Chemistry
1
The wavelength of an electron is $10^{3} \mathrm{~nm}$. What is its momentum in $\mathrm{kg} \mathrm{ms}^{-1}$ ? ( $\left.h=6.625 \times 10^{-34} \mathrm{Js}\right)$
2
Two statements are given below :
Statement I : In H atom, the energy of $2 s$ and $2 p$ orbitals is same.
Statements II : In He atom, the energy of $2 s$ and $2 p$ orbitals is same.
The correct answer is
3
The set containing the elements with positive electron gain enthalpies is
4
Assertion (A) : The ionic radii of $\mathrm{Na}^{+}$and $\mathrm{F}^{-}$are same.
Reason (R) : Both $\mathrm{Na}^{+}$and $\mathrm{F}^{-}$are isoelectronic species.
The correct answer is
5
The number of lone pairs of electrons on central atom of $\mathrm{ClF}_{3}, \mathrm{NF}_{3}, \mathrm{SF}_{4}, \mathrm{XeF}_{4}$ respectively are
6
The hybridisation of central atom of $\mathrm{BF}_{3}, \mathrm{SnCl}_{2}, \mathrm{HgCl}_{2}$ respectively is
7
The variation of volume of an ideal gas with its number of moles $(n)$ is obtained as a graph at 300 K and 1 atm pressure. What is the slope of the graph ?
8
Observe the following reaction,
$ 2 \mathrm{KClO}_{3}(s) \xrightarrow{\Delta} 2 \mathrm{KCl}(\mathrm{~s})+3 \mathrm{O}_{2}(\mathrm{~g}) $
In this reaction
9
The $\Delta_{f} H^{\theta}$ of $\mathrm{AO}(s), \mathrm{BO}_{2}(g)$ and $A B \mathrm{O}_{3}(s)$ is $-635, x$ and $-1210 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively. $\left.\mathrm{ABO}_{3}(s) \rightarrow \mathrm{AO}(s)+\mathrm{BO}_{2}(g): \Delta_{r} H^{\Theta}=175 \mathrm{~kJ} \mathrm{~mol}^{-1}\right)$. What is the value of $x$ (in $\mathrm{kJ} \mathrm{mol}^{-1}$ )?
10
At $27^{\circ} \mathrm{C}, 100 \mathrm{~mL}$ of 0.5 M HCl is mixed with 100 mL of 0.4 M NaOH solution. To this resultant solution, 800 mL of distilled water is added. What is the pH of final solution?
11
The proper conditions of storing $\mathrm{H}_{2} \mathrm{O}_{2}$ are
12
The standard electrode potentials $E^{\circ}(\mathrm{V})$ for $\mathrm{Li}^{+} / \mathrm{Li}, \mathrm{Na}^{+} / \mathrm{Na}$ respectively are
13
The alloy formed by beryllium with ' $X$ ' is used in the preparation of high strength springs. ' $X$ ' is
14
What are $X$ and $Y$ respectively in the following reactions ?
$ X \stackrel{\mathrm{CO}}{\longleftarrow} \mathrm{~B}_{2} \mathrm{H}_{6} \xrightarrow[\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{O}]{\mathrm{NaH}} Y $
15
Which of the following statements are correct?
(i) $\mathrm{CCl}_{4}$ undergoes hydrolysis easily
(ii) Diamond has directional covalent bonds
(iii) Fullerene is thermodynamically most stable allotrope of carbon
(iv) Glass is a man-made silicate
The correct answer is
16
Which of the following industries generate nonbiodegradable wastes?
17
Possible number of isomers including stereoisomers for an organic compound with the molecular formula $\mathrm{C}_{4} \mathrm{H}_{9} \mathrm{Br}$ is
18
The alkane which is next to methane in homologous series can be prepared from which of the following reactions?
$ \text { I. } 2 \mathrm{CH}_{3} \mathrm{Br} \xrightarrow[\text { Dry ether }]{\mathrm{Na}} $
II. $\mathrm{CH}_{3} \mathrm{COOH} \xrightarrow[\mathrm{CaO}, \Delta]{\mathrm{NaOH}}$
III. $\mathrm{CH}_{3} \mathrm{CH}=\mathrm{CH}_{2} \xrightarrow{\mathrm{H}_{2} / \mathrm{Pt}}$ IV. $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Br} \xrightarrow[\mathrm{H}^{+}]{\mathrm{Zn}}$
19
At high pressure and regulated supply of air, methane is heated with catalyst ' $X$ to give methanol and with catalyst ' $Y$ ' to give methanal. $X$ and $Y$ respectively are
20
What is ' $ Y $ ' in the following set of reactions?
