TG EAPCET 2024 (Online) 9th May Morning Shift | TS EAMCET Year Wise Previous Years Questions

Chemistry

Identify the pair of species having same energy from the following. (The number given in the bracket corresponds to principal quantum number ( $n$ ) in which the electron is present.)

Which one of the following corresponds to the wavelength of line spectrum of H atom in its Balmer series ? ( $R=$ Rydberg constant)

Identify the pair of element in which the number of $s$-electrons to $p$-electrons ratio is $2: 3$

Which of the following has the least electron gain enthalpy?

According to Fajan's rules, which of the following is not correct about covalent character?

Consider the following pairs.

$$ \begin{array}{l|l|l} \hline & \text { Order } & \text { Property } \\ \hline \text { (A) } & \mathrm{NO}_2>\mathrm{O}_3>\mathrm{H}_2 \mathrm{O} & \text { Bond angle } \\ \hline \text { (B) } & \mathrm{HF}>\mathrm{H}_2 \mathrm{O}>\mathrm{NH}_3 & \text { Dipole moment } \\ \hline \text { (C) } & \mathrm{I}_2>\mathrm{F}_2>\mathrm{N}_2 & \text { Bond length } \\ \hline \end{array} $$

Which of the above pairs are correctly matched?

An open vessel containing air was heated from $27^{\circ} \mathrm{C}$ to $727^{\circ} \mathrm{C}$. Some air was expelled. What is the fraction of air remaining in the vessel ? (Assume air as an ideal gas.)

12 g of an element reacts with 32 g of oxygen. What is the equivalent weight of the element ?

The standard enthalpy of formation $\left(\Delta_f H^{\varphi}\right)$ of ammonia is $-46.2 \mathrm{~kJ} \mathrm{~mol}^{-1}$, What is the $\Delta_{,} H^{\ominus}$ of the following reaction? $$ \mathrm{N}_2(g)+3 \mathrm{H}_2(g) \longrightarrow 2 \mathrm{NH}_3(g) $$

At $T(\mathrm{~K}), K_c$ for the reaction, $A O_2(g)+B \mathrm{O}_2(g) \rightleftharpoons A \mathrm{O}_3(g)+B \mathrm{O}(g)$ is 16 . One mole each of reactants and products are taken in a IL flask and heated to $T(\mathrm{~K})$, and equilibrium is established. What is the equilibrium concentration of $B O$ ( in $\mathrm{mol} \mathrm{L}^{-1}$ )?

Observe the following reactions.

$$ \begin{aligned} & \text { I. } \mathrm{H}_2 \mathrm{O}(l)+2 \mathrm{Na}(s) \longrightarrow 2 \mathrm{NaOH}(a q)+\mathrm{H}_2(g) \\\\ & \text { II. } 2 \mathrm{H}_2 \mathrm{O}(l)+2 \mathrm{~F}_2(g) \longrightarrow 4 \mathrm{H}^{+}(a q)+4 \mathrm{~F}^{-}(a q)+\mathrm{O}_2(g) \end{aligned} $$

What is the correct stability order of $\mathrm{KO}_2, \mathrm{RbO}_2, \mathrm{CsO}_2$ ?

Assertion (A) $\mathrm{MgO}, \mathrm{CaO}, \mathrm{SrO}$ and BaO are insoluble in water.

Reason ( R ) In aqueous medium the basic strength of $\mathrm{MgO}, \mathrm{CaO}, \mathrm{SrO}$ and BaO increases with increase in the atomic number of metal.

The correct option among the following is

Identify the element for which +1 oxidation state is more stable than +3 oxidation state.

Observe the oxides $\mathrm{CO}, \mathrm{B}_2 \mathrm{O}_3, \mathrm{SiO}_2, \mathrm{CO}_2, \mathrm{Al}_2 \mathrm{O}_3$. $\mathrm{PbO}_2, \mathrm{Tl}_2 \mathrm{O}_3$. The number of acidic oxides in the list is

The common components of photochemical smog are

The electron displacement effect observed in the given structures is known as

An alkene $X\left(\mathrm{C}_4 \mathrm{H}_8\right)$ exhibits geometrical isomerism. Oxidation of $A$ with $\mathrm{KMnO}_4 / \mathrm{H}^{+}$gave $Y$. On heating sodium salt of $Y$ with a mixture of NaOH and CaO gave $Z$. What is $Z$ ?

