Class 12 previous year questions (PYQ)

If a discrete random variable $X$ takes the values $1, 2, 3, 4$ such that $2P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4)$, then $P(X = 4) = ...$ (in $61$)

Consider the following sequence of reactions. The percentage of nitrogen in the yellow product $(X)$ formed is . . . . . . %. (Nearest Integer) (Given Molar mass in g mol$^{-1}$ $H$:$1$, $C$:$12$, $N$:$14$)

Let the system of equations: $2x + 3y + 5z = 9$,$7x + 3y - 2z = 8$,$12x + 3y - (4 + \lambda)z = 16 - \mu$ have infinitely many solutions. Then the radius of the circle centred at $(\lambda, \mu)$ and touching the line $4x = 3y$ is

If $a$ is the determinant of the adjoint of the matrix $A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 3 \end{bmatrix}$ and $b$ is the determinant of the inverse of the matrix $B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & -3 & -1 \\ 2 & 1 & -4 \end{bmatrix}$,then find the value of $\frac{b+1}{18b}$.

$A$ $p-n$ junction diode is connected to a battery of emf $5.7 \ V$ in series with a resistor of $5 \ k\Omega$ such that it is forward biased. If the barrier potential of the diode is $0.7 \ V$,neglecting the diode resistance,the current in the circuit is:

Which one of the following is an amphoteric oxide?

Insulin receptors are

If the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ meets the plane $x+2y+3z=15$ at the point $P$,then the distance of $P$ from the origin is

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx} + \left(\frac{2x^2+11x+13}{x^3+6x^2+11x+6}\right)y = \frac{x+3}{x+1}$,where $x \gt -1$,which passes through the point $(0,1)$. Then $y(1)$ is equal to:

The area bounded by the curves $y = |x^{2} - 1|$ and $y = 1$ is:

Let $\overrightarrow{a} = \hat{i} + 2\hat{j} - 3\hat{k}$ and $\overrightarrow{b} = 2\hat{i} - 3\hat{j} + 5\hat{k}$. If $\overrightarrow{r} \times \overrightarrow{a} = \overrightarrow{b} \times \overrightarrow{r}$,$\overrightarrow{r} \cdot (\alpha\hat{i} + 2\hat{j} + \hat{k}) = 3$ and $\overrightarrow{r} \cdot (2\hat{i} + 5\hat{j} - \alpha\hat{k}) = -1$,where $\alpha \in R$,then the value of $\alpha + |\overrightarrow{r}|^{2}$ is equal to:

If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$,$\vec{b}=-\hat{i}+2\hat{j}+\hat{k}$,$\vec{c}=3\hat{i}+\hat{j}$ and $\vec{a}+\lambda\vec{b}$ is perpendicular to $\vec{c}$,then $\lambda=$

How many primary amines are possible for the formula $C_4H_{11}N$?

$ABCD$ is a parallelogram,$P$ is the mid-point of $AB$. If $R$ is the point of intersection of $AC$ and $DP$,then $R$ divides $AC$ internally in the ratio

Given three vectors $a=2 \hat{i}-\hat{j}+\hat{k}$,$b=\hat{i}+2 \hat{j}-\hat{k}$,and $c=\hat{i}+\hat{j}-2 \hat{k}$,a vector in the plane of $b$ and $c$ whose projection on $a$ is of magnitude $\sqrt{\frac{2}{3}}$ is

If the vectors \(a\hat{i} + \hat{j} + \hat{k}\), \(\hat{i} + b\hat{j} + \hat{k}\) and \(\hat{i} + \hat{j} + c\hat{k}\) are coplanar (\(a \neq b \neq c \neq 1\)), then the value of \(abc - (a+b+c)\) =

A small bulb is placed at the bottom of a tank containing water to a depth of \(\sqrt{7}\, \text{m}\). The refractive index of water is \(\tfrac{4}{3}\). The area of the surface of water through which light from the bulb can emerge out of \(x\pi \, \text{m}^2\). The value of \(x\) is ______.  
 

Question
In the following in each set a conservation approach and an example of method of conservation are given  
(a) In situ conservation – Biosphere Reserve  
(b) Ex situ conservation – Sacred groves  
(c) In situ conservation – Seed bank  
(d) Ex situ conservation – Cryopreservation  
Select the option with correct match of approach and method :  

A rectangular loop of sides 10 cm and 5 cm carrying a current \( I = 12 \, \text{A} \) is placed in different orientations as shown in the figure.  
If there is a uniform magnetic field of \( 0.3 \, T \) in the positive \( z \)-direction, in which orientations the loop would be in  
(i) stable equilibrium and (ii) unstable equilibrium.  

