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Question Number 21223 Answers: 1 Comments: 0

lim_(x→π/2) (π−2x)tan (x)

$$\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\left(\pi−\mathrm{2}{x}\right)\mathrm{tan}\:\left({x}\right) \\ $$

Question Number 21224 Answers: 0 Comments: 1

One mole of a monoatomic real gas satisfies the equation p(V − b) = RT where b is a constant. The relationship of interatomic potential V(r) and interatomic distance r for the gas is given by

$$\mathrm{One}\:\mathrm{mole}\:\mathrm{of}\:\mathrm{a}\:\mathrm{monoatomic}\:\mathrm{real}\:\mathrm{gas} \\ $$$$\mathrm{satisfies}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{p}\left(\mathrm{V}\:−\:\mathrm{b}\right)\:=\:\mathrm{RT} \\ $$$$\mathrm{where}\:\mathrm{b}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}.\:\mathrm{The}\:\mathrm{relationship} \\ $$$$\mathrm{of}\:\mathrm{interatomic}\:\mathrm{potential}\:\mathrm{V}\left(\mathrm{r}\right)\:\mathrm{and} \\ $$$$\mathrm{interatomic}\:\mathrm{distance}\:\mathrm{r}\:\mathrm{for}\:\mathrm{the}\:\mathrm{gas}\:\mathrm{is} \\ $$$$\mathrm{given}\:\mathrm{by} \\ $$

Question Number 21219 Answers: 1 Comments: 0

(A) If ∣w∣ = 2, then the set of points z = w − (1/w) is contained in or equal to (B) If ∣w∣ = 1, then the set of points z = w + (1/w) is contained in or equal to Options for both A and B: (p) An ellipse with eccentricity (4/5) (q) The set of points z satisfying Im z = 0 (r) The set of points z satisfying ∣Im z∣ ≤ 1 (s) The set of points z satisfying ∣Re z∣ ≤ 2 (t) The set of points z satisfying ∣z∣ ≤ 3

$$\left(\mathrm{A}\right)\:\mathrm{If}\:\mid{w}\mid\:=\:\mathrm{2},\:\mathrm{then}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points} \\ $$$${z}\:=\:{w}\:−\:\frac{\mathrm{1}}{{w}}\:\mathrm{is}\:\mathrm{contained}\:\mathrm{in}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{If}\:\mid{w}\mid\:=\:\mathrm{1},\:\mathrm{then}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points} \\ $$$${z}\:=\:{w}\:+\:\frac{\mathrm{1}}{{w}}\:\mathrm{is}\:\mathrm{contained}\:\mathrm{in}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{Options}\:\mathrm{for}\:\mathrm{both}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}: \\ $$$$\left(\mathrm{p}\right)\:\mathrm{An}\:\mathrm{ellipse}\:\mathrm{with}\:\mathrm{eccentricity}\:\frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\left(\mathrm{q}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mathrm{Im}\:{z} \\ $$$$=\:\mathrm{0} \\ $$$$\left(\mathrm{r}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mid\mathrm{Im}\:{z}\mid \\ $$$$\leqslant\:\mathrm{1} \\ $$$$\left(\mathrm{s}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mid\mathrm{Re}\:{z}\mid \\ $$$$\leqslant\:\mathrm{2} \\ $$$$\left(\mathrm{t}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mid{z}\mid\:\leqslant\:\mathrm{3} \\ $$

Question Number 21212 Answers: 0 Comments: 0

Question Number 21205 Answers: 0 Comments: 1

Question Number 21210 Answers: 0 Comments: 0

Question Number 21194 Answers: 0 Comments: 4

. lim_(x→2^+ ) {(([x]^3 )/3) −[ (x/3)]^3 } is equal to ...

$$.\:\underset{{x}\rightarrow\mathrm{2}^{+} } {\mathrm{li}{m}}\:\left\{\frac{\left[{x}\right]^{\mathrm{3}} }{\mathrm{3}}\:−\left[\:\frac{{x}}{\mathrm{3}}\right]^{\mathrm{3}} \right\}\:{is}\:{equal}\:{to}\:... \\ $$$$ \\ $$

