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TG EAPCET 2025 (Online) 4th May Evening Shift

JEE 2025 Previous Year

3 hDuration
159Total Marks
158Questions
3Sections
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Paper Structure

Chemistry

Chemistry

Q1. mcq single +1 / 0
The pair of ions with paramagnetic nature and same number of electrons is
Q2. mcq single +1 / 0
The polymer chains are held together by hydrogen bonding in a polymer $X$. Polymer $X$ is formed from monomers $Y$ and $Z$. What are $Y$ and $Z$ ?
Q3. mcq single +1 / 0
The number of lone pairs of electrons on the central atom of $\mathrm{XeO}_3, \mathrm{XeOF}_4$ and $\mathrm{XeF}_6$ respectively is
Q4. mcq single +1 / 0
The atomic numbers of the elements $X, Y, Z$ are $a, a+1, a+2$ respectively. $Z$ is an alkali metal. The nature of bonding in the compound formed by $X$ and $Z$ is
Q5. mcq single +1 / 0
The sets of molecules in which central atom has no lone pair of electrons are I. $\mathrm{SnCl}_2, \mathrm{NH}_3, \mathrm{SF}_4$ II. $\mathrm{HgCl}_2, \mathrm{SO}_3, \mathrm{SF}_6$ III. $\mathrm{BeCl}_2 \mathrm{BF}_3, \mathrm{PCl}_5$ IV. $\mathrm{ClF}_3, \mathrm{BrF}_5, \mathrm{XeF}_6$
Q6. mcq single +1 / 0
In hydrogen atom, an electron is transferred from an orbit of radius 1.3225 nm to another orbit of radius 0.2116 nm . What is the energy (in J ) of emitted radiation?
Q7. mcq single +1 / 0
Observe the following statements. Statement -I Rutherford model of an atom cannot explain the stability of an atom. Statement-II The wavelength of X-rays is higher than the wavelength of microwaves. The correct answer is
Q8. mcq single +1 / 0
Composition of siderite ore is
Q9. mcq single +1 / 0
$A \rightarrow P$ is a first order reaction. At 300 K this reaction was started with $[A]=0.5 \mathrm{~mol} \mathrm{~L}^{-1}$. The rate constant of reaction was $0.125 \mathrm{~min}^{-1}$. The same reaction was started separately with $[A]=1 \mathrm{molL}^{-1}$ at 300 K . The rate constant (in $\mathrm{min}^{-1}$ ) now is
Q10. mcq single +1 / 0
100 mL of $0.05 \mathrm{M} \mathrm{Cu}^{2+}$ aqueous solution is added to IL of 0.1 M KI solution. The number of moles of $\mathrm{I}_2$ and $\mathrm{Cu}_2 \mathrm{I}_2$ formed are respectively.
Q11. mcq single +1 / 0
Consider the following set of reactions What are $A$ and $B$ respectively?
Q12. mcq single +1 / 0
What are $X$ and $Y$ respectively in the following reactions?
Q13. mcq single +1 / 0
Observe the following reaction I. $\mathrm{CO}(\mathrm{g})+\mathrm{H}_2(\mathrm{~g}) \xrightarrow{X} \mathrm{HCHO}(\mathrm{g})$ II. $\mathrm{CO}(\mathrm{g})+3 \mathrm{H}_2(\mathrm{~g}) \xrightarrow{\gamma} \mathrm{CH}_4(\mathrm{~g})+\mathrm{H}_2 \mathrm{O}(\mathrm{g})$ The catalysts $X$ and $Y$ in the above reactions are respectively
Q14. mcq single +1 / 0
$$ \text { Observe the following reactions } $$ The correct order of reactivity of $X, Y, Z$ towards $\mathrm{S}_{\mathrm{N}} 1$ reaction is
Q15. mcq single +1 / 0
At $T(\mathrm{~K}), K_p$ value for the reaction, $$ 2 \mathrm{AO}_2(\mathrm{~g})+\mathrm{O}_2(\mathrm{~g}) \rightleftharpoons 2 \mathrm{AO}_3(\mathrm{~g}) \text { is } 4 \times 10^{10}, $$ What is the $K_p^{\prime}$ value for $$ 2 \mathrm{AO}_2(\mathrm{~g})+\frac{3}{2} \mathrm{O}_2 \rightleftharpoons 3 \mathrm{AO}_3(\mathrm{~g}) \text { at } T(\mathrm{~K}) $$
Q16. mcq single +1 / 0
Identify the correct orders regarding atomic radii (i) $\mathrm{Cl}>\mathrm{F}>\mathrm{Li}$ (ii) $\mathrm{P}>\mathrm{C}>\mathrm{N}$ (iii) $\mathrm{Tm}>\mathrm{Sm}>\mathrm{Eu}$ (iv) $\mathrm{Sr}>\mathrm{Ca}>\mathrm{Mg}$
Q17. mcq single +1 / 0
$$ \text { Match the following } $$ $$ \begin{array}{cccc} \hline & \text { List-I (Elements) } & & \text { List-II (Group) } \\ \hline \text { A } & \mathrm{Mn}, \mathrm{Tc}, \mathrm{Re} & \text { I } & 12 \\ \hline \text { B } & \mathrm{Zn}, \mathrm{Cd}, \mathrm{Hg} & \text { II } & 4 \\ \hline \text { C } & \mathrm{Ti}, \mathrm{Zr}, \mathrm{Hf} & \text { III } & 17 \\ \hline \text { D } & \mathrm{Ga}, \mathrm{In}, \mathrm{Tl} & \text { IV } & 7 \\ \hline & & \text { V } & 13 \\ \hline \end{array} $$ The correct answer is
Q18. mcq single +1 / 0
The $C_p$ of an ideal gas is $10314 \mathrm{Jmol}^{-1} \mathrm{~K}^{-1}$. One mole of this gas is expanded against a constant pressure of $p \mathrm{~atm}$. The change in temperature during expansion is 1.0 K . The value of $q$ (in J ) and $\Delta H$ (in $\mathrm{Jmol}^{-1}$ ) are respectively.
