Consider the two curves
$$C_1 : y^2 = 4x$$
$$C_2 : x^2 + y^2 - 6x + 1 = 0$$
Then,
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Consider the two curves
$$C_1 : y^2 = 4x$$
$$C_2 : x^2 + y^2 - 6x + 1 = 0$$
Then,
If $$0 < x < 1$$, then
$$\sqrt{1 + x^2}[\left\{x \cos (\cot^{-1} x) + \sin(\cot^{-1} x)\right\}^2 - 1]^{\frac{1}{2}}=$$
The edges of a parallelopiped are of unit length andare parallel to non-coplanar unit vectors $$\hat{a}, \hat{b}, \hat{c}$$ such that
$$\hat{a} . \hat{b} = \hat{b} . \hat{c} = \hat{c} . \hat{a} = \frac{1}{2}$$.
Then, the volume of the parallelopiped is
Let a and b be non-zero real numbers. Then, the equation
$$(ax^2 + by^2 + c)(x^2 - 5xy + 6y^2) = 0$$
represents
Let
$$g(x) = \frac{(x - 1)^n}{\log \cos^m (x - 1)};0 < x < 2$$, m and n are integers, $$m \neq 0, n > 0$$, and let p be the left hand derivative of $$\mid x - 1 \mid$$ at x=1.
If $$\lim_{x \rightarrow 1+}g(x) = p$$, then
The total number of local maxima and local minima of the function
$$f(x) = \begin{cases}(2 + x)^3, & -3<x \leq -1\\x^{\frac{2}{3}} & -1 < x < 2\end{cases}$$ is
A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then
Let $$P(x_1, y_1)$$ and $$Q(x_2, y_2), y_1 < 0, y_2 < 0$$, be the end of the latus rectum of the ellipse $$x^2 + 4y^2 = 4$$. The equations of parabolas with latus rectum PQ are
Let
$$S_n = \sum_{k=1}^{n}\frac{n}{n^2 + kn + k^2}$$ and $$T_n = \sum_{k=0}^{n-1}\frac{n}{n^2 + kn + k^2}$$, for $$n = 1, 2, 3, ..., $$ Then,
Let f(x) be a non-constant twice differentiable function defined on $$(-\infty, \infty)$$ such that f(x) = f(1 - x) and $$f'\left(\frac{1}{4}\right) = 0$$. Then,
Let f and g be real valued functions defined on interval (-1, 1) such that $$g''(x)$$ is continuous, $$g(0) \neq 0, g'(0) = 0, g''(0) \neq 0$$, and $$f(x) = g(x)\sin x$$.
STATEMENT-1: $$\lim_{x \rightarrow 0}[g(x)\cot x - g(0) \cosec x] =f''(0).$$
STATEMENT-2: $$f'(0) = g(0)$$.
Consider three planes
$$P_1 : x - y + z = 1$$
$$P_2 : x + y - z = -1$$
$$P_3 : x - 3y + 3z = 2$$.
Let $$L_1, L_2, L_3$$ be the lines of interaction of the planes $$P_2$$ and $$P_3, P_3$$ and $$P_1, $$ and $$P_1$$ and $$P_2$$, respectively.
STATEMENT-1: At least two of the lines $$L_1, L_2$$, and $$L_3$$ are non-parallel.
and
STATEMENT-2: The three planes do not have a common point.
Consider the system of equations
$$x - 2y + 3z = -1$$
$$-x + y - 2z = k$$
$$x - 3y + 4z = 1$$.
STATEMENT-1: The system of equations has no solution for $$k \neq 3$$.
and
STATEMENT-2: The determinant
$$\neq 0$$, for $$k \neq 3$$.
Consider the system of equations
$$ax + by = 0,cx + dy = 0$$, where $$a, b, c, d \in \left\{0, 1\right\}$$.
STATEMENT-1: The probability that the system of equations has a unique solution is $$\frac{3}{8}$$
and
STATEMENT-2: The probability that the system of equations hasa solution is 1.
A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equation $$\sqrt{3}x + y - 6 = 0$$ and the point D is $$\left(\frac{3\sqrt{3}}{2}, \frac{3}{2}\right)$$. Further, it is given that the origin and the centre of C are on the same side of the line PQ.
The equation of circle C is
Points E and F are given by
Equations of the sides QR, RP are
Consider the functions defined implicitly by the equation $$y^2 - 3y + x = 0$$ on various intervals in the real line. If $$x \in (-\infty, -2) \bigcup (2, \infty)$$, the equation implicitly defines a unique real valued differentiable function y = f(x). If $$x \in (-2, 2)$$, the equation implicitly defines a unique real valued differentiable function y = g(x) satisfying g(0) = 0.