$$ \mathrm{C}_3 \mathrm{H}_4 \xrightarrow[\substack{\mathrm{Hg}^{2+} / \mathrm{H}^{+} \\ 333 \mathrm{~K}}]{\mathrm{H}_2 \mathrm{O}} X \xleftarrow[\substack{\text { (ii) } \mathrm{Zn}+\mathrm{H}_2 \mathrm{O}}]{\text { (i) } \mathrm{O}_3} Y $$
21
The molecular formula of a crystal is $A B_{2} \mathrm{O}_{4}$. Oxygen atoms form ccp lattice. Atoms of $A$ occupy $x \%$ of tetrahedral voids and atoms of $B$ occupy $y \%$ of octahedral voids. $x$ and $y$ are respectively
22
At $T(\mathrm{~K}), 0.1$ mole of a non-volatile solute was dissolved in 0.9 mole of a volatile solvent. The vapour pressure of pure solvent is 0.9 bar. What is the vapour pressure (in bar ) of solution ?
23
Two statements are given below.
Statement I : Molten NaCl is electrolysed using Pt electrodes. $\mathrm{Cl}_{2}$ is liberated at anode.
Statement II : Aqueous $\mathrm{CuSO}_{4}$ is electrolysed using Pt electrodes. $\mathrm{O}_{2}$ is liberated at cathode.
The correct answer is
24
For a first order reaction, the graph between $\log \frac{a}{(a-x)}$ (on $y$-axis) and time (in min, on $x$-axis) gave a straight line passing through origin. The slope is $2 \times 10^{-3} \mathrm{~min}^{-1}$. What is the rate constant (in $\mathrm{min}^{-1}$ )?
25
In Haber's process of manufacture of ammonia, the 'catalyst' the 'promoter' and 'poison for the catalyst' are respectively
26
Among the following the calcination process is
27
The correct order of boiling points of hydrogen halides is
28
Observe the following reactions (unbalanced)
$ \begin{array}{r} \mathrm{P}_{2} \mathrm{O}_{3}+\mathrm{H}_{2} \mathrm{O} \longrightarrow X \\ \mathrm{P}_{4} \mathrm{O}_{10}+\mathrm{H}_{2} \mathrm{O} \longrightarrow Y \end{array} $
The number of $\mathrm{P}=\mathrm{O}$ bonds present in $X, Y$ are respectively
29
Carbon on reaction with hot conc. $\mathrm{H}_{2} \mathrm{SO}_{4}$, gives two oxides along with $\mathrm{H}_{2} \mathrm{O}$. What is the nature of these two oxides?
30
Which of the following orders is correct for the property given?
31
Arrange the following in increasing order of their crystal field splitting energy
I. $\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}$
II. $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$
III. $\left[\mathrm{Co}(\mathrm{CN})_{6}\right]^{3-}$
IV. $\left[\mathrm{CoF}_{6}\right]^{3-}$
32
What are ' $X$ ' and ' $Y$ ' respectively in the following reactions ?
33
Two statements are given below :
I. Milk sugar is disaccharide of $\alpha$-D-galactose and $\beta$-D-glucose
II. Sucrose is disaccharide of $\alpha$-D-glucose and $\beta$-D-fructose
34
The effects that aspirin can produce in the body are
| Anti-Inflammatory | Anti-depressant | Anti-pyretic | Anti-coagulant | Hypnotic |
| A | B | C | D | E |
35
The reagent ' $X$ ' used in the following reaction to obtain good yield of the product is
36
37
IUPAC name of the following compound is
38
The bromides formed by the cleavage of ethers $A$ and $B$ with HBr respectively are
39
Identify the set, in which $X$ and $Y$ are correctly matched
40
What are $X$ and $Y$ respectively in the following reactions?
Mathematics
1
The domain of the real valued function $f(x)=\sin ^{-1}\left(\log _{2}\left(\frac{x^{2}}{2}\right)\right)$ is
2
The range of the real valued function $f(x)=\log _{3}\left(5+4 x-x^{2}\right)$ is
3
If $3^{2 n+2}-8 n-9$ is divisible by $2^{p}, \forall n \in \mathrm{~N}$, then the maximum value of $P$ is
4
$A=\left[a_{i j}\right]$ is a $3 \times 3$ matrix with positive integers as its elements. Elements of $A$ are such that the sum of all elements of each row is equal to 6 and $a_{22}=2$.