Q19. mcq single +1 / 0
The condensed, bond line and complete formulae of $n$-butane are respectively.
Q20. mcq single +1 / 0
In which of the following reactions, hydrogen is evolved? I. Reaction of sodium borohydride with iodine II. Oxidation of diborane III. Reaction of boron trifluoride with sodium hydride IV. Hydrolysis of diborane
Q21. mcq single +1 / 0
Which of the following statements is not correct?
Q22. mcq single +1 / 0
Which of the following gives more number of oxides on reacting with HCl ?
Q23. mcq single +1 / 0
Which of the following statements is not correct regarding the gas evolved by the reaction of dilute HCl on $\mathrm{CaCO}_3$ ?
Q24. mcq single +1 / 0
$$ \text { Observe the following complex ions } $$ $$ \begin{array}{cccc} \hline\left[\mathrm{Mn}(\mathrm{CN})_6\right]^{3-} & {\left[\mathrm{Fe}(\mathrm{CN})_6\right]^{3-}} & {\left[\mathrm{CoF}_6\right]^{3-}} & {\left[\mathrm{Co}\left(\mathrm{C}_2 \mathrm{O}_4\right)_3\right]^{3-}} \\ \hline A & B & C & D \\ \hline \end{array} $$ Identify the option in which the unpaired electrons in the complex ions are in correct increasing order
Q25. mcq single +1 / 0
Observe the following statements Statement I The carbon containing components of photochemical smog are acrolein, methanal and PAN. Statement II The number of greenhouse gases in the list given below is 5 . $$ \mathrm{CH}_4, \mathrm{CO}_2, \mathrm{NO}, \mathrm{H}_2 \mathrm{O}(l), \mathrm{H}_2 \mathrm{O}(g), \mathrm{O}_2, \mathrm{O}_3 $$ The correct answer is
Q26. mcq single +1 / 0
The chemical $X$ is used in the prevention of heart attack. The structure of $X$ is
Q27. mcq single +1 / 0
A sample of water contains $\mathrm{Mg}\left(\mathrm{HCO}_3\right)_2$ and $\mathrm{Ca}\left(\mathrm{HCO}_3\right)_2$ On boiling this water, these hydrogen carbonates are removed as precipitates. The precipitates are
Q28. mcq single +1 / 0
Which of the following statements is not correct?
Q29. mcq single +1 / 0
An alochol $X\left(\mathrm{C}_5 \mathrm{H}_{12} \mathrm{O}\right)$ produces turbidity instantly with conc. $\mathrm{HCl} / \mathrm{ZnCl}_2$. Isomer $(Y)$ of $X$ undergoes dehydration with conc. $\mathrm{H}_2 \mathrm{SO}_4$ at $443 \mathrm{~K}, X$ and $Y$ respectively are
Q30. mcq single +1 / 0
$$ \text { What are } X \text { and } Y \text { in the following reaction sequence? } $$ $$ \text { Isopentane } \xrightarrow{\mathrm{KMnO}_4} X \xrightarrow[358 \mathrm{~K}]{20 \% \mathrm{H}_3 \mathrm{PO}_4} Y $$
Q31. mcq single +1 / 0
' $X$ ' is the isomer of $\mathrm{C}_6 \mathrm{H}_{14}$. It has four primary carbons and two tertiary carbons, ' $X$ ' can be prepared from which of the following reactions?
Q32. mcq single +1 / 0
$$ \begin{aligned} &\text { What are } X \text { and } Y \text { in the following reaction sequence? }\\ &\mathrm{C}_5 \mathrm{H}_{12} \mathrm{O} \xrightarrow[573 \mathrm{~K}]{\mathrm{Cu}} \mathrm{C}_5 \mathrm{H}_{10} \xrightarrow[\text { (ii) } \mathrm{Zn}+\mathrm{H}_2 \mathrm{O}]{\text { (i) } \mathrm{O}_3} X+Y \end{aligned} $$
Q33. mcq single +1 / 0
The isobars of one mole of an ideal gas were obtained at three different pressure ( $p_1, p_2$ and $p_3$ ). The slopes of these isobars are $m_1, m_2$ and $m_3$ respectively. If $p_1 < p_2 < p_3$, then the correct relation of the slopes is
Q34. mcq single +1 / 0
Amino acid ' $X$ ' contains phenolic hydroxy group and amino acid ' $Y$ ' contains amide group. ' $X$ ' and ' $Y$ ' respectively are
Q35. mcq single +1 / 0
A metal ( $M$ ), crystallises in fcc lattice with edge length of $4.242 \mathop {\rm{A}}\limits^{\rm{o}}$. What is the radius of $M$ atom (in $\mathop {\rm{A}}\limits^{\rm{o}}$ )?