Question Number 21179 Answers: 1 Comments: 0

lim_(x→∞) (√(16x^2 + 4x)) − (√x^2 ) − (√(9x^2 + 3x))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{16}{x}^{\mathrm{2}} \:+\:\mathrm{4}{x}}\:−\:\sqrt{{x}^{\mathrm{2}} }\:−\:\sqrt{\mathrm{9}{x}^{\mathrm{2}} \:+\:\mathrm{3}{x}} \\ $$

Question Number 21174 Answers: 1 Comments: 0

The factors of determinant ((x,a,b),(a,x,b),(a,b,x))are

$$\mathrm{The}\:\mathrm{factors}\:\mathrm{of}\:\begin{vmatrix}{{x}}&{{a}}&{{b}}\\{{a}}&{{x}}&{{b}}\\{{a}}&{{b}}&{{x}}\end{vmatrix}\mathrm{are} \\ $$

Question Number 21168 Answers: 1 Comments: 0

Let f(x) = ∣x − 1∣ + ∣x − 2∣ + ∣x − 3∣, then find the value of k for which f(x) = k has 1. no solution 2. only one solution 3. two solutions of same sign 4. two solutions of opposite sign

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:\mid{x}\:−\:\mathrm{1}\mid\:+\:\mid{x}\:−\:\mathrm{2}\mid\:+\:\mid{x}\:−\:\mathrm{3}\mid, \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{for}\:\mathrm{which}\:{f}\left({x}\right) \\ $$$$=\:{k}\:\mathrm{has} \\ $$$$\mathrm{1}.\:\mathrm{no}\:\mathrm{solution} \\ $$$$\mathrm{2}.\:\mathrm{only}\:\mathrm{one}\:\mathrm{solution} \\ $$$$\mathrm{3}.\:\mathrm{two}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{same}\:\mathrm{sign} \\ $$$$\mathrm{4}.\:\mathrm{two}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{opposite}\:\mathrm{sign} \\ $$

Question Number 21154 Answers: 1 Comments: 2

prove: ∀n∈N^∗ ,∀(a,b)∈C^2 , a^(2n+1) +b^(2n+1) = Σ_(k=o) ^(2n) (−1)^k a^k b^(2n−k)

$${prove}:\:\forall{n}\in\mathbb{N}^{\ast} ,\forall\left({a},{b}\right)\in\mathbb{C}^{\mathrm{2}} ,\:{a}^{\mathrm{2}{n}+\mathrm{1}} +{b}^{\mathrm{2}{n}+\mathrm{1}} = \\ $$$$\underset{{k}={o}} {\overset{\mathrm{2}{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} {a}^{{k}} {b}^{\mathrm{2}{n}−{k}} \\ $$

Question Number 21150 Answers: 0 Comments: 8

Two particles of mass m each are tied at the ends of a light string of length 2a. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance ′a′ from the center P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force F. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes 2x, is

$$\mathrm{Two}\:\mathrm{particles}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{each}\:\mathrm{are}\:\mathrm{tied} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{ends}\:\mathrm{of}\:\mathrm{a}\:\mathrm{light}\:\mathrm{string}\:\mathrm{of}\:\mathrm{length}\:\mathrm{2}{a}. \\ $$$$\mathrm{The}\:\mathrm{whole}\:\mathrm{system}\:\mathrm{is}\:\mathrm{kept}\:\mathrm{on}\:\mathrm{a}\:\mathrm{frictionless} \\ $$$$\mathrm{horizontal}\:\mathrm{surface}\:\mathrm{with}\:\mathrm{the}\:\mathrm{string}\:\mathrm{held} \\ $$$$\mathrm{tight}\:\mathrm{so}\:\mathrm{that}\:\mathrm{each}\:\mathrm{mass}\:\mathrm{is}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance} \\ $$$$'{a}'\:\mathrm{from}\:\mathrm{the}\:\mathrm{center}\:{P}\:\left(\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\right. \\ $$$$\left.\mathrm{figure}\right).\:\mathrm{Now},\:\mathrm{the}\:\mathrm{mid}-\mathrm{point}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{string}\:\mathrm{is}\:\mathrm{pulled}\:\mathrm{vertically}\:\mathrm{upwards}\:\mathrm{with} \\ $$$$\mathrm{a}\:\mathrm{small}\:\mathrm{but}\:\mathrm{constant}\:\mathrm{force}\:{F}.\:\mathrm{As}\:\mathrm{a}\:\mathrm{result}, \\ $$$$\mathrm{the}\:\mathrm{particles}\:\mathrm{move}\:\mathrm{towards}\:\mathrm{each}\:\mathrm{other} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{surface}.\:\mathrm{The}\:\mathrm{magnitude}\:\mathrm{of} \\ $$$$\mathrm{acceleration},\:\mathrm{when}\:\mathrm{the}\:\mathrm{separation} \\ $$$$\mathrm{between}\:\mathrm{them}\:\mathrm{becomes}\:\mathrm{2}{x},\:\mathrm{is} \\ $$

