TS EAMCET 2020 (Online) 10th September Evening Shift

A projectile is fired at an angle of $45^{\circ}$ with the horizontal. Elevation angle of the projectile at its highest point as seen from the point of projection is

In five successive measurements, the mass of a ball is measured to be $2.61 \mathrm{~g}, 2.58 \mathrm{~g}, 2.40 \mathrm{~g}, 2.73 \mathrm{~g}$ and 2.80 g . The absolute error in the measurement is

Initial velocity with which a body is projected is $10 \mathrm{~m} / \mathrm{s}$ from the base of an inclined plane as shown in the given figure. If the angle of projection is $60^{\circ}$ with the horizontal, then the range $R$ is [take, $g=10 \mathrm{~m} / \mathrm{s}^2$ ]

A ball of mass $m=1 \mathrm{~kg}$ is thrown from the top of a building with initial velocity $\mathbf{v}=(20 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}+(24 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{j}}$ at time $t=0$. The change in the potential energy of the ball between $t=0$ and $t=6 \mathrm{~s}$, if the ball does not hit the ground, then (assume, $g=10 \mathrm{~m} \mathrm{~s}^2$ )

A particle is moving along the $Y$-axis. The position of the particle from the origin as a function of time $(t)$ is given as $y(t)=10 t e^{-2 t}$. How far is the particle from the origin when it stops momentarily? ( $y$ is given in units of metre and $t$ is in units of second)

A point $P$ is moving in uniform circular motion with radius 3 m . Let at some instant the acceleration of the point is $\quad \mathbf{a}=(6 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^2$, the position vector is $\mathbf{r}$ and velocity vector is $\mathbf{v}$. Choose the correct statement.

Two cars $A$ and $B$ are moving with speeds $v_A=120 \mathrm{km} / \mathrm{h}$ and $v_B=50 \mathrm{~km} / \mathrm{h}$ respectively in the directions as indicated by the arrow in the figure below. What is the relative speed of the car $B$ with respect to car $A$ ?

A block of mass 3 kg is pressed against a vertical wall by applying a force $F$ at an angle $30^{\circ}$ to the horizontal as shown in the figure. As a result, the block is prevented from falling down. If the coefficient of static friction between the block and wall is $\sqrt{3}$, then the value of $F$ is (use, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A force $\mathbf{F}=(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}) \mathrm{N}$ is applied on an object of mass $M$. What is the work done by this force in moving the object horizontally along the $X$-axis by 3 m ?

A solid sphere and a solid cylinder, each of mass $M$ and radius $R$ are rolling with a linear speed on a flat surface without slipping. Let $L_1$ be magnitude of the angular momentum of the sphere with respect to a fixed point along the path of the sphere. Likewise $L_2$ be the magnitude of angular momentum of the cylinder with respect to the same fixed point along its path. The ratio $L_1 / L_2$ is

A point mass oscillates along the $X$-axis according to the law $x=x_0 \cos \left(\omega t-\frac{\pi}{4}\right)$. If the acceleration of the particle is written as $a=A \cos (\omega t-\delta)$, then

A cubical block of wood having mass of 160 g has a metal piece fastened underneath as shown in the figure. Find the maximum mass of the metal piece which will allow the block to float in water. Specific gravity of wood is 0.8 and that metal is 10 and density of water $=1 \mathrm{~g} / \mathrm{cc}$.

An object of mass 2 kg is hanging from a rope that is wrapped around a pulley of radius 25 cm . The mass of pulley is 2 kg . Find the acceleration of the object. (Assume, pulley to be a solid disk $g=10 \mathrm{~m} / \mathrm{s}^2$ )

The graph correctly represents the variation of acceleration due to gravity $(g)$ with radial distance from the centre of the earth (radius of the earth $=R_e$ ) is

The length of a metal wire is found to be $L_1$ and $L_2$ when the tension of $T_1$ and $T_2$ are applied to it respectively. The natural length of the wire is

A solid of 2 kg mass absorbs 50 kJ when its temperature is raised from $20^{\circ} \mathrm{C}$ to $70^{\circ} \mathrm{C}$. The specific heat capacity of this solid in unit of $\mathrm{J} / \mathrm{kg}{ }^{\circ} \mathrm{C}$ is