Q36. mcq single +1 / 0
The order of negative standard potential values of Li, $\mathrm{Na}, \mathrm{K}$ is
Q37. mcq single +1 / 0
At 298 K the equilibrium constant for the reaction $M(s)+2 \mathrm{Ag}^{+}(a q) \longrightarrow M^{2+}(a q)+2 \mathrm{Ag}(s)$ is $10^{15}$. What is the $E_{\text {cell }}^{\ominus}$ (in V) for this reaction? $$ \left(\frac{2.303 R T}{F}\right)=0.06 \mathrm{~V} $$
Q38. mcq single +1 / 0
Toluene on reaction with reagent $A$ gives $X$. This $(X)$ forms 2, 4 dinitrophenylhydrazone and reduces ammoniacal silver nitrate solution. Reaction of toluene with another reagent $B$ forms $Y$, which dissolves in $\mathrm{NaHCO}_3$ with evolution of $\mathrm{CO}_2$. What are $A$ and $B$ respectively?
Q39. mcq single +1 / 0
$$ \text { Observe the following set of reactions } $$ $$ \text { What are } X, Y \text { and } Z \text { respectively } $$
Mathematics

Mathematics

Q1. mcq single +1 / 0
The mean deviation about median of the numbers $3 x, 6 x, 9 x, \ldots .81 x$ is 91 , then $|x|=$
Q2. mcq single +1 / 0
Consider the following Assertion (A) The two lines $\mathbf{r}=\mathbf{a}+t(\mathbf{b})$ and $\mathbf{r}=\mathbf{b}+s(\mathbf{a})$ intersect each other. Reason (R) The shortest distance between the lines $\mathbf{r}=\mathbf{p}+t(\mathbf{q})$ and $\mathbf{r}=\mathbf{c}+s(\mathbf{d})$ is equal to the length of projection of the vector ( $\mathbf{p}-\mathbf{c}$ ) on ( $\mathbf{q} \times \mathbf{d}$ ) The correct answer is
Q3. mcq single +1 / 0
Two adjacent sides of a triangle are represented by the vectors $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $2 \sqrt{3} \hat{\mathbf{i}}-2 \sqrt{3} \hat{\mathbf{j}}+\sqrt{3} \hat{\mathbf{k}}$. Then, the least angle of the triangle and perimeter of the triangle are respectively.
Q4. mcq single +1 / 0
$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three non- coplanar and mutually perpendicular vectors of same magnitude $K . r$ is any vectors satisfying $\mathbf{a} \times((\mathbf{r}-\mathbf{b}) \times \mathbf{a})+\mathbf{b} \times((\mathbf{r}-\mathbf{c}) \times \mathbf{b})+\mathbf{c} \times((\mathbf{r}-\mathbf{a}) \times \mathbf{c})=\mathbf{0}$, then $\mathbf{r}=$
Q5. mcq single +1 / 0
In a quadrilateral $A B C D, \mathbf{A}=\frac{2 \pi}{3}$ and $A C$ is the bisector of angle $\mathbf{A}$. If $15|\mathbf{A C}|=5|\mathbf{A D}|=3|\mathbf{A B}|$, then angle between $\mathbf{A B}$ and $\mathbf{B C}$ is
Q6. mcq single +1 / 0
A plane $\pi_1$ contains the vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$. Another plane $\pi_2$ contains the vectors $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}$ and $3 \hat{\mathbf{i}}+2 \hat{\mathbf{k}}$. $\mathbf{a}$ is a vectors parallel to the line of intersection of $\pi_1$ and $\pi_2$. If the angle $\theta$ between $\mathbf{a}$ and $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is acute, then $\theta=$
Q7. mcq single +1 / 0
If $a, b$ are real numbers and $\alpha$ is a real roots of $x^2+12+3 \sin (a+b x)+6 x=0$, then the value of $\cos (a+b \alpha)$ for the least positive value of $a+b \alpha$ is
Q8. mcq single +1 / 0
In possion distribution, if $\frac{P(x=5)}{P(X=2)}=\frac{1}{7500}$ and $\frac{P(X=5)}{P(X=3)}=\frac{1}{500}$, then the mean of the distribution is
Q9. mcq single +1 / 0
In a shelf there are three mathematics and two physics books. A student takes a book randomly. If he randomly takes, successively for three time by replacing the book already taken every time, then the mean of the number of mathematics books which is treated as random variable is
Q10. mcq single +1 / 0
Functions are formed from the set $A=\left\{a_1, a_2, a_3\right\}$ to another set $B=\left\{b_1, b_2, b_3, b_4, b_5\right\}$. If a function is selected at random, then probability, that it is a non-one function is
Q11. mcq single +1 / 0
There are two boxes each containing 10 balls. In each box, few of them are black balls and rest are white. A ball is drawn at random from one of the boxes and found that it is black. If the probability that the black ball drawn is from the second box is $\frac{1}{5}$, then number of black balls in the first box is
