Choose the correct statements in reference to the photoelectric effect.

(A) There is no time lag between the striking of light and ejection of electrons from the metal surface.

(B) The number of electrons ejected is independent of the intensity of light.

(C) The elements $\mathrm{K}, \mathrm{Rb}$ and Cs can show photoelectric effect when exposed to the beam of light.

Assertion (A) Mo has the ground state electronic configuration $4 d^5 5 s^1$.

Reason (R) Mo has the highest exchange energy among the second row transition elements.

The correct option among the following is

Which of the following sequence is correct for decreasing order of ionic radius?

The successive ionisation energies (starting from the 1 st ) of an element are $801,2430,3660,25,000$ and $32,800 \mathrm{~kJ} \mathrm{~mol}^{-1}$, respectively. The element is

The compounds with $s p^2$ hybridised central atom among the following are

(A) $\mathrm{H}_2 \mathrm{CO}_3$

(B) $\mathrm{SiF}_4$

(C) $\mathrm{BF}_3$

(D) $\mathrm{HClO}_2$

The hybridisation and shape of $I_3^{-}$ion, respectively, are

1 mole of a real gas is kept at high pressure of 100 bar at 300 K . If van der Waals' constant $b$ is $0.005 \mathrm{~L} / \mathrm{mol}$, what are the values of compressibility factor $Z$ of the gas and \% deviation of volume from ideality?

1 L closed flask contains a mixture of 4 g of methane and 4.4 g of carbon dioxide. The pressure inside the flask at $27^{\circ} \mathrm{C}$ is

[assume ideal behaviour of gases].

Round the number 234555359 to 3 significant figure.

Which of the following are disproportionation reactions?

(A) $\mathrm{Cl}_2+2 \mathrm{NaOH} \longrightarrow \mathrm{NaCl}+\mathrm{NaOCl}+\mathrm{H}_2 \mathrm{O}$

(B) $\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{H}_2 \mathrm{O}+\mathrm{O}_2$

(C) $2 \mathrm{KMnO}_4 \rightarrow \mathrm{~K}_2 \mathrm{MnO}_4+\mathrm{MnO}_2+\mathrm{O}_2$

(D) $3 \mathrm{MnO}_4^{2-}+4 \mathrm{H}^{+} \longrightarrow 2 \mathrm{MnO}_4^{-}+\mathrm{MnO}_2+2 \mathrm{H}_2 \mathrm{O}$

The bond enthalpies of heavy hydrogen, $\mathrm{O}-\mathrm{O}$ and $\mathrm{D}-\mathrm{O}$ are $+400,+498$ and $+490 \mathrm{kJmol}^{-1}$, respectively. The $\Delta_r H^{\circ}$ of the reaction to produce $\mathrm{D}_2 \mathrm{O}$ is

Calculate the value of the equilibrium constant $\left(K_p\right)$ for the reaction of oxygen gas oxidising ammonia gas to nitric oxide and water vapour. The pressure of each gas at equilibrium is 0.5 atm .

Ammonia is a Lewis base because it is

$$ \mathrm{CaC}_2+2 \mathrm{D}_2 \mathrm{O} \longrightarrow " P^{\prime \prime}+\mathrm{Ca}(\mathrm{OD})_2 $$

In the above reaction, product " $P$ " is

Assertion (A) The ionic radii of the alkaline earth metals are smaller than those of alkali metals in the same period.

Reason (R) Alkali metals have higher nuclear charge than that of the alkaline earth metals.

The correct option among the following is

A black coloured element with $n s^2 n p^1$ outer electronic configuration cannot react with air in its crystalline form. However, in amorphous form, it gives an oxide in air which is acidic in nature. Identify the element.

The element that does not show catenation is

From the following compounds, the ones which contain both $s p$ and $s p^2$ hybridised carbons are

1-chloro-3- methylbutane on reaction with zinc and dilute hydrochloric acid gives $\_\_\_\_$ product. as the major

The major products $Y$ and $Z$ in the following reactions are

A solid has a structure in which ' W ' atoms are located at the corners of a cubic lattice, oxygen atoms at the edge centre and

Na atom at the body centre. The formula of the compound is

If 2 g of NaOH is dissolved to make 200 mL solution at $25^{\circ} \mathrm{C}$, the molarity ( $M$ ) at $90^{\circ} \mathrm{C}$ is