Question Number 21148 Answers: 0 Comments: 12

Find the compression in the spring if the system shown below is in equilibrium.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{compression}\:\mathrm{in}\:\mathrm{the}\:\mathrm{spring}\:\mathrm{if} \\ $$$$\mathrm{the}\:\mathrm{system}\:\mathrm{shown}\:\mathrm{below}\:\mathrm{is}\:\mathrm{in} \\ $$$$\mathrm{equilibrium}. \\ $$

Question Number 21145 Answers: 0 Comments: 7

Figure shows an arrangement of blocks, pulley and strings. Strings and pulley are massless and frictionless. The relation between acceleration of the blocks as shown in the figure is

$$\mathrm{Figure}\:\mathrm{shows}\:\mathrm{an}\:\mathrm{arrangement}\:\mathrm{of}\:\mathrm{blocks}, \\ $$$$\mathrm{pulley}\:\mathrm{and}\:\mathrm{strings}.\:\mathrm{Strings}\:\mathrm{and}\:\mathrm{pulley} \\ $$$$\mathrm{are}\:\mathrm{massless}\:\mathrm{and}\:\mathrm{frictionless}.\:\mathrm{The} \\ $$$$\mathrm{relation}\:\mathrm{between}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{blocks}\:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\:\mathrm{figure}\:\mathrm{is} \\ $$

Question Number 21142 Answers: 0 Comments: 1

if A+B=(π/4) so proof (1+tan A)(1+tan B)=2

$${if}\:{A}+{B}=\frac{\pi}{\mathrm{4}} \\ $$$${so}\:{proof}\:\left(\mathrm{1}+\mathrm{tan}\:{A}\right)\left(\mathrm{1}+\mathrm{tan}\:{B}\right)=\mathrm{2} \\ $$

Question Number 21141 Answers: 1 Comments: 0

if sin αsin β−cos αcos β+1=0 so proof that 1+cot αtan β=0

$${if}\:\mathrm{sin}\:\alpha\mathrm{sin}\:\beta−\mathrm{cos}\:\alpha\mathrm{cos}\:\beta+\mathrm{1}=\mathrm{0}\: \\ $$$${so}\:{proof}\:{that}\:\mathrm{1}+\mathrm{cot}\:\alpha\mathrm{tan}\:\beta=\mathrm{0} \\ $$

Question Number 21137 Answers: 1 Comments: 0

If the minimum value of ∣z+1+i∣ + ∣z−1−i∣ + ∣2 − z∣ + ∣3 − z∣ is k then (k − 8) equals

$$\mathrm{If}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mid{z}+\mathrm{1}+{i}\mid\:+\:\mid{z}−\mathrm{1}−{i}\mid\:+\:\mid\mathrm{2}\:−\:{z}\mid\:+\:\mid\mathrm{3}\:−\:{z}\mid\:\mathrm{is} \\ $$$${k}\:\mathrm{then}\:\left({k}\:−\:\mathrm{8}\right)\:\mathrm{equals} \\ $$

