TS EAMCET 2023 ONLINE 14TH MAY EVENING SHIFT

Match the values of $\frac{d y}{d x}$ at $x=\frac{\pi}{3}$ for the following system of curves in parametric form given in List-I with those of the items in List-II

A physical quantity $X$ is given by $X=\frac{2 k^3 l^2}{m \sqrt{n}}$. The percentage errors in the measurements of $k, l, m$ and $n$ are $1 \%, 2 \%, 3 \%$ and $4 \%$ respectively. The value of $X$ is uncertain by

The displacement-time graphs of two moving particles make angles of $30^{\circ}$ and $45^{\circ}$ with the time axis. The ratio of their velocities is

A bomb of mass 16 kg explodes into two pieces of masses 4 kg and 12 kg . The velocity of the 12 kg mass is $4 \mathrm{~ms}^{-1}$. The kinetic energy of the second piece is

If the radii of circular path of two particles of same mass are in the ratio of $1: 2$, then to have a constant centripetal force, the ratio of their speeds should be

A constant torque acting on a uniform circular wheel changes its angular momentum from $A_0$ to $4 A_0$ in 4 seconds. The magnitude of the torque is

A projectile is given an initial velocity of $\hat{\mathbf{i}}+2 \hat{\mathbf{j}} \mathrm{~ms}^{-1}$. The cartesian equation of its path is ( $x$ and $y$ are in metres and $g=10 \mathrm{~ms}^{-1}$ )

The angle between force $\mathbf{F}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ and displacement $\mathbf{d}=5 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is

A water drop breaks into 64 identical droplets of each surface area $10^{-7} \mathrm{~m}^2$. If the surface tension of water is $0.07 \mathrm{Nm}^{-1}$, the increase in the surface energy in the process is

When a large bubble rises from the bottom of a lake to the surface, the volume of the bubble becomes 5 times its volume at the bottom of the lake. If $H$ is the atmospheric pressure expressed in terms of water column height, then the depth of the lake is

(The temperature of the water in the lake is same at all points)

The displacement of a particle is given by the relation $x=4(\cos \pi t+\sin \pi t)$. The amplitude of the particle is

The ratio of the areas of cross-sections of three wires is $1: 2: 3$ and the ratio of the Young's modulii of their materials is $3: 2: 1$. If the three wires are of same length and same stretching force is applied to the three wires, then the ratio of the elongations of the three wires is

A particle performs uniform circular motion with an angular momentum $L$. If the frequency of the particle's motion is doubled and its kinetic energy is halved, then its angular momentum becomes

A body of mass $m$ is at height $R$ from the surface of the earth where $R$ is the radius of the earth. If the body is taken from here to a height of $3 R$ from the surface of the earth, the increase in the gravitational potential energy of the body is

( $g$ is acceleration due to gravity on the surface of the earth)

A source emitting sound is tied to one end of a string of length 50 cm and is rotated with an angular speed of $40 \mathrm{rad} \mathrm{s}^{-1}$ in the horizontal plane. The ratio of the maximum and minimum frequencies of the sound heard by an observer standing at a distance of 10 m from the fixed end of the string is

(speed of sound in air $=340 \mathrm{~ms}^{-1}$ )

A blacksmith fixes circular iron frame on the wooden wheel of a bullock cart. The diameter of wooden wheel and circular iron frame are 5.012 m and 5 m respectively at $27^{\circ} \mathrm{C}$. The temperature (in ${ }^{\circ} \mathrm{C}$ ) to which iron ring must be heated so as to fit the wooden wheel is

(Coefficient of linear expansion of iron $\left.=1.2 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)$

Steam at $100^{\circ} \mathrm{C}$ is added to 150 g water to increase its temperature from $20^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$. The total mass of the water at $40^{\circ} \mathrm{C}$ is (specific heat capacity of water $=1 \mathrm{cal} \mathrm{g}^{-1}{ }^{\circ} \mathrm{C}^{-1}$ and latent heat of steam $\left.=540 \mathrm{cal} \mathrm{g}^{-1}\right)$