The rate of a reaction $A$ doubles on increasing the temperature from $300 \, K$ to $310 \, K$. By how much should the temperature of reaction $B$ be increased from $300 \, K$ so that its rate doubles,if the activation energy of reaction $B$ is twice that of reaction $A$ (in $, K$)?

Which among the following carboxylic acids is a tricarboxylic acid?

Two inductors $L_1$ (inductance $1 \text{ mH}$,internal resistance $3 \text{ } \Omega$) and $L_2$ (inductance $2 \text{ mH}$,internal resistance $4 \text{ } \Omega$),and a resistor $R$ (resistance $12 \text{ } \Omega$) are all connected in parallel across a $5 \text{ V}$ battery. The circuit is switched on at time $t=0$. The ratio of the maximum to the minimum current $(I_{\max} / I_{\min})$ drawn from the battery is:

If \( x = \sqrt{a^{\sin^{-1}t}} \) and \( y = \sqrt{a^{\cos^{-1}t}} \), then \( \frac{dy}{dx} = \)  

Let 
A = 
\begin{bmatrix}
3 & -2 & 1 \\
-1 & 3 & -1
\end{bmatrix}
\text{ and }
B = 
\begin{bmatrix}
1 \\
\alpha \\
-1
\end{bmatrix}.
If 
AB = 
\begin{bmatrix}
-2 \\
6
\end{bmatrix},
\text{ then the value of } \alpha \text{ is equal to}

Galvanized iron is coated with which metal?

In the following reaction,the major product $X$ is

Which one of the following is a hallucinogenic drug?

Four persons independently solve a certain problem correctly with probabilities $\frac{1}{2}, \frac{3}{4}, \frac{1}{4}, \frac{1}{8}$. Then the probability that the problem is solved correctly by at least one of them is

Which of the following conditions favours the reduction of a metal oxide to metal?

If $y(x)$ satisfies the differential equation $y^{\prime}-y \tan x=2 x \sec x$ and $y(0)=0$,then which of the following is true?

Of the total incident solar radiation,the proportion of $PAR$ is:

$A$ plane passes through $(2,3,-1)$ and is perpendicular to the line having direction ratios $3,-4,7$. The perpendicular distance from the origin to this plane is

If $I_1 = \int_0^{3 \pi} f(\cos^2 x) dx$ and $I_2 = \int_0^\pi f(\cos^2 x) dx$, then

$A$ plane meets the coordinate axes at $A, B, C$ such that the centroid of the triangle $ABC$ is $(1, 2, 4)$. Then,the equation of the plane is

The number of electrons in $2 \ C$ of charge is:

If $\overrightarrow{a} \cdot \overrightarrow{b} = -|\overrightarrow{a}||\overrightarrow{b}|$,then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is (in $^{\circ}$)

The wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere constructively (see figure).
Electrons accelerated by potential $V$ are diffracted from a crystal. If $d = 1\; \text{\AA}$, and $i = 30^{\circ}$, $V$ should be about ....... $V$.
$(h = 6.6 \times 10^{-34}\; J-s, m = 9.1 \times 10^{-31}\; kg, e = 1.6 \times 10^{-19}\; C)$

Which equation is true to calculate the energy of activation,if the rate of reaction is doubled by increasing temperature from $T_1 \ K$ to $T_2 \ K$?

If $2 x^2-3 x y+y^2+x+2 y-8=0$,then $\frac{d y}{d x}$ is equal to

The largest wavelength in the ultraviolet region of the hydrogen spectrum is $122 \ nm$. The smallest wavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is (in $nm$)

Which of the following units denotes the dimension $\frac{M L^2}{Q^2}$,where $Q$ denotes the electric charge?

Assertion : The water pouch of an instant cold pack for treating athletic injuries breaks when squeezed and $NH_4NO_3$ dissolves,lowering the temperature.
Reason : Addition of a non-volatile solute into a solvent results in the depression of the freezing point of the solvent.

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

The area (in square units) bounded by the curves $y^2=4x$ and $x^2=4y$ in the plane is

Among $K, Ca, Fe,$ and $Zn$,the element which can form more than one binary compound with chlorine is

Which of the following is the strongest base?

Solar energy is mainly caused due to

Assertion : Benzyl bromide when kept in acetone water,it produces benzyl alcohol.
Reason : The reaction follows $S_{N}2$ mechanism.

