Ozone is formed by the reaction

$$ \mathrm{O}_{2(g)}+\mathrm{O}_{(g)} \rightarrow \mathrm{O}_{3(g)}, \Delta \mathrm{H}=-107.2 \mathrm{~kJ} . $$

Given $\mathrm{O}=0$ bond energy is $498.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$, the average bond energy of ozone is:

The number of structural isomers possible for a compound with molecular formula $\mathrm{C}_3 \mathrm{H}_9 \mathrm{~N}$ is:

$$ \text { Match the mixtures listed in Column I with the correct method used for their separation. } $$

An aqueous solution of an unknown solute " $X$ " is prepared by adding 4.0 g of it into 2.0 moles of water. What is the mass percent of " $X$ " in the aqueous solution?

The rate constants for two different reactions, $\mathrm{k}_1$ and $\mathrm{k}_2$ are $10^{16} \cdot \mathrm{e}^{-2000 / \mathrm{T}}$ and $10^{15} \cdot \mathrm{e}^{-1000 / \mathrm{T}}$ respectively. The temperature at which $k_1=k_2$ is

The hybridisation of atomic orbitals of nitrogen in $\mathrm{NO}_2{ }^{+}, \mathrm{NO}_3{ }^{-}$and $\mathrm{NH}_4{ }^{+}$are respectively

An aliphatic compound $[\mathrm{X}]$, Molecular formula $\left(\mathrm{C}_4 \mathrm{H}_{10} \mathrm{O}\right)$ can be prepared from Acetone and $\mathrm{R}-\mathrm{Mg}-\mathrm{X}$.

$[\mathrm{X}]$ with $20 \%$ Phosphoric acid gives an unsaturated compound $\mathrm{C}_4 \mathrm{H}_8[\mathrm{Y}]$. Compound $[\mathrm{Y}]$ is also obtained on heating $[\mathrm{X}]$ with Copper metal at 573 K . $[\mathrm{X}]$ shows no reaction with PCC but on heating with acidified $\mathrm{KMnO}_4$ it forms Acetone $+\mathrm{CO}_2+\mathrm{H}_2 \mathrm{O}$.

Compounds $[\mathrm{X}]$ and $[\mathrm{Y}]$ are $\_\_\_\_$ and $\_\_\_\_$

Rate law can be determined from a balanced chemical equation if

What is the percentage dissociation of 0.8 ml of Acetic acid (density is $1.04 \mathrm{~g} / \mathrm{ml}$ ) which is dissolved in 1.2 L of water if the observed Depression in freezing point is 0.0228 K ? ( $\mathrm{K}_{\mathrm{f}}$ for water $=1.86 \mathrm{Kkg} / \mathrm{mol}$.)

A compound with molecular formula $\mathrm{C}_5 \mathrm{H}_{10}$ that gives acetone on ozonolysis is:

Consider the gaseous equilibrium

$2 \mathrm{AB}_{2(\mathrm{~g})} \rightleftharpoons 2 \mathrm{AB}_{(\mathrm{g})}+\mathrm{B}_{2(\mathrm{~g})}$

The expression relating the degree of dissociation ( $\alpha$ ) and equilibrium constant ( $K_p$ ) and total pressure $P$ is:

Which one of the following statements is wrong?

$60 \%$ of a first order reaction was completed in 60 min , then $50 \%$ of the same reaction can be completed in: $[\log 4=0.60, \log 5=0.69]$

Consider a Galvanic cell in which the following reactions occurs: $\mathrm{Fe}^{2+}(\mathrm{aq})+\mathrm{Ag}^{+}(\mathrm{aq}) \rightarrow \mathrm{Fe}^{3+}(\mathrm{aq})+\mathrm{Ag}(\mathrm{s})$. What is the standard potential of the cell? Given: $\mathrm{E}^0\left(\mathrm{Ag}^{+} / \mathrm{Ag}\right)=\mathrm{aV} \quad \mathrm{E}^0\left(\mathrm{Fe}^{2+} / \mathrm{Fe}\right)=\mathrm{bV} \quad \& \quad \mathrm{E}^0\left(\mathrm{Fe}^{3+} / \mathrm{Fe}\right)=\mathrm{cV}$

Which among the following is NOT in accordance with the property stated against it?

$$ \text { Which one of the following is not the correct IUPAC name of the compound? } $$

$$ \text { Identify the correct order of the type of reactions taking place in this sequence: } $$

Which one of the following species will impart colour to an aqueous solution?

If the equilibrium constant for $\mathrm{N}_{2(g)}+\mathrm{O}_{2(g)} \rightleftharpoons 2 \mathrm{NO}_{(g)}$ is 49 The equilibrium constant for the reaction $N O_{(g)} \rightleftharpoons \frac{1}{2} N_{2(g)}+\frac{1}{2} O_{2(g)}$ is $\_\_\_\_$

Which of the following options represent the correct bond order?

$$ \text { The decreasing order of reactivity towards electrophilic substitutions is: } $$

Identify the correct values to be plotted in the graph, the slope of which can be used to determine the activation energy of a reaction.