The number of activating and deactivating groups of the following are respectively

$$ \begin{aligned} & -\mathrm{OCH}_2 \mathrm{CH}_3,-\mathrm{COCH}_3,-\mathrm{NHCOCH}_3, \\ & -\mathrm{COOCH}_3, \mathrm{SO}_3 \mathrm{H} \end{aligned} $$

$X$ and $Z$ respectively in the following reaction sequence are $\mathrm{C}_3 \mathrm{H}_6 \xrightarrow{X} Y \xrightarrow[\mathrm{AlCl}_3]{\mathrm{C}_6 \mathrm{H}_6} Z$ (Major projuct)

The molecular formula of a compound is $A B_2 \mathrm{O}_4$. Atoms of $O$ form ccp lattice. Atoms of $A$ (cation) occupy $\frac{1}{8}$ th of tetrahedral voids. Atoms of $B$ (cation) occupy a fraction of octahedral voids. What is the fraction of vacant octahedral voids?

Distilled water boils at 373.15 K and freezes at 273.15 K . A solution of glucose in distilled water boils at 373.202 K . What is the freezing point (in K ) of the same solution? (For water, $K_b=0.52 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$, $K_f=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$ )

Identify the correct statements from the following

(A) At 298 K , the potential of hydrogen electrotle placed in a solution of $\mathrm{pH}=10$, is -0.59 V

(B) The limiting molar conductivity of $\mathrm{Ca}^{2+}$ and $\mathrm{Cl}^{-}$is 119 and $76 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$ respectively. The limiting molar conductivity of $\mathrm{CaCl}_2$ is $195 \mathrm{Scm}^2 \mathrm{~mol}^{-1}$

(C) The correct relationship between $K_C$ and $E_{\text {cell }}^{\ominus}$ is $$ E_{\text {cell }}^\theta=\frac{2303 R T}{n F} \log K_C $$

For a first order reaction, a plot of $\ln k\left(Y\right.$-axis) and $\frac{1}{T}$ $(X$-axis) gave the straight line with slope equal to $-10^3 \mathrm{~K}$ and intercept equal to 2.303 ( on $Y$-axis). What is the activation energy ( $E_a$ in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) of the reaction? (Given, $R=8314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ )

Adsorption of a gas ( $A$ ) on an adsorbent follows Freundlich adsorption isotherm. The slope and intercept (on $Y$-axis) of the isotherm are $0: 5$ and 1.0 respectively. What is the value of $\frac{x}{m}$, when the pressure of the gas $(A)$ is 100 atm ?

A low boiling point metal contains high boiling point metal as impurity. The correct refining method is

Which of the following when subjected to thermal decomposition will liberate dinitrogen ?

(i) Sodium nitrate

(ii) Ammonium dichromate

(iii) Barium azide

"Observe the following reaction. This reaction represents

$$ 4 \mathrm{HCl}+\mathrm{O}_2 \xrightarrow[723 \mathrm{~K}]{\mathrm{CuCl}_2} 2 \mathrm{Cl}_2+2 \mathrm{H}_2 \mathrm{O} $$"

Identify the set which is not correctly matched in the following.

Identify the correct statements from the following.

(i) Ti (IV) is more stable than Ti (III) and Ti (II).

(ii) Among $3 d$-series elements (From $Z=22$ to 29). Only copper has positive reduction potential $\left(M^{2+} / M\right)$.

(iii) Both Sc and Zn exhibit +1 oxidation state.

The molecular formula of a coordinate complex is $\mathrm{CoH}_{12} \mathrm{O}_6 \mathrm{Cl}_3$. When one mole of this aqueous solution of complex is reacted with excess of aqueous $\mathrm{AgNO}_3$ solution, three moles of AgCl was formed. What is the correct formula of the complex ?