One end of a string is tied to the ceiling of a lift and a load is attached at the bottom end of the string. When the lift is moving upwards with an acceleration of 2.1 $\mathrm{ms}^{-2}$, the speed of the transverse wave at the lower end of the string is $88 \mathrm{~ms}^{-1}$. If the lift moves downwards with an acceleration of $1.9 \mathrm{~ms}^{-2}$, the speed of the transverse wave at the lower end of the string is $\left(g=10 \mathrm{~ms}^{-2}\right)$

If the temperature of a gas is increased from $27^{\circ} \mathrm{C}$ to $159^{\circ} \mathrm{C}$, then the percentage increase in the rms speed of the gas molecules is

Two moles of triatomic gas $\left(\gamma=\frac{4}{3}\right)$ at temperature $327^{\circ} \mathrm{C}$ expands adiabatically such that its volume becomes 8 times its initial volume. Later the temperature of the gas is doubled in an isochoric process. The total work done in the two processes is

(Where, R is universal gas constant)

A hollow spherical shell of radius $r$ has a uniform charge density $\sigma$. It is kept in a cube of edge $3 r$ such that the centres of the cube and the shell coincide. Then the electric flux coming out of one face of a cube is ( $\varepsilon_0=$ permittivity of free space)

The circuit shows two capacitor $A$ and $B$ of capacitances $C$ and $2 C$ respectively.

When they are fully charged, the cell is removed and the capacitors are connected with their plates of opposite polarities touching each other. Then

(i) Charge on $A$ is $\frac{4 C E}{9}$

(ii) Charge on $B$ is zero

(iii) Loss of energy in this process is $\left(C E^2 / 3\right)$

The correct statement/s is/are

The focal lengths of the objective and the eyepiece of a compound microscope are 2 cm and 3 cm respectively and the distance between them is 15 cm . The final image formed by the eyepiece is at infinity. The distances of the object and the image produced by the object from the objective lens are respectively.

A uniform conducting wire $A B$ of length 5 m and resistance $5 \Omega$ is connected as shown in the circuit. If the balancing point is obtained at 3 m from $A$, then the value of $E$ is

In the given circuit, the equivalent resistance between $A$ and $B$ is

A bar magnet of magnetic moment $2 \mathrm{~A}-\mathrm{m}^2$ lies aligned with the direction of a uniform magnetic field of 0.3 T . The amount of work required by an external torque to turn the magnet so as to align its magnetic moment normal to the field direction is

A current $i$ is flowing through a wire of length ' $L$ '. If it is made into a circular loop of one turn, then its magnetic moment is

A coil has a resistance of $30 \Omega$ and an inductive reactance of $20 \Omega$ at 50 Hz frequency. If an AC source of $200 \mathrm{~V}, 100 \mathrm{~Hz}$ is connected across the coil, the current in the coil is

A conducting rod is moving towards right with a velocity $v$ in a uniform magnetic field $B$. If the direction of induced current $i$ is as shown in the figure, then the direction of $B$ is

A current carrying loop is placed in a uniform magnetic field $B$ in different orientations I, II, III and IV as shown in the figure. The correct order of decreasing potential energy is

( $\hat{\mathbf{n}}=$ unit vector normal to the plane of the loop)

A square loop of side $a$ and carrying a current $I$ is suspended from an insulating hanger of a spring balance as shown in figure. The transverse magnetic field $B$ directed into the paper occurs only at the bottom side of the loop. When direction of current in the loop is reversed, the change in the reading of spring balance is

Match the following.