The ion that has a spin only magnetic moment of 5.9 BM is:

In the reaction,

$$ \mathrm{Cr}_2 \mathrm{O}_7^{2-}+4 \mathrm{H}_2 \mathrm{O}_2+2 \mathrm{H}^{+} \rightarrow 2 \mathrm{CrO}_5+\mathrm{H}_2 \mathrm{O} $$

The change in oxidation state of Cr is:

$$ \text { Match the Coordination compounds in Column I with the type of stereoisomerism given in Column II exhibited by them. } $$

Identify the correct mathematical expression which represents the variation in molar conductivity of a weak acid having concentration C and ionisation constant $\mathrm{K}_{\mathrm{a}}$

( $\lambda_m^{\infty}=$ molar conductivity at infinite dilution, $\lambda_{\mathrm{m}}=$ molar conductivity at concentration C )

Lassaigne's test for the detection of nitrogen fails in

$\Delta \mathrm{H}$ and $\Delta \mathrm{S}$ for a reaction are $35.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $83.6 \mathrm{~J} \mathrm{~K}^{-1}$ respectively. Assuming that $\Delta \mathrm{H}$ and $\Delta \mathrm{S}$ do not vary with temperature, the reaction is spontaneous when:

When the initial concentration of a zero order reaction is doubled, the half-life of the reaction is:

The secondary structure of protein consists of:

What will be the change in the electrode potential of chromium electrode dipping into chromic sulphate solution, when the electrolyte is diluted 10 times at $25^{\circ} \mathrm{C}$ ? $\left[\mathrm{E}^0\left(\mathrm{Cr}^{3+} / \mathrm{Cr}\right)\right]=-0.74 \mathrm{~V}$

Which of the following is a correct match?

$$ \text { Match reactions in Column I with the corresponding products formed as given in Column II. } $$

Identify the wrong statement from the following

Etard reaction is a method of preparation of benzaldehyde by oxidation of toluene. The oxidizing agent used in this reaction is:

When 2-butyne is treated with dilute $\mathrm{H}_2 \mathrm{SO}_4 / \mathrm{HgSO}_4$, the product formed is:

Two statements, one Assertion and the other Reason are given. Choose the right option.

Assertion: The Molar conductivity of KCl increases very slowly with dilution and approaches a limiting value when dilution is infinite.

Reason: In case of KCl there is an increase in the number of ions on dilution due to complete ionisation at infinite dilution.

The statements given below contains assertion and reason. Choose the correct option

Assertion (A): Cr, Mo and W possess the highest melting point in their respective series of elements.

Reason (R): Cr, Mo and W have stable half-filled electrons in the 'd' shell resulting in strong metallic bonding.

$\mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7$ on heating with aqueous NaOH gives $\_\_\_\_$

In the photoelectric emission, the energy of the emitted electron is:

Identify the name of the reaction involved in conversion of $A \rightarrow B$ and name the compound $B$.

$$ \mathrm{CH}_3 \mathrm{CN} \xrightarrow[\mathrm{H}_2 \mathrm{O}]{\mathrm{H}^{+}} \mathrm{A}-\xrightarrow[\mathrm{H}^{+} / \mathrm{H}_2 \mathrm{O}]{\mathrm{Cl}_2 / \text { Red Phosphorous }} \mathrm{B} $$

The Crystal Field Stabilisation energy of (i) $\left[\mathrm{CoF}_6\right]^{3-}$ and (ii) $\left[\mathrm{Fe}\left(\mathrm{H}_2 \mathrm{O}\right)_6\right]^{3+}$ are ⋯and ⋯⋯ respectively.

Identify the correct statement

0.02 M solution of sucrose is isotonic with 0.008 M solution of sodium sulphate. What is the percentage dissociation of the electrolyte?

If the molar masses of two chemically non-reacting gases $\mathrm{A}_2$ and $\mathrm{B}_2$ are 28 amu and 32 amu respectively and 0.8 g of $\mathrm{A}_2$ and 0.4 g of $B_2$ are enclosed in a container with a total pressure of 820 mm of Hg , then the partial pressure of $A_2$ in atmosphere unit is $\_\_\_\_$

The basic character of transition metal monoxides follows the order:

Propanal on reaction with dilute NaOH , undergoes aldol condensation resulting in the formation of a compound A . identify the correct structure of A and name of the compound.

Select the reagents needed to convert Benzene to 4-Bromophenylpropene in the correct sequential order.

(A). $\mathrm{Cl}_2 / \Delta$

(B). $\left(\mathrm{CH}_3\right)_2-\mathrm{CHCl}$ /anhyd. $\mathrm{AlCl}_3$

(C). Alc. KOH

(D). $\mathrm{Br}_2 / \mathrm{Fe}$

$$ \text { Identify the compounds } \mathrm{A} \text { and } \mathrm{B} \text { from the following reaction sequence: } $$

The reagent used in carbylamine reaction is $\_\_\_\_$ and the product of the reaction will be $\_\_\_\_$

Two statements, Assertion and Reason are given. With reference to them, choose the correct option.

Assertion: Hydrolysis of 2-Bromo-2-methylpropane follows $S_N 1$ mechanism which involves 2 steps and the first step is:

Reason: The energy required to cleave $\mathrm{C}-\mathrm{Br}$ is taken from the energy released due to Inductive effect and hyper-conjugative effect.

The heat of combustion of carbon to $\mathrm{CO}_2$ is $-393.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$.

The heat released on the formation of 35.2 g of $\mathrm{CO}_2$ by combustion of C is:

Which among the following is an INCORRECT statement?

$3.482 \times 10^{-1} \mathrm{~g}$ of Fe gets deposited when an aqueous solution of Ferric sulphate is electrolysed for 20 minutes using a current of " $x$ " amperes. Find " $x$ ". (Atomic mass of $\mathrm{Fe}=56 \mathrm{amu}$ ).