Q12. mcq single +1 / 0
$A$ and $B$ are two events of a random experiment such that $P(B)=0.4, P(A \cap \bar{B})=0.5, P(A \cup B)+P\left(\frac{B}{A \cup \bar{B}}\right)=1.15$ then $P(A)=$
Q13. mcq single +1 / 0
If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+\frac{a}{2} x+b=0$ and $(\alpha-\beta)(\alpha-\gamma),(\beta-\alpha)(\beta-\gamma),(\gamma-\alpha),(\gamma-\beta)$ are the roots of the equation $(y+a)^3+K(y+a)^2+L=0$, then $\frac{L}{K}=$
Q14. mcq single +1 / 0
If $\tan \left(\frac{\pi}{4}+\frac{\alpha}{2}\right)=\tan ^3\left(\frac{\pi}{4}+\frac{\beta}{2}\right)$, then $\frac{3+\sin ^2 \beta}{1+3 \sin ^2 \beta}=$
Q15. mcq single +1 / 0
If $\cos \alpha=\frac{l \cos \beta+m}{l+m \cos \beta}$, then $\left(\frac{\tan \frac{\alpha}{2}}{\tan \frac{\beta}{2}}\right)^2=$
Q16. mcq single +1 / 0
If $P=\sin \frac{2 \pi}{7}+\sin \frac{4 \pi}{7}+\sin \frac{8 \pi}{7}$ and $Q=\cos \frac{2 \pi}{7}+\frac{4 \pi}{7}+\cos \frac{8 \pi}{7}$, then the point $(P, Q)$ lies on the circle of radius
Q17. mcq single +1 / 0
Let $f:[-1,2] \rightarrow R$ be defined by $f(x)=\left[x^2-3\right]$ where $[$. denotes greatest integer function, then the number of points of discontinuity for the function $f$ in $(-1,2)$ is
Q18. mcq single +1 / 0
The set of all values of $x$ for which $f(x)=\| x|-1|$ is differentiable is
Q19. mcq single +1 / 0
$$ \mathop {\lim }\limits_{x \to 0} \frac{\sqrt{\cos x}-\sqrt[3]{\cos x}}{\sin ^2 x}= $$
Q20. mcq single +1 / 0
If $f(x)=\left\{\begin{array}{cc}x^2\left|\cos \frac{\pi}{2}\right|, & x \neq 0 \\ 0, & x=0\end{array}\right.$, then at $x=2, f(x)$ is
Q21. mcq single +1 / 0
$f(x)=x^2-2(4 k-1) x+g(k)>0, \forall x \in R$ and for $k \in(a, b)$. If $g(k)=15 k^2-2 k-7$, then
Q22. mcq single +1 / 0
Consider all functions given in List I in the interval [1,3]. The list II has the value of ' $c$ ' obtained by applying Lagrange's mean value theorem on the function of List I . Match the function and values of ' c ' $$ \begin{array}{llll} \hline & \text { List I } & & \text { List II } \\ \hline \text { A } & |x-1| & \text { I } & 2 \log \left(e^3+e^2\right) \\ \hline \text { B } & \log x & \text { II } & 2 \\ \hline \text { C } & x^2+x+1 & \text { III } & \log _3 e^2 \\ \hline \text { D } & e^x & \text { IV } & \sqrt{2} \\ \hline & & \text { V } & \log \left(\frac{e^3-e}{2}\right) \\ \hline \end{array} $$
Q23. mcq single +1 / 0
If the percentage error in the radius of a circle is 3 , then the percentage error in its area is
Q24. mcq single +1 / 0
The function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ where $a>0$ attains its local maximum and local minimum at $p$ and $q$ respectively. If $p^2=q$, then $a=$
Q25. mcq single +1 / 0
For the curve $\frac{x^n}{a^n}+\frac{y^n}{b^n}=2,(n \in N$ and $n>1)$ the line $\frac{x}{a}+\frac{y}{b}=2$ is
Q26. mcq single +1 / 0
The height of a cone with semi-vertical angle $\frac{\pi}{3}$ is increasing at the rate of 2 units $/ \mathrm{min}$. The rate at which the radius of the cone is to be decreased so as to have a fixed volume always is
Q27. mcq single +1 / 0
If local maximum of $f(x)=\frac{a x+b}{(x-1)(x-4)}$ exists at $(2,-1)$, then $a+b=$
Q28. mcq single +1 / 0
The area of the region bounded by $y=x^3, X$-axis, $x=-2$ and $x=4$ is
Q29. mcq single +1 / 0
The number of real values of ' $a$ ' for which the system of equations $2 x+3 y+a z=0, x+a y-2 z=0$ and $3 x+y+3 z=0$ has non-trivial solution is
Q30. mcq single +1 / 0
A is a $3 \times 3$ matrix satisfying $A^3-5 A^2+7 A+I=0$ If $A^5-6 A^4+12 A^3-6 A^2+2 A+2 I=l A+m I$, then $l+m=$
Q31. mcq single +1 / 0
If $A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & x & 1\end{array}\right], A^{-1}=\frac{1}{2}\left[\begin{array}{ccc}1 & -1 & 1 \\ -8 & 6 & 2 y \\ 5 & -3 & 1\end{array}\right]$, then the point $(x, y)$ lies on the curve represented by the equation.
