The velocity of a particle is $$v = (v_0 + gt + Ft^2)$$ m s$$^{-1}$$. Its position is $$x = 0$$ at $$t = 0$$; then its displacement after time ($$t = 1$$ s) is:
JEE WhatsApp Group
Sign in
Please select an account to continue using cracku.in
↓ →
JEE WhatsApp Group
Join Our JEE Preparation Group
Prep with like-minded aspirants; Get access to free daily tests and study material.
The velocity of a particle is $$v = (v_0 + gt + Ft^2)$$ m s$$^{-1}$$. Its position is $$x = 0$$ at $$t = 0$$; then its displacement after time ($$t = 1$$ s) is:
Login to view the detailed solution.
Two identical blocks $$A$$ and $$B$$ each of mass $$m$$ resting on the smooth horizontal floor are connected by a light spring of natural length $$L$$ and spring constant $$K$$. A third block $$C$$ of mass $$m$$ moving with a speed $$v$$ along the line joining $$A$$ and $$B$$ collides with $$A$$. The maximum compression in the spring is:
Login to view the detailed solution.
A rubber ball is released from a height of 5 m above the floor. It bounces back repeatedly, always rising to $$\frac{81}{100}$$ of the height through which it falls. Find the average speed of the ball. (Take $$g = 10$$ m s$$^{-2}$$)
Login to view the detailed solution.
.webp)
A sphere of mass 2 kg and radius 0.5 m is rolling with an initial speed of 1 m s$$^{-1}$$ goes up an inclined plane which makes an angle of 30° with the horizontal plane, without slipping. How low will the sphere take to return to the starting point $$A$$?
Login to view the detailed solution.
A geostationary satellite is orbiting around an arbitrary planet $$P$$ at a height of $$11R$$ above the surface of $$P$$, $$R$$ being the radius of $$P$$. The time period of another satellite in hours at a height of $$2R$$ from the surface of $$P$$ is ________. has the time period of 24 hours.
Login to view the detailed solution.
An object is located at 2 km beneath the surface of the water. If the fractional compression $$\frac{\Delta V}{V}$$ is 1.36%, the ratio of hydraulic stress to the corresponding hydraulic strain will be ________. [Given: density of water is 1000 kg m$$^{-3}$$ and $$g = 9.8$$ m s$$^{-2}$$].
Login to view the detailed solution.
Which one is the correct option for the two different thermodynamic processes?
Login to view the detailed solution.
If one mole of the polyatomic gas is having two vibrational modes and $$\beta$$ is the ratio of molar specific heats for polyatomic gas $$\left(\beta = \frac{C_p}{C_v}\right)$$ then the value of $$\beta$$ is:
Login to view the detailed solution.
A block of mass 1 kg attached to a spring is made to oscillate with an initial amplitude of 12 cm. After 2 minutes the amplitude decreases to 6 cm. Determine the value of the damping constant for this motion. (take $$\ln 2 = 0.693$$)
Login to view the detailed solution.
Two particles $$A$$ and $$B$$ of equal masses are suspended from two massless springs of spring constants $$K_1$$ and $$K_2$$ respectively. If the maximum velocities during oscillations are equal, the ratio of the amplitude of $$A$$ and $$B$$ is:
Login to view the detailed solution.
A sound wave of frequency 245 Hz travels with the speed of 300 m s$$^{-1}$$ along the positive x-axis. Each point of the wave moves to and fro through a total distance of 6 cm. What will be the mathematical expression of this travelling wave?
Login to view the detailed solution.
Two cells of emf $$2E$$ and $$E$$ with internal resistance $$r_1$$ and $$r_2$$ respectively are connected in series to an external resistor $$R$$ (see figure). The value of $$R$$, at which the potential difference across the terminals of the first cell becomes zero is:
Login to view the detailed solution.
The four arms of a Wheatstone bridge have resistances as shown in the figure. A galvanometer of 15$$\Omega$$ resistance is connected across BD. Calculate the current through the galvanometer when a potential difference of 10 V is maintained across AC.