The highest dipole moment is for:

The mass of precipitate formed when 50 mL of $16.9 \%$ aqueous solution of $\mathrm{AgNO}_3$ is mixed with 50 mL of $5.8 \% \mathrm{NaCl}$ solution is-----------$[\mathrm{Ag}=107.8, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{Na}=23, \mathrm{Cl}=35.5]$

$$ \text { Identify X, Y and Z. } $$

Which of the following is the correct reaction of Hinsberg's reagent with primary, secondary and tertiary amines?

The Bohr orbit of the hydrogen atom ( $n=1$ ) is $0.530 \mathop {\rm{A}}\limits^{\rm{o}} $. The radius of the first excited state orbit in $\mathop {\rm{A}}\limits^{\rm{o}} $ is:

Which of the following is NOT TRUE about nucleophilic addition reactions of aldehydes?

$$ \text { The solution of } \boldsymbol{d} \boldsymbol{y}=\boldsymbol{\operatorname { c o s }} \boldsymbol{x}(\mathbf{2}-\boldsymbol{y} \boldsymbol{\operatorname { c o s e c }} \boldsymbol{x}) \boldsymbol{d} \boldsymbol{x} \quad \text { where } y=\sqrt{2} \text { when } x=\frac{\boldsymbol{\pi}}{4} \text { is } $$

$$ \text { The conjugate of } z=\frac{(\mathbf{4}+\boldsymbol{i})(\mathbf{1}-\boldsymbol{i})}{(\mathbf{1}+\boldsymbol{i})(\mathbf{2}-\boldsymbol{i})} $$

If A and B are two square matrices of the same order such that $\mathrm{AB}=\mathrm{A}$ and $\mathrm{BA}=\mathrm{B}$, then $(\boldsymbol{A}+\boldsymbol{B})^2$ is equal to:

If $f(x)=\left\{\begin{array}{l}\frac{\sqrt{1+x}-\sqrt{1-x}}{\sin x} \\ \boldsymbol{k}, x=0\end{array}, x \neq 0\right.$ is continuous at $x=0$, then $\boldsymbol{k}=$

The area of the region bounded by the line $y=x+2$ and the curve $x=-y^2$ is

If $x=4$ is a root of $\left|\begin{array}{cc}x & 3 \\ 1 & x-2\end{array}\right|=5$, then the other root is:

The difference between the distance of any point on the hyperbola from the two foci is $\mathbf{1 6}$ and the eccentricity is $\mathbf{2}$. Then the equation of the hyperbola is

$$ \text { If } x=a(\theta-\sin \theta) \text { and } y=a(1-\cos \theta) \text {, then } \frac{\left(\mathbf{1}+\boldsymbol{y}_{\mathbf{1}}{ }^{\mathbf{2}}\right)^{\mathbf{3} / \mathbf{2}}}{\boldsymbol{y}_{\mathbf{2}}}= $$

The area of the region in the first quadrant enclosed by the $x$-axis, the line $x=\sqrt{3} y$ and the circle $x^2+y^2=4$ is

Which of the following is the simplest form of the expression $\boldsymbol{\operatorname { t a n }}^{-\mathbf{1}}\left(\frac{\sqrt{\mathbf{1 + x ^ { \mathbf { 2 } }}}-\mathbf{1}}{\boldsymbol{x}}\right)$ where $x \neq 0$

$$ \mathop {\lim }\limits_{x \to 0} \frac{(1-\cos 2 x)(3+\cos x)}{x \tan 4 x} \text { is equal to: } $$

A square plate is contracting at a uniform rate of $2 \mathrm{~cm}^2 / \mathrm{min}$. The rate at which the perimeter is decreasing when the side of the square is 16 cm is:

If $A=\{x: x$ is the first three odd numbers $\}$

$B=\{2 x+3: 0 \leq x<5, x \in \mathbb{N}\}$, then which of the following is true

Samhita faces a three-headed dragon. She wins a "Tactical medal" if she manages to defeat exactly one of the three heads.

The battle proceeds head-by-head under the following conditions:

• The probability of defeating the first head is $\frac{\mathbf{1}}{\mathbf{3}}$.

• After a win: if she defeats a head, the probability of defeating the next head is $\frac{2}{3}$.

• After a loss: if she fails to defeat a head, the probability of defeating the next head is $\frac{\mathbf{1}}{\mathbf{4}}$.

What is the probability that Samhita earns the "Tactical medal"?

$$ \int e^{2 x} \cos (5 x+3) d x= $$

Given $A=\left(\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right)$ and $f(x)=x^2-2 x-3$ then $f(A)$ is:

The value of $\mathop {\lim }\limits_{x \to 3}\left[\frac{1}{x-3}+\frac{9 x}{27-x^3}\right]$ is:

$$ \int \sqrt{2 a x-x^2} d x= $$

Consider the following list of ordered pairs: $(1,0),(-2,-1),(7,-6),(-3,4)$ and $(0,2)$

Which of the following options correctly identifies only those pairs that are NOT elements of the relation $R=\{(x, y): y=1-|x| ; x, y \in \mathbb{Q}\}$ ?

Given $P=\left[\begin{array}{lll}2 & \boldsymbol{\alpha} & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 3\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix A and $|A|=3$, then the value of $\boldsymbol{\alpha}$ is:

The radius of a circle is $\mathbf{5 ~ c m}$. A chord of this circle is equal to the radius. Then the length of the arc of this chord is:

If for a distribution of 20 items, $\sum(x-4)=10$ and $\sum(x-4)^2=85$ then the standard deviation is:

The odds against Arjun solving a problem are $\mathbf{5 : 2}$ and the odds in favour of Bhavana solving the same problem are 3:4. What is the probability that the problem is NOT solved by either of them?