(Take the relative strength of the strongest fundamental forces in nature as one)

The graph given in the figure shows the variation of photo current $(I)$ and the applied voltage ( $V$ ) for two different materials and for two different intensities of the incident radiations. Then the curves which represent the same material are

Match the electromagnetic radiations given in List-I with their uses given in List-II

Half-life of a radioactive substance $A$ is two times the half-life of another radioactive substance $B$. Initially the number of nuclei of $A$ and $B$ are $N_A$ and $N_B$ respectively. After three half-lives of $A$, the number of nuclei of both are equal. Then $N_A / N_B$ is

The ratio of longest wavelengths of the spectral lines in the Lyman and Balmer series of hydrogen spectrum is

Energy released in the fission of a single uranium nucleus is 200 MeV . Then the number of fissions per second to produce 5 mW power is

5 logic gates are connected as shown in the figure. If $A$ and $B$ are the inputs, $Y$ is the output then the truth table of the circuit is

Consider the following statements regarding digital signals

(i) provide a continuous set of values

(ii) represent values as discrete steps

(iii) can utilise binary system

(iv) are in the form of rectangular waves

Then the true statements are

When an $n$-type semiconductor is heated

A 100 W bulb emits light of wavelength

$x \mathop {\rm{A}}\limits^{\rm{o}} $. What is the value of $x$, if the number of photons emitted is $2.0 \times 10^{20} \mathrm{~s}^{-1}$ ?

$$ \left(h=6.63 \times 10^{-34} \mathrm{Js}, 1 \mathrm{~W}=1 \mathrm{Js}^{-1}\right) $$

The ratio of the difference in energy between the first and second Bohr orbits to that between the second and third orbit is

In the reaction I and II the covalencies of Be and Al in $X$ and $Y$ are respectively

I. $\mathrm{Be}(\mathrm{OH})_2+\underset{\text { (Excess) }}{\mathrm{NaOH}} \longrightarrow X$

II. $\mathrm{Al}(\mathrm{OH})_3+\underset{\text { (Excess) }}{\mathrm{NaOH}} \longrightarrow Y$

In the structure of a solid, W atoms are located at the cube corners of the unit cell, O atoms are located at the cube edges and Na atoms at the cube centres. The formula of the compound is

At 298 K the molar solubility of $\mathrm{Cd}(\mathrm{OH})_2$ in $0.1 \mathrm{M} \mathrm{KOH}_{\mathrm{OH}}$ solution is $x \times 10^{-y}$. The values of $x$ and $y$ are respectively.

(At $298 \mathrm{~K}, K_{s p}$ of $\mathrm{Cd}(\mathrm{OH})_2=2.5 \times 10^{-14}$ )

Two containers $A$ and $B$ contain $\mathrm{CO}_2$ gas. Pressure, volume and absolute temperature of the gas in $A$ are 4 times more compared to that in $B$. The mass of the gas in $B$ is $x \mathrm{~g}$, then the mass of the gas in $A$ will be

The sol prepared by Bredig's arc method is $X$ and the charge of sol particles of it is $q . X$ and $q$ are respectively

At $T(\mathrm{~K}) 2$ mole of an ideal gas is allowed to expand reversibly and isothermally from a pressure of 10 atmospheres to 1 atmosphere. The work done (in kJ ) is $\left(R=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right)$

An aqueous solution of a non-volatile solute boils at $100.17^{\circ} \mathrm{C}$. The temperature at which this solution will freeze (in ${ }^{\circ} \mathrm{C}$ ) is

$$ \begin{aligned} & \left(K_b\left(\mathrm{H}_2 \mathrm{O}\right)=0.512^{\circ} \mathrm{C} \mathrm{~kg} \mathrm{~mol}^{-1},\right. \\ & \left.K_f\left(\mathrm{H}_2 \mathrm{O}\right)=1.86^{\circ} \mathrm{C} \mathrm{~kg} \mathrm{~mol}^{-1}\right) \end{aligned} $$

The reduction potential of a half-cell consisting of a Pt electrode immersed in $2.0 \mathrm{M} \mathrm{Fe}^{2+}$ and $0.02 \mathrm{M} \mathrm{Fe}^{3+}$ solution (in V) is

Given : $\left(\frac{2.303 R T}{F}=0.059, E_{\mathrm{Fe}^{3+} \mid \mathrm{Fe}^{2+}}^{\circ}=0.771 \mathrm{~V}\right)$

Tha atomic radius of gallium is less than that of aluminium. This is due to

In which of the following, molecules are arranged in the increasing order of their bond angles?

