# Datasets: daman1209arora /jeebench

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gold
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int64
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4 values
question
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1.61k
chem
6.75
35
Numeric
Consider the kinetic data given in the following table for the reaction $\mathrm{A}+\mathrm{B}+\mathrm{C} \rightarrow$ Product. \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline Experiment No. & $\begin{array}{c}{[\mathrm{A}]} \\ \left(\mathrm{mol} \mathrm{dm}^{-3}\right)\end{array}$ & $\begin{array}{c}{[\mathrm{B}]} \\ \left(\mathrm{mol} \mathrm{dm}^{-3}\right)\end{array}$ & $\begin{array}{c}{[\mathrm{C}]} \\ \left(\mathrm{mol} \mathrm{dm}^{-3}\right)\end{array}$ & $\begin{array}{c}\text { Rate of reaction } \\ \left(\mathrm{mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}\right)\end{array}$ \\ \hline 1 & 0.2 & 0.1 & 0.1 & $6.0 \times 10^{-5}$ \\ \hline 2 & 0.2 & 0.2 & 0.1 & $6.0 \times 10^{-5}$ \\ \hline 3 & 0.2 & 0.1 & 0.2 & $1.2 \times 10^{-4}$ \\ \hline 4 & 0.3 & 0.1 & 0.1 & $9.0 \times 10^{-5}$ \\ \hline \end{tabular} \end{center} The rate of the reaction for $[\mathrm{A}]=0.15 \mathrm{~mol} \mathrm{dm}^{-3},[\mathrm{~B}]=0.25 \mathrm{~mol} \mathrm{dm}^{-3}$ and $[\mathrm{C}]=0.15 \mathrm{~mol} \mathrm{dm}^{-3}$ is found to be $\mathbf{Y} \times 10^{-5} \mathrm{~mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}$. What is the value of $\mathbf{Y}$?
math
A
37
MCQ
Let $S$ be the set of all complex numbers $Z$ satisfying $|z-2+i| \geq \sqrt{5}$. If the complex number $Z_{0}$ is such that $\frac{1}{\left|Z_{0}-1\right|}$ is the maximum of the set $\left\{\frac{1}{|z-1|}: z \in S\right\}$, then the principal argument of $\frac{4-z_{0}-\overline{z_{0}}}{Z_{0}-\overline{z_{0}}+2 i}$ is (A) $-\frac{\pi}{2}$ (B) $\frac{\pi}{4}$ (C) $\frac{\pi}{2}$ (D) $\frac{3 \pi}{4}$
math
C
38
MCQ
Let $M=\left[\begin{array}{cc} \sin ^{4} \theta & -1-\sin ^{2} \theta \\ 1+\cos ^{2} \theta & \cos ^{4} \theta \end{array}\right]=\alpha I+\beta M^{-1}$ where $\alpha=\alpha(\theta)$ and $\beta=\beta(\theta)$ are real numbers, and $I$ is the $2 \times 2$ identity matrix. If $\alpha^{*}$ is the minimum of the set $\{\alpha(\theta): \theta \in[0,2 \pi)\}$ and $\beta^{*}$ is the minimum of the set $\{\beta(\theta): \theta \in[0,2 \pi)\}$ then the value of $\alpha^{*}+\beta^{*}$ is (A) $-\frac{37}{16}$ (B) $-\frac{31}{16}$ (C) $-\frac{29}{16}$ (D) $-\frac{17}{16}$
math
B
39
MCQ
A line $y=m x+1$ intersects the circle $(x-3)^{2}+(y+2)^{2}=25$ at the points $P$ and $Q$. If the midpoint of the line segment $P Q$ has $x$-coordinate $-\frac{3}{5}$, then which one of the following options is correct? (A) $-3 \leq m<-1$ (B) $2 \leq m<4$ (C) $4 \leq m<6$ (D) $6 \leq m<8$
math
A
40
MCQ
The area of the region $\left\{(x, y): x y \leq 8,1 \leq y \leq x^{2}\right\}$ is (A) $16 \log _{e} 2-\frac{14}{3}$ (B) $8 \log _{e} 2-\frac{14}{3}$ (C) $16 \log _{e} 2-6$ (D) $8 \log _{e} 2-\frac{7}{3}$
math
ABD
42
MCQ(multiple)
Let $M=\left[\begin{array}{lll} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{array}\right] \quad \text { and adj } M=\left[\begin{array}{rrr} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{array}\right]$ where $a$ and $b$ are real numbers. Which of the following options is/are correct? (A) $a+b=3$ (B) $(\operatorname{adj} M)^{-1}+\operatorname{adj} M^{-1}=-M$ (C) $\operatorname{det}\left(\operatorname{adj} M^{2}\right)=81$ (D) If $M\left[\begin{array}{l}\alpha \\ \beta \\ \gamma\end{array}\right]=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$, then $\alpha-\beta+\gamma=3$
math
AB
43
MCQ(multiple)
There are three bags $B_{1}, B_{2}$ and $B_{3}$. The bag $B_{1}$ contains 5 red and 5 green balls, $B_{2}$ contains 3 red and 5 green balls, and $B_{3}$ contains 5 red and 3 green balls. Bags $B_{1}, B_{2}$ and $B_{3}$ have probabilities $\frac{3}{10}, \frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct? (A) Probability that the chosen ball is green, given that the selected bag is $B_{3}$, equals $\frac{3}{8}$ (B) Probability that the chosen ball is green equals $\frac{39}{80}$ (C) Probability that the selected bag is $B_{3}$, given that the chosen ball is green, equals $\frac{5}{13}$ (D) Probability that the selected bag is $B_{3}$ and the chosen ball is green equals $\frac{3}{10}$
math
ACD
44
MCQ(multiple)
In a non-right-angled triangle $\triangle P Q R$, let $p, q, r$ denote the lengths of the sides opposite to the angles at $P, Q, R$ respectively. The median from $R$ meets the side $P Q$ at $S$, the perpendicular from $P$ meets the side $Q R$ at $E$, and $R S$ and $P E$ intersect at $O$. If $p=\sqrt{3}, q=1$, and the radius of the circumcircle of the $\triangle P Q R$ equals 1 , then which of the following options is/are correct? (A) Length of $R S=\frac{\sqrt{7}}{2}$ (B) Area of $\triangle S O E=\frac{\sqrt{3}}{12}$ (C) Length of $O E=\frac{1}{6}$ (D) Radius of incircle of $\triangle P Q R=\frac{\sqrt{3}}{2}(2-\sqrt{3})$
math
BC
45
MCQ(multiple)
Define the collections $\left\{E_{1}, E_{2}, E_{3}, \ldots\right\}$ of ellipses and $\left\{R_{1}, R_{2}, R_{3}, \ldots\right\}$ of rectangles as follows: $E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ $R_{1}$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E_{1}$; $E_{n}:$ ellipse $\frac{x^{2}}{a_{n}^{2}}+\frac{y^{2}}{b_{n}^{2}}=1$ of largest area inscribed in $R_{n-1}, n>1$; $R_{n}:$ rectangle of largest area, with sides parallel to the axes, inscribed in $E_{n}, n>1$. Then which of the following options is/are correct? (A) The eccentricities of $E_{18}$ and $E_{19}$ are NOT equal (B) $\quad \sum_{n=1}^{N}\left(\right.$ area of $\left.R_{n}\right)<24$, for each positive integer $N$ (C) The length of latus rectum of $E_{9}$ is $\frac{1}{6}$ (D) The distance of a focus from the centre in $E_{9}$ is $\frac{\sqrt{5}}{32}$
math
AB
47
MCQ(multiple)
Let $\Gamma$ denote a curve $y=y(x)$ which is in the first quadrant and let the point $(1,0)$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y_{P}$. If $P Y_{P}$ has length 1 for each point $P$ on $\Gamma$, then which of the following options is/are correct? (A) $y=\log _{e}\left(\frac{1+\sqrt{1-x^{2}}}{x}\right)-\sqrt{1-x^{2}}$ (B) $x y^{\prime}+\sqrt{1-x^{2}}=0$ (C) $y=-\log _{e}\left(\frac{1+\sqrt{1-x^{2}}}{x}\right)+\sqrt{1-x^{2}}$ (D) $x y^{\prime}-\sqrt{1-x^{2}}=0$
math
ABC
48
MCQ(multiple)
Let $L_{1}$ and $L_{2}$ denote the lines $\vec{r}=\hat{i}+\lambda(-\hat{i}+2 \hat{j}+2 \hat{k}), \lambda \in \mathbb{R}$ and $\vec{r}=\mu(2 \hat{i}-\hat{j}+2 \hat{k}), \mu \in \mathbb{R}$ respectively. If $L_{3}$ is a line which is perpendicular to both $L_{1}$ and $L_{2}$ and cuts both of them, then which of the following options describe(s) $L_{3}$ ? (A) $\vec{r}=\frac{2}{9}(4 \hat{i}+\hat{j}+\hat{k})+t(2 \hat{i}+2 \hat{j}-\hat{k}), t \in \mathbb{R}$ (B) $\vec{r}=\frac{2}{9}(2 \hat{i}-\hat{j}+2 \hat{k})+t(2 \hat{i}+2 \hat{j}-\hat{k}), t \in \mathbb{R}$ (C) $\vec{r}=\frac{1}{3}(2 \hat{i}+\hat{k})+t(2 \hat{i}+2 \hat{j}-\hat{k}), t \in \mathbb{R}$ (D) $\vec{r}=t(2 \hat{i}+2 \hat{j}-\hat{k}), t \in \mathbb{R}$
math
3
49
Numeric
Let $\omega \neq 1$ be a cube root of unity. Then what is the minimum of the set $\left\{\left|a+b \omega+c \omega^{2}\right|^{2}: a, b, c \text { distinct non-zero integers }\right\}$ equal?
math
157
50
Numeric
Let $A P(a ; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d>0$. If $A P(1 ; 3) \cap A P(2 ; 5) \cap A P(3 ; 7)=A P(a ; d)$ then what does $a+d$ equal?
math
0.5
51
Numeric
Let $S$ be the sample space of all $3 \times 3$ matrices with entries from the set $\{0,1\}$. Let the events $E_{1}$ and $E_{2}$ be given by \begin{aligned} & E_{1}=\{A \in S: \operatorname{det} A=0\} \text { and } \\ & E_{2}=\{A \in S: \text { sum of entries of } A \text { is } 7\} . \end{aligned} If a matrix is chosen at random from $S$, then what is the conditional probability $P\left(E_{1} \mid E_{2}\right)$?
math
10
52
Numeric
Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8 x-6 y-23=0$. Let $\Gamma_{A}$ and $\Gamma_{B}$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_{A}$ and $\Gamma_{B}$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then what is the length of the line segment $A C$?
math
4
53
Numeric
If $I=\frac{2}{\pi} \int_{-\pi / 4}^{\pi / 4} \frac{d x}{\left(1+e^{\sin x}\right)(2-\cos 2 x)}$ then what does $27 I^{2}$ equal?
math
0.75
54
Numeric
Three lines are given by $\vec{r} & =\lambda \hat{i}, \lambda \in \mathbb{R}$ $\vec{r} & =\mu(\hat{i}+\hat{j}), \mu \in \mathbb{R}$ $\vec{r} =v(\hat{i}+\hat{j}+\hat{k}), v \in \mathbb{R}.$ Let the lines cut the plane $x+y+z=1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $A B C$ is $\triangle$ then what is the value of $(6 \Delta)^{2}$?
phy
ACD
1
MCQ(multiple)
A thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle $60^{\circ}$ with vertical? [ $g$ is the acceleration due to gravity] (A) The angular speed of the rod will be $\sqrt{\frac{3 g}{2 L}}$ (B) The angular acceleration of the rod will be $\frac{2 g}{L}$ (C) The radial acceleration of the rod's center of mass will be $\frac{3 g}{4}$ (D) The normal reaction force from the floor on the rod will be $\frac{M g}{16}$
phy
ACD
5
MCQ(multiple)
A mixture of ideal gas containing 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially at pressure $P_{0}$, volume $V_{0}$, and temperature $T_{0}$. If the gas mixture is adiabatically compressed to a volume $V_{0} / 4$, then the correct statement(s) is/are, (Given $2^{1.2}=2.3 ; 2^{3.2}=9.2 ; R$ is gas constant) (A) The work $|W|$ done during the process is $13 R T_{0}$ (B) The average kinetic energy of the gas mixture after compression is in between $18 R T_{0}$ and $19 R T_{0}$ (C) The final pressure of the gas mixture after compression is in between $9 P_{0}$ and $10 P_{0}$ (D) Adiabatic constant of the gas mixture is 1.6
phy
8
MCQ(multiple)
A free hydrogen atom after absorbing a photon of wavelength $\lambda_{a}$ gets excited from the state $n=1$ to the state $n=4$. Immediately after that the electron jumps to $n=m$ state by emitting a photon of wavelength $\lambda_{e}$. Let the change in momentum of atom due to the absorption and the emission are $\Delta p_{a}$ and $\Delta p_{e}$, respectively. If $\lambda_{a} / \lambda_{e}=\frac{1}{5}$, which of the option(s) is/are correct? [Use $h c=1242 \mathrm{eV} \mathrm{nm} ; 1 \mathrm{~nm}=10^{-9} \mathrm{~m}, h$ and $c$ are Planck's constant and speed of light, respectively] (A) $m=2$ (B) $\lambda_{e}=418 \mathrm{~nm}$ (C) $\Delta p_{a} / \Delta p_{e}=\frac{1}{2}$ (D) The ratio of kinetic energy of the electron in the state $n=m$ to the state $n=1$ is $\frac{1}{4}$
phy
135
13
Numeric
Suppose a ${ }_{88}^{226} R a$ nucleus at rest and in ground state undergoes $\alpha$-decay to a ${ }_{86}^{222} R n$ nucleus in its excited state. The kinetic energy of the emitted $\alpha$ particle is found to be $4.44 \mathrm{MeV}$. ${ }_{86}^{222} R n$ nucleus then goes to its ground state by $\gamma$-decay. What is the energy of the emitted $\gamma$ photon is $\mathrm{keV}$? [Given: atomic mass of ${ }_{88}^{226} R a=226.005 \mathrm{u}$, atomic mass of ${ }_{86}^{222} R n=222.000 \mathrm{u}$, atomic mass of $\alpha$ particle $=4.000 \mathrm{u}, 1 \mathrm{u}=931 \mathrm{MeV} / \mathrm{c}^{2}, \mathrm{c}$ is speed of the light]
phy
0.69
14
Numeric
An optical bench has $1.5 \mathrm{~m}$ long scale having four equal divisions in each $\mathrm{cm}$. While measuring the focal length of a convex lens, the lens is kept at $75 \mathrm{~cm}$ mark of the scale and the object pin is kept at $45 \mathrm{~cm}$ mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at $135 \mathrm{~cm}$ mark. In this experiment, what is the percentage error in the measurement of the focal length of the lens?