If $\mathrm{a}_{i j}=\left\{\begin{array}{cl}\mathrm{a}_{i j}+\mathrm{a}_{j i}, & j=i+1 \text { when } i < 3 \\ \mathrm{a}_{i j}+\mathrm{a}_{j i}, & j=4-i \text { when } i=3\end{array}\right.$ for $i=1,2,3$, then $|\mathrm{A}|=$
5
If $|\operatorname{adj} A|=x$ and $|\operatorname{adj} B|=y$, then $\left|(\operatorname{adj}(A B))^{-1}\right|=$
6
The system of equations $x+3 b y+b z=0, x+2 a y+a z=0$ and $x+4 c y+c z=0$ has
7
$\left|\begin{array}{ccc}\frac{-b c}{a^{2}} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{a c}{b^{2}} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{a b}{c^{2}}\end{array}\right|=$
8
If $z=x+i y$ satisfies the equation $z^{2}+a z+a^{2}=0, a \in R$, then
9
If $z_{1}, z_{2}, z_{3}$ are three complex numbers with unit modulus such that $\left|z_{1}-z_{2}\right|^{2}+\left|z_{1}-z_{3}\right|^{2}=4$, then $z_{1} \bar{z}_{2}+\bar{z}_{1} z_{2}+z_{1} \bar{z}_{3}+\bar{z}_{1} z_{3}=$
10
If $\omega$ is the complex cube root of unity and
$\left(\frac{a+b \omega+c \omega^{2}}{c+a \omega+b \omega^{2}}\right)^{k}+\left(\frac{a+b \omega+c \omega^{2}}{b+a \omega^{2}+c \omega}\right)^{l}=2$, then $2 k+l$ is always
11
If $z_{1}=\sqrt{3}+i \sqrt{3}$ and $z_{2}=\sqrt{3}+i$, and $\left(\frac{z_{1}}{z_{2}}\right)^{50}=x+i y$, then the point $(x, y)$ lies in
12
The solution set of the equation $3^{x}+3^{1-x}-4 < 0$ contained in $R$ is
13
The common solution set of the inequations $x^{2}-4 x \leq 12$ and $x^{2}-2 x \geq 15$ taken together is
14
The roots of the equation $x^{3}-3 x^{2}+3 x+7=0$ are $\alpha, \beta, \lambda$ and $\omega, \omega^{2}$ are complex cube roots of unity, If the terms containing $x^{2}$ and $x$ are missing in the transformed equation when each one of these roots is decreased by $h$, then $\frac{\alpha-h}{\beta-h}+\frac{\beta-h}{\gamma-h}+\frac{\gamma-h}{\alpha-h}=$
15
With respect to the roots of the equation $3 x^{3}+b x^{2}+b x+3=0$, match the items of List I with those fo List II
| List I | List II |
| A All the roots are negative. | I. $(b-3)^2=36+P^2$ for $P \in R$ |
| B Two roots are complex. | II. $-3<b<9$ |
| C Two roots are positive. | III. $b \in(-\infty,-3) \cup(9, \infty)$ |
| D All roots are real and | IV. $b=9$ |
| V. $b=-3$ |
16
The number of ways of arranging all the letters of the word 'COMBINATIONS' around a circle so that no two vowels together is
17
If all the numbers which are greater than 6000 and less than 10000 are formed with the digits, $3,5,6,7,8$ without repetition of the digits, then the difference between the number of odd numbers and the number of even number among them is
18
A man has 7 relatives, 4 of them are ladies and 3 gents; his wife has 7 other relatives, 3 of them are ladies and 4 gents. The number of ways they can invite them to a party of 3 ladies and 3 gents so that the there are 3 of man's relatives and 3 of wife's relatives, is
19
If the coefficient fo $x^{r}$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{100}$ is $a_{r}$ and $S=\sum_{r=0}^{300} a_{r}$ then $\sum_{r=0}^{300} r \cdot a_{r}=$
20
Assertion (A) : $1+\frac{2 \cdot 1}{3 \cdot 2}+\frac{2 \cdot 5}{3 \cdot 6} \frac{1}{4}+\frac{2 \cdot 5 \cdot 8}{3 \cdot 6 \cdot 9} \frac{1}{8}+\ldots \infty=\sqrt[3]{4}$
Reason (R) : |x| < 1,(1-x) $=1+n x+\frac{n(n+1)}{1 \cdot 2} x^2$$+\frac{n(n+1)(n+2)}{1 \cdot 2 \cdot 3} x^{3}+\ldots$
The correct answer is :
21
If $\frac{1}{x^{4}+x^{2}+1}=\frac{A x+B}{x^{2}+a x+1}+\frac{C x+D}{x^{2}-a x+1}$, then $A+B-C+D=$
22
If $0 < \theta < \frac{\pi}{4}$ and $8 \cos \theta+15 \sin \theta=15$, then $15 \cos \theta-8 \sin \theta=$
23
$\sin 20^{\circ}\left(4+\sec 20^{\circ}\right)=$
24
