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Question Number 16277 Answers: 3 Comments: 1

Question Number 16273 Answers: 2 Comments: 4

If in ΔABC r_1 = r_2 + r_3 + r, prove that triangle is right angled.

$$\mathrm{If}\:\mathrm{in}\:\Delta{ABC}\:{r}_{\mathrm{1}} \:=\:{r}_{\mathrm{2}} \:+\:{r}_{\mathrm{3}} \:+\:{r},\:\mathrm{prove} \\ $$$$\mathrm{that}\:\mathrm{triangle}\:\mathrm{is}\:\mathrm{right}\:\mathrm{angled}. \\ $$

Question Number 16271 Answers: 1 Comments: 0

The unemployment rate among workers under 25 in a state went from 8.2% to 7.5% in one year. Assume an average of 1340200 workers and estimate the decrease in the number unemployed.

$${The}\:{unemployment}\:{rate}\:{among}\:{workers} \\ $$$${under}\:\mathrm{25}\:{in}\:{a}\:{state}\:{went}\:{from}\:\mathrm{8}.\mathrm{2\%} \\ $$$${to}\:\mathrm{7}.\mathrm{5\%}\:{in}\:{one}\:{year}.\:{Assume}\:{an}\:{average} \\ $$$${of}\:\mathrm{1340200}\:{workers}\:{and}\:{estimate} \\ $$$${the}\:{decrease}\:{in}\:{the}\:{number}\:{unemployed}. \\ $$

Question Number 16269 Answers: 1 Comments: 0

2^(nd) part of Q. 16214: Prove that r_1 = s tan ((A/2)), r_2 = s tan ((B/2)), r_3 = s tan ((C/2)).

$$\mathrm{2}^{\mathrm{nd}} \:\mathrm{part}\:\mathrm{of}\:\mathrm{Q}.\:\mathrm{16214}:\:\mathrm{Prove}\:\mathrm{that} \\ $$$${r}_{\mathrm{1}} \:=\:{s}\:\mathrm{tan}\:\left(\frac{{A}}{\mathrm{2}}\right),\:{r}_{\mathrm{2}} \:=\:{s}\:\mathrm{tan}\:\left(\frac{{B}}{\mathrm{2}}\right), \\ $$$${r}_{\mathrm{3}} \:=\:{s}\:\mathrm{tan}\:\left(\frac{{C}}{\mathrm{2}}\right). \\ $$

Question Number 16240 Answers: 2 Comments: 1

Question Number 16238 Answers: 0 Comments: 0

we have a^5 +b^5 =1 and u^5 +v^5 =1 find value a^3 u^5 +b^3 v^5 =?

$${we}\:{have}\:{a}^{\mathrm{5}} +{b}^{\mathrm{5}} =\mathrm{1}\:{and}\:{u}^{\mathrm{5}} +{v}^{\mathrm{5}} =\mathrm{1} \\ $$$${find}\:{value}\:\:{a}^{\mathrm{3}} {u}^{\mathrm{5}} +{b}^{\mathrm{3}} {v}^{\mathrm{5}} =? \\ $$

Question Number 16226 Answers: 0 Comments: 1

Question Number 16220 Answers: 1 Comments: 0

Question Number 16214 Answers: 2 Comments: 4

In ΔABC, r_1 , r_2 and r_3 are the exradii as shown. Prove that r_1 = (Δ/(s − a)) , r_2 = (Δ/(s − b)) and r_3 = (Δ/(s − c)) . Here s = ((a + b + c)/2) .

$$\mathrm{In}\:\Delta{ABC},\:{r}_{\mathrm{1}} ,\:{r}_{\mathrm{2}} \:\mathrm{and}\:{r}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{exradii} \\ $$$$\mathrm{as}\:\mathrm{shown}.\:\mathrm{Prove}\:\mathrm{that}\:{r}_{\mathrm{1}} \:=\:\frac{\Delta}{{s}\:−\:{a}}\:, \\ $$$${r}_{\mathrm{2}} \:=\:\frac{\Delta}{{s}\:−\:{b}}\:\mathrm{and}\:{r}_{\mathrm{3}} \:=\:\frac{\Delta}{{s}\:−\:{c}}\:.\:\mathrm{Here} \\ $$$${s}\:=\:\frac{{a}\:+\:{b}\:+\:{c}}{\mathrm{2}}\:. \\ $$