Match the following.

List-I (Monomer/s)	List-II (Name of polymer)
A. CF₂$=$CF₂	I. Neoprene
B. NH₂(CH₂)₆NH₂, HO₂C(CH₂)₄CO₂H	II. Bakelite
C. C₆H₅OH₃ HCHO	III. Teflon
D. CH₂$=$CH(Cl)—CH$=$CH₂	IV. Nylon 6,6

The correct answer is

The functional groups involved in the conversion of glucose to gluconic acid and gluconic acid to saccharic acid respectively are

Among the following the incorrect statement about chloramphenicol is

A halogen compound $X\left(\mathrm{C}_4 \mathrm{H}_9 \mathrm{Br}\right)$ on hydrolysis gave alcohol $Y$. The alcohol $Y$ undergoes dehydration with $20 \% \mathrm{H}_3 \mathrm{PO}_4$ at 358 K . What is $X$ ?

An alcohol $X\left(\mathrm{C}_3 \mathrm{H}_{12} \mathrm{O}\right)$ when reacted with conc. HCl and anhydrous $\mathrm{ZnCl}_2$ produces turbidity instantly. The alcohol $X$ can be prepared from which of the following reactions?

Assertion (A) : Chlorobenzene is not formed in the reaction of phenol with thionyl chloride.

Reason (R) : In phenol, carbon - oxygen bond has partial double bond character.

The correct answer is :

The $\mathrm{p} K_{\mathrm{a}}$ values of $X, Y, Z$ respectively are $8.3,7.1,10.2$. What are $X, Y, Z$ ?

The reagents/ chemicals $X$ and $Y$ that convert cyanobenzene to Schiff's base are

The correct statement(s) of the following is/are (A) Aniline forms a stable benzene diazonium chloride at 285 K . (B) N - phenylethanamide is less reactive towards nitration than aniline. (C) $p-\mathrm{CH}_3 \mathrm{C}_6 \mathrm{H}_4 \mathrm{COCl}$ is Hinsberg reagent.

Mathematics

If $f(x)=\frac{2 x-3}{3 x-2}$ and $f_n(x)=($ fofofo .......n times) $(x)$, then $f_{32}(x)=$

The domain of the real valued function $f(x)=\sqrt{\cos (\sin x)}+\cos ^{-1}\left(\frac{1+x^2}{2 x}\right)$ is

For $n \in N$ the largest positive integer that divides $81^n+20 n-1$ is $k$. If $S$ is the sum of all positive divisors of $k$, then $S-k=$

$A, B, C$ and $D$ are square matrices such that $A+B$ is symmetric, $A-B$ is skew-symmetric and $D$ is the transpose of $C$. If $A=\left[\begin{array}{ccc}-1 & 2 & 3 \\\\ 4 & 3 & -2 \\\\ 3 & -4 & 5\end{array}\right]$ and $C=\left[\begin{array}{ccc}0 & 1 & -2 \\\\ 2 & -1 & 0 \\\\ 0 & 2 & 1\end{array}\right]$, then the matrix $B+D=$

If $A$ is square matrix and $A^2+I=2 A$, then $A^9=$

$\operatorname{det}\left[\begin{array}{ccc}\frac{a^2+b^2}{c} & c & c \\\\ a & \frac{b^2+c^2}{a} & a \\\ b & b & \frac{c^2+a^2}{b}\end{array}\right]=$

The system of simultaneous linear equations

$$ \begin{aligned} & x-2 y+3 z=4,3 x+y-2 z=7 \\ & 2 x+3 y+z=6 \text { has } \end{aligned} $$

If $\sqrt{5}-i \sqrt{15} \doteqdot r(\cos \theta+i \sin \theta),-\pi<\theta<\pi$, then $r^2\left(\sec \theta+3 \operatorname{cosec}^2 \theta\right)=$

The point $P$ denotes the complex number $z=x+i y$ in the argand plane. If $\frac{2 z-i}{z-2}$ is a purely real number, then the equation of the locus of $P$ is

$x$ and $y$ are two complex numbers such that $|x|=|y|=1$.