If $$f(-10\sqrt{2}) = 2\sqrt{2}$$, then $$f''(-10\sqrt{2}) =$$
The area of the region bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b, where $$-\infty < a < b < -2$$, is
$$\int_{-1}^{1} g'(x)dx =$$
Let A, B, C be three sets of complex numbersas defined below
$$A = \left\{z : Imz \geq 1\right\}$$
$$B = \left\{z : \mid z - 2 - i \mid = 3\right\}$$
$$C = \left\{z : Re((1 - i)z) = \sqrt{2}\right\}$$
The number of elements in the set $$A \bigcap B \bigcap C$$ is
Let z be any point in $$A \bigcap B \bigcap C$$. Then, $$\mid z + 1 - i \mid^2 + \mid z - 5 - i \mid^2$$ lies between
Let z be any point in $$A \bigcap B \bigcap C$$ and let w be any point satisfying $$\mid w - 2 - i \mid < 3$$. Then, $$\mid z \mid - \mid w \mid + 3$$ lies between
Students I, II and III perform an experiment for measuring the acceleration due to gravity (g) using a simple pendulum. They use different lengths of the pendulum and/or record time for different number of oscillations. The observations are shown in the table.
Least count for length = 0.1 cm
Least count for time = 0.1 s
If $$E_I, E_{II}$$ and $$E_{III}$$ are the percentage errors in g, i.e., $$\left(\frac{\triangle g}{g} \times 100\right)$$ for students I, II and III, respectively,
Figure shows three resistor configurations R1, R2 and R3 connected to 3 V battery. If the power dissipated by the configuration R1, R2 and R3 is P1, P2 and P3, respectively, then
Which one of the following statements is WRONG in the context of X-rays generated from a X-ray tube?
Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is $$60^\circ$$). In the position of minimum deviation, the angle of refraction will be
An ideal gas is expanding such that $$PT^2 =$$ constant. The coefficient of volume expansion of the gas is
A spherically symmetric gravitational system of particles has a mass density $$\rho = \begin{cases}\rho_0 & for & r \leq R\\0 & for & r > R\end{cases}$$ where $$\rho_0$$ is a constant. A test mass can undergo circular motion underthe influence of the gravitational field of particles. Its speed V as a function of distance $$r(0 < r < \infty)$$ from the center of the system is represented by
Two balls, having linear momenta $$\overrightarrow{p}_1 = p\hat{i}$$ and $$\overrightarrow{p}_1 = -p\hat{i}$$, undergo a collision in free space. There is no external force acting on the balls. Let $$\overrightarrow{p}_1'$$ and $$\overrightarrow{p}_2'$$ be their final momenta. The following option(s) is(are) NOT ALLOWED for any non-zero value of $$p, a_1, a_2, b_1, b_2, c_1$$ and $$c_2$$.
Assume that the nuclear binding energy per nucleon (B/A) versus mass number (A) is as shown in the figure. Use this plot to choose the correct choice(s) given below.
A particle of mass m and charge q, moving with velocity V enters Region II normal to the boundary as shown in the figure. Region II has a uniform magnetic field B perpendicular to the plane of the paper. The length of the Region II is l. Choose the correct choice(s).
In a Young’s double slit experiment, the separation between the two slits is d and the wavelength of the light is $$\lambda$$. The intensity of light falling on slit 1 is four times the
intensity of light falling on slit 2. Choose the correct choice(s).
STATEMENT-1
In a Meter Bridge experiment, null point for an unknown resistance is measured. Now, the unknown resistance is put inside an enclosure maintained at a higher temperature. The null point can be obtained at the same point as before by decreasing the value of the standard resistance.
and
STATEMENT-2
Resistance of a metal increases with increase in temperature.
STATEMENT-1
Anastronaut in an orbiting space station above the Earth experiences weightlessness.
and
STATEMENT-2
An object moving around the Earth underthe influence of Earth’s gravitational force is in a state of ‘free-fall’.
STATEMENT-1
Twocylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the sameheight. The hollow cylinder will reach the bottom of the inclined plane first.
and
STATEMENT-2
By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.
STATEMENT-1
The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down.
and
STATEMENT-2
In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.
A small spherical monoatomic ideal gas bubble $$\left(\lambda = \frac{5}{3}\right)$$ is trapped inside a liquid of density $$\rho_r$$ (see figure), Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is $$T_0$$, the height of the liquid is H and the atmospheric pressure is $$P_0$$, (Neglect surface tension).
As the bubble moves upwards, besides the buoyancy force the following forces are acting on it
When the gas bubble is at a height y from the bottom, its temperature is
The buoyancy force acting on the gas bubble is (Assume R is the universal gas constant)
In a mixture of $$H - He^+$$ gas ($$He^+$$ is singly ionized He atom), H atoms and $$He^+$$ ions are excited to their respective first excited states. Subsequently, H atoms transfer their total excitation energy to $$He^+$$ ions (by collisions). Assume that the Bohr model of atomis exactly valid.
The quantum number n ofthe state finally populated in $$He^+$$ ions is
The wave length of light emitted in the visible region by $$He^+$$ ions after collisions with H atoms is
The ratio of the kinetic energy of the n = 2 electron for the H atom to that of $$He^{+}$$ ion is
A small block of mass M moves on frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from $$60^\circ$$ to $$30^\circ$$ at point B.