Q32. mcq single +1 / 0
Consider a homogeneous system of three linear equations in three unknowns represented by $A X=0$. If $X=\left[\begin{array}{c}l \\ m \\ 0\end{array}\right], l \neq 0, m \neq 0, l, m \in R$ represents an infinite number of solutions of this system, then rank of $A$ is
Q33. mcq single +1 / 0
Let $A(\alpha, 4,7)$ and $B(3, \beta, 8)$ be two points in space. If $Y Z$ plane and $Z X$-plane respectively divide the line segment joining the points $A$ and $B$ in the ratio $2: 3$ and $4: 5$, then the point $C$ which divides $A B$ in the ratio $\alpha: \beta$ externally is
Q34. mcq single +1 / 0
If the plane $-4 x-2 y+2 z+\alpha=0$ is at a distance of two units from the plane $2 x+y-z+1=0$, then the product of all the possible values of $\alpha$ is
Q35. mcq single +1 / 0
The direction ratios of the line bisecting the angle between the $X$-axis and the line having direction ratios $(3,-1,5)$ are
Q36. mcq single +1 / 0
If the angle between the tangents drawn to the parabola $y^2=4 x$ from the points on the line $4 x-y=0$ is $\frac{\pi}{3}$, then the sum of the abscissae of all such points is
Q37. mcq single +1 / 0
The normal at a point on the parabola $y^2=4 x$ passes through a point $P$. Two more normals to this parabola also pass through $P$. If the centroid of the triangle formed by the feet of these three normals is $G(2,0)$, then the abscissa of $P$ is
Q38. mcq single +1 / 0
The number of real solution of $\tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4}$ is
Q39. mcq single +1 / 0
Consider the following Assertion $$ \begin{aligned} & \text { (A) } \begin{array}{r} \sqrt{x-3}\left(\sin ^{-1}(\log x)+\cos ^{-1}\right. \\ (\log x) d x=\frac{\pi}{3}(x-3)^{3 / 2}+c \end{array} \end{aligned} $$ Reason $(\mathrm{R}) \sin ^{-1}(f(x))+\cos ^{-1}(f(x))=\frac{\pi}{2},|f(x)|<1$ The correct answer is
Q40. mcq single +1 / 0
Numerically greatest term in the expansion of $(3 x-4 y)^{23}$ when $x=\frac{1}{6}$ and $y=\frac{1}{8}$ is
Q41. mcq single +1 / 0
Let $K$ be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[3]{3})^{6144}$. If the coefficient of $x^P(P \in N)$ in the expansion of $\frac{1}{(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}$ is $\alpha_p$, then $\alpha_k-\alpha_{k+1}-\alpha_{k-1}=$
Q42. mcq single +1 / 0
Two non-parallel sides of a rhombus are parallel to the lines $x+y-1=0$ and $7 x-y-5=0$. If $(1,3)$ is the centre of the rhombus and one of its vertices $A(\alpha, \beta)$ lies on $15 x-5 y=6$, then one of the possible values of $(\alpha+\beta)$ is
Q43. mcq single +1 / 0
If the equations $3 x^2+2 h x y-3 y^2=0$ and $3 x^2+2 h x y-3 y^2+2 x-4 y+c=0$ represent the four sides of a square, then $\frac{h}{c}=$
Q44. mcq single +1 / 0
Two families of lines are given by $a x+b y+c=0$ and $4 a^2+9 b^2-c^2-12 a b=0$. Then, the line common to both the families is
Q45. mcq single +1 / 0
$A(2,0), B(0,2), C(-2,0)$ are three points. Let $a, b, c$ be the perpendicular distances from a variable point $P$ on to the lines $A B, B C$ and $C A$ respectively. If $a, b, c$ are in arithmetic progression, then the locus of $P$ is
Q46. mcq single +1 / 0
If $\omega \neq 1$ is a cube root of unity, then one root among the 7th roots of $(1+\omega)$ is
Q47. mcq single +1 / 0
If the eight vertices of a regular octagon are given by the complex number $\frac{1}{x_j-2 i}(j=1,2,3,4,5,6,7,8)$, then the radius of the circumcircle of the octagon is
Q48. mcq single +1 / 0
If $\left|Z_1-3-4 i\right|=5$ and $\left|Z_2\right|=15$, then the sum of the maximum and minimum values of $\left|Z_1-Z_2\right|$ is
Q49. mcq single +1 / 0
If $1+2 i$ is a root of the equation $x^4-3 x^3+8 x^2-7 x+5=0$, then sum of the squares of the other roots is
Q50. mcq single +1 / 0
If $Z=r(\cos \theta+i \sin \theta),\left(\theta \neq-\frac{\pi}{2}\right)$ is solution of $x^3=i$, then $r^9(\cos \theta+i \sin \theta)^9=x^{3-}=i$
Q51. mcq single +1 / 0
The number of common tangents that can drawn to the curves $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $x^2+y^2=16$ is
Q52. mcq single +1 / 0
Consider the following statements Statement $\mathrm{I} \cosh ^{-1} x=\tanh ^{-1} x$ has no solution Statement II $\cosh ^{-1} x=\operatorname{coth}^{-1} x$ has only one solution The correct answer is