A solvent freezes at $17^{\circ} \mathrm{C}$ and its latent heat of fusion is $180 \mathrm{Jg}^{-1}$. The molal depression constant of the solvent is [units of $K_f=\mathrm{K} \mathrm{kg} \mathrm{mol}^{-1}$ ]

The $E^{\circ}$ of $\mathrm{Ce}^{4+} / \mathrm{Ce}^{3+}=1.6 \mathrm{~V}, \mathrm{Fe}^{3+} / \mathrm{Fe}^{2+}=0.76 \mathrm{~V}$ the $E^{\circ}$ of $\mathrm{Fe}^{3+}$ oxidising $\mathrm{Ce}^{3+}$ is

For a reaction, the threshold energy is $75 \mathrm{~kJ} /$ mole. If the internal energy of the reactants is 20 $\mathrm{kJ} /$ mole, the activation energy (in $\mathrm{kJ} /$ mole) is

The process in which colloids, when subjected to DC electric field move towards an electrode is

$$ \text { Match the following } $$

$$ \text { The correct match } $$

In the following, the oxoacid with a peroxy bond is

Chlorine is allowed to react with excess of ammonia. In this, 1 mole of chlorine can oxidise ' $Z$ ' moles of $\mathrm{NH}_3$. ' $Z$ ' is

The correct order of enthalpy of vaporisation of noble gases is

Choose the correct statement.

$\mathrm{Fe}^{3+}$ ion is more stable than $\mathrm{Fe}^{2+}$ ion because

Secondary valences of the following complexes based on their reactions with excess $\mathrm{AgNO}_3$ are

Assertion (A) In aqueous solution, amino acids exist in dipolar ion form.

Reason (R) Most of the naturally occurring amino acids have $L$-configuration.

The correct option among the following is

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

$$ \begin{array}{llll} \hline \text { (A) } & \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}_2 \xrightarrow{\mathrm{Aq} . \mathrm{KOH}} & \text { I } & \mathrm{S}_{\mathrm{N}} 1 \\ \hline \text { (B) } & \left(\mathrm{CH}_3\right)_3 \mathrm{Cl} \xrightarrow{\mathrm{H}_2 \mathrm{O}} & \text { II } & \mathrm{E} 2 \\ \hline \text { (C) } & \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}_2 \xrightarrow{\text { Alc. } \mathrm{KOH}} & \text { III } & \mathrm{E} 1 \\ \hline \text { (D) } & \left(\mathrm{CH}_3\right)_3 \mathrm{COH} \xrightarrow{\mathrm{H}^{+}} & \text {IV } & \mathrm{S}_{\mathrm{N}} 2 \\ \hline \end{array} $$

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

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

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

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

Suitable reagents $X$ and $Y$, respectively, in the following reactions are

Among the compounds

(i) $\mathrm{H}-\mathrm{C} \equiv \mathrm{C}-\mathrm{COOH}$

(ii) $\mathrm{CH}_2=\mathrm{CH}-\mathrm{COOH}$

(iii) $\mathrm{CH}_3-\mathrm{CH}_2 \mathrm{COOH}$ and

(iv) $\mathrm{CH}_3-\mathrm{CH}_2-\mathrm{OH}$

The correct order of acid strength is

What is the IUPAC name of the below given compound?

If $[x]$ represents the greatest integer function, then the set of all real values of $x$ for which $f(x)=\sqrt{\frac{[x]-x}{x-[x]}}$ is real is

If $[x]$ denotes the greatest integer $\leq x$, then the range of the real valued function $f(x)=\frac{1}{\sqrt{x-[x]}}$ is

If $A$ is a $2 \times 2$ matrix such that $\operatorname{det} A=-21$ and trace of $A^3$ is 2024 , then the trace of $A$ is

If $\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$ is a skew-symmetric matrix and $b, c$ and $f$ are non-zero real numbers, then $\frac{b}{c}=$

In the matrix $\left[\begin{array}{ccc}-1 & x & 3 \\ -4 & -5 & -6 \\ -7 & y & 9\end{array}\right]$, if the cofactors of -6 and -7 are respectively 22 and 27 , then $5 x+y=$

Consider the simultaneous linear equations $\beta x+\alpha y-z=-1,3 x-\beta y+\alpha z=0 \alpha x+\beta y+z=1$, In the usual notation used in Crammer's rule, given that $\frac{\Delta_1}{\Delta}=-1, \frac{\Delta_2}{\Delta}=1, \frac{\Delta_3}{\Delta}=2$, then $(\alpha, \beta)=$

If $\left|\begin{array}{cc}2+3 i & i \\ 1-2 i & -i\end{array}\right|=x+i y$, then $x+y=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2 x+2=0$, then $\alpha^{2020}+\beta^{2020}=$

If $z=\frac{-1-i \sqrt{3}}{2}$, then $\sum_{k=1}^{2022}\left(z^k+\frac{1}{z^k}\right)^2=$

Statement I The set of solutions of $|x|^2-4|x|+3<0$ is the interval $(-3,3)$

Statement II If $x<3$ or $x>5$, then $x^2-8 x+15>0$

Which of the above statements is (are) true?