If $\arg (x)=2 \alpha, \arg (y)=3 \beta$ and $\alpha+\beta=\frac{\pi}{36}$, then $x^6 y^4+\frac{1}{x^6 y^4}=$

One of the roots of the equation $x^{14}+x^9-x^5-1=0$ is

If the quadratic equation $3 x^2+(2 k+1) x-5 k=0$ has real and equal roots, then the value of $k$ such that

$\frac{1}{2}$ < $k$ < 0 is

The equations $2 x^2+a x-2=0$ and $x^2+x+2 a=0$ have exactly one common root. If $a \neq 0$, then one of the roots of the equation $a x^2-4 x-2 a=0$ is

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $2 x^3-3 x^2+5 x-7=0$, then $\sum \alpha^2 \beta^2=$

The sum of two roots of the equation $x^4-x^3-16 x^2+4 x+48=0$ is zero. If $\alpha, \beta, \gamma$ and $\delta$ are the roots of this equation, then $\alpha^4+\beta^4+\gamma^4+\delta^4=$

The sum of all the 4 -digit numbers formed by taking all the digits from $2,3,5,7$ without repetition, is

The number of ways in which 15 identical gold coins can be distributed among 3 persons such that each one gets atleast 3 gold coins, is

The number of all possible combinations of 4 letters which are taken from the letters of the word 'ACCOMMODATION', is

If ${ }^n c_r=c_r$ and $2 \frac{c_1}{c_0}+4 \frac{c_2}{c_1}+6 \frac{c_3}{c_2}+\ldots .+2 n \frac{c_n}{c_{n-1}}=650$, then ${ }^n C_2=$ $\qquad$

When $|x|<2$, then coefficient of $x^2$ in the power series expansion of $\frac{x}{(x-2)(x-3)}$, is

If $\frac{x^4}{\left(x^2+1\right)(x-2)}=f(x)+\frac{A x+B}{x^2+1}+\frac{C}{x-2}$, then $f(14)+2 A-B=$

If the period of the function $f(x)=2 \cos (3 x+4)-3 \tan (2 x-3)+5 \sin (5 x)-7$ is $k$, then

If $\tan A<0$ and $\tan 2 A=-\frac{4}{3}$, then $\cos 6 A=$

If $m \cos (\alpha+\beta)-n \cos (\alpha-\beta)$ $=m \cos (\alpha-\beta)+n \cos (\alpha+\beta)$, then $\tan \alpha \tan \beta=$

The number of solutions of the equation $\sin 7 \theta-\sin 3 \theta=\sin 4 \theta$ that lie in the interval $(0, \pi)$, is

$\cos ^{-1} \frac{3}{5}+\sin ^{-1} \frac{5}{13}+\tan ^{-1} \frac{16}{63}=$

If $\cosh ^{-1}\left(\frac{5}{3}\right)+\sinh ^{-1}\left(\frac{3}{4}\right)=k$, then $e^k=$

In a $\triangle A B C$, if $(a-b)^2 \cos ^2 \frac{C}{2}+(a+b)^2 \sin ^2 \frac{C}{2}=a^2+b^2$, then $\cos A=$

In a $\triangle A B C$, if $r_1 r_2+r_3=35, r_2 r_3+r_1=63$ and $r_3 r_1+r_2=45$, then $2 s=$

$\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ are the position vectors of the vertices $A, B$ and $C$ of a $\triangle A B C$ respectively. If $D$ and $E$ are the mid points of $B C$ and $C A$ respectively, then the unit vector along DE is

A vector of magnitude $\sqrt{2}$ units along the internal bisector of the angle between the vectors $2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is

If $\theta$ is the angle between the vectors $4 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$, then $\sin 2 \theta=$

$\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are three vectors such that $|a|=3,|b|=2 \sqrt{2},|c|=5$ and $\mathbf{c}$ is perpendicular to the plane of $\mathbf{a}$ and $\mathbf{b}$. If the angle between the vectors a and $\mathbf{b}$ is $\frac{\pi}{4}$, then $|\mathbf{a}+\mathbf{b}+\mathbf{c}|=$

