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Let the function : (0,𝜋)→ℝ be defined by $ (\theta)=(\sin \theta+\cos \theta)^2+(\sin \theta-\cos \theta)^4 $ Suppose the function $f$ has...

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Let 𝑓:ℝ→ℝ be a differentiable function such that its derivative 𝑓′ is continuous and ƒ(𝜋) = −6. If $...

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Let the function $f:[0,1] \rightarrow \mathbb{R}$ be defined by $\mathrm{f}(\mathrm{x})=\frac{4^{\mathrm{x}}}{4^{\mathrm{x}}+2}$ Then the value of $\mathrm{f}\left(\frac{1}{40}\right)+\mathrm{f}\left(\frac{2}{40}\right)+\mathrm{f}\left(\frac{3}{40}\right)+\ldots \ldots \ldots+\mathrm{f}\left(\frac{39}{40}\right)-\mathrm{f}\left(\frac{1}{2}\right)$ is

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Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on...

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In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a...

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An engineer is required to visit a factory for exactly four days during the first 15 days of every month...

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For non-negative integers s and r, let $...

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Let a and b be positive real numbers. Suppose $\overrightarrow{\mathrm{PQ}}=a \hat{\mathrm{i}}+b \hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{PS}}=a \hat{\mathrm{i}}-b \hat{\mathrm{j}}$ are adjacent sides of...

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Let $\alpha, \beta, \gamma, \delta$ be real numbers such that $\alpha^2+\beta^2+\gamma^2 \neq 0$ and $\alpha+\gamma=1$. Suppose the point $(3,2,-1)$ is...

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Let 𝒇:ℝ→ℝ and 𝒈:ℝ→ℝ be functions satisfying 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦)+𝑓(𝑥)𝑓(𝑦) and 𝑓(𝑥)=𝑥𝑔(𝑥) for all 𝑥,𝑦∈ℝ. If $\lim _{x \rightarrow 0} g(x)=1$, then...

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Let b be a nonzero real number. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $(0)=1$. If...

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The value of the limit $...

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Let the functions : (−1,1)→ℝ and 𝑔 : (−1,1)→(−1,1) be defined by ƒ(𝑥) = |2𝑥−1|+|2𝑥+1| and 𝑔(𝑥) = 𝑥−[𝑥], where...

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The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a...

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Let O be the centre of the circle $\mathrm{x}^2+\mathrm{y}^2=\mathrm{r}^2$, where $\mathrm{r}>\frac{\sqrt{5}}{2}$. Suppose $P Q$ is a chord of this circle...

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The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least...

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For a complex number z, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all...

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Let e denote the base of the natural logarithm. The value of the real number a for which the right...

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For a polynomial $\mathrm{g}(\mathrm{x})$ with real coefficient, let $\mathrm{m}_{\mathrm{g}}$ denote the number of distinct real roots of $\mathrm{g}(\mathrm{x})$. Suppose S...

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A container with 1 kg of water in it is kept in sunlight, which causes the water to get warmer...

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In a triangle PQR , let $\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{QR}}, \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{RP}}$ and $\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{PQ}}$. If $|\overrightarrow{\mathrm{a}}|=3,|\overrightarrow{\mathrm{~b}}|=4$ and $\frac{\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{b}})}{\overrightarrow{\mathrm{c}} \cdot(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})}=\frac{|\overrightarrow{\mathrm{a}}|}{|\overrightarrow{\mathrm{a}}|+|\overrightarrow{\mathrm{b}}|}$, then the value of...

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Let f : [0, 2] -> R be the function defined by $...

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The inductors of two $L R$ circuits are placed next to each other, as shown in the figure. The values...

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A cubical solid aluminium (bulk modulus $=-V \frac{\mathrm{dP}}{\mathrm{dV}}=70 \mathrm{GPa}$ ) block has an edge length of 1 m on the...

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Let $a_1, a_2, a_3, \ldots$. be a sequence of positive integers in arithmetic progression with common difference 2 . Also,...

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Two capacitors with capacitance values $\mathrm{C}_1=2000 \pm 10 \mathrm{pF}$ and $C_2=3000 \pm 15 \mathrm{pF}$ are connected in series. The voltage...

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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 $y_1+y_2+y_3=9$....

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In the balanced condition, the values of the resistances of the four arms of a Wheatstone bridge are shown in...