$A$ charged particle $q$ is placed at the centre $O$ of a cube of side length $L$ $(A, B, C, D, E, F, G, H)$. Another identical charge $q$ is placed at a distance $L$ from $O$, outside the cube, as shown in the figure. The electric flux through the face $BGFC$ is

If a current is passed through a spring,then the spring will:

Assertion: In Young's experiment,the fringe width for dark fringes is different from that for bright fringes.
Reason: In Young's double slit experiment,if the fringes are performed with a source of white light,then only black and bright fringes are observed.

Which of the following is prepared by the electrolytic method?

Assertion : Copper reacts with $HCl$ and liberates hydrogen.
Reason : Hydrogen is present above $Cu$ in the reactivity series.

$A$ rectangular glass slab $ABCD$,of refractive index $n_1$,is immersed in water of refractive index $n_2$ $(n_1 \gt n_2)$. $A$ ray of light is incident at the surface $AB$ of the slab as shown. The maximum value of the angle of incidence $\alpha_{max}$,such that the ray comes out only from the other surface $CD$,is given by

Energy in the sun is generated mainly by

$A$ bar magnet of magnetic moment $\overrightarrow{M}$ is placed in a magnetic field of induction $\overrightarrow{B}$. The torque exerted on it is

$A$ dielectric slab of thickness $d$ is inserted in a parallel plate capacitor whose negative plate is at $x = 0$ and positive plate is at $x = 3d$. The slab is equidistant from the plates. The capacitor is given some charge. As one goes from $x = 0$ to $x = 3d$:

$IUPAC$ name of $[Pt(NH_3)_3(Br)(NO_2)Cl]Cl$ is

Let $p, q, r$ be three mutually perpendicular vectors of the same magnitude. If a vector $x$ satisfies the equation $p \times \{(x - q) \times p\} + q \times \{(x - r) \times q\} + r \times \{(x - p) \times r\} = 0$,then $x$ is given by

When chloroform reacts with ethyl amine in the presence of alcoholic $KOH$,the compound formed is

Arrhenotoky is a type of parthenogenetic development found in:

Which of the following is a true fruit?

Which of the following pairs is not correctly matched?

Which of the following endocrine glands stores its secretion in the extracellular space before releasing it into the blood?

The compound which does not react with sodium is

For the adsorption of a gas on a solid,the plot of $\log(x/m)$ versus $\log P$ is linear with slope equal to

The arrangement of nucleotides in $DNA$ can be seen by which of the following techniques?

Let $a = 2i - j + k$,$b = i + 2j - k$,and $c = i + j - 2k$ be three vectors. $A$ vector in the plane of $b$ and $c$ whose projection on $a$ is of magnitude $\sqrt{2/3}$ is

The segregation of Mendelian factors (no linkage,no crossing over) occurs during ............ .

In the presence of $NaOH$,phenol reacts with $CHCl_3$ to form $o$-hydroxybenzaldehyde. This reaction is called:

If $f(x) = \begin{cases} 1, & 0 \lt x \le \frac{3\pi}{4} \\ 2\sin \frac{2}{9}x, & \frac{3\pi}{4} \lt x \lt \pi \end{cases}$,then

Find the odd one out.

$A$ husband and wife both have normal vision,although both of their fathers were colorblind. What is the probability that their daughter will be colorblind (in $\%$)?

The entry of the pollen tube through the micropyle is known as:

An astronomical telescope has an angular magnification of magnitude $5$ for distant objects. The separation between the objective and the eyepiece is $36 \, cm$ and the final image is formed at infinity. The focal length $f_o$ of the objective and the focal length $f_e$ of the eyepiece are:

Which of the following has the lowest boiling point?

Who discovered double fertilization and triple fusion?

$\int_0^{1.5} {[x^2] \, dx}$,where $[.]$ denotes the greatest integer function,equals

The determinant $\left| \begin{array}{ccc} a & b & a\alpha + b \\ b & c & b\alpha + c \\ a\alpha + b & b\alpha + c & 0 \end{array} \right| = 0$,if $a, b, c$ are in

The probability of happening at least one of the events $A$ and $B$ is $0.6$. If the events $A$ and $B$ happen simultaneously with a probability of $0.2$,then $P(\bar{A}) + P(\bar{B}) =$

Amongst the following,$HBr$ reacts fastest with

Biotic potential refers to

The binding energy of the innermost electron in tungsten is $40 \text{ keV}$. To produce characteristic $X$-rays using a tungsten target in an $X$-ray tube,the potential difference $V$ between the cathode and the anti-cathode should be:

$A$ box contains $100$ tickets numbered $1, 2, \dots, 100$. Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than $10$. What is the probability that the minimum number on them is $5$?