Login to view the detailed solution.
A hairpin like shape as shown in figure is made by bending a long current carrying wire. What is the magnitude of a magnetic field at point $$P$$ which lies on the centre of the semicircle?
Login to view the detailed solution.
Match List-I with List-II:
List-I | List-II
a. Phase difference between current and voltage in a purely resistive AC circuit | i. $$\frac{\pi}{2}$$; current leads voltage
b. Phase difference between current and voltage in a pure inductive AC circuit | ii. zero
c. Phase difference between current and voltage in a pure capacitive AC circuit | iii. $$\frac{\pi}{2}$$; current lags voltage
d. Phase difference between current and voltage in an LCR series circuit | iv. $$\tan^{-1}\left(\frac{X_C - X_L}{R}\right)$$
Choose the most appropriate answer from the options given below:
Login to view the detailed solution.
What happens to the inductive reactance and the current in a purely inductive circuit if the frequency is halved?
Login to view the detailed solution.
Two identical photocathodes receive the light of frequencies $$f_1$$ and $$f_2$$ respectively. If the velocities of the photo-electrons coming out are $$v_1$$ and $$v_2$$ respectively, then:
Login to view the detailed solution.
The atomic hydrogen emits a line spectrum consisting of various series. Which series of hydrogen atomic spectra is lying in the visible region?
Login to view the detailed solution.
Which one of the following will be the output of the given circuit?
Login to view the detailed solution.
A carrier signal $$C(t) = 25\sin(2.512 \times 10^{10}t)$$ is amplitude modulated by a message signal $$m(t) = 5\sin(1.57 \times 10^{8}t)$$ and transmitted through an antenna. What will be the bandwidth of the modulated signal?
Login to view the detailed solution.
A body of mass 1 kg rests on a horizontal floor with which it has a coefficient of static friction $$\frac{1}{\sqrt{3}}$$. It is desired to make the body move by applying the minimum possible force $$F$$ N. The value of $$F$$ will be ________. (Round off to the Nearest Integer) [Take $$g = 10$$ m s$$^{-2}$$]
Login to view the detailed solution.
A boy of mass 4 kg is standing on a piece of wood having mass 5 kg. If the coefficient of friction between the wood and the floor is 0.5, the maximum force that the boy can exert on the rope so that the piece of wood does not move from its place is ________ N. (Round off to the Nearest Integer) [Take $$g = 10$$ m s$$^{-2}$$]
Login to view the detailed solution.
The disc of mass $$M$$ with uniform surface mass density $$\sigma$$ is shown in the figure. The center of mass of the quarter disc (the shaded area) is at the position $$\left(\frac{xa}{3\pi}, \frac{xa}{3\pi}\right)$$ where $$x$$ is ________. (Round off to the Nearest Integer) [$$a$$ is an area as shown in the figure]
Login to view the detailed solution.
Suppose you have taken a dilute solution of oleic acid in such a way that its concentration becomes 0.01 cm$$^3$$ of oleic acid per cm$$^3$$ of the solution. Then you make a thin film of this solution (monomolecular thickness) of area 4 cm$$^2$$ by considering 100 spherical drops of radius $$\left(\frac{3}{40\pi}\right)^{1/3} \times 10^{-3}$$ cm. Then the thickness of oleic acid layer will be $$x \times 10^{-14}$$ m. Where $$x$$ is ________.
Login to view the detailed solution.
The electric field intensity produced by the radiation coming from a 100 W bulb at a distance of 3 m is $$E$$. The electric field intensity produced by the radiation coming from 60 W at the same distance is $$\sqrt{\frac{x}{5}}E$$. Where the value of $$x$$ is ________.
Login to view the detailed solution.
The electric field in a region is given by $$\vec{E} = \frac{2}{5}E_0\hat{i} + \frac{3}{5}E_0\hat{j}$$ with $$E_0 = 4.0 \times 10^3$$ N C$$^{-1}$$. The flux of this field through a rectangular surface, area 0.4 m$$^2$$ parallel to the $$Y-Z$$ plane is ________ N m$$^2$$ C$$^{-1}$$.