Question Number 16194 Answers: 0 Comments: 21

Question Number 16179 Answers: 1 Comments: 0

If a > 0, b > 0 and the minimum value of a sin^2 θ + b cosec^2 θ is equal to maximum value of a sin^2 θ + b cos^2 θ, then (a/b) is equal to [Answer: 4]

$$\mathrm{If}\:{a}\:>\:\mathrm{0},\:{b}\:>\:\mathrm{0}\:\mathrm{and}\:\mathrm{the}\:\mathrm{minimum} \\ $$$$\mathrm{value}\:\mathrm{of}\:{a}\:\mathrm{sin}^{\mathrm{2}} \:\theta\:+\:{b}\:\mathrm{cosec}^{\mathrm{2}} \:\theta\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:{a}\:\mathrm{sin}^{\mathrm{2}} \:\theta\:+\:{b}\:\mathrm{cos}^{\mathrm{2}} \:\theta, \\ $$$$\mathrm{then}\:\frac{{a}}{{b}}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\left[\boldsymbol{\mathrm{Answer}}:\:\mathrm{4}\right] \\ $$

Question Number 16616 Answers: 1 Comments: 1

A particle starts from the origin with velocity (√(44)) ms^(−1) on a straight horizontal road. Its acceleration varies with displacement as shown. The velocity of the particle as it passes through the position x = 0.2 km is [Answer: 18 ms^(−1) ]

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{starts}\:\mathrm{from}\:\mathrm{the}\:\mathrm{origin}\:\mathrm{with} \\ $$$$\mathrm{velocity}\:\sqrt{\mathrm{44}}\:\mathrm{ms}^{−\mathrm{1}} \:\mathrm{on}\:\mathrm{a}\:\mathrm{straight} \\ $$$$\mathrm{horizontal}\:\mathrm{road}.\:\mathrm{Its}\:\mathrm{acceleration}\:\mathrm{varies} \\ $$$$\mathrm{with}\:\mathrm{displacement}\:\mathrm{as}\:\mathrm{shown}.\:\mathrm{The} \\ $$$$\mathrm{velocity}\:\mathrm{of}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{as}\:\mathrm{it}\:\mathrm{passes} \\ $$$$\mathrm{through}\:\mathrm{the}\:\mathrm{position}\:{x}\:=\:\mathrm{0}.\mathrm{2}\:\mathrm{km}\:\mathrm{is} \\ $$$$\left[\mathrm{Answer}:\:\mathrm{18}\:\mathrm{ms}^{−\mathrm{1}} \right] \\ $$

Question Number 16156 Answers: 0 Comments: 2

A body in a uniform horizontal circular motion possesses a variable velocity. Does it mean that the K.E. of the body is also variable?

$$\mathrm{A}\:\mathrm{body}\:\mathrm{in}\:\mathrm{a}\:\mathrm{uniform}\:\mathrm{horizontal} \\ $$$$\mathrm{circular}\:\mathrm{motion}\:\mathrm{possesses}\:\mathrm{a}\:\mathrm{variable} \\ $$$$\mathrm{velocity}.\:\mathrm{Does}\:\mathrm{it}\:\mathrm{mean}\:\mathrm{that}\:\mathrm{the}\:\mathrm{K}.\mathrm{E}. \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{body}\:\mathrm{is}\:\mathrm{also}\:\mathrm{variable}? \\ $$

Question Number 16155 Answers: 1 Comments: 0

A body of mass m is projected with a speed v making an angle θ with the vertical. What is the change in momentum of the body along the Y- axis; between the starting point and the highest point of its path?