Assertion (A) First ionisation enthalpy of oxygen is less than that of nitrogen.

Reason (R) Atoms with half-filled or completely filled orbitals are less stable.

The correct option among the following is

Arrange the molecules $\mathrm{B}_2, \mathrm{He}_2, \mathrm{~N}_2$ and $\mathrm{C}_2$ in the increasing order of their bond order values.

A possible mechanism for the gaseous reaction $2 \mathrm{H}_2+2 \mathrm{NO} \longrightarrow 2 \mathrm{H}_2 \mathrm{O}+\mathrm{N}_2$ is

Step 1:2 $\mathrm{NO} \rightleftharpoons \mathrm{N}_2 \mathrm{O}_2$

Step 2 : $\mathrm{N}_2 \mathrm{O}_2+\mathrm{H}_2 \longrightarrow \mathrm{~N}_2 \mathrm{O}+\mathrm{H}_2 \mathrm{O}$ (slow)

Step 3: $\mathrm{N}_2 \mathrm{O}+\mathrm{H}_2 \longrightarrow \mathrm{~N}_2+\mathrm{H}_2 \mathrm{O}$

The rate law for this reaction is

Identify the incorrect reaction from the following.

Identify the incorrect statement about the interhalogen compounds.

Sodium nitrite with hydrochloric acid gives water alongwith two nitrogen oxides. They are

Zeolite is a silicate of two metal ions $X$ and $Y . X$ and $Y$ are respectively

Which of the following has lowest melting point?

Among the hydrides of group 16 elements, the hydride $X$ has lowest boiling point and the hydride $Y$ has highest boiling point. $X$ and $Y$ respectively are

The product formed during thermal decomposition of ammonium dichromate are

The sum of oxidation state and co-ordination number of central metal atom is maximum with respect to which of the following complex?

$$ \mathrm{C}_3 \mathrm{H}_4+\mathrm{H}_2 \mathrm{O} \xrightarrow[333 \mathrm{~K}]{\mathrm{Hg}^{2+} / \mathrm{H}^{+}}[X] \stackrel{\text { Tautomerisation }}{\rightleftharpoons} Y $$

$X$ and $Y$ are respectively

The rate of attack of an electrophile is least when $X$ in the given compound is

The metal which is refined by Mond process is $(X)$, by van Arkel process is $(Y)$ and by zone refining is $(Z), X, Y$ and Z respectively are

In +2 oxidation state, which of the following lanthanoids act as reducing agents?

Arrange the following in the correct order of their acidic strength.

I. $\mathrm{H}_2 \mathrm{C}=\mathrm{CH}_2$

II. $\mathrm{CH} \equiv \mathrm{CH}$

III. $\mathrm{CH}_3-\mathrm{C} \equiv \mathrm{CH}$

IV. $\mathrm{CH}_3-\mathrm{CH}_3$

Addition of HBr to propene in presence of a peroxide takes place contrary to Markownikoff rule. This can be explained by the mechanism involving

Which one of the following gives a foul-smelling substance when treated with chloroform and alcoholic KOH ?

Conversion of $X$ into $Y$ is an example of the reaction

Identify the halogen exchange reaction from the following.

The main reactants involved in Etard reaction are

Identify the major product $Y$ in the given reaction sequence.

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

$$ \begin{array}{llll} \hline & \text { List-I (Polymer type) } & & \text { List-II (Example) } \\ \hline \text { A. } & \text { Fibre } & \text { I. } & \text { Bakelite } \\ \hline \text { B. } & \text { Elastomer } & \text { II. } & \text { Polystyrene } \\ \hline \text { C. } & \text { Thermosetting polymer } & \text { III. } & \text { Neoprene } \\ \hline \text { D. } & \text { Thermosplastic polymer } & \text { IV. } & \text { Dacron } \\ \hline \end{array} $$

The correct answer is

From the following, the correct statements about polysaccharides are

I. Starch is a polymer of $\alpha-\mathrm{D}(+)$-glucose.

II. Amylose component of starch is not soluble in water.

III. Amylose is a branched chain polymer of $\alpha-\mathrm{D}(+)$-glucose.

IV. Cellulose is a straight chain polymer of $\beta-\mathrm{D}(+)$-glucose units.

Which of the following acts as antihistamine?