Login to view the detailed solution.
A 2$$\mu$$F capacitor $$C_1$$ is first charged to a potential difference of 10 V using a battery. Then the battery is removed and the capacitor is connected to an uncharged capacitor $$C_2$$ of 8$$\mu$$F. The charge in $$C_2$$ on equilibrium condition is ________ $$\mu$$C. (Round off to the Nearest Integer)
Login to view the detailed solution.
Seawater at a frequency $$f = 9 \times 10^2$$ Hz, has permittivity $$\varepsilon = 80\varepsilon_0$$ and resistivity $$\rho = 0.25$$ $$\Omega$$ m. Imagine a parallel plate capacitor is immersed in seawater and is driven by an alternating voltage source $$V(t) = V_0 \sin(2\pi ft)$$. Then the conduction current density becomes $$10^x$$ times the displacement current density after time $$t = \frac{1}{800}$$ s. The value of $$x$$ is ________. (Given: $$\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9$$ N m$$^2$$ C$$^{-2}$$)
Login to view the detailed solution.
The image of an object placed in air formed by a convex refracting surface is at a distance of 10 m behind the surface. The image is real and is at $$\frac{2rd}{3}$$ of the distance of the object from the surface. The wavelength of light inside the surface is $$\frac{2}{3}$$ times the wavelength in air. The radius of the curved surface is $$\frac{x}{13}$$ m, the value of $$x$$ is ________.
Login to view the detailed solution.
A particle of mass $$m$$ moves in a circular orbit in a central potential field $$U(r) = U_0 r^4$$. If Bohr's quantization conditions are applied, radii of possible orbitals $$r_n$$ vary with $$n^{1/\alpha}$$, where $$\alpha$$ is ________.
Login to view the detailed solution.
Amongst the following, the linear species is:
Login to view the detailed solution.
During which of the following processes, does entropy decrease?
(A) Freezing of water to ice at 0°C
(B) Freezing of water to ice at -10°C
(C) N$$_2$$(g) + 3H$$_2$$(g) $$\to$$ 2NH$$_3$$(g)
(D) Adsorption of CO(g) and lead surface
(E) Dissolution of NaCl in water
Login to view the detailed solution.
The functional groups that are responsible for the ion-exchange property of cation and anion exchange resins, respectively, are:
Login to view the detailed solution.
The set of elements that differ in mutual relationship from those of the other sets is:
Login to view the detailed solution.
One of the by-products formed during the recovery of NH$$_3$$ from Solvay process is:
Login to view the detailed solution.
The correct pair(s) of the ambident nucleophiles is (are):
(A) AgCN / KCN
(B) RCOOAg / RCOOK
(C) AgNO$$_2$$ / KNO$$_2$$
(D) AgI / KI
Login to view the detailed solution.
Nitrogen can be estimated by Kjeldahl's method for which of the following compound?
Login to view the detailed solution.
Given below are two statements:
Statement-I: 2-methylbutane on oxidation with KMnO$$_4$$ gives 2-methylbutan-2-ol.
Statement-II: n-alkanes can be easily oxidised to corresponding alcohol with KMnO$$_4$$.
Choose the correct option:
Login to view the detailed solution.
Choose the correct statement regarding the formation of carbocations A and B given:
Login to view the detailed solution.
Which of the following statement(s) is (are) incorrect reason for eutrophication?
(A) excess usage of fertilisers
(B) excess usage of detergents
(C) dense plant population in water bodies
(D) lack of nutrients in water bodies that prevent plant growth
Choose the most appropriate answer from the options given below:
Login to view the detailed solution.
For the coagulation of a negative sol, the species below, that has the highest flocculating power is:
Login to view the detailed solution.