If $6 x-x^2+12$ attains its extreme value $\beta$ at $x=\alpha$, then $\beta=$

Let $a$ be a common root of the equations $x^3-2 x-25 \lambda=0,3 x^3-8 x-\frac{175}{3} \lambda=0$ and $\lambda>0$. Then, $\lambda=$

If the sum of two roots of the equation $x^3-7 p x^2+5 q x-6 r=0$ is zero, then

If $\alpha$ and $\beta$ are the irrational roots of the equation $3 p^2 x^3+p x^2+q x+3=0$ when $p=1$ and $q=-7$, then $|\alpha-\beta|=$

The roots of a cubic equation $f(x)=0$ are diminished by $\frac{-3}{2}$ so, as to remove the term containing $x^2$ and the transformed equation is $8 x^3-54 x-78=0$. Then, the equation $f(x)=0$ is

If ${ }^m P_r-{ }^{(m-1)} p_r=a \cdot{ }^{(m-1)} P_s$, then $a-s=$

The total number of ways of selecting 4 letters from all the letters of the word TSEAMCET is

Numerically greatest term in the expansion of $(2 x-3 y)^{11}$ when $x=\frac{1}{3}$ and $y=\frac{1}{2}$ is

If $\frac{x-2}{x^2(2 x-3)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{2 x-3}$, then $2(A-C)=$

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

If $\sin A=\frac{-7}{25}, \cos B=\frac{8}{17}, A$ does not lie in the 3rd quadrant and $B$ does not lie in the 1st quadrant, then $8 \tan A-5 \cot B=$

If $\sin \theta-\cos \theta=\frac{1}{\sqrt{3}}$, then $\sin (2 \theta)+\cos (4 \theta)+\sin (6 \theta)=$

If $a \tan \alpha+b \tan \beta=(a+b) \tan \left(\frac{\alpha+\beta}{2}\right)$ and $\alpha-\beta \neq 2 n \pi$ then $\frac{\cos \beta}{\cos \alpha}=$

Assertion (A) $\operatorname{coth} x=\frac{1-k}{1+k}(0 < k < 2)$.

Reason (R) The graph of $y=\tanh x$ always lies between the lines $y=-1$ and $y=1$

The correct option among the following is

If $\frac{5 \sinh 2 x}{7+6 \cosh 2 x}=\frac{3}{2}$, then $3 \tanh ^2 x+20 \tanh x=$

In $\triangle A B C$, if $a=7, b=8, \tan C=\frac{3 \sqrt{5}}{2}$ and $C$ is an acute angle, then $c=$

In a $\triangle A B C$, if $\frac{a}{\tan A}=\frac{b}{\tan B}=\frac{c}{\tan C}$, then $\cos ^2 A+\cos ^2 B+\cos ^2 C=$

In $\triangle A B C$, if $a=7, b=10$ and $c=11$, then $\frac{R}{r}=$

Let $A B C$ be a triangle and $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be the position vectors of $A, B$ and $C$, respectively. Let $D$ divides $B C$ in the ratio $3: 1$ internally and $E$ divides $A D$ in the ratio $4: 1$ internally. Let $B E$ meet $A C$ in $F$. If $E$ divides $B F$ in the ratio $3: 2$ internally, then the position vector of $F$ is

If $\alpha, \beta$ and $\gamma$ are real numebrs such that

$$ \begin{aligned} & \left(\frac{7}{3}+\beta\right) \hat{\mathbf{i}}-\hat{\mathbf{j}}+(\alpha+\gamma) \hat{\mathbf{k}} \\ & =\frac{5}{3}(\alpha \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\beta(2 \hat{\mathbf{j}}+\hat{\mathbf{k}})+(\hat{\mathbf{i}}+\gamma \hat{\mathbf{j}}+3 \hat{\mathbf{k}}), \text { then } \\ & 5 \alpha-9 \beta+13 \gamma= \end{aligned} $$