The block is initially at rest at A. Assume that collisions between the block and the incline are totally inelastic (g = 10 m/s$$^2$$)
The speed of the block at point B immediately after it strikes the second incline is
The speed of the block at point C, immediately before it leaves the second incline is
If collision between the block and theincline is completely elastic, then the vertical (upward) component of the velocity of the block at point B, immediately after it strikes the second incline is
Hyperconjugation involves overlap of the following orbitals
The major product of the following reaction is
Aqueous solution of $$Na_2S_2O_3$$ on reaction with $$Cl_2$$ gives
Native silver metal forms a water soluble complex with a dilute aqueous solution of NaCN in the presence of
Under the same reaction conditions, initial concentration of 1.386 mol dm$$^{-3}$$ of a substance becomes half in 40 seconds and 20 seconds through first order and zero order kinetics, respectively. Ratio $$\left(\frac{k_1}{k_0}\right)$$ of the rate constants for first order $$(k_1)$$ and zero order $$(k_0)$$ of the reactions is
2.5 mL of $$\frac{2}{5}$$ M weak monoacidic base $$(K_b = 1 \times 10^{-12}$$ at $$25^\circ C)$$ is titrated with $$\frac{2}{15}$$ M HCl in water at $$25^\circ C$$. The concentration of $$H^{+}$$ at equivalence point is $$(K_w = 1 \times 10^{-14}$$ at $$25^\circ C)$$
The correct statement(s) about the compound given below is (are)
The correct statement(s) concerning the structures E, F and G is (are)
A solution of colourless salt H on boiling with excess NaOH produces a non-flammable gas. The gas evolution ceases after sometime. Upon addition of Zn dust to the same solution, the gas evolution restarts. The colourless salt(s) H is (are)
A gas described by van der Waals equation
STATEMENT-1: Bromobenzene upon reaction with Br,/Fe gives 1,4-dibromobenzene as the major product.
and
STATEMENT-2: In bromobenzene, the inductive effect of the bromo group is more dominant than the mesomeric effect in directing the incoming electrophile.
STATEMENT-1: $$Pb^{4+}$$ compounds are stronger oxidizing agents than $$Sn^{4+}$$ compounds.
and
STATEMENT-2: The higher oxidation states for the group 14 elements are more stable for the heavier members of the group due to ‘inert pair effect’.
STATEMENT-1: The plot of atomic number (y-axis) versus number of neutrons (x-axis) for stable nuclei shows a curvature towards x-axis from the line of $$45^\circ$$ slope as the atomic numberis increased.
and
STATEMENT-2: Proton-proton electrostatic repulsions begin to overcome attractive forces involving protons and neutrons in heavier nuclides.
STATEMENT-1: For every chemical reaction at equilibrium, standard Gibbs energy of reaction is zero.
and
STATEMENT-2: At constant temperature and pressure, chemical reactions are spontaneous in the direction of decreasing Gibbs energy.
In the following reaction sequence, products I, J and L are formed. K represents a reagent.
The structure of the product I is
The structures of compounds J and K, respectively, are
The structure of product L is
There are some deposits of nitrates and phosphates in earth’s crust. Nitrates are more soluble in water. Nitrates are difficult to reduce under the laboratory conditions but microbes do it easily. Ammonia forms large number of complexes with transition metal ions. Hybridization easily explains the ease of sigma donation capability of $$NH_3$$ and $$PH_3$$. Phosphine is a flammable gas and is prepared from white phosphorous.
Among the following, the correct statement is
Among the following, the correct statement is
White phosphorus on reaction with $$NaOH$$ gives $$PH_3$$ as one of the products. This is a
Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examplesis the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator of automobiles
A solution M is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtureis 0.9
Given : Freezing point depression constant of water $$\left(K_{f}^{water}\right)$$ = 1.86 K kg mol$$^{-1}$$
Freezing point depression constant of ethanol $$\left(K_{f}^{ethanol}\right)$$ = 2.0 K kg mol$$^{-1}$$
Freezing point elevation constant of water $$\left(K_{b}^{water}\right)$$ = 0.52 K kg mol$$^{-1}$$
Freezing point elevation constant of ethanol $$\left(K_{b}^{ethanol}\right)$$ = 1.2 K kg mol$$^{-1}$$
Standard freezing point of water = 273 K
Standard freezing point of ethanol = 155.7 K
Standard boiling point of water = 373 K
Standard boiling point of ethanol = 351.5 K
Vapour pressure of pure water = 32.8 mm Hg
Vapour pressure of pure ethanol = 40 mm Hg
Molecular weight of water = 18 g mol$$^{-1}$$
Molecular weight of ethanol = 46 g mol$$^{-1}$$
In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.
The freezing point of the solution M is
The vapour pressure of the solution M is
Water is added to the solution M such that the mole fraction of water in the solution becomes 0.9. The boiling point of this solution is
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