$$\mathrm{A}\:\mathrm{body}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{m}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{speed}\:{v}\:\mathrm{making}\:\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the} \\ $$$$\mathrm{vertical}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{change}\:\mathrm{in} \\ $$$$\mathrm{momentum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{body}\:\mathrm{along}\:\mathrm{the}\:\mathrm{Y}- \\ $$$$\mathrm{axis};\:\mathrm{between}\:\mathrm{the}\:\mathrm{starting}\:\mathrm{point}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{highest}\:\mathrm{point}\:\mathrm{of}\:\mathrm{its}\:\mathrm{path}? \\ $$

Question Number 16153 Answers: 1 Comments: 0

Is angular displacement a vector quantity?

$$\mathrm{Is}\:\mathrm{angular}\:\mathrm{displacement}\:\mathrm{a}\:\mathrm{vector} \\ $$$$\mathrm{quantity}? \\ $$

Question Number 16152 Answers: 1 Comments: 0

In long jump, does it matter how high you jump? What factors determine the span of the jump?

$$\mathrm{In}\:\mathrm{long}\:\mathrm{jump},\:\mathrm{does}\:\mathrm{it}\:\mathrm{matter}\:\mathrm{how}\:\mathrm{high} \\ $$$$\mathrm{you}\:\mathrm{jump}?\:\mathrm{What}\:\mathrm{factors}\:\mathrm{determine}\:\mathrm{the} \\ $$$$\mathrm{span}\:\mathrm{of}\:\mathrm{the}\:\mathrm{jump}? \\ $$

Question Number 16150 Answers: 1 Comments: 1

A projectile is fired at an angle θ with the horizontal direction from O. Neglecting the air friction, it hits the ground at B after 3 seconds. What is the height of point A from ground? [Use g = 10 m/s^2 ]

$$\mathrm{A}\:\mathrm{projectile}\:\mathrm{is}\:\mathrm{fired}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{horizontal}\:\mathrm{direction}\:\mathrm{from}\:{O}. \\ $$$$\mathrm{Neglecting}\:\mathrm{the}\:\mathrm{air}\:\mathrm{friction},\:\mathrm{it}\:\mathrm{hits}\:\mathrm{the} \\ $$$$\mathrm{ground}\:\mathrm{at}\:{B}\:\mathrm{after}\:\mathrm{3}\:\mathrm{seconds}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{height}\:\mathrm{of}\:\mathrm{point}\:{A}\:\mathrm{from}\:\mathrm{ground}? \\ $$$$\left[\mathrm{Use}\:{g}\:=\:\mathrm{10}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right] \\ $$

Question Number 16140 Answers: 2 Comments: 0

Question Number 16139 Answers: 2 Comments: 0

Path of the bomb released from an aeroplane moving with uniform velocity at certain height as observed by the pilot is (a) a straight line (b) a parabola (c) a circle (d) none of the above

$$\mathrm{Path}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bomb}\:\mathrm{released}\:\mathrm{from}\:\mathrm{an} \\ $$$$\mathrm{aeroplane}\:\mathrm{moving}\:\mathrm{with}\:\mathrm{uniform} \\ $$$$\mathrm{velocity}\:\mathrm{at}\:\mathrm{certain}\:\mathrm{height}\:\mathrm{as}\:\mathrm{observed} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{pilot}\:\mathrm{is} \\ $$$$\left({a}\right)\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line} \\ $$$$\left({b}\right)\:\mathrm{a}\:\mathrm{parabola} \\ $$$$\left({c}\right)\:\mathrm{a}\:\mathrm{circle} \\ $$$$\left({d}\right)\:\mathrm{none}\:\mathrm{of}\:\mathrm{the}\:\mathrm{above} \\ $$

Question Number 16138 Answers: 0 Comments: 0

How many nodal planes are present in 4d_z^2 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{nodal}\:\mathrm{planes}\:\mathrm{are}\:\mathrm{present}\:\mathrm{in} \\ $$$$\mathrm{4d}_{\mathrm{z}^{\mathrm{2}} } \:? \\ $$