Suppose, $\theta_{1}$ and $\theta_{2}$ are such that $\left(\theta_{1}-\theta_{2}\right)$ lies in 3rd or 4th quadrant. If $\sin \theta_{1}+\sin \theta_{2}=-\frac{21}{65}$ and $\cos \theta_{1}+\cos \theta_{2}=-\frac{27}{65}$, then $\cos \left(\frac{\theta_{1}-\theta_{2}}{2}\right)=$
25
If $A$ is the solution set of the equation $\cos ^{2} x=\cos ^{2} \frac{\pi}{6}$ and $B$ is the solution set of the equation $\cos ^{2} x=\log _{16} P$ where, $P+\frac{16}{P}=10$, then, $B-A=$
26
The trigonometric equation $\sin ^{-1} x=2 \sin ^{-1} a$, has a solution
27
If $\sin h x=\frac{12}{5}$, then $\sin h 3 x+\cos h 3 x=$
28
If $A B C$ is an isosceles triangle with base $B C$, then $r_{1}=$
29
In $\triangle A B C$, if $r_{1}+r_{2}=3 R, r_{2}+r_{3}=2 R$, then
30
$\mathbf{n}$ is a unit vector normal to the plane $\pi$ containing the vectors $\hat{\mathbf{i}}+3 \hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$. If this plane $\pi$ passes through the point $(-3,7,1)$ and $p$ is the perpendicular distance from the origin to this plane $\pi$, then $\sqrt{p^{2}+5}=$
31
If $\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{c}=-\hat{\mathbf{k}}$ are position vectors of two points and $\mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}, \mathbf{d}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ are two vectors, then the lines $\mathbf{r}=\mathbf{a}+t \mathbf{b}, \mathbf{r}=\mathbf{c}+s \mathbf{d}$ are
32
$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three vectors each having $\sqrt{2}$ magnitude such that $(\mathbf{a}, \mathbf{b})=(\mathbf{b}, \mathbf{c})=(\mathbf{c}, \mathbf{a})=\frac{\pi}{3}$. If $\mathbf{x}=\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ and $\mathbf{y}=\mathbf{b} \times(\mathbf{c} \times \mathbf{a})$, then
33
$\mathbf{a}$ is a vector perpendicular to the plane containing non zero vectors $\mathbf{b}$ and $\mathbf{c}$. If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are such that
$|\mathbf{a}+\mathbf{b}+\mathbf{c}|=\sqrt{|\mathbf{a}|^{2}+|\mathbf{b}|^{2}+|\mathbf{c}|^{2}}$, then
$|(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}|+|(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}|=$
34
If $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=3(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})$ and $\mathbf{c}$ is a vector such that $\mathbf{a} \times \mathbf{c}=\mathbf{b}$ and $\mathbf{a} . \mathbf{c}=3$, then $\mathbf{a} \cdot(\mathbf{c} \times \mathbf{b}-\mathbf{b}-\mathbf{c})=$
35
The variance of the first 10 natural numbers which are multiples of 3 is
36
If three numbers are randomly selected from the set $\{1,2,3, \ldots \ldots 50\}$, then the probability that they are in arithmetic progression is
37
The probability that exactly 3 heads appear in six tosses of an unbiased coin, given that first three tosses resulted in 2 or more heads is
38
A student has to write the words ABILITY, PROBABILITY, FACILITY, MOBILITY. He wrote one word and erased all the letters in it except two consecutive letters. If 'LI' is left after erasing then the probability that the boy wrote the word PROBABILITY is
39
Two cards are drawn at random one after the other with replacement from a pack of playing cards. If $X$ is the random variable denoting the number of ace cards drawn, then the mean of the probability distribution of X is
40
If $X \sim B(6, p)$ is a binomial variate and $\frac{P(X=4)}{P(X=2)}=\frac{1}{9}$, then $p=$
41
If the locus of the centroid of the triangle with vertices $A(a, 0), B(a \cos t, a \sin t)$ and $C(b \sin ,-b \cos t)$ ( $t$ is a parameter) is $9 x^{2}+9 y^{2}-6 x \overline{\bar{x}} 49$, then the area of the triangle formed by the line $\frac{x}{a}+\frac{y}{b}=1$ with the coordinate axes is