Question Number 21131 Answers: 0 Comments: 4

Figure shows a small bob of mass m suspended from a point on a thin rod by a light inextensible string of length l. The rod is rigidly fixed on a circular platform. The platform is set into rotation. The minimum angular speed ω, for which the bob loses contact with the vertical rod, is (1) (√(g/l)) (2) (√((2g)/l)) (3) (√(g/(2l))) (4) (√(g/(4l)))

$$\mathrm{Figure}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{small}\:\mathrm{bob}\:\mathrm{of}\:\mathrm{mass}\:{m} \\ $$$$\mathrm{suspended}\:\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{a}\:\mathrm{thin}\:\mathrm{rod} \\ $$$$\mathrm{by}\:\mathrm{a}\:\mathrm{light}\:\mathrm{inextensible}\:\mathrm{string}\:\mathrm{of}\:\mathrm{length} \\ $$$${l}.\:\mathrm{The}\:\mathrm{rod}\:\mathrm{is}\:\mathrm{rigidly}\:\mathrm{fixed}\:\mathrm{on}\:\mathrm{a}\:\mathrm{circular} \\ $$$$\mathrm{platform}.\:\mathrm{The}\:\mathrm{platform}\:\mathrm{is}\:\mathrm{set}\:\mathrm{into} \\ $$$$\mathrm{rotation}.\:\mathrm{The}\:\mathrm{minimum}\:\mathrm{angular}\:\mathrm{speed} \\ $$$$\omega,\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{bob}\:\mathrm{loses}\:\mathrm{contact}\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{vertical}\:\mathrm{rod},\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\sqrt{\frac{{g}}{{l}}} \\ $$$$\left(\mathrm{2}\right)\:\sqrt{\frac{\mathrm{2}{g}}{{l}}} \\ $$$$\left(\mathrm{3}\right)\:\sqrt{\frac{{g}}{\mathrm{2}{l}}} \\ $$$$\left(\mathrm{4}\right)\:\sqrt{\frac{{g}}{\mathrm{4}{l}}} \\ $$

Question Number 21124 Answers: 2 Comments: 0

Question Number 21116 Answers: 1 Comments: 0

Question Number 21112 Answers: 0 Comments: 0

A ball is bouncing elastically with a speed 1 m/s between walls of a railway compartment of size 10 m in a direction perpendicular to walls. The train is moving at a constant velocity of 10 m/s parallel to the direction of motion of the ball. As seen from the ground (a) the direction of motion of the ball changes every 10 seconds. (b) speed of ball changes every 10 seconds. (c) average speed of ball over any 20 second interval is fixed. (d) the acceleration of ball is the same as from the train.

Question Number 21111 Answers: 0 Comments: 0

STATEMENT-1 : z_1 ^2 + z_2 ^2 + z_3 ^2 + z_4 ^2 = 0 where z_1 , z_2 , z_3 and z_4 are the fourth roots of unity. and STATEMENT-2 : (1)^(1/4) = (cos0° + i sin0°)^(1/4) .

$$\mathrm{STATEMENT}-\mathrm{1}\::\:{z}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{2}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{3}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{4}} ^{\mathrm{2}} \:= \\ $$$$\mathrm{0}\:\mathrm{where}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{and}\:{z}_{\mathrm{4}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{fourth} \\ $$$$\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}. \\ $$$$\boldsymbol{\mathrm{and}} \\ $$$$\mathrm{STATEMENT}-\mathrm{2}\::\:\left(\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} \:=\:\left(\mathrm{cos0}°\:+\right. \\ $$$$\left.{i}\:\mathrm{sin0}°\right)^{\frac{\mathrm{1}}{\mathrm{4}}} . \\ $$

Question Number 21109 Answers: 0 Comments: 2

STATEMENT-1 : The locus of z, if arg(((z − 1)/(z + 1))) = (π/2) is a circle. and STATEMENT-2 : ∣((z − 2)/(z + 2))∣ = (π/2), then the locus of z is a circle.