Match List-I and List-II:
List-I List-II
a. Haematite i. Al$$_2$$O$$_3$$ . xH$$_2$$O
b. Bauxite ii. Fe$$_2$$O$$_3$$
c. Magnetite iii. CuCO$$_3$$ . Cu(OH)$$_2$$
d. Malachite iv. Fe$$_3$$O$$_4$$
Choose the correct answer from the options given below:
Login to view the detailed solution.
The set that represents the pair of neutral oxides of nitrogen is:
Login to view the detailed solution.
The common positive oxidation states for an element with atomic number 24, are:
Login to view the detailed solution.
Match List-I with List-II:
List-I List-II
a. [Co(NH$$_3$$)$$_6$$][Cr(CN)$$_6$$] i. Linkage isomerism
b. [Co(NH$$_3$$)$$_3$$(NO$$_2$$)$$_3$$] ii. Solvate isomerism
c. [Cr(H$$_2$$O)$$_6$$]Cl$$_3$$ iii. Co-ordination isomerism
d. $$cis$$-[CrCl$$_2$$(ox)$$_2$$]$$^{3-}$$ iv. Optical isomerism
Choose the correct answer from the options given below:
Login to view the detailed solution.
In the above reaction, the structural formula of (A), "X" and "Y" respectively are:
Login to view the detailed solution.
Primary, secondary and tertiary amines can be separated using:
Login to view the detailed solution.
Match List-I with List-II:
List-I (Chemical Compound) List-II (Used as)
a. Sucralose i. Synthetic detergent
b. Glyceryl ester of stearic acid ii. Artificial sweetener
c. Sodium benzoate iii. Antiseptic
d. Bithional iv. Food preservative
Choose the correct match:
Login to view the detailed solution.
Fructose is an example of:
Login to view the detailed solution.
In the above reactions, the enzyme A and enzyme B respectively are:
Login to view the detailed solution.
Consider the above reaction. The percentage yield of amide product is ________ (Round off to the Nearest Integer).
(Given: Atomic mass: C: 12.0u, H: 1.0u, N: 14.0u, O: 16.0u, Cl: 35.5u)
Login to view the detailed solution.
The number of chlorine atoms in 20 mL of chlorine gas at STP is ________ $$\times 10^{21}$$. (Round off to the Nearest Integer).
[Assume chlorine is an ideal gas at STP. R = 0.083 L bar mol$$^{-1}$$ K$$^{-1}$$, $$N_A = 6.023 \times 10^{23}$$]
Login to view the detailed solution.
Consider the reaction $$N_2O_4(g) \rightleftharpoons 2NO_2(g)$$. The temperature at which $$K_C = 20.4$$ and $$K_P = 600.1$$, is ________ K. (Round off to the Nearest Integer). [Assume all gases are ideal and R = 0.0831 L bar K$$^{-1}$$ mol$$^{-1}$$]
Login to view the detailed solution.
The total number of C-C sigma bond/s in mesityl oxide ($$C_6H_{10}O$$) is ________ (Round off to the Nearest Integer).
Login to view the detailed solution.
KBr is doped with $$10^{-5}$$ mole percent of SrBr$$_2$$. The number of cationic vacancies in 1 g of KBr crystal is $$10^{14}$$ ________. (Round off to the Nearest Integer). [Atomic Mass: K: 39.1u, Br: 79.9u, $$N_A = 6.023 \times 10^{23}$$]
Login to view the detailed solution.
A 1 molal $$K_4Fe(CN)_6$$ solution has a degree of dissociation of 0.4. Its boiling point is equal to that of another solution which contains 18.1 weight percent of a non electrolytic solute A. The molar mass of A is ________ u. (Round off to the Nearest Integer). [Density of water = 1.0 g cm$$^{-3}$$]
Login to view the detailed solution.
A KCl solution of conductivity 0.14 S m$$^{-1}$$ shows a resistance of 4.19$$\Omega$$ in a conductivity cell. If the same cell is filled with an HCl solution, the resistance drops to 1.03$$\Omega$$. The conductivity of the HCl solution is ________ $$\times 10^{-2}$$ S m$$^{-1}$$. (Round off to the Nearest Integer).