42
By shifting the origin to the point $(h, 5)$ by the translation of coordinate axes, if the equation $y=x^{3}-9 x^{2}+c x-d$ transforms to $Y=X^{3}$, then $\left(d-\frac{c}{h}\right)=$
43
The equation of the straight line whose slope is $\frac{-2}{3}$ and which divides the line segment joining $(1,2),(-3,5)$ in the ratio $4: 3$ externally is
44
$7 x+y-24=0$ and $x+7 y-24=0$ represent the equal sides of an isosceles triangle. If the third side passes through $(-1,1)$ then, a possible equation for the third side is
45
The combined equation of a possible pair of adjacent sides of a square with area 16 square units whose centre is the point of intresection of the lines $x+2 y-3=0$ and $2 x-y-1=0$ is
46
If the line $2 x+b y+5=0$ forms an equilateral to triangle with $a x^{2}-96 b x y+k y^{2}=0$, then $a+3 k=$
47
A rhombus is inscribed in the region common to the two circles $x^{2}+y^{2}-4 x-12=0$ and $x^{2}+y^{2}+4 x-12=0$. If the line joining the centres of these circles and the common chord of them are the diagonals of this rhombus, then the area (in sq units) of the rhombus is
48
If $m$ is the slope and $P(8, \beta)$ is the mid-point of a chord of contact of the circle $x^{2}+y^{2}=125$, then the number of values of $\beta$ such that $\beta$ and $m$ are integers is
49
A rectangle is formed by the lines $x=4, x=-2, y=5, y=-2$ and a circle is drawn through the vertices of this rectangle. The pole of the line $y+2=0$ with respect to this circle is
50
The equation of a circle which passes through the points of intersection of the circles $2 x^{2}+2 y^{2}-2 x+6 y-3=0, x^{2}+y^{2}+4 x+2 y+1=0$ and whose centre lies on the common chord of these circles is
51
If the equation of the circle which cuts each of the circles $x^{2}+y^{2}=4, x^{2}+y^{2}-6 x-8 y+10=0$ and $x^{2}+y^{2}+2 x-4 y-2=0$ at the extremities of a diameter of these circles is $x^{2}+y^{2}+2 g x+2 f y+c=0$, then $g+f+c=$
52
The equation of the circle passing through the origin and cutting the circles $x^{2}+y^{2}+6 x-15=0$ and $x^{2}+y^{2}-8 y-10=0$ orthogonally is
53
$S=(-1,1)$ is the focus, $2 x-3 y+1=0$ is the directrix corresponding, to $S$ and $\frac{1}{2}$ is the eccentricity of an ellipse, If $(a, b)$ is the centre of the ellipse, then $3 a+2 b$ :
54
$S=y^{2}-4 a x=0, S^{\prime}=y^{2}+a x=0$ are two parabolas and $P(t)$ is a point on the parabola $S^{\prime}=0$. If $A$ and $B$ are the feet of the perpendiculars from $P$ on to coordinate $2 x_{4}$ and $A B$ is a tangent to the parabola $S=0$ at the point $Q\left(t_{1}\right)$, then $t_{1}=$
55
$a$ and $b$ are the semi-major and semi-minor axes of an ellipse whose axes are along the coordinate axes, If its latus rectum is of length 4 units and the distance between its foci is $4 \sqrt{2}$, then $a^{2}+b^{2}=$
56
If the extremities of the latus recta having positive ordinate of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a > b)$ lie on the parabola $x^{2}+2 a y-4=0$, then the points $(a, b)$ lie on the curve
57
If the tangent drawn at a point $P(t)$ on the hyperbola $x^{2}-y^{2}=c^{2}$ cuts $X$-axis at $T$ and the normal drawn at the same point $P$ cuts the $Y$-axis at $N$, then the equation of the locus of the mid-point of $T N$ is
58
If the harmonic conjugate of $P(2,3,4)$ with respect to the line segment joining the points $A(3,-2,2)$ and $B(6,-17,-4)$ is $Q(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma=$
59
If $L$ is the line of intersection of two planes $x+2 y+2 z=15$ and $x-y+z=4$ and the direction ratio of the line $L$ are $(a, b, c)$, then $\frac{\left(a^{2}+b^{2}+c^{2}\right)}{b^{2}}=$
60