If $\mathbf{3 ~ c m} / \mathbf{s}$ is the rate at which the side of an equilateral triangle increases, then the rate of change of area, when the side is $\mathbf{1 2 ~ c m}$ is:

If $\sin A+\sin 2 A=x$ and $\cos A+\cos 2 A=y$ then the value of the expression $\left(x^2+y^2\right)\left(x^2+y^2-3\right)$ equals

Given the matrices $A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{lll}2 & 1 & 0 \\ 1 & 1 & 2 \\ 0 & 2 & 1\end{array}\right]$, then the minor $\boldsymbol{M}_{\mathbf{2 3}}$ of the matrix $\left(A B^{-1}\right)^{-1}$ is:

The order and degree of the differential equation $\left(\frac{d y}{d x}\right)^2+\frac{d x}{d y}=x$ is:

A teacher has two jars of candy on her desk:

Jar 1: Contains 3 Strawberry candies and 2 Orange candies.

Jar 2: Contains 1 Strawberry candy and 4 Orange candies.

The teacher randomly picks two candies from Jar 1 and drops them into Jar 2.

Then, a student reaches into Jar 2 and picks two candies.

What is the probability that the student picks two Strawberry candies?

$$ \text { The direction ratios of the vector }(\hat{\imath}+\hat{\jmath}) \times(\hat{\jmath}+\hat{k}) \text { are } $$

The product of three numbers in geometric progression is 8 and the sum of the product of the numbers taken in pairs is 14 . Find the numbers.

$$ \int \frac{x+1}{x\left(1+x e^x\right)} d x= $$

Let $\vec{p}$ and $\vec{q}$ be the position vectors of P and Q with respect to the origin. If points R and S divide PQ internally and externally in the ratio 2:3 respectively, then $\overrightarrow{O R}$ and $\overrightarrow{O S}$ are perpendicular when

The length of the latus rectum of the curve represented by $x=3(\cos t+\sin t)$ and $y=4(\cos t-\sin t)$ is:

If the function $f(x)=x^4-31 x^2+\boldsymbol{a} x+5$ has a turning point at $x=1$, then the value of ' $\boldsymbol{a}$ ' is $\_\_\_\_$ and the function attains a $\_\_\_\_$ at $x=1$

Advika chooses one of three scarves every morning: Red, Blue, or Green.

• The probability she chooses Red is $20 \%$.

• The probability she chooses Blue is twice the probability of choosing Red.

• On the remaining days she wears a Green scarf.

Once a scarf is chosen, she decides whether to wear a Hat (H) and Sunglasses (S).

These choices are independent of each other but depend on the scarf colour:

$$ \begin{array}{|l|l|l|} \hline \text { Scarf colour } & \mathbf{P ( H )} & \mathbf{P ( S )} \\ \hline \text { Red } & 0.5 & 0.8 \\ \hline \text { Blue } & 0.4 & 0.5 \\ \hline \text { Green } & 0.1 & 0.5 \\ \hline \end{array} $$

Advika is spotted outdoors wearing both a Hat and Sunglasses.

What is the probability that she is wearing the Red scarf?

$$ \int \frac{e^{\log \left(1+\frac{1}{x^2}\right)}}{x^2+\frac{1}{x^2}} d x= $$

Which of the following is NOT a comer point of the feasible region determined by the constraints:

$$ \begin{aligned} & x+2 y \leq 4 \\ & x+y \geq 2 \\ & x \geq 0 \text { and } y \geq 0 \end{aligned} $$

The angle between the two lines whose direction cosines satisfy the relations $\boldsymbol{l}+\boldsymbol{m}+\boldsymbol{n}=\mathbf{0}$ and $\boldsymbol{l}^{\mathbf{2}}=\boldsymbol{m}^{\mathbf{2}}+\boldsymbol{n}^{\mathbf{2}}$ is

In how many ways can the squares of a $\mathbf{4} \times \mathbf{2}$ grid ( 4 rows and 2 columns) be filled with the letters of the word 'SPHERE' such that each row contains at least one letter?

The value of the expression $\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)+\sin ^{-1}\left(\sin \frac{22 \pi}{3}\right)+\tan ^{-1}\left(\tan \frac{4 \pi}{5}\right)$ is:

A straight line passes through the point $P\left(\log _2 16, \log _3 27\right)$ such that the portion of the line intercepted between the co-ordinate axes is divided by $P$ in the ratio $1: 2$ internally (starting from the $x$-axis). Then the equation of the line is:

Let L be the foot of the perpendicular drawn from the point $P(5,3 k-7,-4)$ to the YZ - plane. If the distance of point L from the origin is $\sqrt{41}$ units, then the possible value of ' $\boldsymbol{k}$ ' is:

Distance between $8 x+15 y-20=0$ and $8 x+15 y+14=0$ is:

In a bank the principal increases continuously at the rate of $4 \%$ per annum. In how many years will ₹ 1000 triple itself?

$$ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^5 x \cos ^7 x d x= $$

A batch of $\mathbf{1 0}$ cupcakes consists of $\mathbf{5}$ chocolate, $\mathbf{3}$ vanilla, and $\mathbf{2}$ strawberry. If 4 cupcakes are selected to be put into a gift box, find the number of different ways they can be chosen if the selection must include at least $\mathbf{2}$ chocolate, at most $\mathbf{1}$ vanilla, and exactly $\mathbf{1}$ strawberry cupcake.