Five moles of an ideal gas has pressure $p_0$, volume $V_0$ and temperature $T_0$. The gas is expanded to volume $3 V_0$ along a path, so that the pressure $p$ is changed as function of volume $V$ as $p=p_0\left(V / V_0\right)$. The pressure is then reduced to $p_0$ maintaining the volume constant. The gas undergoes an isobaric compression till the volume and temperature become $V_0$ and $T_0$, respectively. The total work done by the gas during the entire process is

A solid cylinder of radius $r_1=2.5 \mathrm{~cm}$, length $l_1=5.0 \mathrm{~cm}$ and temperature $40^{\circ} \mathrm{C}$ is suspended in an environment of temperature $60^{\circ} \mathrm{C}$. The thermal radiation transfer rate for cylinder is 1.0 W . If the cylinder is stretched until its radius becomes $r_2=0.50 \mathrm{~cm}$, the thermal radiation transfer rate is changed to

The transverse displacement $y(x, t)$ of a wave on a string is given by $y(x, t)=e^{-\left(a x^2+b t^2+2 \sqrt{a b x} t\right)}$. This represents a

How many rotational degrees of freedom does a rigid diatomic molecule have?

A telescope has an objective of focal length 100 cm and an eye-piece of focal length 5 cm . The magnifying power of the telescope is

A convex lens and a concave lens, each with focal length of 4 cm are separated by a distance of 6 cm along their axis. An object is placed 8 cm before the convex lens. The distance between the object and its image is

An infinite non-conducting sheet has a surface charge density $2 \times 10^{-7} \mathrm{C} / \mathrm{m}^2$ on one side. The distance between two equipotential surfaces whose potential differ by 90 V is (assume, $\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \frac{\mathrm{Nm}^2}{\mathrm{C}^2}$ )

The limit of resolution of a telescope is $3.0 \times 10^{-7} \mathrm{rad}$. Assuming that it is used to see the light of wavelength 525 nm from a star, what should be the diameter of the objective?

Find the current in the circuit.

Two negative charges of equal magnitude are located in $x y$-plane as shown below in the figure. The direction of the electric field at point $P$ is

A varying current in a coil changes from 10 A to zero in 1.5 s . If the average emf induced in the coil is 200 V , the self-inductance of the coil is

A cylindrical wire $P$ has resistance $10 \Omega$. A second wire $Q$ has length and diameter half that of $P$. If the material of both the wires is same, then resistance of wire $Q$ is

A particle of charge $q$ and mass $m$ moves in a circular orbit of radius $r$ with angular speed $\omega$. The ratio of the magnitude of its magnetic moment to that of its angular momentum is

An alternating voltage $\varepsilon=30 \sin 200 t$ (in volts) is applied to the circuit below. The amplitude of the current through the circuit is

Two tangent galvanometers $A$ and $B$ have coils of radii 8 cm and 16 cm respectively and having resistance of 8 $\Omega$ each. They are connected in parallel with a cell of emf 4 V and negligible internal resistance. The deflections produced in the tangent galvanometers $A$ and $B$ are $30^{\circ}$ and $60^{\circ}$, respectively. If $A$ has 2 turns, then $B$ must have

Two short magnets of equal dipole moments $M$ are fastened perpendicularly at their centres which lies at origin. Let two magnets lie along $X$-axis and $Y$-axis, respectively.

The magnitude of the magnetic field at a distance $R$ from the centre on the $Y$-axis is $\frac{\mu_0}{4 \pi} \frac{M_0}{R^3}$. Assuming, $R \gg l$ (magnet length), the magnitude of $M$ is

A semiconductor is doped with phosphorous atoms as impurity. The impurity levels created in the semiconductor are close to the

In atomic scale the weakest force in nature is

The wavelength of a spectral line emitted by hydrogen atom in the Balmer series is $\frac{16}{3 R}$

( $R$ is Rydberg constant). What is the value of the principal quantum number of the state from which the transition takes place?