If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are non-coplanar vectors and the points $\lambda \mathbf{a}+3 \mathbf{b}-\mathbf{c}, \mathbf{a}-\lambda \mathbf{b}+3 \mathbf{c}, 3 \mathbf{a}+4 \mathbf{b}-\lambda \mathbf{c}$ and $\mathbf{a}-6 b+6 \mathbf{c}$ are coplanar, then one of the values of $\lambda$ is

The mean deviation about the mean for the following data is \begin{array}{c|l|l|l|l|l} \hline \text { Class interval } & 0-2 & 2-4 & 4-6 & 6-8 & 8-10 \\\\ \hline \text { Frequency } & 1 & 3 & 5 & 3 & 1 \\\\ \hline \end{array}

When 2 dice are thrown, it is observed that the sum of the numbers appeared on the top faces of both the dice is a prime number. Then, the probability of having a multiple of 3 among the pair of numbers thus obtained is

If 2 cards are drawn at random from a well shuffled pack of 52 playing cards from the same suit, then the probability of getting a face card and a card having a prime number is

A dealer gets refrigerators from 3 different manufacturing companies $C_1, C_2$ and $C_3 .25 \%$ of his stock is from $C_1, 35 \%$ from $C_2$ and $40 \%$ from $C_3$. The percentages of receiving defective refrigerators from $C_1, C_2$ and $C_3$ are $3 \% 2 \%, 1 \%$ respectively. If a refrigerator sold at random is found to be defective by a customer, then the probability that it is from $\mathrm{C}_2$ is

If the probability that a student selected at random from a particular college is good at mathematics is 0.6 , then the probability of having two students who are good at Mathematics in a group of 8 students of that college standing in front of the college, is

If on an average 4 customers visit a shop in an hour, then the probability that more than 2 customers visit the shop in a specific hour is

The centroid of a variable $\triangle A B C$ is at the distance of 5 units from the origin. If $A=(2,3)$ and $B=(3,2)$, then the locus of $C$ is

When the origin is shifted to the point $(2, b)$ by translation of axes, the coordinates of the point $(a, 4)$ have changed to $(6,8)$. When the origin is shifted to $(a, b)$ by translation of axes, if the transformed equation of $x^2+4 x y+y^2=0$ is $X^2+2 H X Y+Y^2+2 G X+2 F Y+C=0$, then $2 H(G+F)=$

The slope of a line $L$ passing through the point $(-2,-3)$ is not defined. If the angle between the lines $L$ and $a x-2 y+3=0(a>0)$ is $45^{\circ}$, then the angle made by the line $x+a y-4 \doteq 0$ with positive $X$-axis in the anti-clockwise direction is

$(a, b)$ is the point of concurrency of the lines $x-3 y+3=0, k x+y+k=0$ and $2 x+y-8=0$. If the perpendicular distance from the origin to the line $L=a x-b y+2 k=0$ is $p$, then the perpendicular distance from the point $(2,3)$ to $L=0$ is

If $(4,3)$ and $(1,-2)$ are the end points of a diagonal of a square, then the equation of one of its sides is

Area of the triangle bounded by the lines given by the equations $12 x^2-20 x y+7 y^2=0$ and $x+y-1=0$ is

If $(1,1),(-2,2)$ and $(2,-2)$ are 3 points on a circle $S$, then the perpendicular distance from the centre of the circle $S$ to the line $3 x-4 y+1=0$ is

If the line $4 x-3 y+p=0(p+3>0)$ touches the circle $x^2+y^2-4 x+6 y+4=0$ at the point $(h, k)$, then $h-2 k=$

If the inverse point of the point $P(3,3)$ with respect to the circle $x^2+y^2-4 x+4 y+4=0$ is $Q(a, b)$, then $a+5 b=$

If the equation of the transverse common tangent of the circles $x^2+y^2-4 x+6 y+4=0$ and $x^2+y^2+2 x-2 y-2=0$ is $a x+b y+c=0$, then $\frac{a}{c}=$

A circle $S \equiv x^2+y^2+2 g x+2 f y+6=0$ cuts another circle $x^2+y^2-6 x-6 y-6=0$ orthogonally. If the angle between the circles $S=0$ and $x^2+y^2+6 x+6 y+2=0$ is $60^{\circ}$, then the radius of the circle $S=0$ is

If $m_1$ and $m_2$ are the slopes of the direct common tangents drawn to the circles $x^2+y^2-2 x-8 y+8=0$ and $x^2+y^2-8 x+15=0$, then $m_1+m_2=$

If $(2,3)$ is the focus and $x-y+3=0$ is the directrix of a parabola, then the equation of the tangent drawn at the vertex of the parabola is

The equation of the common tangent to the parabola $y^2=8 x$ and the circle $x^2+y^2=2$ is $a x+b y+2=0$. If $-\frac{a}{b}>0$, then $3 a^2+2 b+1=$

Consider the parabola $25\left[(x-2)^2+(y+5)^2\right]=(3 x+4 y-1)^2$, match the characteristic of this parabola given in List I with its corresponding item in List II.

$$ \begin{array}{lll} \hline & \text { List I } & \text { List II } \\\\ \hline \text { I } & \text { Vertex } & \text { (A) } 8 \\\\ \hline \text { II } & \text { length of latus rectum } & \text { (B) }\left(\frac{29}{10}, \frac{-38}{10}\right) \\\\ \hline \text { III } & \text { Directrix } & \text { (C) } 3 x+4 y-1=0 \\\\ \hline \text { IV } & \begin{array}{l} \text { One end of the latus } \\\\ \text { rectum } \end{array} & \text { (D) }\left(\frac{-2}{5}, \frac{-16}{5}\right) \\\\ \hline \end{array} $$

The correct answer is

If $6 x-5 y-20=0$ is a normal to the ellipse $x^2+3 y^2=K$, then $K=$

The point of intersection of two tangents drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{4}=1$ lie on the circle $x^2+y^2=5$. If these tangents are perpendicular to each other, then $a=$

If the ratio of the perpendicular distances of a variable point $P(x, y, z)$ from the $X$-axis and from the $Y Z$ - plane is $2: 3$, then the equation of the locus of $P$ is

The direction cosines of two lines are connected by the relations $l-m+n=0$ and $2 l m-3 m n+n l=0$. If $\theta$ is the angle between these two lines, then $\cos \theta=$

A plane $\pi$ passes through the points $(5,1,2),(3,-4,6)$ and $(7,0,-1)$. If $p$ is the perpendicular distance from the origin to the plane $\pi$ and $l, m$ and $n$ are the direction cosines of a normal to the plane $\pi$, the $|3 l+2 m+5 n|=$

$\lim _{x \rightarrow 0} \frac{3^{\sin x}-2^{\tan x}}{\sin x}=$

If the function

$$ f(x)=\left\{\begin{array}{cc} \frac{\cos a x-\cos 9 x}{x^2} & \text {, if } x \neq 0 \\ 16 & \text {, if } x=0 \end{array}\right. $$

is continuous at $x=0$, then $a=$

If $ f(x)=\left\{\begin{array}{ll}\frac{8}{x^{3}}-6 x & \text {, if } 0 < x \leq 1 \\\\ \frac{x-1}{\sqrt{x}-1} & \text {,if } x > 1\end{array}\right. $ is a real valued function, then at $ x=1, f $ is

If $2 x^2-3 x y+4 y^2+2 x-3 y+4=0$, then $\left(\frac{d y}{d x}\right)_{(3,2)}=$

If $x=\frac{9 t^2}{1+t^4}$ and $y=\frac{16 t^2}{1-t^4}$ then $\frac{d y}{d x}=$

If $y=\sin a x+\cos b x$, then $y^{\prime \prime}+b^2 y=$

The radius of a sphere is 7 cm . If an error of 0.08 sq cm is made in measuring it, then the approximate error (in cubic cm ) found in its volume is

The curve $y=x^3-2 x^2+3 x-4$ intersects the horizontal line $y=-2$ at the point $P(h, k)$. If the tangent drawn to this curve at $P$ meets the $X$-axis at $\left(x_1, y_1\right)$, then $x_1=$

A particle moving from a fixed point on a straight line travels a distance $S$ metres in $t \mathrm{sec}$. If $S=t^3-t^2-t+3$, then the distance (in mts) travelled by the particle when it comes to rest, is

If $f(x)=(2 x-1)(3 x+2)(4 x-3)$ is a real valued function defined on $\left[\frac{1}{2}, \frac{3}{4}\right]$, then the value(s) of $c$ as defined in the statement of Rolle's theorem

If the interval in which the real valued function $f(x)=\log \left(\frac{1+x}{1-x}\right)-2 x-\frac{x^3}{1-x^2}$ is decreasing in $(a, b)$, where $|b-a|$ is maximum, then $\frac{a}{b}=$

$\int(\sqrt{1-\sin x}+\sqrt{1+\sin x}) d x=f(x)+c$, where $c$ is the constant of integration. If $\frac{5 \pi}{2}$<$x<\frac{7 \pi}{2}$ and $$ f\left(\frac{8 \pi}{3}\right)=-2, \text { then } f^{\prime}\left(\frac{8 \pi}{3}\right)= $$

If $f(x)=\int \frac{\sin 2 x+2 \cos x}{4 \sin ^2 x+5 \sin x+1} d x$ and $f(0)=0$, then $f(\pi / 6)=$

$\int \frac{\left(1-4 \sin ^2 x\right) \cos x}{\cos (3 x+2)} d x=$

$\lim \limits_{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right) \ldots .(2)\right]^{1 / n}=$

$\int_{-2}^2 x^4\left(4-x^2\right)^{\frac{7}{2}} d x=$

Area of the region enclosed between the curves $y^2=4(x+7)$ and $y^2=5(2-x)$ is

If the slope of the tangent drawn at any point $(x, y)$ on the curve $y=f(x)$ is $\left(6 x^2+10 x-9\right)$ and $f(2)=0$, then $f(-2)=$

The general solution of the differential equation $\left(3 x^2-2 x y\right) d y+\left(y^2-2 x y\right) d x=0$ is

Physics

Regarding fundamental forces in nature, the correct statement is

The equation of motion of a damped oscillator is given by $m \frac{d^2 x}{d t^2}+b \frac{d x}{d t}+k x=0$. The dimensional formula of $\frac{b}{\sqrt{k m}}$ is

A body is falling freely from the top of a tower of height 125 m . The distance covered by the body during the last second of its motion is $x \%$ of the height of the tower. Then, $x$ is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

A body $P$ is projected at an angle of $30^{\circ}$ with the horizontal and another body $Q$ is projected at an angle of $30^{\circ}$ with the vertical. If the ratio of the horizontal ranges of the bodies $P$ and $Q$ is $1: 2$, then the ratio of the maximum heights reached by the bodies $P$ and $Q$ is

A car is moving on circular track banked at an angle of $45^{\circ}$. If the maximum permissible speed of the car to avoid slipping is twice the optimum speed of the car to avoid the wear and tear of the tyres, then the coefficient of static friction between the wheels of the car and the road is

A block of mass 0.5 kg is at rest on a horizontal table. The coefficient of kinetic friction between the table and the block is 0.2 . If a horizontal force of 5 N is applied on the block, the kinetic energy of the block in a time of 4 s is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

The sphere $A$ of mass $m$ moving with a constant velocity hits another sphere $B$ of mass $2 m$ at rest. If the coefficient of restitution is 0.4 the ratio of the velocities of the spheres $A$ and $B$ after collision is

A solid sphere rolls down without slipping from the top of an inclined plane of height 28 m and angle of inclination $30^{\circ}$. The velocity of the sphere, when it reaches the bottom of the plane is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

Four identical particles each of mass $m$ are kept at the four corners of a square of side $a$. If one of the particles is removed, the shift in the position of the centre of mass is

In a time $t$ amplitude of vibrations of a damped oscillator becomes half of its initial value, then the mechanical energy of the oscillator decreases by