If a straight line passing through a fixed point $(a, b)$, where $\boldsymbol{a}, \boldsymbol{b}>\mathbf{0}$, makes positive intercepts OA and OB on the coordinate axes, then the least value of $\mathbf{O A}+\mathbf{O B}$ is:

Let $\mathbf{P}$ be a point on the line $L_1: \frac{x-2}{2}=y+1=\frac{z-1}{2}$ such that its distance from the point $A(2,-1,1)$ is 6 units.

Given that $\boldsymbol{x}$-coordinate of $\mathbf{P}$ is greater than $\mathbf{2}$,

Find the coordinates of point Q on the line $L_2: x-1=\frac{y-2}{2}=\frac{z-2}{2}$ such that $\mathbf{Q}$ is the closest point to $\mathbf{P}$.

The range of the function $f(x)={ }^{(7-x)} P_{(x-3)}$ is

The set expression $A \cup\left(B \cap\left(A^{\prime} \cup B^{\prime}\right)\right)$ is equivalent to

\text { The remainder when } \mathbf{7}^{\mathbf{1 0 3}} \text { is divided by } \mathbf{2 5} \text { is }

The function $y=||x|-1|$ is differentiable for all values of ' $x$ ' except

"A storage room must be kept at a temperature (T) such that triple the temperature is at least $\mathbf{1 5}^{\circ} \mathbf{C}$, but the temperature plus $\mathbf{8}$ is strictly not more than $\mathbf{2 0}^{\circ} \mathbf{C}$. What is the range of safe temperatures?"

The derivative of $y=\sin ^2\left[\cot ^{-1}\left(\sqrt{\frac{\mathbf{1}-\boldsymbol{x}}{\mathbf{1}+\boldsymbol{x}}}\right)\right]$ is

An engineering team is testing a new prototype drone. The drone has constant success rate of $\frac{\mathbf{2}}{\mathbf{7}}$ for every autonomous landing attempt. Two engineers, Sarah and Swarna, take turns initiating the landing sequence, with Swarna going first.

If they continue the process until a landing is successful, what is the probability that Sarah is the one who initiates the successful landing?

Let ' $\boldsymbol{a}$ ' and ' $\mathbf{b}$ ' be two numbers where $\boldsymbol{a}<\boldsymbol{b}$. The geometric mean of these numbers exceeds the smaller number by 12 and the arithmetic mean is smaller than the larger number by 24 . Then the value of $|\boldsymbol{b}-\boldsymbol{a}|$ is:

$$ \text { The expression } \frac{1-\tan ^2\left(\frac{\pi}{4}-A\right)}{1+\tan ^2\left(\frac{\pi}{4}-A\right)} \text { equals } $$

The behaviour of the function $f(x)=\sin \left(2 x+\frac{\pi}{4}\right)$ on $\left(\frac{3 \pi}{8}, \frac{5 \pi}{8}\right)$ is:

$$ \text { The particular solution of the equation } \sin \left(\frac{d y}{d x}\right)=a \text {, where } a \in \mathbb{R} \text { and } y=2 \text { when } x=0 \text { is } $$

$$ \begin{aligned} &\text { Find the co-ordinates of the orthocentre of the triangle formed by the lines }\\ &\begin{aligned} & L_1: y-x=2 \\ & L_2: y+2 x=8 \\ & L_3: 3 y-x=18 \end{aligned} \end{aligned} $$

An atom has a single electron. Its ground state energy is -30 eV and its first excited state energy is -8 eV . The atom is bombarded with a stream of photons, each of energy 15 eV .

Assuming the atom being in the ground state, which of the following statements is correct?

A. Atom gets excited to the first excited state and later emit photons of 22 eV

B. Atom absorbs energy continuously until 22 eV is accumulated and then gets excited

C. Atom will not get excited, and the transmitted light will have the same frequency as the incident light

D. Atom will absorb the photon and re-emit a photon of lower energy

Paramagnetic substances

A. Move from a region of strong magnetic field to weak magnetic field

B. Has susceptibility less than zero

C. Attract strongly towards external magnetic field

D. Align themselves along the directions of external magnetic field

A wire of length 1 m has a resistance of $20 \Omega$ at $0^{\circ} \mathrm{C}$. It is uniformly stretched so that its length increases by $21 \%$. Assuming the volume of the wire remains constant, the percentage change in resistance is $n \%$. Alternatively, if the wire is heated [without stretching] through a temperature of $27^{\circ} \mathrm{C}$ and if the temperature coefficient of resistance of the material of wire is $0.004 K^{-1}$, the percentage change in resistance is $m \%$. The values of $m$ and $n$ are:

An object is placed 25 cm in front of a fully silvered concave mirror of focal length 15 cm . A plane mirror is placed 35 cm behind the concave mirror on the side opposite to the object. The final image after reflection first from the concave mirror and then from the plane mirror is formed at:

A bar magnet of length 12 cm is placed such that its north pole points towards the geographic north. Two neutral points which are separated by 16 cm are obtained on the equatorial axis of the bar magnet. What is the pole strength of the bar magnet if the horizontal component of the earth's field is $1.25 \times 10^{-5} \mathrm{~T}$ ?