Let $v_1$ and $v_2$ be the maximum velocities of the emitted electrons when the surface of a metal is illuminated with light waves of energy $E_1=4 \mathrm{eV}$ and $E_2=2.5 \mathrm{eV}$,respectively. If the work function of the metal is 2 eV , then the ratio $\frac{v_1}{v_2}$ is

The half-life of a radioactive sample is 5 s . If the initial mass of the sample is 60 g , then the time required to reduce the sample to 7.5 g is

A parallel-plate capacitor with circular plates is being discharged. The radius of the circular plate is 10 cm . A circular loop of radius 20 cm is concentric with the capacitor and located halfway between the plates. If the electric field between the plates is charging at the rate $3.6 \times 10^{12} \mathrm{~V} /(\mathrm{ms})$, then the displacement current through the loop is

$$ \text { (Assume } \frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 / \mathrm{C}^2 \text { ) } $$

Identify the correct observation with respect to the given graphs.

A Zener diode is connected to battery and a load resistance as shown below

The currents $I, I_Z$ and $I_L$ respectively are

Which of the following statements is not true about Thomson's model of atom?

A gas is present at a pressure of 2 atm . What should be the increase in pressure, so that the volume of the gas can be decreased to $\frac{1}{4}$ th of the initial volume at constant temperature?

Assertion (A) Television signals are received through sky-wave propagation.

Reason (R) The ionosphere reflects electromagnetic waves of frequencies in the range ( $3-30) \mathrm{MHz}$.

Choose the correct option.

Which of the following statements is not the result of cathode ray discharge tube experiment with perforated anode?

What will be the $\Delta U$ value, when one mole of oxygen $\left(\mathrm{O}_2\right)$ is going from $-20^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$ at constant volume? (Molar heat capacity for oxygen $\simeq 20.8 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ )

What is the equivalent weight of $\mathrm{KMnO}_4$ in acidic medium? (Molecular weight of $\mathrm{KMnO}_4=158 \mathrm{~g}$ )

An aqueous solution of $98 \%(w / w) \mathrm{H}_2 \mathrm{SO}_4$ has density of $1.02 \mathrm{~g} / \mathrm{cc}$. The molality of the solution is

Match the given ionisation constant values with the corresponding acids.

$$ \begin{array}{lll} \hline \text { A. } & \mathrm{HI} & \text { (i) } 3.2 \times 10^9 \\ \hline \text { B. } & \mathrm{HF} & \text { (ii) } 3.5 \times 10^{-4} \\ \hline \text { C. } & \mathrm{HCl} & \text { (iii) } 1.3 \times 10^6 \\ \hline \text { D. } & \mathrm{HBr} & \text { (iv) } 1.0 \times 10^9 \\ \hline \end{array} $$

$$ \text { The correct match is } $$

Calculate the molar solubility of calcium hydroxide $\mathrm{Ca}(\mathrm{OH})_2$ in 0.10 M NaOH solution. The ionic product of calcium hydroxide is $5.5 \times 10^{-6}$.

With reference to the redox properties of hydrogen peroxide $\left(\mathrm{H}_2 \mathrm{O}_2\right)$, which of these reactions are feasible?

(i) $2 \mathrm{Fe}^{2+} 2 \mathrm{H}^{+}+\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{H}_2 \mathrm{O}$

(ii) $2 \mathrm{MnO}_4^{-}+6 \mathrm{H}^{+}+5 \mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Mn}^{2+}+8 \mathrm{H}_2 \mathrm{O}+5 \mathrm{O}_2$

(iii) $2 \mathrm{Fe}^{2+}+\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{OH}^{-}$

(iv) $2 \mathrm{MnO}_4^{-}+3 \mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{MnO}_2+2 \mathrm{H}_2 \mathrm{O} +3 \mathrm{O}_2+2 \mathrm{OH}^{-} $

The specific rate constant of decomposition of a compound is given by $\ln k=5.0-\frac{12000}{T}$. The activation energy of decomposition for this compound at 300 K is

Based on the quantum numbers, what will be the maximum number of element for sixth period of the periodic table?

The maximum work that can be obtained from the following cells is

$$ X\left|X^{2+}(a q) \| Y^{+}(a q)\right| Y $$

Given, $E_{X^{2+} / X}^{\circ}=-1.7 \mathrm{~V}, E_{Y^{2+} / Y}^{\circ}=0.8 \mathrm{~V}$

Let's assume the $\mathrm{C}_1 \equiv \mathrm{C}_2$ bond is acetylene is along $Z$-axis. Find out the correct combination of atomic orbitals with non-zero overlapping.

In a bcc lattice having the edge length of 200 pm , the cation has the radius of 70 pm . The radius ratio of $\frac{r^{+}}{r^{-}}$is (Given, $\sqrt{2}=1.4, \sqrt{3}=1.7$ and $\sqrt{6}=2.4$ )

Which of the following statements is correct for chemisorption?

Which halide of alkaline earth metals is covalent in nature and can be soluble in organic solvent, such as ethanol?

The molecular formula of metaphosphoric acid is

The final acid product obtained during the synthesis of $\mathrm{H}_2 \mathrm{SO}_4$ by contact process is

An aqueous solution of borax is

Which of the following molecules is not paramagnetic in nature?

Which among the following is/are primary component(s) of the synthesis gas?

Which of the following is air pollutant?

The correct match for complex with its magnetic behaviour in the following is

Among the following series of transition metal ions, the one in which all the metal ions have $3 d^2, 3 p^6$ electronic configuration is

(Atomic number, $\mathrm{Ti}=22, \mathrm{~V}=23, \mathrm{Cr}=24$, $\mathrm{Mn}=25$ )

$$ \text { Match the following. } $$

$$ \begin{array}{llll} \hline & \text { Ores } & & \text { Composition } \\ \hline \text { A. } & \text { Calamine } & \text { (i) } & \mathrm{CuFeS}_2 \\ \hline \text { B. } & \text { Chalcopyrite } & \text { (ii) } & \mathrm{ZnCO}_3 \\ \hline \text { C. } & \text { Bauxite } & \text { (iii) } & \mathrm{Fe}_2 \mathrm{O}_3 \\ \hline \text { D. } & \text { Haematite } & \text { (iv) } & \mathrm{Al}_2 \mathrm{O}_3 \cdot 2 \mathrm{H}_2 \mathrm{O} \\ \hline \end{array} $$

$$ \text { The correct match is } $$

The highest oxidation state observed in first row transition metals is

$$ \text { Hybridisations of carbon-2 in } P \text { and } Q \text { are respectively. } $$

$$ \text { The major product formed in the following is } $$

Which of the following can be used as the test for unsaturation with regard to colour change of reaction?

A benzene derivative did not produce white precipitate with the ammonical silver nitrate solution but decolorised the cold dilute alkaline $\mathrm{KMnO}_4$ solution. The compound is

The major products $P$ and $Q$ formed in the following reactions are respectively.

$$ \text { The major product formed in the following reactions is :} $$

What is the product $R$ in the following reaction sequence?

$$ \text { Choose the correct Zwitter ionic form for aspartic acid. } $$

Which of the following are synthetic rubbers?

(i) Terylene

(ii) Buna-S

(iii) Buna-N

(iv) Neoprene

(v) Polyacrylonitrile

$$ \text { The major product formed in the following reactions is } $$

Assertion (A) If $z$ is a complex number such that $|z| \geq 3$, then the least value of $\left|z+\frac{3}{z}\right|$ is 1 .

Reason (R) $\left|z_1-z_2\right| \leq\left|z_1\right|+\left|z_2\right|$, for any two complex numbers $z_1, z_2$

The correct option among the following is

Paracetamol is

If $z_1=x_1+i y_1, z_2=x_2+i y_2, z_3=x_1+\frac{i x_2}{2}, z_4=2 y_1+i y_2$ are complex numbers such that $\left|z_1\right|=1,\left|z_2\right|=2$ and $\operatorname{Re} \left(\begin{array}{ll}z_1 & z_2\end{array}\right)=0$, then

If $\omega$ is a complex cube root of unity, then $\sum_{x=1}^{10}\left((\omega x+2)\left(\omega^2 x+2\right)-3\right)$

$$ \text { If }\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2020}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2021}=x+i y, $$

then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

$$ \log (9+3 \sqrt{2}(2+\sqrt{5})+4 \sqrt{5})= $$

If $\alpha$ and $\beta$ are the real roots of the equation $\sqrt{\frac{5 x}{x-2}}+\sqrt{\frac{x-2}{5 x}}=\frac{29}{10}$ and $\alpha>\beta$, then $\sqrt{\alpha^2-11^4 \beta^2}=$

$p$ is non-zero real number. If the equation whose roots are the squares of the roots of the equation $x^3-p x^2+p x-1=0$ is identical with the given equation, then $p=$

The minimum value of $\frac{9 \cdot 3^{2 x}+6 \cdot 3^x+4}{9 \cdot 3^{2 x}-6 \cdot 3^x+4}$ is

Numbers between 1 and 10,000 are formed using the digits 2 and 3 only once and the digit 4 twice. If the numbers thus formed are arranged in increasing order and $x, y$ represent the ranks of 4324 and 324 respectively then $x-y=$

If the roots of the equation, $8 x^3+6 p x^2+3 q x-27=0$ are in a geometric progression, then $q^2+9 p^2+6 p q+q / p=$

If $S_n$ is the sum of the first $n$ terms of the series $1^2+2 \times 2^2+3^2+2 \times 4^2+5^2+2 \times 6^2+\ldots \infty$, then, when $n$ is even $S_n=$

If $A_1, A_2, \ldots, A_{15}$ are the events of a random experiment, then which one of the following is true?

In a hospital, on an average if there are 35 births in a weak, then the probability that there will be less than 3 births in a day, is

If $x$ and $y$ represent the number of arrangements of the letters of word ATRAPATRAM such that (i) all A's are together and (ii) no two A's are together respectively, then $x+y$

In an examination there are four Yes/No type of questions. The probability that the answer by the student to a question without guess to be correct is $2 / 3$. The probability that a student guesses a correct answer is $1 / 2$. A student writes the examination either by without guessing answers to all the 4 questions or by guessing answers to all 4 questions. The probability that he attempt the exam by guessing answers to all questions is $3 / 7$. Given that a student answered at least 3 questions correctly, the probability that he answered all the questions without guessing is

If the 9th and 10th terms are the numerically greatest terms in the expansion of $(5 x-6 y)^n$ when $x=2 / 5$ and $y=1 / 2$, then the absolute value of the middle terms of that expansion is

A die is thrown thrice. If getting 1 or 6 in a single throw is considered as success, then the variance of the number of successes is

If the orthocentre of the triangle whose vertices are $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}, 5 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is $x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$, then

If $12 \hat{\mathbf{i}}-12 \hat{\mathbf{j}}-18 \hat{\mathbf{k}},-3 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}-9 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-24 \hat{\mathbf{k}}$ be the position vectors of the vertices $A, B$ and $C$ respectively of $\triangle A B C$, then the position vector of the incentre of $\triangle A B C$ is

$$ 1-\frac{3}{16}+\frac{1 \cdot 4}{1 \cdot 2}\left(\frac{3}{16}\right)^2-\frac{1 \cdot 4 \cdot 7}{1 \cdot 2 \cdot 3}\left(\frac{3}{16}\right)^3+\ldots $$

If the vectors $\mathbf{A B}=p \hat{\mathbf{i}}+q \hat{\mathbf{j}}+r \hat{\mathbf{k}}, \mathbf{A C}=s \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$, $\mathbf{C B}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ from $\triangle A B C$, then the values of $p, q, r$ and $s$ such that the area of that $\triangle A B C$ is $5 \sqrt{6}$ are

For non-coplanar vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, if the point of intersection of the line $\mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{c})$ and the plane $\mathbf{r}=\mathbf{b}+\mathbf{c}+x(\mathbf{a}-\mathbf{b})+y(\mathbf{c}+\mathbf{a})$ is $l \mathbf{a}+m \mathbf{b}+n \mathbf{c}$, then $3 l+4 m+2 n=$

Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be three unit vectors such that $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{1}{\sqrt{2}}(\mathbf{b}+\mathbf{c})$ and $\mathbf{b}$ is not parallel to $\mathbf{c}$. If $\alpha$ and $\beta$ are the angles between $\mathbf{a}, \mathbf{b}$ and $\mathbf{a}, \mathbf{c}$ respectively then $\alpha-\beta=$

The direction cosines of the normal to the plane containing the lines having direction ratios $1,2,1$ and 4,5, -3 are

If the line passing through the points $(a, 2,-4)$ and $(5,3, b)$ crosses the $Z X$-plane at the point $(-a+2 b, 0, a+b)$, then $14 a+7 b$

Let $\Pi$ be a plane containing the points $(0,-5,-1),(1,-2,5),(-3,5,0)$ and $L$ be a line passing through the point $(0,-5,-1)$ and parallel to the vector $\hat{\mathbf{i}}+5 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}$. Then the length of the projection of the unit normal vector to the plane $\Pi$ on the line $L$ is

Let $A=\left[\begin{array}{ccc}1 & 4 & 2 \\ 2 & -1 & 4 \\ -3 & 7 & -6\end{array}\right]$ and $B=\left[b_{i j}\right]_{3 \times 3}$ with $b_{11}=2$, $b_{13}=-2, b_{12}=0$ is such that $A B=\left[\begin{array}{ccc}2 & 14 & -4 \\ 4 & 1 & -8 \\ -6 & 15 & 12\end{array}\right]$, then $|B|+\operatorname{trace}(B)=$

The foot of the perpendicular drawn from the point $(1,1,1)$ to the plane $\pi_1$ is $(1,3,5)$. If $(2,2,-1),(3,4,2)$, $(3,3,0)$ are three points on the plane $\pi_2$, then the angle between the planes $\pi_1$ and $\pi_2$ is

A is a $m \times n$ matrix of rank 4 . If A contains an $m$ th order non singular sub matrix and $A^T A$ is a $7 \times 7$ matrix, then the number of rows of $A$ is

If $\alpha, \beta$ are respectively the mean deviation about the mean and variance of the first five prime numbers, then the ordered pair ( $\alpha, \beta$ )

If $C$ and $D$ are two $n \times n$ non-singular matrices over the set of real number $\mathbf{R}$ such that $C D=-D C$, then $n$ is

Assertion (A) Variance of $4 x_1, 4 x_2, \ldots, 4 x_n$ is 16 times the variance of $x_1, x_2, x_3, \ldots, x_n$

Reason (R) If $y=a x+b$, then variance of $y$ is a $($ variance of $x)+b$

The correct option among the following is

The general solution of the equation $(\sqrt{3}-1) \sin \theta+(\sqrt{3}+1) \cos \theta=2$ is

$$ \begin{aligned} & \text { If } f(x)=\tan ^{-1}\left(\frac{1}{\sin ^2 x+\sin x+1}\right) \\ & \quad+\tan ^{-1}\left(\frac{1}{\sin ^2 x+3 \sin x+3}\right)+\tan ^{-1} \end{aligned} $$

$\left(\frac{1}{\sin ^2 x+5 \sin x+7}\right)+\ldots+$ upto 10 terms, then $f^{\prime}(0)=$

If $\sin ^{-1}\left(\frac{12}{x}\right)+\sin ^{-1}\left(\frac{5}{x}\right)=\frac{\pi}{2}$, then $x=$

In a $\triangle A B C, \frac{\Delta^2}{a^2+b^2+c^2}\left(\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}+\frac{1}{r^2}\right)=$

In a $\triangle A B C,\left(b^2-c^2\right) \cot A+\left(c^2-a^2\right) \cot B=$

If $R: r_1: r=5: 12: 2$, then $r+r_3+r_2-r_1=$

If $A$ does not belong to the first quadrant, $B$ does not belong to the second quadrant, $\sin A=\frac{11}{61}$ and $\cos B=\frac{-7}{25}$, then $A-B$ and $A+B$ lie respectively in the quadrants

Let $a$ be maximum value of $(3 \cos \theta-4 \sin \theta)$ and $\theta \neq \frac{n \pi}{2}$. If $\alpha=a \sin ^2 \theta \cdot \cos ^3 \theta$ and $\beta=a \sin ^3 \theta \cdot \cos ^2 \theta$, then $\sqrt{\frac{\left(\alpha^2+\beta^2\right)^5}{(\alpha \beta)^4}}=$

If $\cos \left(\frac{\pi}{4}-x\right) \cos 2 x+\sin x \sin 2 x \sec x =\cos x \sin 2 x \sec x+\cos \left(\frac{\pi}{4}+x\right) \cos 2 x$, then a possible value of $\sec x$ is

The value of ' $a$ ' for which the function

$f(x)=\left\{\begin{array}{cl}\frac{1-\cos 4 x}{x^2}, & x<0 \\ \frac{a}{\sqrt{x}}, & x=0 \text { is continuous at } x=0, \text { is } \\ \frac{\sqrt{16+\sqrt{x}}-4}{\sqrt{16+}} & \end{array}\right.$

$$ \int_0^{\pi / 2} \frac{d x}{4+5 \sin x} $$

$$\mathop {\lim }\limits_{x \to \infty } \frac{\pi}{2 n}\left[\sin \frac{\pi}{2 n}+\sin \frac{2 \pi}{2 n}+\ldots+\sin \frac{\pi}{2}\right]= $$