Q53. mcq single +1 / 0
If $2^{4 n+3}+3^{3 n+1}$ is divisible by $P$ for all natural numbers $n$, then $P$ is
Q54. mcq single +1 / 0
A function $f: R \rightarrow R$ defined by $$ f(x)=\left\{\begin{array}{c} 2 x+3, x \leq \frac{4}{3} \\ -3 x^2+8 x, x>\frac{4}{3} \end{array}\right. \text { is } $$
Q55. mcq single +1 / 0
The domain and range of $f(x)=\frac{1}{\sqrt{|x|-x^2}}$ are $A$ and $B$ respectively. Then $A \cup B=$
Q56. mcq single +1 / 0
The differential equation of the family of all circles of radius ' $a$ ' is
Q57. mcq single +1 / 0
If the general solution of $\left(1+y^2\right) d x=\left(\tan ^{-1} y-x\right) d y$ is $x=f(y)+c e^{-\tan ^{-1} y}$, then $f(y)=$
Q58. mcq single +1 / 0
The radius of the circle having three chords along Y-axis, the line $y=x$ and the line $2 x+3 y=10$
Q59. mcq single +1 / 0
If $P\left(\frac{7}{5}, \frac{6}{5}\right)$ is the inverse point of $A(1,2)$ with respect to a circle with centre $C(2,0)$, then the radius of that circle is
Q60. mcq single +1 / 0
If a tangent of the circle $x^2+y^2+2 x+2 y+1=0$ is radical axis of the circles $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y^2+3 x+8 y+2 c=0$, then
Q61. mcq single +1 / 0
The equation of the circle which touches the circle $S \equiv x^2+y^2-10 x-4 y+19=0$ at the point $(2,3)$ internally and having radius equal to half of the radius of the circle $S=0$ is
Q62. mcq single +1 / 0
Among the chords of the circle $x^2+y^2=75$, the number of chords having their mid-points on the line $x=8$ and having their slopes as integers is
Q63. mcq single +1 / 0
If the circle $S=0$ intersect the three circle $$ \begin{aligned} & S_1 \equiv x^2+y^2+4 x-7=0 \\ & S_2 \equiv x^2+y^2+y=0 \text { and } S_3 \equiv x^2+y^2+\frac{3}{2} x+\frac{5}{2} y-\frac{9}{2}=0 \end{aligned} $$ orthogonally, then radical axis of $S=0$ and $S_1=0$ is
Q64. mcq single +1 / 0
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(b>a)$ is an ellipse with eccentricity $\frac{1}{\sqrt{2}}$. If the angle of intersection between the ellipse and parabola $y^2=4 a x$ is $\theta$, then the coordinates of the point $\frac{2 \theta}{3}$ on the ellipse is
Q65. mcq single +1 / 0
When the coordinate axes are rotated about the origin through an angle $\frac{\pi}{4}$ in the positive direction, the equation $a x^2+2 h x y+b y^2=c$ is transformed to $25 x^2+9 y^2=225$, then $(a+2 h+b-\sqrt{c})^2=$
Q66. mcq single +1 / 0
The circumcenter of the equilateral triangle having the three points $\theta_1, \theta_2, \theta_3$ lying on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as its vertices is $(r, s)$. Then, the average of $\cos \left(\theta_1-\theta_2\right)$, $\cos \left(\theta_2-\theta_3\right)$ and $\cos \left(\theta_3-\theta_1\right)$ is
Q67. mcq single +1 / 0
If $x=t-\sin t, y=1-\cos t$ and $\frac{d^2 y}{d x^2}=-1$ at $t=k, k>0$ then $\lim _{i \rightarrow K} \frac{y}{x}=$
Q68. mcq single +1 / 0
The number of positive integral solution of $\frac{1}{x}+\frac{1}{y}=\frac{1}{2025}$ is
Q69. mcq single +1 / 0
The number of positive integral solutions of $x y z=60$ is
Q70. mcq single +1 / 0
All the letters of the word MOTHER are arranged in all possible ways and the resulting words (may or may not have meaning) are arranged as in the dictionary. The number of words that appear after the word MOTHER is
Q71. mcq single +1 / 0
If $\int_0^{\frac{\pi}{2}} \tan ^{14}\left(\frac{x}{2}\right) d x=2\left[\sum_{n=1}^7 f(n)-\frac{\pi}{4}\right]$, then $f(n)=$
Q72. mcq single +1 / 0
If the angular bisector of the angle $A$ of the $\triangle A B C$ meets its circumcircle at $E$ and the opposite side $B C$ at $D$, then $D E \cos \frac{A}{2}=$
Q73. mcq single +1 / 0
$y-x=0$ is the equation of a side of a $\triangle A B C$. The orthocentre and circumcentre of the $\triangle A B C$ are respectively $(5,8)$ and $(2,3)$. The reflection of orthocentre with respect to any side of the triangle lies on its circumcircle. Then, the radius of the circumcircle of the triangle is
Q74. mcq single +1 / 0
In a $\triangle A B C, a=5, b=4$ and $\tan \frac{C}{2}=\sqrt{\frac{7}{9}}$, then its inradius $r=$
Q75. mcq single +1 / 0