Login to view the detailed solution.
The reaction $$2A + B_2 \to 2AB$$ is an elementary reaction. For a certain quantity of reactants, if the volume of the reaction vessel is reduced by a factor of 3, the rate of the reaction increases by a factor of ________. (Round off to the Nearest Integer).
Login to view the detailed solution.
In the ground state of atomic Fe (Z = 26), the spin-only magnetic moment is ________ $$\times 10^{-1}$$ BM. (Round off to the Nearest Integer).
[Given: $$\sqrt{3} = 1.73, \sqrt{2} = 1.41$$]
Login to view the detailed solution.
On complete reaction of FeCl$$_3$$ with oxalic acid in aqueous solution containing KOH, resulted in the formation of product A. The secondary valency of Fe in the product A is ________. (Round off to the Nearest Integer).
Login to view the detailed solution.
Let $$S_1, S_2$$ and $$S_3$$ be three sets defined as
$$S_1 = \{z \in \mathbb{C} : |z-1| \leq \sqrt{2}\}$$,
$$S_2 = \{z \in \mathbb{C} : Re((1-i)z) \geq 1\}$$ and
$$S_3 = \{z \in \mathbb{C} : Im(z) \leq 1\}$$.
Then, the set $$S_1 \cap S_2 \cap S_3$$:
Login to view the detailed solution.
If the sides $$AB$$, $$BC$$ and $$CA$$ of a triangle $$ABC$$ have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:
Login to view the detailed solution.
The value of $$\sum_{r=0}^{6} {^{6} C_{r}} \cdot {^{6} C_{6-r}}$$ is equal to:
Login to view the detailed solution.
The number of solutions of the equation $$x + 2\tan x = \frac{\pi}{2}$$ in the interval $$[0, 2\pi]$$ is:
Login to view the detailed solution.
Two tangents are drawn from a point $$P$$ to the circle $$x^2 + y^2 - 2x - 4y + 4 = 0$$, such that the angle between these tangents is $$\tan^{-1}\left(\frac{12}{5}\right)$$, where $$\tan^{-1}\left(\frac{12}{5}\right) \in (0, \pi)$$. If the centre of the circle is denoted by $$C$$ and these tangents touch the circle at points $$A$$ and $$B$$, then the ratio of the areas of $$\triangle PAB$$ and $$\triangle CAB$$ is:
Login to view the detailed solution.
Let the tangent to the circle $$x^2 + y^2 = 25$$ at the point $$R(3, 4)$$ meet $$x$$-axis and $$y$$-axis at point $$P$$ and $$Q$$, respectively. If $$r$$ is the radius of the circle passing through the origin $$O$$ and having centre at the incentre of the triangle $$OPQ$$, then $$r^2$$ is equal to:
Login to view the detailed solution.
Let $$L$$ be a tangent line to the parabola $$y^2 = 4x - 20$$ at $$(6, 2)$$. If $$L$$ is also a tangent to the ellipse $$\frac{x^2}{2} + \frac{y^2}{b} = 1$$, then the value of $$b$$ is equal to:
Login to view the detailed solution.
The value of $$\lim_{n \to \infty} \frac{[r] + [2r] + \ldots + [nr]}{n^2}$$, where $$r$$ is non-zero real number and $$[r]$$ denotes the greatest integer less than or equal to $$r$$, is equal to:
Login to view the detailed solution.
The value of the limit $$\lim_{\theta \to 0} \frac{\tan(\pi\cos^2\theta)}{\sin(2\pi\sin^2\theta)}$$ is equal to:
Login to view the detailed solution.
If the Boolean expression $$(p \wedge q) \circledast (p \otimes q)$$ is a tautology, then $$\circledast$$ and $$\otimes$$ are respectively given by:
Login to view the detailed solution.