If $\mathbf{r}=(2-\lambda+\mu) \hat{\mathbf{i}}+(1-\mu) \hat{\mathbf{j}}+(2-3 \lambda+2 \mu) \hat{\mathbf{k}}$ is the vector equation of a plane, then the equivalent cartesian equation of the plane is

If $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{x}=\left(\frac{\mathbf{a b}}{|\mathbf{b}|^2}\right) \mathbf{b}, \mathbf{y}=\left(\frac{\mathbf{a b}}{|\mathbf{a}|^2}\right) \mathbf{a}$ and $\theta$ is angle between $\mathbf{a}$ and $\mathbf{b}$, then $x^2+y^2=$

Let $\pi_1$ be a plane passing through the point $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and perpendicular to the vector $-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$. Let the line $L$ passing through the points $3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $-\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ be a normal to the plane $\pi_2$. If the angle between the planes $\pi_1$ and $\pi_2$ is $\theta$, then $\cos \theta=$

Three non-coplanar vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are the coterminous edges of a parallelopiped. If $\mathbf{a}$ and $\mathbf{b}$ determine the base of the parallelopiped, then its height is

Statement I The range of the ungrouped data does not change even if certain intermediate observations are removed

Statement II The value of the mean deviation of an ungrouped data about the median is always less than or equal to the value of the mean deviation computed about any other measure of central tendency

Statement III For a grouped data, range is approximated as the difference between the lower limit of the largest class and the upper limit of the smallest class

The probability of getting a king and a spade card when two cards are drawn simultaneously from a pack of 52 playing card is

Two cards are drawn from a pack of 52 playing cards one after the other. If $p_1$ is the probability of getting a queen in the first draw and a diamond card in the second draw when the first card drawn is replaced and $p_2$ is the probability of the same event when the first card drawn is not replaced. Then $\frac{p_1}{p_2}=$

Bag $A$ contains 4 white and 2 black balls, bag $B$ contains 3 white and 3 black balls and bag $C$ contains 2 white and 4 black balls. If a bag is chosen at random and a ball is chosen at random from it, then the probability that the ball drawn is black is

A random variable $X$ has the following probability distribution

$$ \begin{array}{llllllllll} \hline X=\mathbf{x}_i & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline P\left(X=\mathbf{x}_i\right) & 10 k & 9 k & 8 k & 8 k & 6 k & 5 k & 4 k & 3 k & k \\ \hline \end{array} $$

where $k$ is a real number.

If $A=\left\{x_i \mid x_i\right.$ is a prime number $\}$ and $B=\left\{x_i \mid x_i>5\right\}$ are two events, then $P(A \cup B)=$

If $X$ is a Poisson variate such that $\frac{5}{3} k=P(X=2) =P(X=3)$, then $P(X=5)=$

$A(-4,0)$ and $B(4,0)$ are two fixed points. $C$ and $D$ are two points on $Y$ - axis such that $C D=4$ and $C$ is a point below $D$. Then, the locus of the point of intersection of the lines $A C$ and $B D$ is

By rotating the axes through an angle of $30^{\circ}$ in the anti-clockwise direction about the origin, the equation $4 x^2+12 x y+9 y^2+6 x+9 y+2=0$ becomes $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ becomes, then

In an isosceles triangle the ends of its base are $(2 a, 0),(0, a)$ and one of its two other sides is a horizontal line other than $X$-axis. If the third vertex is $\left(x_1, y_1\right)$, then $x_1+y_1=$

If the lines $L_1 \equiv x-2 y+3=0, L_2 \equiv 2 x+y+1=0$ and $L_3 \equiv 3 x+y+c=0$ are concurrent and $\theta$ is the acute angle between the lines $L_1=0$ and $L_3=0$, then $\tan \theta=$

If the lengths of the perpendiculars drawn from a point $(a, b)$ to the lines $2 x+3 y+4=0$ and $3 x-2 y+4=0$ are same, then the point $(a, b)$ lies on the line

If $3 x+6 y+2=0, x+y+1=0,2 x-y+3=0$ are three given lines, then the point $\left(\frac{-4}{3}, \frac{1}{3}\right)$ is

If $\theta$ is the acute angle between the pair of lines $12 x^2+2 h x y+7 y^2=0$ and $\tan \theta=\frac{8}{19}$, then $h=$