chem
ABC
19
MCQ(multiple)
The cyanide process of gold extraction involves leaching out gold from its ore with $\mathrm{CN}^{-}$in the presence of $\mathbf{Q}$ in water to form $\mathrm{R}$. Subsequently, $\mathrm{R}$ is treated with $\mathrm{T}$ to obtain $\mathrm{Au}$ and $\mathrm{Z}$. Choose the correct option(s) (A) $\mathrm{Q}$ is $\mathrm{O}_{2}$ (B) $\mathrm{T}$ is $\mathrm{Zn}$ (C) $\mathrm{Z}$ is $\left[\mathrm{Zn}(\mathrm{CN})_{4}\right]^{2-}$ (D) $\mathrm{R}$ is $\left[\mathrm{Au}(\mathrm{CN})_{4}\right]^{-}$
chem
ABD
20
MCQ(multiple)
With reference to aqua regia, choose the correct option(s) (A) Aqua regia is prepared by mixing conc. $\mathrm{HCl}$ and conc. $\mathrm{HNO}_{3}$ in $3: 1(v / v)$ ratio (B) Reaction of gold with aqua regia produces an anion having Au in +3 oxidation state (C) Reaction of gold with aqua regia produces $\mathrm{NO}_{2}$ in the absence of air (D) The yellow colour of aqua regia is due to the presence of $\mathrm{NOCl}$ and $\mathrm{Cl}_{2}$
chem
ACD
21
MCQ(multiple)
Consider the following reactions (unbalanced) $\mathrm{Zn}+$ hot conc. $\mathrm{H}_{2} \mathrm{SO}_{4} \rightarrow \mathrm{G}+\mathrm{R}+\mathrm{X}$ $\mathrm{Zn}+$ conc. $\mathrm{NaOH} \rightarrow \mathrm{T}+\mathbf{Q}$ $\mathbf{G}+\mathrm{H}_{2} \mathrm{~S}+\mathrm{NH}_{4} \mathrm{OH} \rightarrow \mathbf{Z}$ (a precipitate) $+\mathbf{X}+\mathrm{Y}$ Choose the correct option(s) (A) $\mathrm{Z}$ is dirty white in colour (B) The oxidation state of $\mathrm{Zn}$ in $\mathrm{T}$ is +1 (C) $\mathrm{R}$ is a V-shaped molecule (D) Bond order of $\mathbf{Q}$ is 1 in its ground state
chem
AB
22
MCQ(multiple)
The ground state energy of hydrogen atom is $-13.6 \mathrm{eV}$. Consider an electronic state $\Psi$ of $\mathrm{He}^{+}$ whose energy, azimuthal quantum number and magnetic quantum number are $-3.4 \mathrm{eV}, 2$ and 0 , respectively. Which of the following statement(s) is(are) true for the state $\Psi$ ? (A) It is a $4 d$ state (B) It has 2 angular nodes (C) It has 3 radial nodes (D) The nuclear charge experienced by the electron in this state is less than $2 e$, where $e$ is the magnitude of the electronic charge
chem
BC
26
MCQ(multiple)
Choose the correct option(s) from the following (A) Natural rubber is polyisoprene containing trans alkene units (B) Nylon-6 has amide linkages (C) Teflon is prepared by heating tetrafluoroethene in presence of a persulphate catalyst at high pressure (D) Cellulose has only $\alpha$-D-glucose units that are joined by glycosidic linkages
chem
288
27
Numeric
What is the amount of water produced (in g) in the oxidation of 1 mole of rhombic sulphur by conc. $\mathrm{HNO}_{3}$ to a compound with the highest oxidation state of sulphur? (Given data: Molar mass of water $=18 \mathrm{~g} \mathrm{~mol}^{-1}$ )
chem
6
28
Numeric
What is the total number of cis $\mathrm{N}-\mathrm{Mn}-\mathrm{Cl}$ bond angles (that is, $\mathrm{Mn}-\mathrm{N}$ and $\mathrm{Mn}-\mathrm{Cl}$ bonds in cis positions) present in a molecule of cis-[Mn(en $\left.)_{2} \mathrm{Cl}_{2}\right]$ complex? (en $=\mathrm{NH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{2}$ )
chem
2.3
29
Numeric
The decomposition reaction $2 \mathrm{~N}_{2} \mathrm{O}_{5}(g) \stackrel{\Delta}{\rightarrow} 2 \mathrm{~N}_{2} \mathrm{O}_{4}(g)+\mathrm{O}_{2}(g)$ is started in a closed cylinder under isothermal isochoric condition at an initial pressure of $1 \mathrm{~atm}$. After $\mathrm{Y} \times 10^{3} \mathrm{~s}$, the pressure inside the cylinder is found to be $1.45 \mathrm{~atm}$. If the rate constant of the reaction is $5 \times 10^{-4} \mathrm{~s}^{-1}$, assuming ideal gas behavior, what is the value of $\mathrm{Y}$?
chem
2.98
30
Numeric
The mole fraction of urea in an aqueous urea solution containing $900 \mathrm{~g}$ of water is 0.05 . If the density of the solution is $1.2 \mathrm{~g} \mathrm{~cm}^{-3}$, what is the molarity of urea solution? (Given data: Molar masses of urea and water are $60 \mathrm{~g} \mathrm{~mol}^{-1}$ and $18 \mathrm{~g} \mathrm{~mol}^{-1}$, respectively)
chem
10
32
Numeric
What is the total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the molecular formula $\mathrm{C}_{4} \mathrm{H}_{8} \mathrm{O}$?