The foot of the perpendicular drawn from $A(1,2,2)$ oril the the plane $x+2 y+2 z-5=0$ is $B(\alpha, \beta, \gamma)$. If $\pi(x, y, z)$ $=x+2 y+2 z+5=0$ is a plane, then $-\pi(A): \pi(B)=$
61
If $0 \leq x \leq \frac{\pi}{2}$, then $\lim _{x \rightarrow a} \frac{|2 \cos x-1|}{2 \cos x-1}$
62
The real valued function $f(x)=\frac{|x-a|}{x-a}$ is
63
If $f(x)=3 x^{15}-5 x^{10}+7 x^{5}+50 \cos (x-1)$, then $\lim\limits_{h \rightarrow 0} \frac{f(1-h)-f(1)}{h^{3}+3 h}$
64
If the function $f(x)=\left\{\begin{array}{cl}\frac{\left(e^{k x}-1\right) \sin k x}{4 \tan x} & x \neq 0 \\ P & x=0\end{array}\right.$ is differentiable at $x=0$, then
65
If $y=\log \left(x-\sqrt{x^{2}-1}\right)$, then $\left(x^{2}-1\right) y^{\prime \prime}+x y^{\prime}+e^{y}+\sqrt{x^{2}-1}=$
66
The maximum interval in which the slopes of the tangents drawn to the curve $y=x^{4}+5 x^{3}+9 x^{2}+6 x+2$ increase is
67
If $A=\{P(\alpha, \beta) /$ the tangent drawn at $P$ to the curve $y^{3}-3 x y+2=0$ is horizontal line $\}$ and $B=\{Q(a, b) /$ the tangent drawn at $Q$ to the curve $y^{3}-3 x y+2=0$ is a vertical line $\}$, then $n(A)+n(B)=$
68
In a $\triangle A B C$, the sides $b, c$ are fixed. In measuring angle $A$, if there is an error of $\delta A$, then the percentage error in measuring the length of the side $a$ is
69
$y=f(x)$ and $x=g(y)$ are two curves and $P(x, y)$ is a common point of the two curves. If at $P$ on the curve $y=f(x), \frac{d y}{d x}=Q(x)$ and at the same point $P$ on the curve $x=g(y), \frac{d x}{d y}=-Q(x)$, then
70
If Rolle's Theorem is applicable for the function $f(x)=\left\{\begin{array}{cl}x^{p} \log x, & x \neq 0 \\ 0, & x=0\end{array}\right.$ on the interval $[0,1]$, then a possible value of $p$ is
71
The sum of the maximum and minimum values of the function $f(x)=\frac{x^{2}-x+1}{x^{2}+x+1}$ is
72
If $\int \frac{1}{x^{4}+8 x^{2}+9} d x=\frac{1}{k}$$\left[\frac{1}{\sqrt{14}} \tan ^{-1}(f(x))-\frac{1}{\sqrt{2}} \tan ^{-1}(g(x))\right]+c$ then,
$\sqrt{\frac{k}{2}+f(\sqrt{3})+g(1)}=$
73
If $\int\left(1+x-x^{-1}\right) e^{\left(x+x^{-1}\right)} d x=f(x)+C$, then $f(1)-f(-1)=$
74
$ \int \frac{1}{x^{m} \sqrt[m]{x^{m}+1}} d x =$
75
If $\int(\sqrt{\operatorname{cosec} x+1}) d x=k \tan ^{-1}(f(x))+C$, then $\frac{1}{k} f\left(\frac{\pi}{6}\right)=$
76
$\frac{3}{25} \int_{0}^{25 \pi} \sqrt{\left|\cos x-\cos ^{3} x\right|} d x=$
77
If the area of the region enclosed by the curve $a y=x^{2}$ and the line $x+y=2 a$ is $k a^{2}$, then $k=$
78
If $m, l, r, s, n$ are integers such that $9 > m > l > s > n > r > 2$ and $\int_{-2 \pi}^{2 \pi} \sin ^{m} x \cos ^{n} x d x=4 \int_{0}^{\pi} \sin ^{m} x \cos ^{n} x d x, \int_{-\pi}^{\pi} \sin ^{r} x \cos ^{s} x d x$ $=4 \int_{0}^{\pi / 2} \sin ^{r} x \cos ^{s} x d x$ and $\int_{-\pi / 2}^{\pi / 2} \sin ^{l} x \cos ^{m} x d x=0$, then
79
The order and degree of the differential equation
$ \frac{d y}{d x}=\left(\frac{d^{2} y}{d x^{2}}+2\right)^{\frac{1}{2}}+\frac{d^{2} y}{d x}+5 \text { are respectively } $
80
If $y=\sin x+A \cos x$ is general solution of $\frac{d x}{d y}+f(x) y=\sec x$, then an integrating factor of the differential equation is
Physics
1
Wave picture of light has failed to explain
2
A capacitor of capacitance $(4.0 \pm 0.2) \mu \mathrm{F}$ is charged to a potential of $(10.0 \pm 0 \mathrm{l}) \mathrm{V}$. The charge on the capacitor is
3
A body is thrown vertically upwards with a velocity of $35 \mathrm{~ms}^{-1}$ from the ground. The ratio of the speeds of the body at times 3 s and 4 s of its motion is (acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )
4