Forces A and B act at a point. The sum of their magnitudes is 50 N and the magnitude of their resultant is 20 N . If the resultant is at $90^{\circ}$ with the smaller force, the magnitudes of A and B , in N are

A nucleus of uranium -235 absorbs a slow neutron and undergoes nuclear fission according to the reaction: ${ }_{92}^{235} U+{ }_0^1 n \rightarrow{ }_{56}^{141} B a+{ }_{36}^{92} K r+3{ }_0^1 n+Q$

If the average energy released per fission is 202 MeV , the energy released when 2.35 g of $U^{235}$ undergoes complete fission is approximately;

[Given $1 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J}$, Avogadro number $=6.02 \times 10^{23}$ ]

Two bodies of specific heats $C_1$ and $C_2$, having the same heat capacities are combined to form a single composite body. The specific heat capacity of the composite body is

In a PN junction diode, the forward bias is increased gradually from 0 Volt to 1 Volt. Which of the following statements is correct?

A. The depletion width increases, and barrier potential increases

B. The depletion width decreases, but the electric field inside the junction increases

C. The depletion width remains unchanged, but current increases

D. The depletion width decreases and barrier potential decreases

Two charges 1 C and 2 C are placed at coordinates $(0,3)$ and $(4,3)$ respectively in an XY plane. What is the work done to put a 30 C charge at the origin of the coordinate system?

Two long parallel conductors $X$ and $Y$ are placed vertically at a distance $d$ apart. Conductor X carries a current I upwards and conductor Y carries a current 21 downwards. A third long conductor Z is placed parallel to both X and Y and between X and Y . If Z carries a current I upwards and is at a distance $x$ from conductor X , then:

A. All the conductors are in equilibrium and the net force on Z is zero

B. The net force on Z is $\frac{\mu_0 I^2}{2 x \pi}\left[\frac{d+x}{d-x}\right]$ towards X

C. The net force on Z is $\frac{\mu_0 I^2}{2 x \pi}\left[\frac{d+x}{d-x}\right]$ towards Y

D. The net force on Z is $\frac{\mu_0 I^2}{2 x \pi}\left[\frac{d-x}{d+x}\right]$ towards Y

A $50 \Omega$ galvanometer is shunted by a resistance of $S \Omega$. If $8 \%$ of total current passes through the galvanometer, the value of $S$ is:

A current of 3 A enters one vertex P of an equilateral triangle PQR having three resistors of $1 \Omega$ each forming the sides of the equilateral triangle as shown. The value of $i_2$ in amperes is:

Current in a conductor is expressed as $I=8 t^3+3 t^2+2$, where current I is measured in amperes and time t is measured in seconds. What is the charge that flows through a cross-section of the conductor between time $t=1 \mathrm{~s}$ to $t=2 \mathrm{~s}$ ?

A diffraction pattern due to a single slit of width 0.12 mm is obtained with a blue green light of wavelength 500 nm . The angular separation between central maximum and second order secondary maximum of the diffraction pattern is

A square loop of wire 2.2 cm on each side contains 2 turns and has a total resistance of $0.0002 \Omega$. It is located 22 cm from a long straight current carrying wire. If the current in the straight wire is increased at a steady rate from 20 A to 50 A in 5 s , determine the magnitude of the current induced in the squared loop.

What is the charge on $15 \mu F$ in the circuit given?

The speed of transverse wave in aluminium wire is $\frac{1}{10}$ times the speed of longitudinal wave in the wire. The stress in the wire is (Young's Modulus of Al=10 ${ }^{10} \mathrm{~Pa}$ )

Two spheres of masses $M_1$ and $M_2$ are in air separated by a distance of d m . The gravitational force of attraction between them is $F$. If the space around the masses is filled with a liquid of density $\rho$, the force between the masses will be

Which of the following is incorrect for displacement current

A. Displacement current exists only when electric field changes with time

B. Displacement current obeys ohm's law

C. Displacement current produces magnetic field just like conduction current

D. Unlike conduction current, displacement current does not involve the flow of electrons through a conductor

In a Young's double slit experiment, the slit separation is 1.5 mm . The setup is illuminated simultaneously by light of wavelengths $6000 \mathop {\rm{A}}\limits^{\rm{o}}$ and $8000 \mathop {\rm{A}}\limits^{\rm{o}}$. The screen is placed at a distance of 1.5 m from the slits. It is observed that at a certain point $P$ on the screen which is 4.8 mm from the central maximum, fringes due to both the wavelengths coincide.

Which of the following options are correct?

A. Light of wavelength $6000\mathop {\rm{A}}\limits^{\rm{o}}$ produces a dark fringe and light of wavelength $8000 \mathop {\rm{A}}\limits^{\rm{o}}$ produces a bright fringe at P

B. Light of wavelength $6000 \mathop {\rm{A}}\limits^{\rm{o}}$ produces a bright fringe and light of wavelength $8000 \mathop {\rm{A}}\limits^{\rm{o}}$ produces a dark fringe at P

C. Light of wavelength $6000\mathop {\rm{A}}\limits^{\rm{o}}$ and light of wavelength $8000 \mathop {\rm{A}}\limits^{\rm{o}}$ both produce a dark fringe at P

D. Light of wavelength $6000 \mathop {\rm{A}}\limits^{\rm{o}}$ and light of wavelength $8000\mathop {\rm{A}}\limits^{\rm{o}}$ both produce a bright fringe at P

A small hollow vessel which has a small circular hole of radius $r$ in its base, is immersed in a tank of oil of density $\rho$ and surface tension $T$. The oil will penetrate into the vessel at a depth of

Two-point charges of equal magnitude 0.01 C and opposite in sign are separated by 0.2 mm , forming an electric dipole. The electric dipole moment of the dipole is:

1 kg of water at $100^{\circ} \mathrm{C}$ is converted to steam at the same temperature. Volume of 1 cc of water changes to $1671 \times 10^3 \mathrm{cc}$ on boiling. The change in internal energy of the system is (Latent Heat of vaporisation of water is $22.68 \times 10^5 \mathrm{~J} \mathrm{~kg}^{-1} ; 1 \mathrm{~atm}=1.0 \times 10^5 \mathrm{~Pa}$ )

A mercury-198 nucleus is bombarded by a neutron, which causes a nuclear reaction

$$ n_0^1+\mathrm{Hg}_{80}^{198} \longrightarrow A u_{79}^{197}+X $$

What is the unknown product particle $X$ ?