The number of real values of $\alpha$ for which the pair of lines represented by $\left(\alpha^2+12|\alpha|\right) x^2+6 x y+(18-21|\alpha|) y^2=0$ are at right angles to each other, is

The line $x+2 y-c=0$ meets the curve $x^2+y^2-3 x-6 y+3=0$ at two points $P$ and $Q$ and $\angle P O Q=\frac{\pi}{2}$, where $O$ is the origin. Then, $2 c^2-15 c=$

The line $4 x+3 y-4=0$ divides the circumference of a circle in the ratio $1: 2$. If $C(5,3)$ is the centre of that circle, then equation of the circle is

Two sides of a square are along the lines $x=-5$ and $y=4$. The point of intersection of the diagonals is $(3,-4)$. The point of intersection of the tangents drawn to the circumcircle of the square at the two consecutive vertices lying on $x=-5$ is

If $L_1, L_2$ and $L_3$ are the chords of contact of the three points $(2,0),(1,-2)$ and $(4,4)$ respectively with respect to the circle $x^2+y^2=3$, then $L_1, L_2$ and $L_3$ are

The combined equation of the direct common tangents of the circles $x^2+y^2+2 x=0$ and $x^2+y^2-2 y-3=0$

If $(h, k)$ is the centre of the circle which passes through the origin and cuts the circles $x^2+y^2+4 x+6 y+12=0$ and $x^2+y^2+4 x-6 y+9=0$ orthogonally, then $k-2 h=$

If $(-1,-1)$ is the radical centre of the circles $x^2+y^2+2 g x-4 y+4=0, x^2+y^2+6 x+2 f y+12=0$ and $x^2+y^2+10 y+20=0$, then $g-f=$

The equation of the given curve is $x^2-4 x+4 y-8=0$. Match the following

$$ \begin{array}{lll} \hline & \text { List I } & \text { List II } \\ \hline \text { (A) } & \text { Focus } & \text { (I) }(4,2) \\ \hline \text { (B) } & \text { Vertex } & \text { (II) }(3,2) \\ \hline \text { (C) } & \begin{array}{l} \text { One end of the } \\ \text { latusrectum } \end{array} & \text { (III) }(2,3) \\ \hline \text { (D) } & \begin{array}{l} \text { point of intersection of the } \\ \text { axis and directrix } \end{array} & \text { (IV) }(2,4) \\ \hline & & \text { (V) }(2,2) \\ \hline \end{array} $$

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

If one end of a focal chord of the parabola $y^2=\frac{8}{a} \times(a>0)$ is at $(1,4)$, then the length of this focal chord is

If $m$ is the length of the latusrectum and $n$ is the length of the major-axis of the ellipse $25 x^2+16 y^2-150 x-64 y-111=0$, then the ordered pair $(m, n)=$

If $P(\theta)$ and $Q\left(\frac{\pi}{2}+\theta\right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the locus of mid-point of $P Q$ is $\frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}=1$, then $\frac{a+b}{\alpha+\beta}=$

Let $S$ be the focus of the hyperbola $x^2-2 y^2=1$ lying on the positive $X$-axis. Let $P(-1,1)$ be a given point. Then, the area of the triangle formed by the line $P S$ with the coordinate axes is (in sq. units)

If $P\left(\frac{\pi}{6}\right)$ is a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, S, S$ are its foci and $S P+S P=2 | S P-S P$|, then $e=$

Let $A=(1,2,0), B=(2,0,-1), C=(0,-2,3)$ and $D=(-1,2,-3)$ be four points in the space. Let $G_1$ be the centroid of $\triangle A B C$ and $G_2$ be the centroid of tetrahedron $A B C D$. If $P$ divides, $G_1 G_2$ in the ratio $4: 3$ internally, then $P=$

If the d.r.'s of two lines are connected by the relations $a-b+c=0, a^2-b^2+2 c^2=0$ and $\theta$ is the angle between these lines, then $\cos \theta=$

If $l, m$ and $n$ are the d.c.'s of a normal to the plane passing through the points $(0,1,2)$, $(3,0,2)$ and $(4,5,0)$, then $|I|+|m|+|n|=$

$$ \lim _{x \rightarrow 2} \frac{x^3-x^2-x-2}{2 x^3-3 x^2-3 x+2}= $$

$$ \lim _{x \rightarrow 0} \frac{4[\sin (2022 x)-\sin (2020 x)]}{x[\cos (2022 x)+2 \cos (2021 x)+\cos (2020 x)]}= $$