If $$x, y, z$$ are in arithmetic progression with common difference $$d$$, $$x \neq 3d$$, and the determinant of the matrix $$\begin{bmatrix} 3 & 4\sqrt{2} & x \\ 4 & 5\sqrt{2} & y \\ 5 & k & z \end{bmatrix}$$ is zero, then the value of $$k^2$$ is:
Login to view the detailed solution.
The number of solutions of the equation $$\sin^{-1}\left[x^2 + \frac{1}{3}\right] + \cos^{-1}\left[x^2 - \frac{2}{3}\right] = x^2$$ for $$x \in [-1, 1]$$, and $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is:
Login to view the detailed solution.
Consider the function $$f : R \to R$$ defined by $$f(x) = \begin{cases} \left(2 - \sin\left(\frac{1}{x}\right)\right)|x|, & x \neq 0 \\ 0, & x = 0 \end{cases}$$. Then $$f$$ is:
Login to view the detailed solution.
Let $$f : R \to R$$ be defined as $$f(x) = e^{-x}\sin x$$. If $$F : [0, 1] \to R$$ is a differentiable function such that $$F(x) = \int_0^x f(t)dt$$, then the value of $$\int_0^1 (F'(x) + f(x))e^x dx$$ lies in the interval:
Login to view the detailed solution.
If the integral $$\int_0^{10} \frac{|\sin 2\pi x|}{e^{[x]}}dx = \alpha e^{-1} + \beta e^{-\frac{1}{2}} + \gamma$$, where $$\alpha, \beta, \gamma$$ are integers and $$[x]$$ denotes the greatest integer less than or equal to $$x$$, then the value of $$\alpha + \beta + \gamma$$ is equal to:
Login to view the detailed solution.
Let $$y = y(x)$$ be the solution of the differential equation $$\cos x(3\sin x + \cos x + 3)dy = (1 + y\sin x(3\sin x + \cos x + 3))dx$$, $$0 \leq x \leq \frac{\pi}{2}$$, $$y(0) = 0$$. Then, $$y\left(\frac{\pi}{3}\right)$$ is equal to:
Login to view the detailed solution.
If the curve $$y = y(x)$$ is the solution of the differential equation $$2(x^2 + x^{5/4})dy - y(x + x^{1/4})dx = 2x^{9/4}dx$$, $$x > 0$$ which passes through the point $$\left(1, 1 - \frac{4}{3}\log_e 2\right)$$, then the value of $$y(16)$$ is equal to:
Login to view the detailed solution.
Let $$O$$ be the origin. Let $$\vec{OP} = x\hat{i} + y\hat{j} - \hat{k}$$ and $$\vec{OQ} = -\hat{i} + 2\hat{j} + 3x\hat{k}$$, $$x, y \in R$$, $$x > 0$$, be such that $$|\vec{PQ}| = \sqrt{20}$$ and the vector $$\vec{OP}$$ is perpendicular to $$\vec{OQ}$$. If $$\vec{OR} = 3\hat{i} + z\hat{j} - 7\hat{k}$$, $$z \in R$$, is coplanar with $$\vec{OP}$$ and $$\vec{OQ}$$, then the value of $$x^2 + y^2 + z^2$$ is equal to:
Login to view the detailed solution.
If the equation of plane passing through the mirror image of a point $$(2, 3, 1)$$ with respect to line $$\frac{x+1}{2} = \frac{y-3}{1} = \frac{z+2}{-1}$$ and containing the line $$\frac{x-2}{3} = \frac{1-y}{2} = \frac{z+1}{1}$$ is $$\alpha x + \beta y + \gamma z = 24$$ then $$\alpha + \beta + \gamma$$ is equal to:
Login to view the detailed solution.
Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be $$\frac{1}{2}$$ and probability of occurrence of 0 at the odd place be $$\frac{1}{3}$$. Then the probability that 10 is followed by 01 is equal to:
Login to view the detailed solution.
If $$1, \log_{10}(4^x - 2)$$ and $$\log_{10}\left(4^x + \frac{18}{5}\right)$$ are in arithmetic progression for a real number $$x$$ then the value of the determinant $$\begin{vmatrix} 2(x-\frac{1}{2}) & x-1 & x^2 \\ 1 & 0 & x \\ x & 1 & 0 \end{vmatrix}$$ is equal to ________.