$$ \begin{aligned} & \int \frac{3 x+2}{4 x^2+4 x+5} d x=A \log \\ & \left(4 x^2+4 x+5\right)+B \tan ^{-1}\left(\frac{2 x+1}{2}\right)+C, \text { then } A+B= \end{aligned} $$
Q76. mcq single +1 / 0
If $\frac{3 x+1}{(x-1)^2\left(x^2+1\right)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C x+D}{x^2+1}$, then $2(A-C+B+D)=$
Q77. mcq single +1 / 0
$$ \begin{aligned} I_1 & =\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, I_2 \\ & =\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x, \text { then } I_2-I_1= \end{aligned} $$
Q78. mcq single +1 / 0
If $\int \frac{1}{x} \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} d x=2 f(x)-2 \sin ^{-1} \sqrt{x}+c$, then $f(x)=$
Q79. mcq single +1 / 0
$$ \begin{aligned} &\text { If } y=f(x)^{g(x)} \text { and } \frac{d y}{d x}=y\left[H(x) f^{\prime}(x)+G(x) g^{\prime}(x)\right] \text {, then }\\ &\int \frac{G(x) H(x) f^{\prime}(x)}{g(x)} d x= \end{aligned} $$
Physics

Physics

Q1. mcq single +1 / 0
A steel rod with a circular cross-section of diameter 1 cm and another steel rod with a square cross-section of side 1 cm have equal mass. If the two rods are subjected to same tension, the ratio of the elongations of the two rods is
Q2. mcq single +1 / 0
The distance for which ray optics becomes a good approximation for an aperture of 0.3 cm and a light of wavelength $6000 \mathop {\rm{A}}\limits^{\rm{o}}$ is
Q3. mcq single +1 / 0
The force ( $F$ in newton) acting on a particle of mass 90 g executing simple harmonic motion is given by $F+0.04 \pi^2 y=0$, where $y$ is displacement of the particle in metre. If the amplitude of the particle is $\frac{6}{\pi} \mathrm{~m}$, then the maximum velocity of the particle is
Q4. mcq single +1 / 0
A solid sphere of mass 2 kg and radius 0.5 m is rolling without slipping on a horizontal surface. The ratio of the rotational and translational kinetic energies of the sphere is
Q5. mcq single +1 / 0
If the length of a thin uniform rod is ' $L$ ' and the radius of gyration of the rod about an axis perpendicular to its length and passing through one end is $K$, then $K: L=$
Q6. mcq single +1 / 0
The small energy losses in transformers due to eddy currents can be reduced by
Q7. mcq single +1 / 0
A coil of resistance $16 \Omega$ is placed with its plane perpendicular to a uniform magnetic field whose flux ( $\phi$ in $10^{-3}$ weber) changes with time ( $t$ in second) as $\phi=5 t^2+4 t+2$. The induced current at time $t=6$ second is
Q8. mcq single +1 / 0
The ratio of the average translational kinetic energies of hydrogen and oxygen at the same temperature is
Q9. mcq single +1 / 0
A steel pendulum clock manufactured at $32^{\circ} \mathrm{C}$ and working at $47^{\circ} \mathrm{C}$ is nearly (Coefficient of linear expansion of steel $=12 \times 10^{-6} /{ }^{\circ} \mathrm{C}$ )
Q10. mcq single +1 / 0
A Carnot engine uses diatomic gas as a working substance. During the adiabatic expansion part of the cycle, if the volume of the gas becomes 32 times its initial volume, then the efficiency of the engine is
Q11. mcq single +1 / 0
A metal metre scale that is accurate up to 0.5 mm is made at a temperature of $25^{\circ} \mathrm{C}$. The range of temperatures within which it can be used is (Coefficient of linear expansion of the metal $=10^{-5} /{ }^{\circ} \mathrm{C}$ )
Q12. mcq single +1 / 0
In a photoelectric experiment, the slope of the graph drawn between stopping potential along $Y$-axis and frequency of incident radiation along $X$-axis is (Planck's constant $=6.6 \times 10^{-34} \mathrm{Js}$ )
Q13. mcq single +1 / 0
729 small identical spheres each charged to an electric potential 3V combine to form a bigger sphere. The electric potential of the bigger sphere is
Q14. mcq single +1 / 0
The electrostatic force between two charges kept in air is $F$. If $30 \%$ of the space between the charges is filled with a medium, then the electrostatic force between the charges becomes $\frac{F}{2.56}$. The dielectric constant of the medium is
Q15. mcq single +1 / 0
The force of mutual attraction between any two objects by virtue of their masses is
Q16. mcq single +1 / 0
Which of the following is incorrect about the gravitational force between two bodies?