If $f(x)=\sum_{p=1}^7 p^2 \sin ^{-1}\left(\frac{4}{5} \sin (p x)-\frac{3}{5} \cos (p x)\right)$, then the value of $\frac{d f}{d x}$ at $x=1$ is [given that $\sin ^{-1}(\sin x)=x$ ])

If $y=\frac{a x+b}{c x+d}$, then $\frac{d x}{d y}=$

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

The equation of the tangent to the curve $x^2+y-7=4 x$ at the point $(1,10)$ is

If $\theta$ is the angle between the curves $x^2-y^2=4$ and $y^2=3 x$, then $\tan \theta=$

The absolute maximum value of the function $f(x)=2 x^3-3 x^2-36 x+9$ defined on $[-3,3]$ is

If $f(x)=\int \frac{2-3 \sin ^2 x}{1+\cos 2 x} d x$ and $f\left(\frac{\pi}{4}\right)=1$, then $f(0)=$

If $x \neq(2 n+1) \frac{\pi}{2}, n \in Z$ and $\cos x \neq \frac{-1}{2}$, then

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

Given that $\int \frac{1}{x^2+a^2} d x=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+c$.

$$ \begin{aligned} & \text { If } \int \frac{1}{x^4+3 x^2+1} d x=a \tan ^{-1}\left(\frac{b\left(x^2-1\right)}{x}\right) \\ & +c \tan ^{-1}\left(\frac{d\left(x^2+1\right)}{x}\right)+k \end{aligned} $$

where $k$ is a constant of integration, then $5(c+d+a b)=$

$$ \int_0^4| | x-2|-x| d x= $$

If $\int_{-a}^a f(x) d x=\int_0^a f(x) d x+\int_0^a g(x) d x$, then $g(x)=$

$f\left(x, y, c_1, c_2\right)=0$ is an equation containing two arbitrary constants $c_1$ and $c_2$. If the differential equation having $f\left(x, y, c_1, c_2\right)=0$ as its general solution is of $k$ th order, then the differential equation corresponding to $x^k+y^k=c^2$ ( $c$ is an arbitrary constant) is

If $l$ and $m$ are respectively the order and the degree of the differential equation $f(x) y^{\prime \prime}+g(x) y^{\prime}=\frac{4 y}{x}$ whose general solution is $y=a x^2+b x^2 \log x$, then $f(m)+g(m)=$

The general solution of the differential equation $d x=(2 x+3 y-4) d y$ is

Among the fundamental forces, which one of the following is the strongest force?

Which of the following is the unit of mobility of a electron in a conductor?

A car starts at time $t=0$ from an initial speed of $10 \mathrm{~m} / \mathrm{s}$ and accelerates uniformly with $2 \mathrm{~m} / \mathrm{s}^2$ on a straight road for time $0 \leq t \leq 10 \mathrm{~s}$. Let $s_1$ and $s_2$ be the distance covered by the car in time $3 \leq t \leq 4 \mathrm{~s}$ and $4 \leq t \leq 5 \mathrm{~s}$, respectively. The ratio $\frac{s_2}{s_1}$ is

Particle $A$ (which was located at the origin at time $t=0$ ) is moving along the $X$-axis with a constant speed of $1 \mathrm{~m} / \mathrm{s}$. Location of particle $B$ which is moving along the $Y$-axis is given by $y=c t^2$, where $c=1 \mathrm{~m} / \mathrm{s}^2$. Find the speed of particle $A$ relative to particle $B$ at $t=1 \mathrm{~s}$

A particle is moving in $X Y$-plane as $\mathbf{x}=\left(4 t+t^2\right) \hat{\mathbf{i}}$, $\mathbf{y}=\left(2 t+\frac{t^2}{2}\right) \hat{\mathbf{j}}$, where $\mathbf{x}$ and $\mathbf{y}$ are displacements measured along $X$ and $Y$-axes respectively, in metres and $t$ in seconds, What is the velocity of the particle?