Login to view the detailed solution.
Let the coefficients of third, fourth and fifth terms in the expansion of $$\left(x + \frac{a}{x^2}\right)^n$$, $$x \neq 0$$, be in the ratio 12 : 8 : 3. Then the term independent of $$x$$ in the expansion, is equal to ________.
Login to view the detailed solution.
Let $$\tan\alpha$$, $$\tan\beta$$ and $$\tan\gamma$$; $$\alpha, \beta, \gamma \neq \frac{(2n-1)\pi}{2}$$, $$n \in N$$ be the slopes of the three line segments $$OA$$, $$OB$$ and $$OC$$, respectively, where $$O$$ is origin. If circumcentre of $$\triangle ABC$$ coincides with origin and its orthocentre lies on $$y$$-axis, then the value of $$\left(\frac{\cos 3\alpha + \cos 3\beta + \cos 3\gamma}{\cos\alpha \cdot \cos\beta \cdot \cos\gamma}\right)^2$$ is equal to ________.
Login to view the detailed solution.
Consider a set of $$3n$$ numbers having variance 4. In this set, the mean of first $$2n$$ numbers is 6 and the mean of the remaining $$n$$ numbers is 3. A new set is constructed by adding 1 into each of the first $$2n$$ numbers, and subtracting 1 from each of the remaining $$n$$ numbers. If the variance of the new set is $$k$$, then $$9k$$ is equal to ________.
Login to view the detailed solution.
Let $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ and $$B = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \end{bmatrix}$$ such that $$AB = B$$ and $$a + d = 2021$$, then the value of $$ad - bc$$ is equal to ________.
Login to view the detailed solution.
Let $$f : [-1, 1] \to R$$ be defined as $$f(x) = ax^2 + bx + c$$ for all $$x \in [-1, 1]$$, where $$a, b, c \in R$$ such that $$f(-1) = 2$$, $$f'(-1) = 1$$ and for $$x \in (-1, 1)$$ the maximum value of $$f''(x)$$ is $$\frac{1}{2}$$. If $$f(x) \leq \alpha$$, $$x \in [-1, 1]$$, then the least value of $$\alpha$$ is equal to ________.
Login to view the detailed solution.
Let $$I_n = \int_1^e x^{19}(\log|x|)^n dx$$, where $$n \in N$$. If $$(20)I_{10} = \alpha I_9 + \beta I_8$$, for natural numbers $$\alpha$$ and $$\beta$$, then $$\alpha - \beta$$ is equal to ________.
Login to view the detailed solution.
Let $$f : [-3, 1] \to R$$ be given as $$f(x) = \begin{cases} \min\{(x+6), x^2\}, & -3 \leq x \leq 0 \\ \max\{\sqrt{x}, x^2\}, & 0 \leq x \leq 1 \end{cases}$$. If the area bounded by $$y = f(x)$$ and $$x$$-axis is $$A$$ sq units, then the value of $$6A$$ is equal to ________.
Login to view the detailed solution.
Let $$\vec{x}$$ be a vector in the plane containing vectors $$\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$$ and $$\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$$. If the vector $$\vec{x}$$ is perpendicular to $$(3\hat{i} + 2\hat{j} - \hat{k})$$ and its projection on $$\vec{a}$$ is $$\frac{17\sqrt{6}}{2}$$, then the value of $$|\vec{x}|^2$$ is equal to ________.
Login to view the detailed solution.
Let $$P$$ be an arbitrary point having sum of the squares of the distances from the planes $$x + y + z = 0$$, $$lx - nz = 0$$ and $$x - 2y + z = 0$$ equal to 9 units. If the locus of the point $$P$$ is $$x^2 + y^2 + z^2 = 9$$, then the value of $$l - n$$ is equal to ________.
Login to view the detailed solution.
Educational materials for JEE preparation
Share