From a height of $h$ above the ground, a ball is projected up at an angle $30^{\circ}$ with the horizontal. If the ball strikes the ground with a speed of 1.25 times its initial speed of $40 \mathrm{~ms}^{-1}$, the value of $h$ is (acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )
5
A block is kept on a rough horizontal surface. The acceleration of the block increases from $6 \mathrm{~ms}^{-2}$ to $11 \mathrm{~ms}^{-2}$ when the horizontal force acting on it increases from 20 N to 30 N . The coefficient of kinetic friction between the block and the surface is
(acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )
6
The kinetic energy of a body of mass 4 kg moving with a velocity of $(2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-\hat{\mathbf{k}}) \mathrm{ms}^{-1}$ is
7
A ball $P$ of mass 0.5 kg moving with a velocity of $10 \mathrm{~ms}^{-1}$ collides with another ball $Q$ of mass 1 kg at rest. If the coefficient of restitution is 0.4 , the ratio of the velocities of the balls $P$ and $Q$ after the collision is
8
A circular plate of radius $r$ is removed from a uniform circular plate $P$ of radius $4 r$ to form a hole. If the distance between the centre of the hole formed and the centre of the plate $P$ is $2 r$, then the distance of mass of the remaining portion from the centre of the plate $P$ is
9
A hollow cylinder and a solid cylinder initially at restat the top of an inclined plane are rolling down without slipping. If the time taken by the hollow cylinder to reach the bottom of the inclined plane is 2 s , the time taken by the solid cylinder to reach the bottom of the inclined plane is
10
A block kept on a frictionless horizontal surface is connected to one end of a horizontal spring of constant $100 \mathrm{Nm}^{-1}$ whose other end is fixed to a rigid vertical wall. Initially the block is at its equilibrium position. The block is pulled to a distance of 8 cm and released.The kinetic energy of the block when it is a distance of 3 cm from the mean position is
11
The ratio of the radii of a planet and the earth is $1: 2$, the ratio of their mean densities is $4: 1$. If the acceleration due to gravity on the surface of the earth is $9.8 \mathrm{~ms}^{-2}$, then the acceleration due to gravity on the surface of the planet is
12
A wire of cross-sectional area $10^{-6} \mathrm{~m}^{2}$ is elongated by $0.1 \%$ when the tension in it is 1000 N . The Young's modulus of the material of the wire is (assume radius of the wire is constant)
13
The work done in blowing a soap bubble of volume $V$ is $W$. The work done in blowing the bubble of volume 2 V from the same soap solution is
14
Three identical vessels are filled up to the same height with three different liquids $A, B$ and $C$ of densities $\rho_{A} \rho_{B}$ and $\rho_{C}$, respectively. If $\rho_{A} > \rho_{B} > \rho_{C}$, then the pressure at the bottom of the vessels is
15
Steam of mass 60 g at a temperature $100^{\circ} \mathrm{C}$ is mixed with water of mass 360 g at a temperature $40^{\circ} \mathrm{C}$. The ratio of the masses of steam and water in equilibrium is (Latent heat of steam is $540 \mathrm{cal} \mathrm{g}^{-1}$ and specific heat capacity of water is $1 \mathrm{cal} \mathrm{g}^{-1}{ }^{\circ} \mathrm{C}^{-1}$ )
16
The temperature difference between the ends of two cylindrical rods $A$ and $B$ of the same material is $2: 3$. In steady state the ratio of the rates of flow of heat through the rods $A$ and $B$ is $5: 9$. If the radii of the rods $A$ and $B$ are in the ratio $1: 2$, then the ratio of lengths of the rods $A$ and $B$ is
17
When $Q_{1}$ amount of heat supplied to a monoatomic gas, the work done by the gas is $W$. When $Q_{2}$ amount of heat is supplied to a diatomic gas, the work done by the gas is $2 W$. Then, $Q_{1}: Q_{2}$.