The surface of a hill inclined at an angle $30^{\circ}$ to the horizontal. A stone is thrown from the summit of the hill (point $A$ ) at an initial speed $10 \mathrm{~m} / \mathrm{s}$ at angle $60^{\circ}$ to the vertical. If the stone strikes the hill at point $B$ as shown in the figure, the distance between $A$ and $B$ is (take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A spherical bob of mass 250 g is attached to the end of a string having length 50 cm . The bob is rotated on a horizontal circular path about a vertical axis. The maximum tension that the string can bear is 72 N . The maximum possible value of angular velocity of bob (in $\mathrm{rad} / \mathrm{s}$ ) is

A boat of mass 1000 kg goes from rest to speed 20.0 $\mathrm{m} / \mathrm{s}$ in 5.0 s . The water exerts a constant drag force and the acceleration of the boat is constant. If the average power required by the boat is 45000 W , then the magnitude of the drag force is

A pump on the ground floor of a building can pump up water to fill a tank of volume $36 \mathrm{~m}^3$ in 30 min . If the tank is 50 m above the ground, and the electric power consumed by the pump is 40 k W , the efficiency of the pump is

(use $g=10 \mathrm{~m} / \mathrm{s}^2$ and density of water $=1000 \mathrm{~kg} / \mathrm{m}^3$ )

A cyclist is riding with a speed of $36 \mathrm{~km} / \mathrm{h}$. As he approaches a circular turn on the road of radius 50 m , he applies brakes and reduces his speed at the constant rate of $0.5 \mathrm{~m} / \mathrm{s}$ every second. The magnitude and direction of the net acceleration of the cyclist on the circular turn respectively, are

A block is in simple harmonic motion (SHM) on the end of the spring with position given by $x=5 \cos \left(\omega t+\frac{\pi}{4}\right) \mathrm{cm}$. If the total mechanical energy used is 100 J to achieve maximum displacement, then the potential energy at time, $t=0$ is

Statement I The force of attraction due to a hollow spherical shell of uniform density on a point mass situated inside it is always positive.

Statement II The force of attraction between a hollow spherical shell of uniform density and a point mass situated outside is same just as, if the entire mass of the shell is at the centre of the shell.

Which of the following is correct?

What is the work done in stretching a uniform metal wire of length from 2 m to 2.004 m with an area of cross-section $10^{-6} \mathrm{~m}^2$ ?

[Young's modulus of the wire $=2 \times 10^{11} \mathrm{~N} / \mathrm{m}^2$ ]

A wide cylindrical vessel 50 cm in height is filled with water and rests on a table. Assuming the viscosity to be negligible. Find at what height from the bottom of the vessel a small hole should be made for the water jet coming out of it to hit the surface of the table at the maximum horizontal distance from the vessel

A spherical drop of radius $r$ is divided into 8 equal droplets. If the surface tension is $S$, then the work done in the process will be

A circular copper ring at $30^{\circ} \mathrm{C}$ has a hole with an area of $9.98 \mathrm{~cm}^2$. It is made to slip onto a steel rod of cross-sectional area of $10 \mathrm{~cm}^2$, by raising the temperature of both ring and rod simultaneously by an amount $\Delta T$. If the coefficient of linear expansion of copper and steel are $17 \times 10^{-6} \rho \mathrm{C}$ and $11 \times 10^{-6} \rho \mathrm{C}$, then minimum value of $\Delta T$ should be

Statement I A device in which heat measurement can be made is called calorimeter.

Statement II Skating is possible on snow due to the formation of water below the skates. Water is formed due to the increase of temperature and ice melts.

Statement III Two bodies at different temperature are mixed in a calorimeter. Total internal energy of the two bodies remains conserved.

Which of the following is correct?

Which of the following statements is not true?

A gas system is taken through the thermodynamic cyclic process $1 \rightarrow 2 \rightarrow 3 \rightarrow 1$ as shown below. The amount of heat released by the system is

An ideal gas at pressure $p$ is enclosed in a container that is placed in a reservoir at temperature $T$. If the volume of the gas is increased to two times its original value, then the new pressure $p^{\prime}=$ $\_\_\_\_$ $p$

A cylindrical tube open at both ends has a fundamental frequency $f$ in air. The tube is dipped vertically in water, so that half of it is in water. The new fundamental frequency is

A convex lens of focal length 25 cm and made of glass with refractive index 1.5 is immersed in water. the absolute change in focal length of the glass is [use refractive index of water $=4 / 3$ ]

In a Young's double slit experiment, if the distance between two slits is reduced by a factor of 2 and the wavelength of light is increased 4 times then the distance between two maxima will become $\_\_\_\_$ times the original value

A small block of mass 5 g and charge $5 \mu \mathrm{C}$ is placed on insulated, frictionless, inclined plane of angle $60^{\circ}$. An electric field is applied parallel to the inclined plane. If the block remains at rest, then the magnitude of electric field is (take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

