3 \mathrm{~m}$$ high from the ground and second from the second floor which is $$6 \mathrm{~m}$$ high from the ground, of a multi storey building, with the same speed of $$6 \mathrm{~ms}^{-1}$$. Calculate the distance that will separate the two balls when they hit the ground.

In intrinsic semiconductors at room temperature, number of electrons and holes are

Figure shows three arrangements of electric field lines. In each arrangement, a proton is released from rest at point $$\mathrm{P}$$ and then accelerated through point $$\mathrm{Q}$$ by the electric field. Points $$\mathrm{P}$$ and $$\mathrm{Q}$$ have equal separations in the three arrangements. If $$p_1 p_2$$ and $$p_3$$ are linear momentum of the proton at point $$\mathrm{Q}$$ in the three arrangement respectively, then

A nucleus with mass number 190 initially at rest emits an alpha particle. If the $$\mathrm{Q}$$ value of the reaction is $$4.5 \mathrm{~MeV}$$, the kinetic energy of the alpha particle is

In a container of height $$21 \mathrm{~cm}$$, certain transparent liquid is taken to a height of $$12 \mathrm{~cm}$$. When seen from above, it appears half filled. The refractive index of the liquid is

A man standing in a truck moving with constant velocity throws a ball vertically into air, ball falls

A charge of $$+1 \mathrm{C}$$ is moving with velocity $$\vec{V}=(2 \mathrm{i}+2 \mathrm{j}-\mathrm{k}) \mathrm{ms}^{-1}$$ through a region in which electric field $$\vec{E}=(\mathrm{i}+\mathrm{j}-3 \mathrm{k}) \quad \mathrm{NC}^{-1}$$ and magnetic field $$\vec{B}=(\mathrm{i}-2 \mathrm{j}+3 \mathrm{k}) \mathrm{T}$$ are present. The force experienced by the charge is

Modulus of rigidity of an incompressible liquid is

In a hydrogen atom, if electron is replaced by a particle which is 40 times heavier but has the same charge, then, the ratio of the radius of the first excited state of a normal hydrogen atom to the ground state of the above atom is

An AC voltage source of variable angular frequency $$\omega$$ and fixed amplitude $$\mathrm{V}_0$$ is connected in series with a capacitance $$\mathrm{C}$$ and an electric bulb of resistance $$\mathrm{R}$$ (inductance zero). When $$\omega$$ is decreased

3 point charges each of $$-\mathrm{q}$$ are placed on the circumference of a circle of diameter $$2 \mathrm{a}$$ at $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}$$ respectively as shown in figure. The electric field at $$\mathrm{O}$$ is

To increase the current sensitivity of a moving coil galvanometer by $$25 \%$$, its resistance is increased so that the new resistance becomes twice its initial resistance. By what factor does the voltage sensitivity change?

A uniformly charged solid sphere of radius $$\mathrm{R}$$ has potential $$\mathrm{V}_0$$ (measured with respect to infinity) on its surface. For this sphere the equipotential surfaces with potentials $$\frac{3 \mathrm{~V}_0}{2}, \frac{\mathrm{V}_0}{1}, \frac{3 \mathrm{~V}_0}{4}$$ and $$\frac{\mathrm{V}_0}{4}$$ have radius $$\mathrm{R}_1, \mathrm{R}_2, \mathrm{R}_3$$ and $$\mathrm{R}_4$$ and respectively, then

Energy required for moving a body of mass $$\mathrm{m}$$ from a circular orbit of radius 3R to a higher orbit of radius 4R around the earth is.

The shortest wavelengths of Paschen, Lymen and Balmer series are in the ratio

Two identical moving coil galvanometers have $$10 \Omega$$ resistance and full-scale deflection at $$2 \mu \mathrm{A}$$ current. One of them is converted into a voltmeter of range $$10 \mathrm{~mV}$$ and the other into an ammeter of range $$1 \mathrm{~mA}$$ using appropriate resistors. The ratio of resistance of the converted voltmeter to that of the ammeter is

The velocity of an electron so that its momentum is equal to that of a photon of wavelength $$660 \mathrm{~nm}$$ is

A wire of uniform cross section and resistance 4 ohms is bent in the shape of square ABCD. Point A is connected to a point $$\mathrm{P}$$ on DC by a wire AP of resistance $$1 \mathrm{ohm}$$. When a potential difference is applied between $$\mathrm{A}$$ and $$\mathrm{C}$$, the points $$\mathrm{B}$$ and $$\mathrm{P}$$ are seen to be in same potential. What is the resistance of part $$\mathrm{DP}$$ ?

The percentage increase in magnetic field $$\mathrm{B}$$ when the space within a current carrying solenoid is filled with a medium of susceptibility 0.004 is

A storage battery of emf $$28.0 \mathrm{~V}$$ and internal resistance $$0.5 \Omega$$ is being charged by a $$140 \mathrm{~V}$$ dc supply using a series resistor of $$27.5 \Omega$$. The terminal voltage of the battery during charging is

A metallic rod of length '$$a$$' is rotated with an angular frequency of $$0.2 \mathrm{~rads}^{-1}$$ about an axis normal to the rod passing through its one end. A constant and uniform magnetic field of '$$\mathrm{B}$$' T parallel to the axis exists everywhere. The emf developed across the ends of the rod is

A semiconductor $$\mathrm{X}$$ is made by doping silicon with phosphorous. A second semiconductor $$\mathrm{Y}$$ is made by doping silicon with aluminium. The two are joined by a suitable technique to form a $$\mathrm{p}$$-$$\mathrm{n}$$ junction and is connected to a battery such that $$\mathrm{Y}$$ is joined to negative of the battery and $$\mathrm{X}$$ to the positive of the battery. Which of the following statements is correct?

If $$\alpha, \beta$$ and $$\gamma$$ are the angles between the vectors $$\overrightarrow{\mathrm{P}}, \overrightarrow{\mathrm{Q}}$$, and $$\overrightarrow{\mathrm{R}}$$ and $$\alpha=90^{\circ}$$ as shown in figure. the product of $$(\vec{Q} \times \vec{R}) \cdot \vec{Q}$$ is equal to

In Young's double slit experiment, the ratio of intensities of light from one slit to the other is $$9: 1$$. If Im is the maximum intensity, what is the resultant intensity when they interfere at phase difference $$\phi$$ ?

In Young's double slit experiment, the intensity of light at a point on the screen where the path difference is $$\lambda$$ is $$\mathrm{K}$$ units ($$\lambda$$ is the wavelength of light used). The percentage change in intensity at a point where the path difference is $$\frac{\lambda}{6}$$ and the above point is

$$\mathrm{K}_1$$ and $$\mathrm{K}_2$$ are maximum kinetic energies of photoelectrons emitted when lights of wavelength $$\lambda_1$$ and $$\lambda_2$$ respectively are incident on a metallic surface. If $$\lambda_1=3 \lambda_2$$, then

An object of mass $$3 \mathrm{~kg}$$ moves due to an applied constant force such that its position along $$\mathrm{X}$$ axis is given by $$x=\frac{t^3}{3}$$ where $$x$$ is in meters and $t$ in seconds. The work done in 1 second is

The radius of a nucleus as measured by electron scattering is $$4.8 \mathrm{~fm}$$. The mass number of nucleus is most likely to be

Light enters at an angle of incidence in a transparent rod of refractive index '$$n$$'. The least value of '$$n$$' for which the light once entered into it will not leave it through its lateral face whatsoever be the value of the angle of incidence is

A cubical box of side $$2 \mathrm{~m}$$ contains helium gas. It was observed that in a time of 1 second, an atom travelling with the root-mean-square speed parallel to one of the edges of the cube, made 250 hits with one of the walls, without any collision with other atoms. The average kinetic energy of the helium gas is Take $$R=\frac{25}{3} \mathrm{~J} / \mathrm{mol}-\mathrm{K}$$ and $$\mathrm{kB}=1.38 \times 10^{-23} \mathrm{JK}{ }^{-1}$$

A circular disc of mass $$20 \mathrm{~kg}$$, having radius $$10 \mathrm{~cm}$$ is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The time period of torsional oscillations is found to be $$1 \mathrm{~s}$$. The torsional spring constant of the wire is