The energy required to take a body from the surface of the earth to a height equal to the radius of the earth is $W$. The energy required to take this body from the surface of the earth to a height equal to twice the radius of the earth is

A steel wire of length 3 m and a copper wire of length 2.2 m are connected end to end. When the combination is stretched by a force, the net elongation is 1.05 mm . If the area of cross-section of each wire is $6 \mathrm{~mm}^2$, then the load applied is (Young's moduli of steel and copper are respectively $2 \times 10^{11} \mathrm{Nm}^{-2}$ and $1.1 \times 10^{11} \mathrm{Nm}^{-2}$ )

The height of water level in a tank of uniform cross-section is 5 m . The volume of the water leaked in 5 s through a hole of area $2.4 \mathrm{~mm}^2$ miade at the bottom of the tank is (Assume the level of the water in the tank remains constant and acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )

The work done in increasing the diameter of a soap bubble from 2 cm to 4 cm is (Surface tension of soap solution $=3.5 \times 10^{-2} \mathrm{Nm}^{-1}$ )

The temperature on a fahrenheit temperature scale that is twice the temperature on a celsius temperature scale is

The temperatures of equal masses of three different liquids $A, B$ and $C$ are $15^{\circ} \mathrm{C}, 24^{\circ} \mathrm{C}$ and $30^{\circ} \mathrm{C}$, respectively. The resultant temperature when liquids $A$ and $B$ are mixed is $20^{\circ} \mathrm{C}$ and when liquids $B$ and $C$ are mixed is $26^{\circ} \mathrm{C}$. Then, the ratio of specific heat capacities of the liquids $A, B$ and $C$ is

The efficiency of a reversible heat engine working between two temperatures is $50 \%$. The coefficient of performance of a refrigerator working between the same two temperatures but in reverse direction is

The total internal energy of 4 moles of a diatomic gas at a temperature of $27^{\circ} \mathrm{C}$ is (gas constant $=831$ $\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}$ )

A car travelling at a speed of 54 kmph towards a wall sounds horn of frequency 400 Hz . The difference in the frequencies of two sounds, one received directly from the car and other reflected from the wall noticed by a person standing between the car and the wall is (speed of sound in air is $335 \mathrm{~m} / \mathrm{s}$ )

The speed of a transverse wave in a stretched string $A$ is $v$. Another string $B$ of same length and same radius is subjected to same tension. If the density of the material of the string $B$ is $2 \%$ more than that of $A$ then the speed of the transverse wave in string $B$ is

For a combination of two convex lenses of focal lengths $f_1$ and $f_2$ to act as a glass slab, the distance of separation between them is

If a ray of light passes through an equilateral prism such that the angle of incidence and the angle of emergence are both equal to $70 \%$ of the angle of the prism, then the angle of minimum deviation is

Young's double slit experiment is performed with monochromatic light of wavelength $6000 \mathring{A}$. If the intensity of light at a point on the screen, where path difference of $2000 \mathring{A}$ is $I_1$ and the intensity of light at a I point on the screen, where the path difference is $1000 \mathring{A}$ is $I_2$, then $I_1: I_2=$

Two positive point charges are separated by a distance of 4 m in air. If the sum of the two charges is $36 \mu \mathrm{C}$ and the electrostatic force between them is 0.18 N , then the bigger charge is

Three capacitors of capacitances $10 \mu \mathrm{~F}, 5 \mu \mathrm{~F}$ and $20 \mu \mathrm{~F}$ are connected in series with a 14 V DC supply. The charge on $5 \mu \mathrm{~F}$ capacitor is

When the temperature of a wire is increased from 303 K to 356 K , the resistance of the wire increases by $10 \%$. The temperature coefficient of resistance of the material of the wire is

Three resistors of resistances $10 \Omega, 20 \Omega$ and $30 \Omega$ are connected as shown in the figure. If the points $A, B$ and $C$ are at potentials $10 \mathrm{~V}, 6 \mathrm{~V}$ and 5 V respectively, then the ratio of the magnitudes of the currents through $10 \Omega$ and $30 \Omega$ resistors is