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A spherical bubble inside water has radius $R$. Take the pressure inside the bubble and the water pressure to be...

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Starting at time $t=0$ from the origin with speed $1 \mathrm{~ms}^{-1}$, a particle follows a two-dimensional trajectory in the $x-y$...

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Let $\mathrm{L}_1$ and $\mathrm{L}_2$ be the following straight line. $\mathbf{L}_1: \frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}$ and $\mathrm{L}_2: \frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}$ Suppose the straight line $\mathrm{L}: \frac{\mathrm{x}-\alpha}{l}=\frac{\mathrm{y}-1}{\mathrm{~m}}=\frac{\mathrm{z}-\gamma}{-2}$...

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Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two...

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Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are lengths of the sides of...

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Let S be the set of all complex numbers z satisfying $\left|\mathrm{z}^2+\mathrm{z}+1\right|=1$. Then which of the following statements is/are TRUE...

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Let M be a 3 X 3 invertible matrix with real entries and let I denote the $3 \times 3$...

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Let the function $f: R \rightarrow R$ be defined by $f(x)=x^3-x^2+(x-1) \sin x$ and let $g: R \rightarrow R$ be...

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In an X-ray tube, electrons emitted from a filament (cathode) carrying current I hit a target (anode) at a distance...

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A rod of mass $m$ and length $L$, pivoted at one of its ends, is hanging vertically. A bullet of...

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Consider all rectangles lying in the region $...

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Let C1 and C2 be two biased coins such that the probabilities of getting head in a single toss are...

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A student skates up a ramp that makes an angle $30^{\circ}$ with the horizontal. $\mathrm{He} /$ she starts (as shown...

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A beaker of radius $r$ is filled with water (refractive index $\frac{4}{3}$ ) up to a height $H$ as shown...

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A thermally isolated cylindrical closed vessel of height 8 m is kept vertically. It is divided into two equal parts...

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A point charge q of mass m is suspended vertically by a string of length $l$. A point dipole of...

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Let $\mathrm{a}, \mathrm{b}$ and $\lambda$ be positive real numbers. Suppose P is an end point of the latus rectum of...

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Let the functions $f: \mathrm{R} \rightarrow \mathrm{R}$ and $\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be defined by $...

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A hot air balloon is carrying some passengers, and a few sandbags of mass 1 kg each so that its...

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Two large circular discs separated by a distance of 0.01 m are connected to a battery via a switch as...

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If the function $f: \mathrm{R} \rightarrow \mathrm{R}$ is defined by $\mathrm{f}(\mathrm{x})=|\mathrm{x}|(\mathrm{x}-\sin \mathrm{x})$, then which of the following statements is TRUE?

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A train with cross-sectional area $\mathrm{S}_{\mathrm{t}}$ is moving with speed $v_t$ inside a long tunnel of cross-sectional area $\mathrm{S}_0\left(\mathrm{~S}_0=4 \mathrm{~S}_{\mathrm{t}}\right)$....

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Suppose $a$, $b$ denote the distinct real roots of the quadratic polynomial $x^2+20 x-2020$ and suppose $c, d$ denote the...

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A large square container with thin transparent vertical walls and filled with water (refractive index $\frac{4}{3}$ ) is kept on...

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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...

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A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved...

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Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally...

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One end of a spring of negligible unstretched length and spring constant k is fixed at the origin $(0,0)$. A...

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When water is filled carefully in a glass, one can fill it to a height $h$ above the rim of...

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Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the...

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As shown schematically in the figure, two vessels contain water solutions (at temperature $T$ ) of potassium permanganate $\left(\mathrm{KMnO}_4\right)$ of...

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Shown in the figure is a semicircular metallic strip that has thickness $t$ and resistivity $\rho$. Its inner radius is...

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A uniform electric field, $\vec{E}=-400 \sqrt{3} \hat{y} \mathrm{NC}^{-1}$ is applied in a region. A charged particle of mass m carrying...

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Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of...

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The filament of a light bulb has surface area $64 \mathrm{~mm}^2$. The filament can be considered as a black body...

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A particle of mass m moves in circular orbits with potential energy $\mathrm{V}(\mathrm{r})=F r$, where F is a positive constant...

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An open-ended U-tube of uniform cross-sectional area contains water (density $10^3 \mathrm{~kg} \mathrm{~m}^{-3}$ ). Initially the water level stands at...

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A parallel beam of light strikes a piece of transparent glass having cross section as shown in the figure below....

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Therefore relative displacement (with respect to centre of roller) is $(75-50) \mathrm{cm}=25 \mathrm{~cm}$. A circular coil of radius R and...

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A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left)....

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A light disc made of aluminium (a nonmagnetic material) is kept horizontally and is free to rotate about its axis...

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A football of radius R is kept on a hole of radius $\mathrm{r}(r < R)$ made on a plank kept...

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A solution of 0.1 M weak base (B) is titrated with 0.1 M of a strong acid (HA). The variation...

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Choose the correct statement(s) among the following :

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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...

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Tin is obtained from cassiterite by reduction with coke. Use the data given below to determine the minimum temperature (in...

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Consider the reaction sequence from $\mathbf{P}$ to $\mathbf{Q}$ shown below. The overall yield of the major product $\mathbf{Q}$ from $\mathbf{P}$...

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The figure below is the plot of potential energy versus internuclear distance $(d)$ of $\mathrm{H}_2$ molecule in the electronic ground...

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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...

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Consider the following transformations of a compound P.

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Consider the following four compounds I, II, III, and IV.

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Choose the correct statement(s) among the following.

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Which among the following statement(s) is(are) true for the extraction of aluminium from bauxite?

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Which of the following plots is(are) correct for the given reaction? $\left([\mathrm{P}]_0\right.$ is the initial concentration of $\left.\mathbf{P}\right)$

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In an experiment, $m$ grams of a compound $\mathbf{X}$ (gas/liquid/solid) taken in a container is loaded in a balance as...

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In the following reaction, compound Q is obtained from compound P via an ionic intermediate What is the degree of...

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${ }^{238}$ Uis known to undergo radioactive decay to form ${ }_{82}^{206} \mathrm{~Pb}$ by emitting alpha and beta particles. A...

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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...

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The structure of a peptide is given below If the absolute values of the net charge of the peptide at...

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An acidified solution of potassium chromate was layered with an equal volume of amyl alcohol. When it was shaken after...

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Aluminium reacts with sulfuric acid to form aluminium sulfate and hydrogen. What is the volume of hydrogen gas in liters...

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In the chemical reaction between stoichiometric quantities of $\mathrm{KMnO}_4$ and KI in weakly basic solution, what is the number of...

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Consider the following compounds in the liquid form : $...

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The $1^{\text {st }}, 2^{\text {nd }}$ and the $3^{\text {rd }}$ ionization enthalpies $\mathrm{I}_1, \mathrm{I}_2$ and $\mathrm{I}_3$, of four...

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Consider a 70% efficient hydrogen-oxygen fuel cell working under standard conditions at 1 bar and 298 K. Its cell reaction...

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Consider the reaction $\mathbf{A} \rightleftharpoons \mathbf{B}$ at 1000 K . At time $\mathrm{t}^{\prime}$, the temperature of the system was increased...

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5.00 mL of 0.10 M oxalic acid solution taken in a conical flask is titrated against NaOH from a burette...

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The cubic unit cell structure of a compound containing cation M and anion X is shown below. When compared to...

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With respect to hypochlorite, chlorate and perchlorate ions, choose the correct statement(s).

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In the reaction scheme shown below Q, R and S are the major products The correct structure of

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With respect to the compounds I-V, choose the correct statement(s).

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In thermodynamics the P-V work done is given by $ \mathrm{w}=-\int \mathrm{dVP}_{\mathrm{ext}} $ For a system undergoing a particular process,...

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The Fischer projection of D-erythrose is shown below. D-Erythrose and its isomers are listed as P, Q, R, and S...

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Which one of the following structures has the IUPAC name 3-ethynyl-2-hydroxy-4-methylhex-3-en-5-ynoic acid ?

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Newman projections P, Q, R and S are shown below : Which one of the following options represents identical molecules...

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A colorless aqueous solution contains nitrates of two metals, $\mathbf{X}$ and $\mathbf{Y}$. When it was added to an aqueous solution...

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Which of the following liberates $\mathrm{O}_2$ upon hydrolysis?

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If the distribution of molecular speeds of a gas is as per the figure shown below, then the ratio of...

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