A ball falls under gravity from a height of 10 m with an initial downward velocity u . It loses one third of its energy in collision and then rises back to 5 m . The initial velocity u is $\left(\mathrm{g}=10 \mathrm{~ms}^{-2}\right)$

In the given circuit, an ideal voltmeter connected across $6 \Omega$ reads 5 V . The internal resistance $r$ of each cell is:

A machine gun fires a bullet of mass m with a velocity of $1000 \mathrm{~m} / \mathrm{min}$. The man holding the gun can exert a force of 200 N on the gun. The product of mass of each bullet and the number of bullets fired in one sec is

Four masses each 2 kg are placed at the corners A, B, C, D of a mass less square frame. 40 kg mass is at the centre O of a square frame of side 0.2 m . It is to be rotated about an axis passing through the centre O and perpendicular to the plane of the frame. Calculate the torque in $\mathrm{N}-\mathrm{m}$ required to produce an angular acceleration of $\frac{\pi}{2} \mathrm{rads}^{-2}$.

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of $45^{\circ}$ with each other. When the system is immersed in a liquid of relative density 0.8 , the angle between the strings remains unchanged. If the density of the material of the sphere is $2.4 \mathrm{gcm}^{-3}$, what is the dielectric constant of the liquid?

A car, starting from rest, accelerates at a rate of $\beta$ through a distance S , then continues at constant speed for time t and then decelerates at a rate of $\frac{\beta}{2}$ to come to rest. If the total distance travelled is $10 S$, then ratio $\frac{S}{t^2}$ is

A glass slab of 16 cm thickness has its bottom surface silvered. An air bubble is located inside the glass slab at a distance of 8 cm from the top surface. The refractive index of glass is 1.6 . Which of the following correctly represents the number of images seen by an observer looking vertically downwards from air onto the top surface?

A. Only one image is seen at a depth of 5 cm

B. Multiple images are formed due to multiple reflections and refractions

C. Only one image is seen at a depth of 16 cm

D. Two images are seen at depths 5 cm and 15 cm

A body is projected vertically upwards from the surface of earth with a velocity ' $v$ ' to reach a height of $10 R$, where $R$ is the radius of the earth, then $v$ is

An AC generator having 400 turns and an area of cross section of $2 \times 10^{-3} \mathrm{~m}^2$ rotates with an angular speed of $200 \pi \mathrm{rad} \mathrm{s}^{-1}$ in a uniform magnetic field of strength 0.4 T . The generator is connected to the primary of an ideal transformer having 500 turns in the primary and 2000 turns in the secondary. The secondary is connected to a $400 \Omega$ resistive load. What is the rms current in the secondary of the transformer? Assume, the transformer is ideal and the resistance of the coil is negligible

An object of mass 10 g executes SHM along the $x$-axis with frequency of $\left(\frac{10}{\pi}\right) \mathrm{Hz}$. At the point $x=2 \mathrm{~cm}$ the object has KE 1.2 J and PE 0.4 J . The amplitude of oscillation is

A circular coil of radius 7 cm and 40 turns is rotated about its vertical diameter with an angular speed of 40 radians per second in a uniform horizontal magnetic field of $4 \times 10^{-2} T$. What is the maximum current induced in the coil if the resistance of the coil is $11 \Omega$ ?

A light rod of length 1 m is suspended from ceiling horizontally by means of two vertical wires of equal length tied to its ends. One of the wires is made of material $X$ and is of cross-section $0.1 \mathrm{~cm}^2$. and the other of material Y of cross-section $0.3 \mathrm{~cm}^2 . \mathrm{A}$ weight is hung from the wire at a point to produce equal strain in the wires. The ratio of Young's moduli of wires A to B is $3: 1$. The location of the point from one end of the wire is

An ideal gas has molar specific heat $\frac{5 R}{2}$ at constant pressure. If 1662 J of heat brings about 50 K temperature change, the number of moles of gas is

A silicon sample is doped simultaneously with donor impurity phosphorus at a concentration of $N_D=3 \times 10^{22} \mathrm{~m}^{-3}$ and acceptor impurity boron at a concentration of $N_D=2.8 \times 10^{22} \mathrm{~m}^{-3}$. The intrinsic carrier concentration of silicon at room temperature is $n_i=1.5 \times 10^{16} \mathrm{~m}^{-3}$. Assuming complete ionization, the hole concentration is :

The velocity of a body moving in a viscous medium is given by $v=\frac{A}{B+C}\left[1-e^{\frac{-t}{B}}\right]$ where $t$ is time; dimensions of $A$ and $C$ are

A proton is moving along negative $X$ axis. If a uniform magnetic field is applied parallel to the positive $Z$ axis, then choose the correct answer from the following options;

1. The proton will experience magnetic force along positive Y direction and will move along a circular path in the XY plane

2. The proton will experience magnetic force along negative Y direction and will move along a circular path in the XY plane