Arrange the following in the order of decreasing basicity.

I. $\mathrm{RN}=\mathrm{CH} R^1$

II. $\mathrm{RC} \equiv \mathrm{N}$

III. $\mathrm{RNH}_2$

$$ \text { The major product ' } Z \text { ' in the reaction sequence is } $$

If $x=a+b, y=a \alpha+b \beta, z=a \beta+b \alpha$ and $\alpha, \beta$ are the complex cube roots of unity, then $x^3+y^3+z^3=$

If $z=x+i y$ is a complex number such that $z \bar{z}^3+\bar{z} z^3=350$ and $x, y$ are integers, then $|z|=$

The least positive integral value of $n$ such that $\left[\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right]^n=1$ is

If a polynomial $P(x)$ given by

$P(x)=2 x^4+a x^3+b x^2+c x+d$ is such that $P(1)=4$,

$P(2)=7, P(3)=12$ and $P(4)=19$, then $P(5)=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then $(\alpha+\beta)^2+\left(\alpha^2+\beta^2\right)^2+\left(\alpha^3+\beta^3\right)^2+\ldots+\left(\alpha^{12}+\beta^{12}\right)^2=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+x^2+x+1=0$, then match the items of List I with those of List II

Then, the correct match is

If $\alpha, \beta$ are the roots of $x^2+a x+2=0$ and $1 / \alpha, 1 / \beta$ are the roots of $x^2-b x+c=0$, then

$$ \left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\frac{1}{\beta}\right)= $$

The number of odd numbers greater than 600000 that can be formed by using the digits $3,6,7,8,9,0$ without repetition is

If the roots of the equation $k x^3-18 x^2-36 x+8=0$ are in harmonic progression, then $k=$

The sum of all the real values of $x$ satisfying the equation $\left(x^2-7 x+11\right)^{x^2-6 x-7}=1$ is

There are three sections in a question paper, each section containing 4 questions. If a candidate has to answer only 5 questions from this paper without leaving any section, then the number of ways in which a candidate can make the choice of questions is

The number of ways in which 6 men and 4 women can be seated around a table, so that a particular man and a particular woman never sit adjacent to each other is

There are 10 coins in a box out of which 8 are normal and the remaining are with heads on both sides. A coin is chosen at random from the box and tossed 6 times. If it shows heads each time, then the probability that the selected coin has head on both sides is

If a matrix is chosen at random from the set of all $3 \times 3$ non-zero matrices whose entries are the elements of the set $\{-1,0,1\}$, then the probability that the matrix is skew-symmetric is

$$ \text { A random variable } X \text { has the following distribution, } $$

$$ \begin{array}{lllllll} \hline X=x_i & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline P\left(X=x_i\right) & 0.1 & k & 0.2 & 2 k & 3 k & k \\ \hline \end{array} $$

Then, the variance of this distribution is

If $x=\frac{2 \cdot 5}{2!3}+\frac{2 \cdot 5 \cdot 7}{3!3^2}+\frac{2 \cdot 5 \cdot 7 \cdot 9}{4!3^3}+\ldots$, then $x^2+8 x+8=$

A boy throws an unbiased die. Whenever he gets 1 on the die he has a further chance to throw it once again immediately. The probability that the boy gets a score of 7 in this process is

The term independent of $x$ in the expansion of $\left(1-3 x+2 x^3\right)\left(\frac{3 x^2}{2}-\frac{1}{3 x}\right)^9$ is

If $\sum_{r=0}^{20}{ }^{20+r} C_r=\frac{p}{q}{ }^{40} C_{20}$ and GCD of $(p, q)=1$, then $p^2-q^2=$