If $y=e^{a x}(\cos b x+\sin b x)$ satisfies the equation $\frac{d^2 y}{d x^2}-K \frac{d y}{d x}+L y=0$, then $L+b K=$

$$ \mathop {\lim }\limits_{x \to 0} \frac{1-\cos \left(x^2+\pi(x+2)\right)}{x^2}= $$

If $\alpha$ is such a minimum value for which the inverse of $f(x)=x^2+3 x-3$ exists in $[\alpha, \infty)$ and $g$ is the inverse of the $f$, then at $x=\alpha+\frac{5}{2}, \frac{d g}{d x}$

A tank in the shape of a rectangular parallelopiped has volume 27 cubic meters. This tank is filled with water such that the rate of change of level of the water is thrice the rate of change water quantity falling in the tank, then the height of the tank (in meters) is

The general solution of the differential equation $x \cos \frac{y}{x}(y d x+x d y)=y \sin \frac{y}{x}(x d y-y d x)$ is

Let $f:[2,5] \rightarrow \mathbf{R}$ be a differentiatiable function and $\frac{f(5)}{f(2)}=1$. If there is a $c \in(2,5)$ such that $c f^{\prime}(c)=2 f(c)-2 c^3$, then $f(x)=$

If $\frac{k}{\alpha^3}$ is the length of the sub normal at any point $P(\alpha, y)$ on the curve $x^2-a^2=\frac{x^2 y^2}{a^2}$, then $k=$

The differential equation for which $y=a x^2+b x+c$ is the general solution is

The area (in square units) of the region enclosed between the parabola $y^2=2 x$ and the line $y=4 x-1$

$$ \text { Match the functions of List I with the items of List II. } $$

A function $f: \mathbf{R} \rightarrow \mathbf{R}$ is such that $f(\mathrm{l})=2$ and $f(x+y)=f(x) \cdot f(y) \forall x, y$. The area (in square units) enclosed by the lines $2|x|+5|y| \leq 4$ expressed interms of $f(1), f(2)$ and $f(4)$ is

For $x \in\left(\frac{3 \pi}{4}, \pi\right), \int(\sqrt{1+\sin 2 x}+\sqrt{1-\sin 2 x}) d x=$

If $\frac{2 x+1}{(x-1)^2\left(x^2+1\right)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C x+D}{x^2+1}$, then $A+B+C+D=$

Let $[\cdot]$ denote greatest integer function. If $f(x)=[x]$ and $g(x)=3\left[\frac{x}{3}\right]$, then the set of all real $x$ such that $f(x)=g(x)$ is

If $f(x)=x-\frac{1}{x}, x \neq 0$, then $3 f(x)=$

$$ \begin{aligned} & \text { If } \int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x=A \log (|x \sin x+\cos x|) \\ & +B \frac{f(x)}{(x \tan x+1)}+C \text {, then } f(A+B)= \end{aligned} $$

$$ \text { If } \begin{aligned} & \int x^3(\log x)^2 d x=x^4\left[A(\log x)^2+B(\log x)\right. \\ &+C \log e]+K, \text { then } A+B+C \end{aligned} $$

If $\alpha \neq-4$ and $(2, \alpha)$ is the mid-point of a chord of the circle $x^2+y^2-4 x+8 y+6=0$, then the values of the $y$-intercept of the chord lie in the interval

$$ \begin{aligned} & \text { If } \int \frac{9 x+15}{x^3-6 x-9} d x=A \log |g(x)| \\ & \quad+B \log |f(x)|+C, \text { then } \frac{(A-B) g(4)}{f(-1)}= \end{aligned} $$

$C_1$ and $C_2$ are the external and internal centres of similitude of the circles $x^2+y^2-2 x+4 y+1=0$ and $x^2+y^2+4 x-6 y+12=0$. If the radius of the circle having $C_1 C_2$ as its diameters is $r$, then $\frac{9}{2} r=$