math
ABC
37
MCQ(multiple)
Let \begin{aligned} & P_{1}=I=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right], \quad P_{2}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right], \quad P_{3}=\left[\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right], \\ & P_{4}=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right], \quad P_{5}=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right], \quad P_{6}=\left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right] \\ & \text { and } X=\sum_{k=1}^{6} P_{k}\left[\begin{array}{lll} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{array}\right] P_{k}^{T} \end{aligned} where $P_{k}^{T}$ denotes the transpose of the matrix $P_{k}$. Then which of the following options is/are correct? (A) If $X\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\alpha\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$, then $\alpha=30$ (B) $X$ is a symmetric matrix (C) The sum of diagonal entries of $X$ is 18 (D) $X-30 I$ is an invertible matrix
math
AB
40
MCQ(multiple)
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. We say that $f$ has PROPERTY 1 if $\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{\sqrt{|h|}}$ exists and is finite, and PROPERTY 2 if $\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h^{2}}$ exists and is finite. Then which of the following options is/are correct? (A) $f(x)=|x|$ has PROPERTY 1 (B) $f(x)=x^{2 / 3}$ has PROPERTY 1 (C) $f(x)=x|x|$ has PROPERTY 2 (D) $f(x)=\sin x$ has PROPERTY 2
math
BCD
41
MCQ(multiple)
Let $f(x)=\frac{\sin \pi x}{x^{2}}, \quad x>0$ Let $x_{1}<x_{2}<x_{3}<\cdots<x_{n}<\cdots$ be all the points of local maximum of $f$ and $y_{1}<y_{2}<y_{3}<\cdots<y_{n}<\cdots$ be all the points of local minimum of $f$. Then which of the following options is/are correct? (A) $x_{1}<y_{1}$ (B) $x_{n+1}-x_{n}>2$ for every $n$ (C) $\quad x_{n} \in\left(2 n, 2 n+\frac{1}{2}\right)$ for every $n$ (D) $\left|x_{n}-y_{n}\right|>1$ for every $n$
math
42
MCQ(multiple)
For $a \in \mathbb{R},|a|>1$, let $\lim _{n \rightarrow \infty}\left(\frac{1+\sqrt[3]{2}+\cdots+\sqrt[3]{n}}{n^{7 / 3}\left(\frac{1}{(a n+1)^{2}}+\frac{1}{(a n+2)^{2}}+\cdots+\frac{1}{(a n+n)^{2}}\right)}\right)=54$ Then the possible value(s) of $a$ is/are (A) -9 (B) -6 (C) 7 (D) 8
math
ABD
43
MCQ(multiple)
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x)=(x-1)(x-2)(x-5)$. Define $F(x)=\int_{0}^{x} f(t) d t, \quad x>0 .$ Then which of the following options is/are correct? (A) $F$ has a local minimum at $x=1$ (B) $F$ has a local maximum at $x=2$ (C) $F$ has two local maxima and one local minimum in $(0, \infty)$ (D) $\quad F(x) \neq 0$ for all $x \in(0,5)$
math
AC
44
MCQ(multiple)
Three lines \begin{aligned} L_{1}: & \vec{r}=\lambda \hat{i}, \lambda \in \mathbb{R}, \\ L_{2}: & \vec{r}=\hat{k}+\mu \hat{j}, \mu \in \mathbb{R} \text { and } \\ L_{3}: & \vec{r}=\hat{i}+\hat{j}+v \hat{k}, \quad v \in \mathbb{R} \end{aligned} are given. For which point(s) $Q$ on $L_{2}$ can we find a point $P$ on $L_{1}$ and a point $R$ on $L_{3}$ so that $P, Q$ and $R$ are collinear? (A) $\hat{k}-\frac{1}{2} \hat{j}$ (B) $\hat{k}$ (C) $\hat{k}+\frac{1}{2} \hat{j}$ (D) $\hat{k}+\hat{j}$
math
6.2
45
Numeric
Suppose $\operatorname{det}\left[\begin{array}{cc} \sum_{k=0}^{n} k & \sum_{k=0}^{n}{ }^{n} C_{k} k^{2} \\ \sum_{k=0}^{n}{ }^{n} C_{k} k & \sum_{k=0}^{n}{ }^{n} C_{k} 3^{k} \end{array}\right]=0$ holds for some positive integer $n$. Then what does $\sum_{k=0}^{n} \frac{{ }^{n} C_{k}}{k+1}$?
math
30
46
Numeric
Five persons $A, B, C, D$ and $E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then what is the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats?
math
422
47
Numeric
Let $|X|$ denote the number of elements in a set $X$. Let $S=\{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then what is the number of ordered pairs $(A, B)$ such that $1 \leq|B|<|A|$?
math
0
48
Numeric
What is the value of $\sec ^{-1}\left(\frac{1}{4} \sum_{k=0}^{10} \sec \left(\frac{7 \pi}{12}+\frac{k \pi}{2}\right) \sec \left(\frac{7 \pi}{12}+\frac{(k+1) \pi}{2}\right)\right)$ in the interval $\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]$?
math
0.5
49
Numeric
What is the value of the integral $\int_{0}^{\pi / 2} \frac{3 \sqrt{\cos \theta}}{(\sqrt{\cos \theta}+\sqrt{\sin \theta})^{5}} d \theta$?
phy
BC
7
MCQ(multiple)
A particle of mass $m$ moves in circular orbits with potential energy $V(r)=F r$, where $F$ is a positive constant and $r$ is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle's orbit is denoted by $R$ and its speed and energy are denoted by $v$ and $E$, respectively, then for the $n^{\text {th }}$ orbit (here $h$ is the Planck's constant) (A) $R \propto n^{1 / 3}$ and $v \propto n^{2 / 3}$ (B) $R \propto n^{2 / 3}$ and $\mathrm{v} \propto n^{1 / 3}$ (C) $E=\frac{3}{2}\left(\frac{n^{2} h^{2} F^{2}}{4 \pi^{2} m}\right)^{1 / 3}$ (D) $E=2\left(\frac{n^{2} h^{2} F^{2}}{4 \pi^{2} m}\right)^{1 / 3}$
phy
BCD
8
MCQ(multiple)
The filament of a light bulb has surface area $64 \mathrm{~mm}^{2}$. The filament can be considered as a black body at temperature $2500 \mathrm{~K}$ emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of $100 \mathrm{~m}$. Assume the pupil of the eyes of the observer to be circular with radius $3 \mathrm{~mm}$. Then (Take Stefan-Boltzmann constant $=5.67 \times 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^{-4}$, Wien's displacement constant $=$ $2.90 \times 10^{-3} \mathrm{~m}-\mathrm{K}$, Planck's constant $=6.63 \times 10^{-34} \mathrm{Js}$, speed of light in vacuum $=3.00 \times$ $\left.10^{8} \mathrm{~ms}^{-1}\right)$ (A) power radiated by the filament is in the range $642 \mathrm{~W}$ to $645 \mathrm{~W}$ (B) radiated power entering into one eye of the observer is in the range $3.15 \times 10^{-8} \mathrm{~W}$ to $3.25 \times 10^{-8} \mathrm{~W}$ (C) the wavelength corresponding to the maximum intensity of light is $1160 \mathrm{~nm}$ (D) taking the average wavelength of emitted radiation to be $1740 \mathrm{~nm}$, the total number of photons entering per second into one eye of the observer is in the range $2.75 \times 10^{11}$ to $2.85 \times 10^{11}$
phy
AB
9
MCQ(multiple)
Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: [position $]=\left[X^{\alpha}\right]$; [speed $]=\left[X^{\beta}\right]$; [acceleration $]=\left[X^{p}\right]$; $[$ linear momentum $]=\left[X^{q}\right] ;$ force $]=\left[X^{r}\right]$. Then (A) $\alpha+p=2 \beta$ (B) $p+q-r=\beta$ (C) $p-q+r=\alpha$ (D) $p+q+r=\beta$
phy
25.6
13
Numeric
Put a uniform meter scale horizontally on your extended index fingers with the left one at $0.00 \mathrm{~cm}$ and the right one at $90.00 \mathrm{~cm}$. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_{R}$ from the center $(50.00 \mathrm{~cm})$ of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32 , respectively, what is the value of $x_{R}$ (in $\mathrm{cm})$?