If $y = f(x) = \frac{x + 2}{x - 1}$,then $x =$

The magnitudes of mutually perpendicular forces $a, b$ and $c$ are $2, 10$ and $11$ respectively. Then the magnitude of its resultant is

$\int {\frac{{x - 1}}{{{{(x + 1)}^3}}}{e^x}\,dx = }$

The highest catenation ability is shown by

$|(a \times b) \cdot c| = |a| |b| |c|$,if

An electron moves on a straight line path $XY$ as shown. The $abcd$ is a coil adjacent to the path of the electron. What will be the direction of current,if any,induced in the coil?

Let $a, b, c$ be non-zero real numbers such that $\int_0^1 {(1 + \cos^8 x)(ax^2 + bx + c) \, dx} = \int_0^2 {(1 + \cos^8 x)(ax^2 + bx + c) \, dx}$. Then the quadratic equation $ax^2 + bx + c = 0$ has:

The standard reduction potentials at $298 \ K$ for the following half reactions are given against each:
$Zn^{2+}(aq.) + 2e^- \rightleftharpoons Zn_{(s)}$; $E^\circ = -0.762 \ V$
$Cr^{3+}(aq.) + 3e^- \rightleftharpoons Cr_{(s)}$; $E^\circ = -0.740 \ V$
$2H^{+}(aq.) + 2e^- \rightleftharpoons H_{2(g)}$; $E^\circ = 0.00 \ V$
$Fe^{3+}(aq.) + e^- \rightleftharpoons Fe^{2+}(aq.)$; $E^\circ = 0.770 \ V$
Which is the strongest reducing agent?

Which one of the following nuclear transformations is of the $(n, p)$ type?

$A$ sample of chloroform being used as an anaesthetic is tested by:

$A$ charged particle is free to move in an electric field. It will travel

Two fair dice are tossed. Let $A$ be the event that the first die shows an even number and $B$ be the event that the second die shows an odd number. The two events $A$ and $B$ are

If $x = cy + bz$,$y = az + cx$,$z = bx + ay$ (where $x, y, z$ are not all zero) have a solution other than $x = 0, y = 0, z = 0$,then $a, b$,and $c$ are connected by the relation:

$\mathop {\lim }\limits_{x \to a} \frac{{\sqrt {a + 2x} - \sqrt {3x} }}{{\sqrt {3a + x} - 2\sqrt x }} = \dots$ (where $a \ne 0$)

$\int \cos \sqrt{x} \, dx =$

$\mathop {\lim }\limits_{x \to 1} \frac{(2x - 3)(\sqrt{x} - 1)}{2x^2 + x - 3} =$

$A$ bag $A$ contains $2$ white and $3$ red balls and bag $B$ contains $4$ white and $5$ red balls. One ball is drawn at random from a randomly chosen bag and is found to be red. The probability that it was drawn from bag $B$ is:

$\mathop {\lim }\limits_{x \to 1} \frac{x - 1}{2x^2 - 7x + 5} =$

$\int {\frac{{{x^5}dx}}{{\sqrt {1 + {x^3}} }}} =$

Silver and copper voltameters are connected in parallel with a battery of $e.m.f.$ $12\, V$. In $30$ minutes,$1\, g$ of silver and $1.8\, g$ of copper are liberated. The power supplied by the battery is ...................... $J/s$ ($Z_{Cu} = 6.6 \times 10^{-4}\,g/C$ and $Z_{Ag} = 11.2 \times 10^{-4}\,g/C$).

$\int x^2 \sin 2x \, dx =$

$\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - \sin x}}{{{x^3}}} =$

$\mathop {\lim }\limits_{x \to \pi /2} \frac{2x - \pi}{\cos x} =$

$\int x \sin^2 x \, dx =$

$\int x \cos^2 x \, dx =$

$\mathop {\lim }\limits_{x \to 0} \frac{{\tan 2x - x}}{{3x - \sin x}} =$

$\int \sqrt{\frac{1-x}{1+x}} \, dx =$

The ratio between the total intensity of the magnetic field at the equator to the poles is:

The area of the triangle formed by the points $(a, b + c)$,$(b, c + a)$,and $(c, a + b)$ is: