TS EAMCET 2020 (Online) 10th September Morning Shift

A car travelling at $15 \mathrm{~m} / \mathrm{s}$ overtake another car travelling at $10 \mathrm{~m} / \mathrm{s}$. Assuming, each car is 4 m long. What is the time taken during the overtake?

A projectile is launched from point $A$ of the given landscape with a water body as shown in the diagram. The launching angle is $15^{\circ}$. From the following, identify the right initial velocity of the projectile with which it will fall somewhere in between the points $C$ and $D$. [Assume, $g=10 \mathrm{~m} / \mathrm{s}^2$ ]

Due to an explosion underneath water, a bubble started oscillating. If this oscillation has time period $T$,which is proportional to $p^\alpha S^\beta E^\gamma$, where $p$ is static pressure, $S$ is density of water and $E$ is total energy of explosion. Determine $\alpha, \beta$ and $\gamma$.

When a bullet is fired from a rifle its momentum becomes $20 \mathrm{~kg}-\mathrm{ms}^{-1}$. If the velocity of the bullet is $1000 \mathrm{~ms}^{-1}$, then what is its mass?

A block is between two surfaces as shown in the figure. Find the normal reaction at both surfaces. [Assume, $g=10 \mathrm{~m} / \mathrm{s}^2$ ]

If $0.5 \hat{\mathbf{i}}+0.8 \hat{\mathbf{j}}+c \hat{\mathbf{k}}$ is a unit vector, then $c$ is

When a body is acted upon by a resultant force, then the work done by the resultant force is equal to

If a body moving in a circular path maintains constant speed of $10 \mathrm{~ms}^{-1}$, then which of the following correctly describes the relation between acceleration (a) and radius $(r)$ ?

A circular hole of radius 3 cm is cut out from a uniform circular disc of radius 6 cm . The centre of the hole is at 3 cm , from the centre of the original disc. The distance of centre of gravity of the resulting flat body from the centre of the original disc is

Choose the correct statement.

For a particle executing SHM, determine the ratio of average acceleration of the particle between extreme position and equilibrium position w.r.t. the maximum acceleration.

A bus moving with an uniform speed of $72 \mathrm{~km} / \mathrm{h}$ towards a building blows a horn of frequency 1.7 kHz . If speed of sound in air is $340 \mathrm{~m} / \mathrm{s}$, what will be the frequency of echo heard by bus driver?

A meniscus drop of radius 1 cm is sprayed into $10^6$ droplets of equal size. Calculate the energy expended if surface tension of mercury is $435 \times 10^{-3} \mathrm{~N} / \mathrm{m}$.

A bullet of mass 25 g moves horizontally at a speed of $250 \mathrm{~m} / \mathrm{s}$ is fired into a wooden block of mass 1 kg suspended by a long string. The bullet crosses the block and emerges on the other side. If the centre of the mass of the block rises through a height of 20 cm . The speed of the bullet as it emerges from the block is (take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A slab of side 50 cm and thickness 10 cm is subjected to a shearing force of $10^5 \mathrm{~N}$ on its narrow edge. If the lower edge is riveted to the floor and upper edge is displaced by 0.2 mm , then shear modulus of the material of the slab is

If the image of an object is at the focal point $f$ to the right side of a convex lens, the position of the object on the left of the lens is at

Young's double slit experiment is carried out by using green, red and blue light, one colour at a time. The fringe width recorded are $\beta_{G^{\prime}} \beta_{R^{\prime}} \beta_B$ respectively, then

If a gas has $n$ degrees of freedom, then the ratio of $\frac{C_p}{C_V}$ is

The specific heat of helium at constant volume is 12.6 J $\mathrm{mol}^{-1} \mathrm{~K}^{-1}$. The specific heat of helium at constant pressure in $\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}$ is approximately (assume, the universal gas constant, $R=8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ )

Work done on heating one mole of monoatomic gas adiabatically through $20^{\circ} \mathrm{C}$ is $W$. Then, the work done on heating 6 moles of rigid diatomic gas through the same change in temperature

A composite slab is prepared with two different materials $A$ and $B$. The relation between their coefficient of thermal conductivity and thickness is given as $K_A=\frac{K_B}{2}$ and $X_A=2 X_B$, respectively. If the temperature of faces of $A$ and $B$ are $75^{\circ} \mathrm{C}$ and $50^{\circ} \mathrm{C}$ respectively, what will be the temperature of common surface?

Assume each oil drop consists of a capacitance of $C$. If combine $n$ drops to form a bigger drop, then the capacitance of bigger drop $C^{\prime}$ would be

If the Wheatstone's bridge with four resistors $R_1, R_2$ and $R_3, R_4$ is balanced, then the correct expression is

A conductor of length 100 cm and area of cross-section $1 \mathrm{~mm}^2$ carries a current of 5 A . If the resistivity of the material of the conductor is $3.0 \times 10^{-8} \Omega-\mathrm{m}$, then the electric field across the conductor is

If a proton is moved against the coulomb force of an electric field, then

A moving coil galvanometer of resistance $100 \Omega$ is used as an ammeter using a resistance $0.1 \Omega$. The maximum deflection current in the galvanometer is $100 \mu \mathrm{~A}$. Find the minimum current in the circuit, so that ammeter shows maximum deflection?

On using red light $(\lambda=6600 \mathop {\rm{A}}\limits^{\rm{o}})$ in Young's double slit experiment, 60 fringes are observed in the field of view. If violet light ( $\lambda=4400 \mathop {\rm{A}}\limits^{\rm{o}}$ ) is used, the number of fringes observed will be

A 50 cm long solenoid has winding of 400 turns. What current must pass through it to produce a magnetic field of induction $4 \pi \times 10^{-3} \mathrm{~T}$ at the centre?

What is the amplitude of the electric field in a parallel beam of light intensity $\left(\frac{15}{\pi}\right) \frac{\mathrm{W}}{\mathrm{m}^2}$ ?

$$ \left[\text { Assume }, \frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \frac{\mathrm{Nm}^2}{\mathrm{C}^2}\right] $$

A circular coil of 10 turns and radius 10 cm is placed in a uniform magnetic field of 0.1 T normal to the plane of the coil. If the current in the coil is 5 A , then the magnitude of the torque on the coil is

The ratio of maximum to minimum wavelength in Balmer series of an hydrogenic atom is

If relative permeability of iron is 5500 , then its susceptibility is

In $C R$-circuit the growth of charge on the capacitor is

Alpha rays emitted from a radioactive substance are

Which of the following depicts the output of the full wave rectifier with capacitor filter for the following AC input?

Photons of energy 2.4 eV and wavelength $\lambda$ fall on a metal plate and release photoelectrons with a maximum velocity $v$. By decreasing $\lambda$ by $50 \%$, the maximum velocity of photoelectrons becomes $3 v$. The work function of the material of the metal plate is

The Boolean expression of the circuit given in figure is

The nuclear forces are

A message signal of frequency 10 kHz and peak voltage of 15 V is used to modulate a carrier frequency of 1 MHz and peak voltage of 30 V . Determine the modulation index.

$\Delta H$ and $\Delta S$ for a reaction are $+30.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and 0.06 $\mathrm{kJK}^{-1} \mathrm{~mol}^{-1}$ at 1 atm pressure. The temperature at which free energy change is equal to zero and nature of the reaction below this temperature are

The vapour density of $\mathrm{N}_2 \mathrm{O}_4$ in $\mathrm{N}_2 \mathrm{O}_4 \rightleftharpoons 2 \mathrm{NO}_2$ is 40 . The degree of dissociation is

The degeneracy of the level of hydrogen atoms that contain the energy of $\left(\frac{-R_{\mathrm{H}}}{16}\right)$ is

Root mean square ( rms ) speed of $\mathrm{O}_2$ is $500 \mathrm{~m} / \mathrm{s}$ at a constant temperature. Calculate the rms speed and the average kinetic energy of $\mathrm{H}_2$ at the same temperature. (Consider, $R=8.33 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ )

Which of the following describes an ideal gas?

(i) The volume occupied by a gas molecule is negligible.

(ii) The collision between ideal gases are elastic.

(iii) Particles are very small compared to the distance between each other.

Wavelength of $\mathrm{H}^{+}$ion with kinetic energy 1.65 eV is (mass of proton $=1.6726 \times 10^{-27} \mathrm{~kg}$ )

$\mathrm{KMnO}_4$ oxidises $\mathrm{C}_2 \mathrm{H}_2 \mathrm{O}_4$ to form $\mathrm{CO}_2$. In which of the following, the reaction will be faster?

Copper crystallises in ccp arrangement and accepted value of metal ion radius was found to be $1.14 \mathop {\rm{A}}\limits^{\rm{o}}$. Calculate the density of copper in grams per cubic centimetre. (Atomic weight of copper is 64 , $N_A=6 \times 10^{23}$ )

The freezing point of equimolal aqueous solution will be highest for

The standard electrode potentials of $\mathrm{Ag}^{+} / \mathrm{Ag}$ is +0.80 V and $\mathrm{Cu}^{+} / \mathrm{Cu}$ is +0.34 V . If these electrodes are connected through a salt-bridge, which of the following statements is correct?

What is the equilibrium constant $\left(K_C\right)$ for the given reaction?

$$ \mathrm{N}_2+\mathrm{O}_2 \rightleftharpoons 2 \mathrm{NO} $$

Where the equilibrium concentration of $\mathrm{N}_2, \mathrm{O}_2$ and NO are found to be $4 \times 10^{-3}, 3 \times 10^{-3}$ and $3 \times 10^{-3} \mathrm{M}$ respectively.

The gold numbers of gelatin, haemoglobin and sodium acetate are $5 \times 10^{-3}, 5 \times 10^{-2}$ and $7 \times 10^{-1}$, respectively. The protective actions will be in the order

What is the $\%$ strength of 22.4 volume of $\mathrm{H}_2 \mathrm{O}_2$ solution?

For a zero order reaction, the plot of concentration of reactant vs time is (Hint: Consider the intercept on the concentration axis)

Hard water contains ion of

Which of the following molecules does not exist according to molecular orbital theory?

Assertion (A) $\mathrm{Mg}^{2+}$ and $\mathrm{Al}^{3+}$ are isoelectronic but the magnitude of ionic radius of $\mathrm{Al}^{3+}$ is less than that in $\mathrm{Mg}^{2+}$.

Reason (R) The effective nuclear charge on the outermost electrons in $\mathrm{Al}^{3+}$ is greater than that in $\mathrm{Mg}^{2+}$.The correct option among the following is

The successive ionisation energy values for an element ' $X$ ' are given below :

(i) 1st ionisation energy $=410 \mathrm{~kJ} \mathrm{~mol}^{-1}$

(ii) 2nd ionisation energy $=820 \mathrm{~kJ} \mathrm{~mol}^{-1}$

(iii) 3rd ionisation energy $=1100 \mathrm{~kJ} \mathrm{~mol}^{-1}$

(iv) 4th ionisation energy $=1500 \mathrm{~kJ} \mathrm{~mol}^{-1}$

(v) 5th ionisation energy $=3200 \mathrm{~kJ} \mathrm{~mol}^{-1}$

What is the nature of the bonding in anhydrous $\mathrm{AlCl}_3$ and hydrated $\mathrm{AlCl}_3$ respectively?

When copper metal is treated with cold and dilute nitric acid, it forms

Predict the feasibility of the given reactions in aqueous solution

(i) $\mathrm{Be}(\mathrm{OH})_2+2 \mathrm{OH}^{-} \longrightarrow\left[\mathrm{Be}(\mathrm{OH})_4\right]^{2-}$

(ii) $\mathrm{Be}(\mathrm{OH})_2+2 \mathrm{H}^{+} \longrightarrow\left[\mathrm{Be}(\mathrm{OH})_4\right]^{2+}$

Which of the following elements reacts with water?

Which one of the following is not a colourless compound?

Which of the following statement is not true about interstitial complexes?

Biochemical oxygen demand (BOD) is a measure of organic materials present in water, BOD value less than 5 ppm indicates a water sample to be

Which of the following molecules is colourless?

Which one of the following ores does not contain iron?

Which one of the following methods is suitable to generate aromatic compound(s) from linear aliphatic saturated hydrocarbons with at least six carbon atoms?

Ethyl phenyl acetylene (1-phenyl-1-butyne) on reduction with partially deactiveated palladised charcoal (Lindlar's catalyst) gives

$A^{2+}, B^{2+}$ and $C^{-}$form an ionic complex like $A_{x-2}\left[B(C)_x\right]_2$. If the complex is $75 \%$ dissociated in a solvent with $i=4$, the coordination number of $B$ is

$$ \text { Which of the following gives alcohol/phenol products? } $$

$$ \text { In the following reactions, } P, Q \text { and } R \text { are } $$

$$ \text { The major products } P \text { and } Q \text { in the following reactions are } $$

$$ \text { Following transformation can be accomplished by } $$

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

Which of the following vinyl derivatives is the most reactive towards anionic polymerisation?

Sodium fusion extract of aniline when heated with ferrous sulphate solution and then acidified with concentrated $\mathrm{H}_2 \mathrm{SO}_4$ from which of the following complexes?

Which among the following is an arsenic based antibiotic drug, for which Paul Ehrlich was awarded Noble prize in 1908.

The solutions of the equation $z^2\left(1-z^2\right)=16, z \in \mathbf{C}$, lie on the curve

If $z, \bar{z},-z,-\bar{z}$ forms a rectangle of area $2 \sqrt{3}$ square units, then one such $z$ is

Let $z=x+i y$ be a complex number, $A=\{z /|z| \leq 2\}$ and $B=\{z /(1-i) z+(1+i) \bar{z} \geq 4\}$ Then which one of the following options belongs to $A \cap B$ ?

Amino acids containing heterocyclic ring are

(i) Histidine

(ii) Valine

(iii) Arginine

(iv) Proline

$$ \left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^8+\left(\frac{1+\cos \theta-i \sin \theta}{1+\cos \theta+i \sin \theta}\right)^{16}= $$

Let $S$ be the set of all possible integral values of $\lambda$ in the interval $(-3,7)$ for which the roots of the quadratic equation $\lambda x^2+13 x+7=0$ are all rational numbers. Then the sum of the elements in $S$ is

For the equation $x^4+x^3-4 x^2+x-1=0$ the ratio of the sum of the squares of all the roots to the product of the distinct roots is

Let the greatest common divisor of $m, n$ be 1 . If $\frac{1}{1 \cdot 7}+\frac{1}{7 \cdot 13}+\frac{1}{13 \cdot 19}+\ldots \ldots$. upto 20 terms $=\frac{m}{n}$, then $5 m+2 n=$

$\alpha$ is the maximum value of $1-2 x-5 x^2$ and $\beta$ is the minimum value of $x^2-2 x+r$. If $5 \alpha x^2+\beta x+6>0$ for all real values $x$, then the interval in which $r$ lies is

If $\alpha_1, \beta_1, \gamma_1, \delta_1$ are the roots of the equation $a x^4+b x^3+c x^2+d x+e=0$ and $\alpha_2, \beta_2, \gamma_2, \delta_2$ are the roots of the equation $e x^4+d x^3+c x^2+b x+a=0$ such that $0<\alpha_1<\beta_1<\gamma_1<\delta_1, 0<\alpha_2<\beta_2<\gamma_2<\delta_2$, $\alpha_1-\delta_2=2=\beta_1-\gamma_2 ; \gamma_1-\beta_2=\delta_1-\alpha_2=4$, then $a+b+c+d+e=$

The total number of three digit and five digit integers which can be formed by using the digits $0,1,2,3,4,5$ but using each digit not more than once in each number is

If $A$ and $B$ are events of a sample space such that $P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{2}{3}$, then $P(\bar{A} \cap B)$ is

Let $X$ and $Y$ be two events of a sample space such that $P(X)=\frac{1}{3}, P(X / Y)=\frac{1}{2}$ and $P(Y / X)=\frac{2}{5}$ then

Let $A$ and $B$ be not mutually exclusive events. If $P(A)=\frac{4}{9}, P(A \cap \bar{B})=\frac{3}{7}$ then $P\left(\frac{B}{A}\right)=$

If $20 \%$ of the bolts produced by a machine are defective then the probability that out of 4 bolts chosen at random, less than 2 bolts will be defective, is

At an election a voter may vote for any number of candidates not exceeding the number to be elected. If 4 candidates are to be elected out of the 12 contested in the election and voter votes for at least one candidate, then the number of ways in which a voter can vote is

In a book consisting of 600 pages, there are 60 typographical errors. The probability that a randomly chosen page will contain at most two errors, is

$\mathbf{p}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{q}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$. If the vectors $\mathbf{a}$ and $\mathbf{b}$ are the orthogonal projections of $\mathbf{p}$ on $\mathbf{q}$ and $\mathbf{q}$ on $\mathbf{p}$ respectively, then $\frac{\mathbf{a} \times \mathbf{b}}{\mathbf{a} \cdot \mathbf{b}}=$

For $0 < x < 1$, the expansion of $\left(1+\frac{1}{x}\right)^{\frac{1}{2}}$ is

Let $x \in \mathbf{R}$ be so small that the powers of $x$ beyond two are insignificant and negligibly small. For such $x$, if $(1-x)^3(2+x)^6$ is approximated by $a+b x+c x^2$, then $a+b+c=$

In a quadrilateral $A B C D$, the point $P$ divides $D C$ in the ratio $1: 3$ internally and $Q$ is the mid-point of $A C$. If $\mathbf{A B}+\mathbf{A D}+\mathbf{B C}-2 \mathbf{D C}=\lambda \mathbf{P Q}$, then the value of $\lambda$ is

Let $\mathbf{O A}=\mathbf{a}, \mathbf{O B}=\mathbf{b}$ be two non collinear vectors,

$\mathbf{O P}=x_1 \mathbf{a}+y_1 \mathbf{b}, \mathbf{O Q}=x_2 \mathbf{a}+y_2 \mathbf{b}$ and $\mathbf{A}^{\prime} \mathbf{O}=\mathbf{O A}$,

$\mathbf{B}^{\prime} \mathbf{O}=\mathbf{O B}$. If $x_1=\frac{-3}{4}, x_2=\frac{1}{3}, y_1=\frac{7}{4}, y_2=\frac{5}{3}$, then

Let $\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. The vector $\mathbf{x}$ such that $\mathbf{x} \cdot \mathbf{c}=60$ and perpendicular to both $\mathbf{a}, \mathbf{b}$ is

If the points $A(-1,0,7), B(3,2, t), C(5, k,-2)$ are collinear, then the ratio in which the point $P(t, k-2 t, t+k)$ divides the line segment $B C$ is

A tetrahedron has vertices $O(0,0,0), A(1,2,1)$, $B(2,1,3), C(-1,1,2)$. If $\theta$ is the angle between the faces $O A B$ and $A B C$, then $\cos \theta=$

The position vector of a point $P$ is $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{a}=-\hat{\mathbf{i}}-2 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are two vectors which determine a plane $\pi$. The equation of a line through $P$ normal to $\mathbf{b}$ and lying on the plane $\pi$ is

The shortest distance between the line $\mathbf{r}=2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}})$ and the plane $\mathbf{r} \cdot(\hat{\mathbf{i}}+5 \hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ is

The direction cosines $l, m, n$ of two lines are satisfying $3 l+m+5 n=0$ and $6 m n-2 n l+5 l m=0$. If $\theta$ is the angle between those lines then $|\cos \theta|=$

The solution set of the trigonometric equation $\tan \theta+5 \cot \theta=\sec \theta$ is

All the real values of $p, q$ so that the system of equations

$$ 2 x+p y+6 z=8, x+2 y+q z=5 $$

and $\quad x+y+3 z=4$

may have no solution are

If $A, B$ are two non singular matrices of order $3,|B|=k$, a positive integer, then match the items of list-I with the items of list-II.

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

For the following frequency distribution, the variance is approximately equal to

$$ \begin{array}{cccccc} \hline \begin{array}{c} \text { Class } \\ \text { Interval } \end{array} & 0-5 & 5-10 & 10-15 & 15-20 & 20-25 \\ \hline \text { Frequency } & 4 & 1 & 10 & 3 & 2 \\ \hline \end{array} $$

If the mean of the discrete distribution $8,9,6,5, x, 4$, 6, 5 is 6 , then its standard deviation (nearest to two decimal places) is

$$ \operatorname{cosec}^{-1}\left[\left(\frac{\tan ^2\left(\frac{\alpha-\pi}{4}\right)-1}{\tan ^2\left(\frac{\alpha-\pi}{4}\right)+1}+\cos \frac{\alpha}{2} \cdot \cot 5 \alpha\right) \sec \frac{11 \alpha}{2}\right] $$

If $p$ and $q$ are two distinct real values of $\lambda$ for which the system of equations

$$ \begin{array}{r} (\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0 \\ (\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0 \\ 2 x+(3 \lambda+1) y+3(\lambda-1) z=0 \end{array} $$

has non-zero solution, then $p^2+q^2-p q=$

If $\tan ^{-1} \frac{1}{5}+\frac{1}{2} \sec ^{-1} x+\tan ^{-1} \frac{1}{8}=\frac{\pi}{8}$, then $x^2=$

In $\triangle A B C, \frac{1+\cos C}{r_1+r_2}+\frac{1+\cos A}{r_2+r_3}+\frac{1+\cos B}{r_1+r_3}=$

$$ \begin{aligned} \sin ^4 \frac{\pi}{8}+\cos ^4 \frac{3 \pi}{8}-\sin ^4 \frac{3 \pi}{8} & +\sin ^4 \frac{5 \pi}{8} +\cos ^4 \frac{7 \pi}{8}-\sin ^4 \frac{7 \pi}{8}= \end{aligned} $$

In a triangle $A B C$, if $\cos A \cos B+\sin A \sin B \sin C=1$, then $a: b: c=$

In a $\triangle A B C$ if $\angle A=3 \angle B, C A=9$ and $B C=16$, then the length of $A B$ is

Assertion (A) If $A=15^{\circ}, B=17^{\circ}$ and $C=13^{\circ}$, then $\cot 2 A+\cot 2 B+\cot 2 C=\cot 2 A \cot 2 B \cot 2 C$

Reason (R) In a $\triangle P Q R$,

$$ \tan \frac{P}{2} \tan \frac{Q}{2}+\tan \frac{Q}{2} \tan \frac{R}{2}+\tan \frac{P}{2} \tan \frac{R}{2}=1 $$

The correct option among the following is

Assertion $(\mathrm{A}) \operatorname{cosech}^{-1}(3)=\log \left(\frac{1+\sqrt{10}}{3}\right)$

Reason (R) $e^{\operatorname{cosech}^{-1} x}$ is a root of the quadratic equation $x p^2-2 p-x=0$

The correct option among the following is

let $g(x) \neq 0, g^{\prime}(x) \neq 0, f(x) \neq 0, f^{\prime}(x) \neq 0$. If

$F(x)=f(x) g(x), G(x)=f^{\prime}(x) g^{\prime}(x)$ and

$F^{\prime}(x)=G(x) H(x)=F(x) K(x)$, then $H(x)+K(x)=$

If $\log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots \ldots \infty$ and $\mathop {\lim }\limits_{x \to 0} \frac{\log (1+x)^{1+x}}{x^2}-\frac{1}{x}=k$, then $12 k=$

Let $f(x)$ and $g(x)$ be twice differentiable functions such that $f(x)=x^2+g^{\prime}(1) x+g^{\prime \prime}(2)$ and $g(x)=f(1) x^2+x f^{\prime}(x)+f^{\prime \prime}(x)$. Then $f(x)-g(x)=$

If $f(x)=\left\{\begin{array}{ll}k, & \text { for } x=1 \\ \frac{(9 x-1)(\sqrt{x}-1)}{3 x^2+2 x-5}, & \text { for } x \neq 1\end{array}\right.$ is continuous on $[0, \infty)$, then $k=$

If $y=\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\log \sqrt{1-x^2}$, then $\frac{d y}{d x}=$

$$ \int_{\pi / 4}^{\pi / 2} \frac{3 d x}{1+e^{\sqrt{8} \sin \left(x-\frac{3 \pi}{8}\right)}}= $$

If the area of the region bounded by $y=\cos x, y=\sin x$, $x=\pi / 4$ and $x=\pi$ is bisected by the line $x=a$, then $\sin \left(a+\frac{\pi}{4}\right)=$

$$ \mathop {\lim }\limits_{x \to \infty }\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \ldots \ldots\left(1+\frac{n^2}{n^2}\right)\right]^{1 / n}= $$

If the family of curves $y=a e^{4 x}+b e^{-x}$, where $a, b$ are arbitrary constants represents the general solution of the differential equation

$$ f\left(x, y \frac{d y}{d x}, \frac{d^2 y}{d x^2}\right)=0, \text { then } \frac{d f}{d x}= $$

If the area of a circle increases at the rate of $\frac{1}{\sqrt{\pi}}$ sq. units/sec, then the rate (in units/sec) at which the perimeter of the circle changes, when perimeter is $\sqrt{\pi}$ units, is

If the general solution of the differential equation $(y-x+1) d y-(y+x+2) d x=0$ is $f(x, y, c)=0$, then the value of $c$ such that $f(1,1, c)=0$ is

If the length of the sub tangent at any point $p(x, y)$ on a curve $f(x, y)=0$ is $x+7 y^2$, then $f(x, y)=$

In each of the choices given below, a function and an interval are given. The correct choice having a function and the associated interval for which the Lagrange's mean value theorem is not valid is

Let $f:[0,10] \rightarrow[1,20]$ be a function defined as

$$ f(x)=\left\{\begin{array}{ll} \frac{60-5 x}{3}, & 0 \leq x \leq 6 \\ 10, & 6 \leq x \leq 7 \\ 31-3 x, & 7 \leq x \leq 10 \end{array} \text { then } f\right. \text { is } $$

Let $P(x)$ be a polynomial of degree 3 having extreme value at $x=1$. If $\mathop {\lim }\limits_{x \to 0}\left(\frac{P(x)+4}{x^2}+2\right)=6$, then $\left(\frac{d P}{d x}\right)_{x=\frac{1}{2}}=$

Let $a$ be a fixed positive real number and $n$ be an arbitrary constant. For the curve $y=\frac{x^n}{a^{n-1}}$, if the length of the subnormal at any point $(\alpha, \beta)$ is proportional to $a^2$, then $n=$

The domain of the function, $f(x)=\sqrt{\log _{10}\left(\frac{5 x-x^2}{4}\right)}$ is

If $\frac{4 x^2+5 x^4+7}{\left(x^2+1\right)\left(x^4+x^2+1\right)}=\frac{A x+B}{x^2+1} +\frac{C x^3+D x^2+E x+F}{x^4+x^2+1}$, then $B+2(D+F+E)-C \cdot A=$

$$ \int e^{-3 x}\left(x^2+\sin 4 x\right) d x= $$

The circle $S=0$ cuts the circles

$C_1=x^2+y^2-8 x-2 y+16=0$ and $C_2=x^2+y^2-4 x-4 y-1=0$ orthogonally. If the common chord of $S=0$ and $C_1=0$ is $2 x+13 y-15=0$, then the centre of $S=0$ is