18
The temperature at which the rms speed of oxygen molecules is $75 \%$ or rms speed of nitrogen molecules at a temperature of $287^{\circ} \mathrm{C}$
19
The path difference between two particles of a sound wave is 50 cm and the phase difference between them is $1.8 \pi$. If the speed of sound in air is $340 \mathrm{~ms}^{-1}$, the frequency of the sound wave is
20
A source at rest emits sound waves of frequency 102 Hz . Two observers are moving away from the source of sound in opposite directions each with a speed of $10 \%$ of the speed of sound. The ratio of the frequencies of sound heard by the observes is
21
The power of a thin convex lens placed in air is +4 D . The refractive index of the material of the convex lens is $\frac{3}{2}$. If this convex lens is immersed in a liquid of refractive index $\frac{5}{3}$, then
22
The refractive index of the material of a small angled prism is 1.6. If the angle of minimum deviation is $4.2^{\circ}$, the angle of the prism is
23
The Brewster angle for air to glass transition of light is (Refractive index of glass $=15$ )
24
A proton and an $\alpha$-particle are both accelerated from rest in a uniform electric field. The ratio of work done by the electric field on the proton and the $\alpha$-particle in a given time is
25
Two capacitors of capacitances $1 \mu \mathrm{~F}$ and $2 \mu \mathrm{~F}$ can separately withstand potentials of 6 kV and 4 kV respectively. The total potential, they together can withstand when they are connected in series is
26
The resistance of a wire is $2.5 \Omega$ at a temperature 373 K . If the temperature coeffficient of resistance of the material of the wire is $3.6 \times 10^{-3} \mathrm{~K}^{-1}$, its resistance at a temperatrue 273 K is nearly.
27
When two identical resistors are connected in series to an ideal cell, the current through each resistor is 2 A . If the resistors are connected in parallel to the cell, the current through each resistor is
28
An electron falling freely under the influence of gravity enters a uniform magnetic field directed towards south. The electron is initially deflected towards
29
Two long straight parallel wires $A$ and $B$ separated by 5 m carry currents 2 A and 6 A respectively in the same direction. The resultant magnetic field due to the two wires at a point of 2 m distance from the wire $A$ in between the two wires is
30
A short bar magnet placed in a uniform magnetic field making an angle with the field experiences a torque.If the angle made by the magnet with field is changed from $30^{\circ}$ to $45^{\circ}$, the torque on the magnet
31
The mutual inductance of two coils is 8 mH . The current in one coil changes according to the equation $I=12 \sin 100 t$, where $I$ is in ampere and $t$ is time in second. The maximum value of emf induced in the second coil is
32
An inductor of inductive reactance $R$, a capacitor of capacitive reactance $2 R$ and a resistor of resistance $R$ are connected in series to an AC source. The power factor of the series $L-C-R$ circuit is
33
The efficiency of a bulb of power 60 W is $16 \%$. The peak value of the electric field produced by the electromagnetic radiation from the bulb at a distance of 2 m from the bulb is $\left(\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} \mathrm{Nm}^{2} \mathrm{C}^{-2}\right)$
34
The work function of a photosensitive metal surface is 1.1 eV . Two light beams of energies 1.5 eV and 2 eV incident on the metal surface. The ratio of the maximum velocities of the emitted photoelectrons is
35
The ground state energy of hydrogen atom is -13.6 cv . The potential energy of the electron in the first excited state of hydrogen is
36
After the decay of a single $\beta$-particle, the parent and daughter nuclei are
37
$\mathrm{A}_{92} \mathrm{U}^{238}$ nucleus decays to a ${ }_{82} \mathrm{~Pb}^{214}$ nucleus. The number of $\alpha$ and $\beta^{-}$particles emitted are
38
In an $n$-type semiconductor, electrons are majority charge carriers and holes are minority charge carriers. The charge of an $n$-type semiconductor is
39
The region in the output voltage versus input voltage graph where a transistor can be used as an amplifier is
40
For an amplitude modulated wave, the maximum and minimum amplitudes are found to be 10 V and 2 V respectively. Then, the modulation index is
1
TG EAPCET 2024 (Online) 10th May Evening Shift
MCQ (Single Correct Answer)
+1
-0
If three numbers are randomly selected from the set $\{1,2,3, \ldots \ldots 50\}$, then the probability that they are in arithmetic progression is
2
TG EAPCET 2024 (Online) 10th May Evening Shift
MCQ (Single Correct Answer)
+1
-0
The probability that exactly 3 heads appear in six tosses of an unbiased coin, given that first three tosses resulted in 2 or more heads is
3
TG EAPCET 2024 (Online) 10th May Evening Shift
MCQ (Single Correct Answer)
+1
-0
A student has to write the words ABILITY, PROBABILITY, FACILITY, MOBILITY. He wrote one word and erased all the letters in it except two consecutive letters. If 'LI' is left after erasing then the probability that the boy wrote the word PROBABILITY is
4
TG EAPCET 2024 (Online) 10th May Evening Shift
MCQ (Single Correct Answer)
+1
-0
Two cards are drawn at random one after the other with replacement from a pack of playing cards. If $X$ is the random variable denoting the number of ace cards drawn, then the mean of the probability distribution of X is
Subject
Chemistry
40
Mathematics
80
Physics
40
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