phy
1.77
16
Numeric
Consider one mole of helium gas enclosed in a container at initial pressure $P_{1}$ and volume $V_{1}$. It expands isothermally to volume $4 V_{1}$. After this, the gas expands adiabatically and its volume becomes $32 V_{1}$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\text {iso }}$ and $W_{\text {adia }}$, respectively. If the ratio $\frac{W_{\text {iso }}}{W_{\text {adia }}}=f \ln 2$, then what is the value of $f$?
phy
0.62
17
Numeric
A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of $2 \mathrm{~ms}^{-1}$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is $320 \mathrm{~ms}^{-1}$, what is the smallest value of the percentage change required in the length of the pipe?
phy
6.4
18
Numeric
A circular disc of radius $R$ carries surface charge density $\sigma(r)=\sigma_{0}\left(1-\frac{r}{R}\right)$, where $\sigma_{0}$ is a constant and $r$ is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is $\phi_{0}$. Electric flux through another spherical surface of radius $\frac{R}{4}$ and concentric with the disc is $\phi$. Then what is the ratio $\frac{\phi_{0}}{\phi}$?
chem
B
20
MCQ
Which of the following liberates $\mathrm{O}_{2}$ upon hydrolysis? (A) $\mathrm{Pb}_{3} \mathrm{O}_{4}$ (B) $\mathrm{KO}_{2}$ (C) $\mathrm{Na}_{2} \mathrm{O}_{2}$ (D) $\mathrm{Li}_{2} \mathrm{O}_{2}$
chem
A
21
MCQ
A colorless aqueous solution contains nitrates of two metals, $\mathbf{X}$ and $\mathbf{Y}$. When it was added to an aqueous solution of $\mathrm{NaCl}$, a white precipitate was formed. This precipitate was found to be partly soluble in hot water to give a residue $\mathbf{P}$ and a solution $\mathbf{Q}$. The residue $\mathbf{P}$ was soluble in aq. $\mathrm{NH}_{3}$ and also in excess sodium thiosulfate. The hot solution $\mathbf{Q}$ gave a yellow precipitate with KI. The metals $\mathbf{X}$ and $\mathbf{Y}$, respectively, are (A) Ag and $\mathrm{Pb}$ (B) Ag and Cd (C) $\mathrm{Cd}$ and $\mathrm{Pb}$ (D) Cd and Zn
chem
ABC
25
MCQ(multiple)
In thermodynamics, the $P-V$ work done is given by $w=-\int d V P_{\mathrm{ext}}$ For a system undergoing a particular process, the work done is, This equation is applicable to a $w=-\int d V\left(\frac{R T}{V-b}-\frac{a}{V^{2}}\right)$ (A) system that satisfies the van der Waals equation of state. (B) process that is reversible and isothermal. (C) process that is reversible and adiabatic. (D) process that is irreversible and at constant pressure.
chem
AC
28
MCQ(multiple)
Choose the correct statement(s) among the following: (A) $\left[\mathrm{FeCl}_{4}\right]^{-}$has tetrahedral geometry. (B) $\left[\mathrm{Co}(\mathrm{en})\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{2}\right]^{+}$has 2 geometrical isomers. (C) $\left[\mathrm{FeCl}_{4}\right]^{-}$has higher spin-only magnetic moment than $\left[\mathrm{Co}(\mathrm{en})\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{2}\right]^{+}$. (D) The cobalt ion in $\left[\mathrm{Co}(\mathrm{en})\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{2}\right]^{+}$has $\mathrm{sp}^{3} d^{2}$ hybridization.
chem
ABD
29
MCQ(multiple)
With respect to hypochlorite, chlorate and perchlorate ions, choose the correct statement(s). (A) The hypochlorite ion is the strongest conjugate base. (B) The molecular shape of only chlorate ion is influenced by the lone pair of electrons of $\mathrm{Cl}$. (C) The hypochlorite and chlorate ions disproportionate to give rise to identical set of ions. (D) The hypochlorite ion oxidizes the sulfite ion.
chem
0.11
31
Numeric
$5.00 \mathrm{~mL}$ of $0.10 \mathrm{M}$ oxalic acid solution taken in a conical flask is titrated against $\mathrm{NaOH}$ from a burette using phenolphthalein indicator. The volume of $\mathrm{NaOH}$ required for the appearance of permanent faint pink color is tabulated below for five experiments. What is the concentration, in molarity, of the $\mathrm{NaOH}$ solution? \begin{center} \begin{tabular}{|c|c|} \hline Exp. No. & Vol. of NaOH (mL) \\ \hline $\mathbf{1}$ & 12.5 \\ \hline $\mathbf{2}$ & 10.5 \\ \hline $\mathbf{3}$ & 9.0 \\ \hline $\mathbf{4}$ & 9.0 \\ \hline $\mathbf{5}$ & 9.0 \\ \hline \end{tabular} \end{center}
chem
13.32
33
Numeric
Consider a 70\% efficient hydrogen-oxygen fuel cell working under standard conditions at 1 bar and $298 \mathrm{~K}$. Its cell reaction is $\mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \rightarrow \mathrm{H}_{2} \mathrm{O}(l)$ The work derived from the cell on the consumption of $1.0 \times 10^{-3} \mathrm{~mol} \mathrm{of}_{2}(g)$ is used to compress $1.00 \mathrm{~mol}$ of a monoatomic ideal gas in a thermally insulated container. What is the change in the temperature (in K) of the ideal gas? The standard reduction potentials for the two half-cells are given below. $\begin{gathered} \mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q)+4 e^{-} \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(l), \quad E^{0}=1.23 \mathrm{~V}, \\ 2 \mathrm{H}^{+}(a q)+2 e^{-} \rightarrow \mathrm{H}_{2}(g), \quad E^{0}=0.00 \mathrm{~V} \end{gathered}$ Use $F=96500 \mathrm{C} \mathrm{mol}^{-1}, R=8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$.
chem
6.15
34
Numeric
Aluminium reacts with sulfuric acid to form aluminium sulfate and hydrogen. What is the volume of hydrogen gas in liters (L) produced at $300 \mathrm{~K}$ and $1.0 \mathrm{~atm}$ pressure, when $5.4 \mathrm{~g}$ of aluminium and $50.0 \mathrm{~mL}$ of $5.0 \mathrm{M}$ sulfuric acid are combined for the reaction? (Use molar mass of aluminium as $27.0 \mathrm{~g} \mathrm{~mol}^{-1}, R=0.082 \mathrm{~atm} \mathrm{~L} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ )
chem
1.2
35
Numeric
${ }_{92}^{238} \mathrm{U}$ is known to undergo radioactive decay to form ${ }_{82}^{206} \mathrm{~Pb}$ by emitting alpha and beta particles. A rock initially contained $68 \times 10^{-6} \mathrm{~g}$ of ${ }_{92}^{238} \mathrm{U}$. If the number of alpha particles that it would emit during its radioactive decay of ${ }_{92}^{238} \mathrm{U}$ to ${ }_{82}^{206} \mathrm{~Pb}$ in three half-lives is $Z \times 10^{18}$, then what is the value of $Z$ ?
math
D
37
MCQ
Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^{2}+20 x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^{2}-20 x+2020$. Then the value of $a c(a-c)+a d(a-d)+b c(b-c)+b d(b-d)$ is (A) 0 (B) 8000 (C) 8080 (D) 16000
math
C
38
MCQ
If the function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined by $f(x)=|x|(x-\sin x)$, then which of the following statements is TRUE? (A) $f$ is one-one, but NOT onto (B) $f$ is onto, but NOT one-one (C) $f$ is BOTH one-one and onto (D) $f$ is NEITHER one-one NOR onto
math
A
39
MCQ
Let the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=e^{x-1}-e^{-|x-1|} \quad \text { and } \quad g(x)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right)$ Then the area of the region in the first quadrant bounded by the curves $y=f(x), y=g(x)$ and $x=0$ is (A) $(2-\sqrt{3})+\frac{1}{2}\left(e-e^{-1}\right)$ (B) $(2+\sqrt{3})+\frac{1}{2}\left(e-e^{-1}\right)$ (C) $(2-\sqrt{3})+\frac{1}{2}\left(e+e^{-1}\right)$ (D) $(2+\sqrt{3})+\frac{1}{2}\left(e+e^{-1}\right)$
math
A
40
MCQ
Let $a, b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^{2}=4 \lambda x$, and suppose the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is (A) $\frac{1}{\sqrt{2}}$ (B) $\frac{1}{2}$ (C) $\frac{1}{3}$ (D) $\frac{2}{5}$
math
B
41
MCQ
Let $C_{1}$ and $C_{2}$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_{1}$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_{2}$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^{2}-\alpha x+\beta$ are real and equal, is (A) $\frac{40}{81}$ (B) $\frac{20}{81}$ (C) $\frac{1}{2}$ (D) $\frac{1}{4}$
math
C
42
MCQ
Consider all rectangles lying in the region $\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: 0 \leq x \leq \frac{\pi}{2} \text { and } 0 \leq y \leq 2 \sin (2 x)\right\}$ and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is (A) $\frac{3 \pi}{2}$ (B) $\pi$ (C) $\frac{\pi}{2 \sqrt{3}}$ (D) $\frac{\pi \sqrt{3}}{2}$
math
AC
43
MCQ(multiple)
Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=x^{3}-x^{2}+(x-1) \sin x$ and let $g: \mathbb{R} \rightarrow \mathbb{R}$ be an arbitrary function. Let $f g: \mathbb{R} \rightarrow \mathbb{R}$ be the product function defined by $(f g)(x)=f(x) g(x)$. Then which of the following statements is/are TRUE? (A) If $g$ is continuous at $x=1$, then $f g$ is differentiable at $x=1$ (B) If $f g$ is differentiable at $x=1$, then $g$ is continuous at $x=1$ (C) If $g$ is differentiable at $x=1$, then $f g$ is differentiable at $x=1$ (D) If $f g$ is differentiable at $x=1$, then $g$ is differentiable at $x=1$
math
BCD
44
MCQ(multiple)
Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M^{-1}=\operatorname{adj}(\operatorname{adj} M)$, then which of the following statements is/are ALWAYS TRUE? (A) $M=I$ (B) $\operatorname{det} M=1$ (C) $M^{2}=I$ (D) $(\operatorname{adj} M)^{2}=I$
math
BC
45
MCQ(multiple)
Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^{2}+z+1\right|=1$. Then which of the following statements is/are TRUE? (A) $\left|z+\frac{1}{2}\right| \leq \frac{1}{2}$ for all $z \in S$ (B) $|z| \leq 2$ for all $z \in S$ (C) $\left|z+\frac{1}{2}\right| \geq \frac{1}{2}$ for all $z \in S$ (D) The set $S$ has exactly four elements
math
BC
46
MCQ(multiple)
Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$, respectively. If $\tan \frac{X}{2}+\tan \frac{Z}{2}=\frac{2 y}{x+y+z}$ then which of the following statements is/are TRUE? (A) $2 Y=X+Z$ (B) $Y=X+Z$ (C) $\tan \frac{x}{2}=\frac{x}{y+z}$ (D) $x^{2}+z^{2}-y^{2}=x z$
math
AB
47
MCQ(multiple)
Let $L_{1}$ and $L_{2}$ be the following straight lines. $L_{1}: \frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3} \text { and } L_{2}: \frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}$ Suppose the straight line $L: \frac{x-\alpha}{l}=\frac{y-1}{m}=\frac{z-\gamma}{-2}$ lies in the plane containing $L_{1}$ and $L_{2}$, and passes through the point of intersection of $L_{1}$ and $L_{2}$. If the line $L$ bisects the acute angle between the lines $L_{1}$ and $L_{2}$, then which of the following statements is/are TRUE? (A) $\alpha-\gamma=3$ (B) $l+m=2$ (C) $\alpha-\gamma=1$ (D) $l+m=0$
math
ABD
48
MCQ(multiple)
Which of the following inequalities is/are TRUE? (A) $\int_{0}^{1} x \cos x d x \geq \frac{3}{8}$ (B) $\int_{0}^{1} x \sin x d x \geq \frac{3}{10}$ (C) $\int_{0}^{1} x^{2} \cos x d x \geq \frac{1}{2}$ (D) $\int_{0}^{1} x^{2} \sin x d x \geq \frac{2}{9}$
math
8
49
Numeric
Let $m$ be the minimum possible value of $\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)$, where $y_{1}, y_{2}, y_{3}$ are real numbers for which $y_{1}+y_{2}+y_{3}=9$. Let $M$ be the maximum possible value of $\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)$, where $x_{1}, x_{2}, x_{3}$ are positive real numbers for which $x_{1}+x_{2}+x_{3}=9$. Then what is the value of $\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)$?
math
1
50
Numeric
Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b_{1}, b_{2}, b_{3}, \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a_{1}=b_{1}=c$, then what is the number of all possible values of $c$, for which the equality $2\left(a_{1}+a_{2}+\cdots+a_{n}\right)=b_{1}+b_{2}+\cdots+b_{n}$ holds for some positive integer $n$?
math
1
51
Numeric
Let $f:[0,2] \rightarrow \mathbb{R}$ be the function defined by $f(x)=(3-\sin (2 \pi x)) \sin \left(\pi x-\frac{\pi}{4}\right)-\sin \left(3 \pi x+\frac{\pi}{4}\right)$ If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$, then what is the value of $\beta-\alpha$?
math
108
52
Numeric
In a triangle $P Q R$, let $\vec{a}=\overrightarrow{Q R}, \vec{b}=\overrightarrow{R P}$ and $\vec{c}=\overrightarrow{P Q}$. If $|\vec{a}|=3, \quad|\vec{b}|=4 \quad \text { and } \quad \frac{\vec{a} \cdot(\vec{c}-\vec{b})}{\vec{c} \cdot(\vec{a}-\vec{b})}=\frac{|\vec{a}|}{|\vec{a}|+|\vec{b}|},$ then what is the value of $|\vec{a} \times \vec{b}|^{2}$?
math
3
53
Numeric
For a polynomial $g(x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by $S=\left\{\left(x^{2}-1\right)^{2}\left(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\right): a_{0}, a_{1}, a_{2}, a_{3} \in \mathbb{R}\right\}$ For a polynomial $f$, let $f^{\prime}$ and $f^{\prime \prime}$ denote its first and second order derivatives, respectively. Then what is the minimum possible value of $\left(m_{f^{\prime}}+m_{f^{\prime \prime}}\right)$, where $f \in S$?
math
1
54
Numeric
Let $e$ denote the base of the natural logarithm. What is the value of the real number $a$ for which the right hand limit $\lim _{x \rightarrow 0^{+}} \frac{(1-x)^{\frac{1}{x}}-e^{-1}}{x^{a}}$ is equal to a nonzero real number?
phy
9
2
Integer
A train with cross-sectional area $S_{t}$ is moving with speed $v_{t}$ inside a long tunnel of cross-sectional area $S_{0}\left(S_{0}=4 S_{t}\right)$. Assume that almost all the air (density $\rho$ ) in front of the train flows back between its sides and the walls of the tunnel. Also, the air flow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be $p_{0}$. If the pressure in the region between the sides of the train and the tunnel walls is $p$, then $p_{0}-p=\frac{7}{2 N} \rho v_{t}^{2}$. What is the value of $N$?
phy
4
4
Integer
A hot air balloon is carrying some passengers, and a few sandbags of mass $1 \mathrm{~kg}$ each so that its total mass is $480 \mathrm{~kg}$. Its effective volume giving the balloon its buoyancy is $V$. The balloon is floating at an equilibrium height of $100 \mathrm{~m}$. When $N$ number of sandbags are thrown out, the balloon rises to a new equilibrium height close to $150 \mathrm{~m}$ with its volume $V$ remaining unchanged. If the variation of the density of air with height $h$ from the ground is $\rho(h)=\rho_{0} e^{-\frac{h}{h_{0}}}$, where $\rho_{0}=1.25 \mathrm{~kg} \mathrm{~m}^{-3}$ and $h_{0}=6000 \mathrm{~m}$, what is the value of $N$?
phy
AC
10
MCQ(multiple)
In an X-ray tube, electrons emitted from a filament (cathode) carrying current I hit a target (anode) at a distance $d$ from the cathode. The target is kept at a potential $V$ higher than the cathode resulting in emission of continuous and characteristic X-rays. If the filament current $I$ is decreased to $\frac{I}{2}$, the potential difference $V$ is increased to $2 V$, and the separation distance $d$ is reduced to $\frac{d}{2}$, then (A) the cut-off wavelength will reduce to half, and the wavelengths of the characteristic $\mathrm{X}$-rays will remain the same (B) the cut-off wavelength as well as the wavelengths of the characteristic X-rays will remain the same (C) the cut-off wavelength will reduce to half, and the intensities of all the $\mathrm{X}$-rays will decrease (D) the cut-off wavelength will become two times larger, and the intensity of all the X-rays will decrease
phy
AC
11
MCQ(multiple)
Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800 \mathrm{~kg} \mathrm{~m}^{-3}$ and dielectric constant 21. If the angle between the strings remains the same after the immersion, then (A) electric force between the spheres remains unchanged (B) electric force between the spheres reduces (C) mass density of the spheres is $840 \mathrm{~kg} \mathrm{~m}^{-3}$ (D) the tension in the strings holding the spheres remains unchanged
phy
BCD
12
MCQ(multiple)
Starting at time $t=0$ from the origin with speed $1 \mathrm{~ms}^{-1}$, a particle follows a two-dimensional trajectory in the $x-y$ plane so that its coordinates are related by the equation $y=\frac{x^{2}}{2}$. The $x$ and $y$ components of its acceleration are denoted by $a_{x}$ and $a_{y}$, respectively. Then (A) $a_{x}=1 \mathrm{~ms}^{-2}$ implies that when the particle is at the origin, $a_{y}=1 \mathrm{~ms}^{-2}$ (B) $a_{x}=0$ implies $a_{y}=1 \mathrm{~ms}^{-2}$ at all times (C) at $t=0$, the particle's velocity points in the $x$-direction (D) $a_{x}=0$ implies that at $t=1 \mathrm{~s}$, the angle between the particle's velocity and the $x$ axis is $45^{\circ}$
phy
1.3
15
Numeric
Two capacitors with capacitance values $C_{1}=2000 \pm 10 \mathrm{pF}$ and $C_{2}=3000 \pm 15 \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=5.00 \pm 0.02 \mathrm{~V}$. What is the percentage error in the calculation of the energy stored in this combination of capacitors?
phy
0.24
16
Numeric
A cubical solid aluminium (bulk modulus $=-V \frac{d P}{d V}=70 \mathrm{GPa}$ block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a $5 \mathrm{~km}$ deep ocean. Taking the average density of water and the acceleration due to gravity to be $10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$ and $10 \mathrm{~ms}^{-2}$, respectively, what is the change in the edge length of the block in $\mathrm{mm}$?
phy
8.33
18
Numeric
A container with $1 \mathrm{~kg}$ of water in it is kept in sunlight, which causes the water to get warmer than the surroundings. The average energy per unit time per unit area received due to the sunlight is $700 \mathrm{Wm}^{-2}$ and it is absorbed by the water over an effective area of $0.05 \mathrm{~m}^{2}$. Assuming that the heat loss from the water to the surroundings is governed by Newton's law of cooling, what will be the difference (in ${ }^{\circ} \mathrm{C}$ ) in the temperature of water and the surroundings after a long time? (Ignore effect of the container, and take constant for Newton's law of cooling $=0.001 \mathrm{~s}^{-1}$, Heat capacity of water $=4200 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$ )
chem
9
19
Integer
The $1^{\text {st }}, 2^{\text {nd }}$, and the $3^{\text {rd }}$ ionization enthalpies, $I_{1}, I_{2}$, and $I_{3}$, of four atoms with atomic numbers $n, n+$ 1, $n+2$, and $n+3$, where $n<10$, are tabulated below. What is the value of $n$ ? \begin{center} \begin{tabular}{|c|c|c|c|} \hline {$\begin{array}{c}\text { Atomic } \\ \text { number }\end{array}$} & \multicolumn{3}{|c|}{Ionization Enthalpy $(\mathrm{kJ} / \mathrm{mol})$} \\ \cline { 2 - 4 } & $I_{1}$ & $I_{2}$ & $I_{3}$ \\ \hline $n$ & 1681 & 3374 & 6050 \\ \hline $n+1$ & 2081 & 3952 & 6122 \\ \hline $n+2$ & 496 & 4562 & 6910 \\ \hline $n+3$ & 738 & 1451 & 7733 \\ \hline \end{tabular} \end{center}
chem
6
21
Integer
In the chemical reaction between stoichiometric quantities of $\mathrm{KMnO}_{4}$ and $\mathrm{KI}$ in weakly basic solution, what is the number of moles of $\mathrm{I}_{2}$ released for 4 moles of $\mathrm{KMnO}_{4}$ consumed?
chem
4
22
Integer
An acidified solution of potassium chromate was layered with an equal volume of amyl alcohol. When it was shaken after the addition of $1 \mathrm{~mL}$ of $3 \% \mathrm{H}_{2} \mathrm{O}_{2}$, a blue alcohol layer was obtained. The blue color is due to the formation of a chromium (VI) compound ' $\mathbf{X}$ '. What is the number of oxygen atoms bonded to chromium through only single bonds in a molecule of $\mathbf{X}$ ?
chem
6
24
Integer
An organic compound $\left(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{O}_{2}\right)$ rotates plane-polarized light. It produces pink color with neutral $\mathrm{FeCl}_{3}$ solution. What is the total number of all the possible isomers for this compound?
chem
ABCD
27
MCQ(multiple)
Which among the following statement(s) is(are) true for the extraction of aluminium from bauxite? (A) Hydrated $\mathrm{Al}_{2} \mathrm{O}_{3}$ precipitates, when $\mathrm{CO}_{2}$ is bubbled through a solution of sodium aluminate. (B) Addition of $\mathrm{Na}_{3} \mathrm{AlF}_{6}$ lowers the melting point of alumina. (C) $\mathrm{CO}_{2}$ is evolved at the anode during electrolysis. (D) The cathode is a steel vessel with a lining of carbon.
chem
AB
28
MCQ(multiple)
Choose the correct statement(s) among the following. (A) $\mathrm{SnCl}_{2} \cdot 2 \mathrm{H}_{2} \mathrm{O}$ is a reducing agent. (B) $\mathrm{SnO}_{2}$ reacts with $\mathrm{KOH}$ to form $\mathrm{K}_{2}\left[\mathrm{Sn}(\mathrm{OH})_{6}\right]$. (C) A solution of $\mathrm{PbCl}_{2}$ in $\mathrm{HCl}$ contains $\mathrm{Pb}^{2+}$ and $\mathrm{Cl}^{-}$ions. (D) The reaction of $\mathrm{Pb}_{3} \mathrm{O}_{4}$ with hot dilute nitric acid to give $\mathrm{PbO}_{2}$ is a redox reaction.
chem
0.2
32
Numeric
Liquids $\mathbf{A}$ and $\mathbf{B}$ form ideal solution for all compositions of $\mathbf{A}$ and $\mathbf{B}$ at $25^{\circ} \mathrm{C}$. Two such solutions with 0.25 and 0.50 mole fractions of $\mathbf{A}$ have the total vapor pressures of 0.3 and 0.4 bar, respectively. What is the vapor pressure of pure liquid $\mathbf{B}$ in bar?
chem
935
35
Numeric
Tin is obtained from cassiterite by reduction with coke. Use the data given below to determine the minimum temperature (in K) at which the reduction of cassiterite by coke would take place. At $298 \mathrm{~K}: \Delta_{f} H^{0}\left(\mathrm{SnO}_{2}(s)\right)=-581.0 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta_{f} H^{0}\left(\mathrm{CO}_{2}(g)\right)=-394.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$, $S^{0}\left(\mathrm{SnO}_{2}(\mathrm{~s})\right)=56.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}, S^{0}(\mathrm{Sn}(\mathrm{s}))=52.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, $S^{0}(\mathrm{C}(s))=6.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}, S^{0}\left(\mathrm{CO}_{2}(g)\right)=210.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$. Assume that the enthalpies and the entropies are temperature independent.
chem
0.2
36
Numeric
An acidified solution of $0.05 \mathrm{M} \mathrm{Zn}^{2+}$ is saturated with $0.1 \mathrm{M} \mathrm{H}_{2} \mathrm{~S}$. What is the minimum molar concentration (M) of $\mathrm{H}^{+}$required to prevent the precipitation of $\mathrm{ZnS}$ ? Use $K_{\mathrm{sp}}(\mathrm{ZnS})=1.25 \times 10^{-22}$ and overall dissociation constant of $\mathrm{H}_{2} \mathrm{~S}, K_{\mathrm{NET}}=K_{1} K_{2}=1 \times 10^{-21}$.
math
8
37
Integer
For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^{4}-|z|^{4}=4 i z^{2}$, where $i=\sqrt{-1}$. Then what is the minimum possible value of $\left|z_{1}-z_{2}\right|^{2}$, where $z_{1}, z_{2} \in S$ with $\operatorname{Re}\left(z_{1}\right)>0$ and $\operatorname{Re}\left(z_{2}\right)<0$?
math
6
38
Integer
The probability that a missile hits a target successfully is 0.75 . In order to destroy the target completely, at least three successful hits are required. Then what is the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95?
math
2
39
Integer
Let $O$ be the centre of the circle $x^{2}+y^{2}=r^{2}$, where $r>\frac{\sqrt{5}}{2}$. Suppose $P Q$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2 x+4 y=5$. If the centre of the circumcircle of the triangle $O P Q$ lies on the line $x+2 y=4$, then what is the value of $r$?
math
5
40
Integer
The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is 3 and the trace of $A^{3}$ is -18 , then what is the value of the determinant of $A$?
math
Let the functions $f:(-1,1) \rightarrow \mathbb{R}$ and $g:(-1,1) \rightarrow(-1,1)$ be defined by $f(x)=|2 x-1|+|2 x+1| \text { and } g(x)=x-[x] .$ where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f \circ g:(-1,1) \rightarrow \mathbb{R}$ be the composite function defined by $(f \circ g)(x)=f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT differentiable. Then what is the value of $c+d$?
What is the value of the limit $\lim _{x \rightarrow \frac{\pi}{2}} \frac{4 \sqrt{2}(\sin 3 x+\sin x)}{\left(2 \sin 2 x \sin \frac{3 x}{2}+\cos \frac{5 x}{2}\right)-\left(\sqrt{2}+\sqrt{2} \cos 2 x+\cos \frac{3 x}{2}\right)}$?