If $\int \frac{2 x^{12}+5 x^9}{\left(1+x^3+x^5\right)^3} d x=\frac{x^m}{l\left(1+x^3+x^5\right)^r}+C$, then $\frac{m-l}{r}=$

If the polar of a point $P$ with respect to a circle of radius $r$ which touches the coordinate axes and lies in the first quadrant is $x+2 y=4 r$, then the point $P$ is

If the circles $x^2+y^2-2 x-2(3+\sqrt{7}) y+8+6 \sqrt{7}=0$ and $x^2+y^2-8 x-6 y+k^2=0, k \in \mathbf{Z}$, have exactly two common tangents, then the number of possible values of $k$ is

The normal at a point on the parabola $y^2=4 x$ passes through $(5,0)$. If there are two more normals to this parabola which pass through $(5,0)$, the centroid of the triangle formed by the feet of these three normals is

The equation of the hyperbola, whose eccentricity is $\sqrt{2}$ and whose foci are 16 units apart, is

If $M$ is the foot of the perpendicular drawn from the origin $O$ on to the variable line $L$, passing through a fixed point $(a, b)$, then the locus of the mid-point of $O M$ is

The equation of the circle passing through the points of intersection of the two orthogonal circles $S_1=x^2+y^2+k x-4 y-1=0$, $S_2=3 x^2+3 y^2-14 x+23 y-15=0$ and passing through the point $(-1,-1)$ is

Consider the parabola $y^2+2 x+2 y-3=0$ and match the items of List-I with those of the List-II.

$$ \begin{array}{llll} \hline & \text { List-I } & & \text { List-II } \\ \hline \text { A. } & 2 x-5=0 & \text { I. } & \text { Vertex } \\ \hline \text { B. } & \left(\frac{3}{2},-1\right) & \text { II. } & \text { Focus } \\ \hline \text { C. } & y+1=0 & \text { III. } & \text { Equation of directrix } \\ \hline \text { D. } & (2,-1) & \text { IV. } & \text { Equation of the axis } \\ \hline & & \text { V. } & \text { Equation of the Latus rectum } \\ \hline \end{array} $$

For $k \in(1, \infty), \int \frac{1}{1+k \cos x} d x=$

Angles made with the $X$-axis by the two lines passing through the point $P(1,2)$ and cutting the line $x+y=4$ at a distance $\frac{\sqrt{6}}{3}$ units from the point $P$ are

When the origin is shifted to the point $\left(\frac{3}{2}, \frac{3}{2}\right)$ by the translation of coordinate axes, then the transformed equation of $32 x^2+8 x y+32 y^2-108 x-108 y+99=0$ is

A line $L_1$ passing through $A(3,4)$ and having slope 1 cuts another line $L_2$ passing through $C$ at $B$, such that $A B=A C$. If the equation of line $B C$ is $2 x-y+4=0$, then the equation of $A C$ is

The straight lines $x+3 y-9=0,4 x+5 y-1=0$, $p x+q y+10=0$ are concurrent, if the line $5 x+6 y+10=0$ passes through the point

If the product of the lengths of the perpendiculars drawn from the foci to the tangent $y=\frac{-3}{4} x+3 \sqrt{2}$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is 9 , then the eccentricity of that ellipse is

The centroid of the triangle formed by the lines $x+y=1$ and $2 y^2-x y-6 x^2=0$ is

The eccentricity of an ellipse passing through $(3 \sqrt{2}, \sqrt{10})$ with foci at $(-4,0)$ and $(4,0)$ is

$$ \int \frac{y^2+\sqrt[3]{y^4}+\sqrt[6]{y^2}}{y\left(1+\sqrt[3]{y^2}\right)} d y= $$

A force acts on a body of mass 10 kg , resulting in its displacement given as $x=\left(\frac{t^3}{25}\right) \mathrm{m}$, where $t$ is the time in seconds. The work done by the force in 5 s is

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

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