3. The proton will experience magnetic force along positive Y direction and will move along a circular path in the XZ plane

4. The proton will experience magnetic force along negative Y direction and will move along a circular path in the XZ plane

The magnetic field at the centre of a wire loop formed by two semicircular wires of radii $r_1=3.14 \mathrm{~m}$ and $r_2=6.28 \mathrm{~m}$ and carrying a current of $\mathrm{I}=2 \mathrm{~A}$ is:

Two vessels A and B contain same mass of Oxygen and Hydrogen respectively at the same temperature. The volume of $B$ is twice that of $A$. The ratio of gas pressure in $A$ to that in $B$ is

Two conducting spherical shells $A$ and $B$ of radii 4 cm and 6 cm respectively are placed with their centres 17 cm apart in air. Initially, sphere A was given a -20 nC charge while sphere B was uncharged. The spheres are then connected by a long thin conducting wire and allowed to reach electrostatic equilibrium. Assuming no charge is lost to the surroundings, the final charge on sphere A and the ratio of magnitude of electric field intensity at the surface of sphere $B$ to that of sphere $A$ is given by;

A uniform electric field $E=3 \hat{i}+6 \hat{j}+\hat{k}$ passes through a closed cuboidal surface. One face of the cuboid has an area $4 m^2$ and an outward unit normal given by $\frac{2 \hat{i}+2 \hat{j}+3 \hat{k}}{\sqrt{17}}$. If the electric flux through the remaining 5 faces is zero, the charge enclosed by the cuboid is:

A particle starts rotating from rest. The instantaneous angular displacement is $\theta=3 t^3-t^2$, where $\theta$ is in radian and $t$ in s; The angular velocity at $t=1 \mathrm{~s}$ is

Point charge $\sqrt{2} C, \sqrt{2} C$, and $-2 C$ are placed at the three vertices of a right-angled triangle in air. [as shown in the figure below]

What is the electric field at a point $P$ on the hypotenuse that is equidistant from all three charges.

Given distances $X P=Y P=Z P=0.5 \mathrm{~m}$

A solenoid having resistance $R=60 \Omega$ and inductance $L=0.4 \mathrm{H}$ is connected to an AC source $V=100 \sqrt{2} \sin 200 t$.

Find the maximum current.

A motor cyclist starts from the top of an inclined plane of height $h$ to go around a globe of death trap of radius $r$. The ratio of minimum height ' $h$ ' of the inclined plane to the radius ' $r$ ' of the death globe in order to go around the death globe successfully is

An object is placed at an unknown distance from a convex objective lens of focal length 5 cm . The objective lens forms a real image which acts as an object for a convex eyepiece of focal length 6.25 cm . The distance between the objective and eyepiece is 20 cm . The microscope is adjusted so that the final image is formed at the least distance of distinct vision $(25 \mathrm{~cm})$. Which of the following is correct?

A. Object distance $=7.5 \mathrm{~cm}$; Total magnification $=10$

B. Object distance = 10 cm ; Total magnification = 20

C. Object distance $=5 \mathrm{~cm}$; Total magnification $=20$

D. Object distance = 2.5 cm ; Total magnification = 10

From the same point, two stones A and B are thrown simultaneously, A is thrown up vertically with a velocity of $10 \mathrm{~ms}^{-1}$ and $B$ is thrown up with the same velocity at $30^{\circ}$ with the horizontal. The separation between the balls at 2 s is

If a clear liquid has a refractive index of 1.45 and a transparent solid has a refractive index of 2.9 , then for total internal reflection to occur at the interface between the two media; which of the following is correct?

A. The ray must travel from transparent solid to liquid at an angle of $15^{\circ}$

B. The ray must travel from liquid to transparent solid at an angle of $45^{\circ}$

C. The ray must travel from transparent solid to liquid at an angle of $30^{\circ}$

D. The ray must travel from liquid to transparent solid at an angle of $60^{\circ}$

Two deuterons are fused to form one alpha particle. If binding energy per nucleon of deuterium is 1.05 MeV and that of alpha particle is 7 MeV , what is the energy released in the formation of one alpha particle from the fusing of two deuterons?

If wattless current flows in an AC circuit, then the circuit is:

A. LR circuit

B. Purely capacitive circuit

C. Purely resistive circuit

D. LCR circuit

In the circuit given, the reverse breakdown voltage of the Zener diode is 4.8 V . The current through the Zener and the power dissipation in Zener is:

In a Young's double slit experiment, the slits are separated by 0.5 mm . Fringes are obtained on a screen which is placed at distance 1 m away from the slits. When the screen is moved 7 cm farther away, the fringe width changes by $63 \mu \mathrm{~m}$. The wavelength of light used in the experiment will be:

What is the ratio of de Broglie wavelength of an electron to that of proton if the velocity of proton is $\frac{1}{6}$ the velocity of electron?

An atom with one electron has ionization energy of 24 eV . An electron in this atom makes a transition from an excited energy level, where $\mathrm{E}=-15 \mathrm{eV}$, to the ground state. What is the wavelength of the emitted photon from this transition?

Genetic Engineering is related to the principle of

A monochromatic beam of photons of intensity to $1.5 \mathrm{Wm}^{-2}$ and energy 11.2 eV is incident on a material of work function 4.8 eV . What is the maximum speed of photoelectrons emitted due to photoelectric effect if, only $0.53 \%$ of the incident photons eject photoelectrons. Given mass of electron $m=9 \times 10^{-31} \mathrm{~kg}$.