Let $\mathbf{c}$ be a vector coplanar with the unit vectors $\mathbf{a}, \mathbf{b}$ and let $\mathbf{d}$ be the unit vector perpendicular to $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$. If $[\mathbf{a} \mathbf{b} \mathbf{d}] \mathbf{c}-[\mathbf{a} \mathbf{b} \mathbf{c}] \mathbf{d}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $30^{\circ}$, then $|\mathbf{c}|=$

Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be three vectors such that $\mathbf{a} \cdot \mathbf{a}=\mathbf{b} \cdot \mathbf{b}=\mathbf{c} \cdot \mathbf{c}=5$ and $|\mathbf{a}+\mathbf{b}-\mathbf{c}|^2+|\mathbf{b}+\mathbf{c}-\mathbf{a}|^2+|\mathbf{c}+\mathbf{a}-\mathbf{b}|^2=50$, then $\mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a}=$

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors such that $\mathbf{a}$ is perpendicular to both $\mathbf{b}, \mathbf{c}$ and angle between $\mathbf{b}, \mathbf{c}$ is $2 \pi / 3$, then $|a+3 b-4 c|^2=$

Let $\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ be the position vector of a point $A$. Let $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ be two vectors and $\mathbf{r}$ be a vector passing through the point $A(\mathbf{a})$ and parallel to the vector $\mathbf{b}$. If the projection of $\mathbf{r}$ on $\mathbf{c}$ is $\frac{9}{\sqrt{6}}$, then $|\mathbf{r}|=$

If the circumcenter of the triangle formed by the points $(1,2,3),(3,-1,5)$ and $(4,0,-3)$ is $(\alpha, \beta, \gamma)$, then $|\alpha|+|\beta|=$

If the foot of the perpendicular drawn from the point $(1,0,-2)$ to the plane $\pi$ is $(2,0,-1)$ and the equation of the plane $\pi$ is $a x+b y+c z=2$, then $a^2+b^2+c^2=$

The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively. Later on it was found that one of the observations was taken as 50 in the place of 40 . If the wrong entry is replaced by the correct one, then the sum of the squares of all the observations is

If $\theta$ is the acute angle between the two lines whose direction cosines are connected by the relations $l+m+n=0$ and $2 l m+2 n l-m n=0$, then $\cos \theta=$

If $X_{4 \times 3}, Y_{4 \times 3}$ and $P_{2 \times 3}$ are the matrices, then the order of the matrix $\left[P\left(X^T Y\right)^{-1} P^T\right]^T$ is

If $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right]$ and $\alpha, \beta \in R$ are such that $\alpha A^2-\beta A=2 I$, then $\alpha^2+\beta=$

7. If the system of equations $x+y+z=5, x+2 y+2 z=6$ and $x+3 y+\lambda z=\mu(\lambda, \mu \in R)$ is solvable by Matrix Inversion Method, then

If $\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5}$, then $27 \sec ^6 \alpha+8 \operatorname{cosec}^6 \alpha=$

If $\sin x \cdot \cosh y=\cos \theta$ and $\cos x \cdot \sinh y=\sin \theta$, then $\sin ^2 x+\cosh ^2 y=$

If $\cos A+\cos B+\cos C=0=\sin A+\sin B+\sin C$, then $\cos (A-B)+\cos (B-C)+\cos (C-A)=$

If $f(x)=\sqrt{x}(x \geq 0)$ and $g(x)=1+x^2$, then $(f \circ g)^{\prime}(1)=$

If $\tan \beta=\frac{n \sin \alpha \cos \alpha}{1-n \cos ^2 \alpha}$, then $\tan (\alpha+\beta) \cdot \cot \alpha=$

In $\triangle A B C$, if $a, b, c$ are in arithmetic progression and $A=2 C$, then $b: c=$

If $y=x \sin x$ and $\frac{\frac{d y}{d x}-\frac{y}{x}}{x \frac{d y}{d x}-y}$ at $x=\alpha$ is 1 , then $\alpha=$

$$ \int_0^{\pi / 2} \frac{x \tan x \sec ^2 x}{\tan ^4 x+1} d x= $$

The area (in sq units) of the region bounded by the circle $x^2+y^2=64$, positive $X$-axis and the line $y=\sqrt{3} x$ is

$$ \lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \ldots(2)\right]^{1 / n}= $$

$$ \int_3^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x= $$

If a function $f$ is defined by $f(x)=\frac{\cot ^3 x-\tan x}{\cos (x+\pi / 4)},(x \neq \pi / 4)$, then $\lim _{x \rightarrow \pi / 4} f(x)=$

$$ \begin{array}{r} \lim _{x \rightarrow 0} \frac{2 \tan x+\cos x-1+x}{\sqrt{4 \sin ^2 x+2 \tan x+1}}= \\ -\sqrt{3 \tan ^2 x+\sin x+1} \end{array} $$

If the solution for the differential equation $y^2 d x+\left(x^2-x y-y^2\right) d y=0$ at $(2,1)$ is $x+y=k\left(x y^2-y^3\right)$, then $k=$

If $a$ and $b$ are the arbitrary constants, then the differential equation corresponding to the family of curves given by $y=x[a \cos (\log x)+b \sin (\log x)]$ is

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

A ladder of length 13 m has one end resting against a vertical wall and the other on the ground. If the lower end moves away from the wall at a speed of $2 \mathrm{~m} / \mathrm{min}$ then the speed (in $\mathrm{m} / \mathrm{min}$ ) at which upper end falls when the bottom is 5 m away from the wall is

The maximum volume (in cu. units) of the cylinder which can be inscribed in a sphere of radius 12 units is

An angle between the curves $x^2-y^2=4$ and $x^2+y^2=4 \sqrt{2}$ is

$\int \frac{1+\sqrt{3} \cot x}{1-\sqrt{3} \cot x} d x=$

If $f(x)$ is a real valued function defined by $f(x)=\frac{a x^{10}+b x^8+c x^6+d x^4+e x^2+12 x+15}{x}(x \neq 0)$ and $f(4)=-4$, then $f(-4)=$

If $f(x)$ and $g(x)$ are two real valued functions such that $f(x)=3 x-2$ and $g(x)=x^2+2$, then $[(g \circ f)+(f \circ g)](x)=$

$$ \int \frac{\tan x}{\sec ^2 x\left(1+\sec ^6 x\right)^{\frac{2}{3}}} d x= $$

$$ \int \frac{1}{(x-1)^{\frac{5}{7}}(x+1)^{\frac{9}{7}}} d x= $$

Let $f: R \rightarrow R$ be a function defined by

$$ f(x)=\left\{\begin{array}{cc} x^2-4 x+3, & \text { if } x<2 \\ x-3, & \text { if } x \geq 2 \end{array}\right. $$

Then, the number of real numbers $x$ for which $f(x)=8$ is

The equation of a circle passing through $(-6,3)$ and touching both the coordinates axes is

$$ \begin{aligned} & \text { If } \int \frac{1}{\operatorname{cosec} x+\cos x} d x=\frac{1}{2 \sqrt{3}} \log |f(x)| \\ & -\int \frac{\cos x-\sin x}{2+\sin 2 x} d x+c, \text { then at } x=\frac{\pi}{3},|f(x)|= \end{aligned} $$

If the equation of the circle whose radius is $\sqrt{10}$ and which touches the circle $x^2+y^2+2 x+8 y-23=0$ externally at the point $(1,2)$ is $x^2+y^2+a x+b y+c=0$, then $|a+b+c|=$

The number of common tangents of the circles $x^2+y^2-4=0$ and $x^2+y^2-6 x-8 y-24=0$ is

If the circumcenter of the triangle formed by the points $A(a, 3), B(b, 5)$ and $C(a, b)$ is $(1,1)$, then out of all the possible coordinates of $C$ the sum of the absolute values of the distinct coordinates of $C$ is

The area (in sq units) of the triangle formed by the $x$-axis, the tangent and the normal drawn to the circle $x^2+y^2=10 x$ at the point $(9,3)$ is

If $\mathbf{A B}$ is the focal chord of the parabola $y^2=16 x$ and $A=(1,-4)$, then the equation of the normal to the parabola at the point $B$ is

If the lines $x+y-1=0, k x+2 y+1=0$ and $4 x+2 k y+7=0$ are concurrent, then $k=$

The equation of the circle passing through the points of intersection of the circles $x^2+y^2+6 x+4 y-12=0$, $x^2+y^2-4 x-6 y-12=0$ and having radius $\sqrt{13}$ is

If one of the vertices of an equilateral triangle inscribed in the parabola $y^2=12 x$ coincides with the vertex of the parabola, then the area (in sq units) of that triangle is

If a circle ' $S$ ' passing through the origin and having its centre on the line $x-y=0$ cuts the circle $x^2+y^2-4 x-6 y+10=0$ orthogonally, then the diameter of ' $S$ ' is

If the line $2 x+\sqrt{6} y=2$ touches the hyperbola $x^2-2 y^2=4$, then the coordinates of the point of contact are

If $\alpha, \beta(\alpha>\beta)$ are two values of $k$ such that the equations $2 x+(3-2 k) y+(2 k+1)=0$ and $k x+(k-1) y-4=0$ represents two perpendicular lines, then $\alpha^2+2 \beta=$

The values of $c$ such that the line $y=4 x+c$ touches the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ is

The point of concurrence of all the chords of the curve $3 x^2-y^2-2 x+4 y=0$ which subtend a right angle at the origin is

If an ellipse with its axes as coordinate axes, $2 a$ and $2 b$ as the lengths of its major and minor axes respectively passes through the points $(2,2)$ and $(3,1)$, then $3 a^2+5 b^2=$

When the origin is shifted to the point $P$ by translation of axes, the equation $2 x^2+y^2-4 x+4 y=0$ is transformed to $2 x^2+y^2-8 x+8 y+18=0$. Then, the transformed equation of the straight line $x+2 y+2=0$, if the origin is shifted to the same point $P$ is

If $k=\frac{a+b}{a b}$ is a non-zero constant, then the point which lies on the straight line $\frac{x}{a}+\frac{y}{b}=1$ is

Assertion (A) In $\triangle A B C$, if $r=6, r_2=36, R=15$, then $c^2+a^2=b^2$.

Reason (R) In $\triangle A B C$, if $r: R: r_2=1: 2.5: 6$, then $B=90^{\circ}$. The correct option among the following is

If $z=\frac{3+2 i \cos \theta}{1-2 i \sin \theta}$ is a purely imaginary number, then

$$ \sin ^2 \theta+\cos ^2 3 \theta= $$

If the coefficient of $x^4$ in the expansion of $\frac{x}{(x-1)^2(x-2)}$ is $\frac{m}{n}$ and $|m|,|n|$ are coprimes, then $\sqrt{|m+n|}=$

If $\left|\begin{array}{ccc}(1+\alpha)^2 & (1+2 \alpha)^2 & (1+3 \alpha)^2 \\ (2+\alpha)^2 & (2+2 \alpha)^2 & (2+3 \alpha)^2 \\ (3+\alpha)^2 & (3+2 \alpha)^2 & (3+3 \alpha)^2\end{array}\right|=k$ and $\alpha=-2$, then $k=$

A straight line passing through a fixed point $(-3,4)$ intersects the coordinate axes at $A$ and $B$. If $O$ is the origin and $O A B C$ forms a rectangle, then the locus of $C$ is

The oxidation states of three carbon atoms in carbon suboxide $\left(\mathrm{C}_3 \mathrm{O}_2\right)$ respectively are

If $S$ is the circumcentre, $O$ is the orthocentre and $G$ is the centroid of a $\triangle A B C$, then match the items of the List-I with those of the items of List-II given below.

Then, the correct match is