Q17. mcq single +1 / 0
A ray of light incidents at an angle of $9.3^{\circ}$ on one face of a small angle prism of refracting angle $6^{\circ}$. If the ray of light emerges normally from the second face, the refractive index of the material of the prism is
Q18. mcq single +1 / 0
A straight metal rod of length 6 cm is placed along the principal axis of a concave mirror of focal length 9 cm such that the end of the rod closer to the mirror is at a distance of 15 cm from the pole of the mirror. The length of the image of the rod is
Q19. mcq single +1 / 0
As shown in the figure, a uniform straight wire of length $30 \sqrt{3} \mathrm{~cm}$ is bent in the form of an equilateral triangle $A B C$. A uniform magnetic field $2 T$ is applied parallel to the side $B C$. If the current through the wire is 2 A , the magnitude of the force on the side $A C$ is ( $\bar{B}$ represents the direction of the magnetic field)
Q20. mcq single +1 / 0
A proton moving with a velocity of $8 \times 10^5 \mathrm{~ms}^{-1}$ enters a uniform magnetic fleld normal to the direction of the magnetic field. If the radius of the circular path of the proton in the magnetic field is 8.3 cm , then the magnitude of the magnetic field is (Charge of proton $=1.6 \times 10^{-19} \mathrm{C}$ and mass of the proton $=1.66 \times 10^{-27} \mathrm{~kg}$ )
Q21. mcq single +1 / 0
If the frequencies of the carrier wave and message signal are 1 MHz and 28 kHz respectively, then the frequencies of the side bands are
Q22. mcq single +1 / 0
The maximum length of water column that can stay without falling in a vertically held capillary tube of diameter 1 mm and open at both the ends is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ and surface tension of water $=0.07 \mathrm{Nm}^{-1}$ )
Q23. mcq single +1 / 0
A cube of side 40 cm is floating with $\frac{1}{4}$ th of its volume immersed in water. When a circular disc is placed on the cube, it floats with $\frac{2}{5}$ th of its volume immersed in water. The mass of the disc is
Q24. mcq single +1 / 0
For a particle moving along a straight line path, the displacements in third and fifth seconds of its motion are 10 m and 18 m respectively. The speed of the particle at time $t=4 \mathrm{~s}$ is
Q25. mcq single +1 / 0
When an element ${ }_{90}^{232} \mathrm{Th}$ decays into ${ }_{82}^{208} \mathrm{~Pb}$, the number of $\alpha$ and $\beta^{-}$particles emitted respectively are
Q26. mcq single +1 / 0
During the disintegration of a radioactive nucleus of mass number 208 at rest, two alpha particles each with kinetic energy $E$ are emitted. The total kinetic energy of the emitted alpha particles and the daughter nucleus after the disintegration is
Q27. mcq single +1 / 0
The maximum wavelength of incident radiation required to ionize a hydrogen atom in its ground state is nearly
Q28. mcq single +1 / 0
The vertical displacement ( $y$ in metre) of a projectile in term of its horizontal displacement ( $x$ in metre) is given by $y=\left(\sqrt{3} x-0.2 x^2\right)$. The time of flight of the projectile is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )
Q29. mcq single +1 / 0
The air columns in two tubes closed at one end vibrating in their fundamental modes produce 2 beats per second. The number of beats produced per second when the same tubes are vibrated in their fundamental mode with their both ends open are
Q30. mcq single +1 / 0
A car moving towards a cliff emits sound of frequency ' $n$ '. If the difference in frequencies of the horn and its echo heard by the driver of the car is $10 \%$ of ' $n$ ', then the speed of the car is nearly (Speed of sound in air is $336 \mathrm{~ms}^{-1}$ )
Q31. mcq single +1 / 0
A block of mass $\sqrt{2} \mathrm{~kg}$ is placed on a rough horizontal surface. A force ' $F$ ' acting upwards at an angle of $45^{\circ}$ with the horizontal causes the block to start motion. If the coefficient of static friction between the surface and the block is 0.25 , the magnitude of the force ' $F$ ' is (Acceleration due to gravity $=10 \mathrm{~ms}^{-2}$ )
Q32. mcq single +1 / 0
At a certain place in the magnetic meridian the Earth's magnetic field is twice its vertical component. The ratio of horizontal component of Earth's magnetic field and the total magnetic field of the Earth at the place is
Q33. mcq single +1 / 0
If the kinetic energy of a body moving with a velocity of $(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}) \mathrm{ms}^{-1}$ is 87 J , then the mass of the body is
Q34. mcq single +1 / 0
A body of mass 0.5 kg is supplied with a power ' $P$ ' (in watt) which varies with time ' $f$ ' (in second) as $P=3 t^2+3$. If the velocity of the body at time $t=0$ is zero, then the velocity of the body at time $t=3 \mathrm{~s}$ is
Q35. mcq single +1 / 0
The area of cross-section of a potentiometer wire is $6 \times 10^{-7} \mathrm{~m}^2$. The potential difference per unit length of the potentiometer wire when it is connected to a cell of negligible internal resistance and a resistor in series is $0.15 \mathrm{Vm}^{-1}$. If the current through potentiometer wire is 0.3 A , then the resistivity of the material of the potentiometer wire is
Q36. mcq single +1 / 0
For the circuit shown in the figure, the current through $6 \Omega$ resistor connected between the junctions $A$ and $B$ is
Q37. mcq single +1 / 0
The error in the measurement of force acting normally on a square plate is $3 \%$. If the error in the measurement of the side of the plate is $1 \%$, then the error in the determination of the pressure acting on the plate is
Q38. mcq single +1 / 0
If the electric field of a plane electromagnetic wave is $E_z=60 \sin \left(0.5 \times 10^3 x+1.5 \times 10^{11} t\right) \mathrm{Vm}^{-1}$, then the magnetic field of the wave is
Q39. mcq single +1 / 0
The current amplification factor of a transistor in common emitter configuration is 80 . If the emitter current is 2.43 mA , then the base current is
Q40. mcq single +1 / 0
The negative feedback in an amplifier