Two metal spheres have their radii in the ratio of $4: 7$. They are put in contact and a charge $8.8 \times 10^{-7} \mathrm{C}$ is given to the system. Then they are separated, so that each can exert no influence on the other. The potential due to the smaller sphere at 60 m from it in V is

The time required to raise the temperature 3 litre of water from $0^{\circ} \mathrm{C}$ to $80^{\circ} \mathrm{C}$ by a heater operated under 200 V having resistance of $50 \Omega$ is [specific heat capacity of water is $4200 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$ ] [density of water $=1000 \mathrm{~kg} / \mathrm{m}^3$ ]

The current density in a circular wire is given by $J(r)=\left(1 \times 10^5 \mathrm{~A} / \mathrm{m}^3\right) r$, where $r$ is the radial distance and the wire's radius is 2 mm . If the potential applied across the wire is 70 V , then the energy consumed by the wire in 1000 s is

Two infinitely long thin wires are placed at $(1 \mathrm{~cm}, 0 \mathrm{~cm})$ and $(2 \mathrm{~cm}, 0 \mathrm{~cm})$ as shown in the figure.

The same current $i$ flows in both the wires in the same direction, say, into the page. Let the magnetic field at the origin due to these wires is $\mathbf{B}$. If $B_0$ is the magnitude of the magnetic field, if only the wire at $(1 \mathrm{~cm}, 0 \mathrm{~cm})$ was present, then the value of $\frac{B}{B_0}$ is

A toroid core has inner radius of 0.24 m and outer radius of 0.26 m . A current of 10 A flows through the wire having 2500 turns around it. Find the magnetic field inside the core of the toroid

In the magnetic meridian of a certain place, the horizontal component of the earth's magnetic field is 86.6 G (Gauss) and the magnetic field of earth is 100 G (Gauss). The the dip angle is

A wheel of 20 metallic spokes each 40 cm long is rotated with a speed of $180 \mathrm{rev} / \mathrm{min}$ in a plane normal to the horizontal component of earth's magnetic field $H_c$ at a place. If $H_c=0.4 \mathrm{G}$ (Gauss) at that place, the induced emf between the axle and the rim of the wheel is

A generator produces a current of 100 A at 4000 V . The voltage is stepped up to $2 \times 10^5 \mathrm{~V}$ by a transformer before being sent on a high voltage transmission line of resistance $50 \Omega$. The percentage of power loss in the transmission line is

In a plane EM wave, the electric field oscillates sinusoidally at a frequency of 30 MHz and amplitude $150 \mathrm{~V} / \mathrm{m}$, Identify the correct expression of $\mathbf{B}$ assuming the wave is propagating along $X$-axis and is oscillating along $Y$-axis.

When monochromatic light falls on a photo sensitive metal, an electron is emitted with maximum velocity $1.6 \times 10^6 \mathrm{~m} / \mathrm{s}$. Find the stopping potential.

[charge of electron $=1.6 \times 10^{-19} \mathrm{C}$, mass of electron $\left.=9 \times 10^{-31} \mathrm{~kg}\right]$

A lamp of power 942 W radiates energy uniformly in all direction. The wavelength of radiation is 660 nm .The photon flux on a small screen 5.0 m from the lamp in units of photon $/ \mathrm{m}^2 \mathrm{~s} Q$ is

(take Planck's constant, $h=6.6 \times 10^{-34}$ SI unit)

The shortest wavelength in Balmer series of hydrogen atom spectrum is approximately equal to (use $R_H=1.097 \times 10^7 \mathrm{~m}^{-1}$ )

What will be the energy released in joule, in the process of fission by 1 mg of ${ }_{92}^{240} \mathrm{U}$. Assume energy release per fission is 200 MeV .

[use Avogadro's number as $6 \times 10^{23}$ and 1 eV $=1.6 \times 10^{-19} \mathrm{~J}$ ]

The band gap in a semiconductor is 0.6 eV . The maximum wavelength of electromagnetic radiation which can create a hole electron pair in the semiconductor is equal to

[use $h c=1242 \mathrm{eV}-\mathrm{nm}$ ]

Identify the logic gate from the following with the same truth table characteristics of the logic circuit below

For an amplitude modulated wave, the modulation index is found to be 0.5 . If the maximum amplitude is found to be 10.0 V , then the minimum amplitude is