$$\mathrm{STATEMENT}-\mathrm{1}\::\:\mathrm{The}\:\mathrm{locus}\:\mathrm{of}\:{z},\:\mathrm{if} \\ $$$$\mathrm{arg}\left(\frac{{z}\:−\:\mathrm{1}}{{z}\:+\:\mathrm{1}}\right)\:=\:\frac{\pi}{\mathrm{2}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}. \\ $$$$\boldsymbol{\mathrm{and}} \\ $$$$\mathrm{STATEMENT}-\mathrm{2}\::\:\mid\frac{{z}\:−\:\mathrm{2}}{{z}\:+\:\mathrm{2}}\mid\:=\:\frac{\pi}{\mathrm{2}},\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}. \\ $$

Question Number 21108 Answers: 1 Comments: 0

Let A, B, C be three sets of complex numbers as defined below A = {z : Im z ≥ 1} B = {z : ∣z − 2 − i∣ = 3} C = {z : Re((1 − i)z) = (√2)}. Let z be any point in A ∩ B ∩ C and let w be any point satisfying ∣w − 2 − i∣ < 3. Then, ∣z∣ − ∣w∣ + 3 lies between (1) −6 and 3 (2) −3 and 6 (3) −6 and 6 (4) −3 and 9

$$\mathrm{Let}\:{A},\:{B},\:{C}\:\mathrm{be}\:\mathrm{three}\:\mathrm{sets}\:\mathrm{of}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\mathrm{as}\:\mathrm{defined}\:\mathrm{below} \\ $$$${A}\:=\:\left\{{z}\::\:\mathrm{Im}\:{z}\:\geqslant\:\mathrm{1}\right\} \\ $$$${B}\:=\:\left\{{z}\::\:\mid{z}\:−\:\mathrm{2}\:−\:{i}\mid\:=\:\mathrm{3}\right\} \\ $$$${C}\:=\:\left\{{z}\::\:\mathrm{Re}\left(\left(\mathrm{1}\:−\:{i}\right){z}\right)\:=\:\sqrt{\mathrm{2}}\right\}. \\ $$$$\mathrm{Let}\:{z}\:\mathrm{be}\:\mathrm{any}\:\mathrm{point}\:\mathrm{in}\:{A}\:\cap\:{B}\:\cap\:{C}\:\mathrm{and}\:\mathrm{let} \\ $$$${w}\:\mathrm{be}\:\mathrm{any}\:\mathrm{point}\:\mathrm{satisfying}\:\mid{w}\:−\:\mathrm{2}\:−\:{i}\mid\:< \\ $$$$\mathrm{3}.\:\mathrm{Then},\:\mid{z}\mid\:−\:\mid{w}\mid\:+\:\mathrm{3}\:\mathrm{lies}\:\mathrm{between} \\ $$$$\left(\mathrm{1}\right)\:−\mathrm{6}\:\mathrm{and}\:\mathrm{3} \\ $$$$\left(\mathrm{2}\right)\:−\mathrm{3}\:\mathrm{and}\:\mathrm{6} \\ $$$$\left(\mathrm{3}\right)\:−\mathrm{6}\:\mathrm{and}\:\mathrm{6} \\ $$$$\left(\mathrm{4}\right)\:−\mathrm{3}\:\mathrm{and}\:\mathrm{9} \\ $$

Question Number 21107 Answers: 0 Comments: 0

Find out the value of nth derivative of y=e^(msin^(−1) x ) at x=0

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{nth}\:\mathrm{derivative}\:\mathrm{of}\:\mathrm{y}=\mathrm{e}^{\mathrm{msin}^{−\mathrm{1}} \mathrm{x}\:} \:\mathrm{at}\:\mathrm{x}=\mathrm{0} \\ $$

Question Number 21200 Answers: 1 Comments: 1

Suppose an integer x, a natural number n and a prime number p satisfy the equation 7x^2 − 44x + 12 = p^n . Find the largest value of p.

$$\mathrm{Suppose}\:\mathrm{an}\:\mathrm{integer}\:{x},\:\mathrm{a}\:\mathrm{natural} \\ $$$$\mathrm{number}\:{n}\:\mathrm{and}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:{p} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{7}{x}^{\mathrm{2}} \:−\:\mathrm{44}{x}\:+\:\mathrm{12}\:=\:{p}^{{n}} . \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:{p}. \\ $$

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