If the origin lies on a diameter of the circle $x^2+y^2-4 x-2 y-4=0$, then the equation of the circle passing through the end points of that diameter and the point $(1,2)$ is

For the parabola $y=\frac{h^3}{3} x^2+\frac{h^2}{2} x-h+\frac{3}{4 h^3}$, if the equation of directrix is $y=k$, then $k: h$

Let $A(2,1)$ be a point and equation of the straight line $L$ be $x-y=0$. Let $a$ and $b$ respectively represent the distances from a variable point $P(\alpha, \beta)$ to $A$ and to the line $L$. If $C$ is distance of the point $A$ from origin such that $a=b c$, then locus of $P$ is

If the circle $x^2+y^2=a^2$ intersects the hyperbola $x y=b^2$ at four points $\left(x_1, y_1\right),\left(x_2, y_2\right),\left(x_3, y_3\right),\left(x_4, y_4\right)$, then $y_1 \quad y_2 \quad y_3 y_4=$

The equation of the common tangent of the parabolas $x^2=108 y$ and $y^2=32 x$ is

If the circle $S_1: x^2+y^2=16$ intersects another circle $S_2$ of radius 5 units such that the common chord is of maximum length and slope $\frac{3}{4}$, then the centre of the circle $S_2$ is

Suppose the circle $S: x^2+y^2+2 g x+2 f y+c=0$ cuts orthogonally the two circles $S^{\prime}: x^2+y^2-4 x-6 y+11=0$ and $S^{\prime \prime}: x^2+y^2-10 x-4 y+21=0$. If the centre of $S=0$ lies on the bisector of the angle between the positive coordinate axes, then $2 g+2 f+c=$

Let $P$ be the pair of lines represented by $2 x^2-5 x y+2 y^2+6 x-3 y=0$ and consider the following independent statements

(i) $\alpha$ is the $x$ coordinate of the point of intersection of the pair of lines $P$.

(ii) $\beta$ is the slope of one of the lines of $P$ passing through origin.

(iii) $\gamma$ is the constant term in the equation of the pair of angular bisectors of $P$.

Then,

Two straight lines $3 x+4 y=5$ and $4 x-3 y=15$ intersect at the point $A$. The equations of the lines passing through $(1,2)$ and intersecting the given lines at $B$ and $C$ such that $A B=A C$ are

The equation of a line making an angle $60^{\circ}$ with the line $x+y-3=0$ and passing through the point $(1,1)$ is

The ellipse having its foci $(0, \pm 1)$ and major axis of length $\sqrt{5}$ is

The point $(4,1)$ undergoes the following transformations successively :

(i) Reflection is the line $x-y=0$

(ii) Shifting through a distance of 2 units along the positive $X$-axis

(iii) Projection on $X$-axis

The coordinates of the point in its final position are

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{2 \sqrt{2}}{3}$ is inscribed in a circle $x^2+y^2=18$ such that the length of its major axis is equal to the diameter of this circle. The locus of the poles of all the tangents of the circle with respect to the ellipse is

The combined equation of the diagonals of the parallelogram formed by the lines

$$ \left(7 x^2-4 x y+8 y^2\right)^2+(4 x-8 y-32)\left(7 x^2-4 x y+8 y^2\right)=0 $$

is

Four boxes $A, B, C$ and $D$ contain 5000, 3000, 2000 and 1000 fuses respectively. The percentages of defective fuses in these boxes are $3 \%, 2 \%, 1 \%$ and $0.5 \%$ respectively. If a fuse selected at random from one of the boxes is found to be defective, then the probability that it has come from box $D$ is

The general solution of the differential equation $(3 y-7 x+7) d x+(7 y-3 x+3) d y=0$ is

Identify the law for which the following statement is true.

"Equal volume of all gases at same temperature and pressure should contain equal number of molecules".

25 mL of 0.1 N NaOH solution neutralises 12.5 mL of HCl solution. The amount of water needed to convert 500 mL of such HCl solution to 0.1 N is

Let $f:[2,5] \rightarrow \mathbf{R}$ be a differentiatiable function and $\frac{f(5)}{f(2)}=1$. If there is a $c \in(2,5)$ such that $c f^{\prime}(c)=2 f(c)-2 c^3$, then $f(x)=$

The general solution of the differential equation

$(3 y-7 x+7) d x+(7 y-3 x+3) d y=0$ is