Question Number 16137 Answers: 0 Comments: 0

For 2s orbital Ψ_r = (1/(√8))((z/a_0 ))^(3/2) (2 − ((zr)/a_0 ))e^(−((zr)/(2a_0 ))) then, hydrogen radial node will be at the distance of (1) a_0 (2) 2a_0 (3) (a_0 /2) (4) (a_0 /3)

$$\mathrm{For}\:\mathrm{2}{s}\:\mathrm{orbital}\:\Psi_{\mathrm{r}} \:=\:\frac{\mathrm{1}}{\sqrt{\mathrm{8}}}\left(\frac{\mathrm{z}}{\mathrm{a}_{\mathrm{0}} }\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \left(\mathrm{2}\:−\:\frac{\mathrm{zr}}{\mathrm{a}_{\mathrm{0}} }\right)\mathrm{e}^{−\frac{\mathrm{zr}}{\mathrm{2a}_{\mathrm{0}} }} \\ $$$$\mathrm{then},\:\mathrm{hydrogen}\:\mathrm{radial}\:\mathrm{node}\:\mathrm{will}\:\mathrm{be}\:\mathrm{at} \\ $$$$\mathrm{the}\:\mathrm{distance}\:\mathrm{of} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{a}_{\mathrm{0}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{2a}_{\mathrm{0}} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{a}_{\mathrm{0}} }{\mathrm{2}} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{a}_{\mathrm{0}} }{\mathrm{3}} \\ $$

Question Number 16136 Answers: 1 Comments: 0

Photoelectric emission is observed from a surface when lights of frequency n_1 and n_2 incident. If the ratio of maximum kinetic energy in two cases is K : 1 then (Assume n_1 > n_2 ) threshold frequency is (1) (K − 1) × (Kn_2 − n_1 ) (2) ((Kn_1 − n_2 )/(1 − K)) (3) ((K − 1)/(Kn_1 − n_2 )) (4) ((Kn_2 − n_1 )/(K − 1))

$$\mathrm{Photoelectric}\:\mathrm{emission}\:\mathrm{is}\:\mathrm{observed}\:\mathrm{from} \\ $$$$\mathrm{a}\:\mathrm{surface}\:\mathrm{when}\:\mathrm{lights}\:\mathrm{of}\:\mathrm{frequency}\:{n}_{\mathrm{1}} \\ $$$$\mathrm{and}\:{n}_{\mathrm{2}} \:\mathrm{incident}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{maximum} \\ $$$$\mathrm{kinetic}\:\mathrm{energy}\:\mathrm{in}\:\mathrm{two}\:\mathrm{cases}\:\mathrm{is}\:\mathrm{K}\::\:\mathrm{1} \\ $$$$\mathrm{then}\:\left(\mathrm{Assume}\:{n}_{\mathrm{1}} \:>\:{n}_{\mathrm{2}} \right)\:\mathrm{threshold} \\ $$$$\mathrm{frequency}\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\left(\mathrm{K}\:−\:\mathrm{1}\right)\:×\:\left(\mathrm{K}{n}_{\mathrm{2}} \:−\:{n}_{\mathrm{1}} \right) \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{K}{n}_{\mathrm{1}} \:−\:{n}_{\mathrm{2}} }{\mathrm{1}\:−\:\mathrm{K}} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{K}\:−\:\mathrm{1}}{\mathrm{K}{n}_{\mathrm{1}} \:−\:{n}_{\mathrm{2}} } \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{K}{n}_{\mathrm{2}} \:−\:{n}_{\mathrm{1}} }{\mathrm{K}\:−\:\mathrm{1}} \\ $$

Question Number 16135 Answers: 0 Comments: 0

An electron is moving in 3^(rd) orbit of Hydrogen atom. The frequency of moving electron is (1) 2.19 × 10^(14) rps (2) 7.3 × 10^(14) rps (3) 2.44 × 10^(14) rps (4) 7.3 × 10^(10) rps

$$\mathrm{An}\:\mathrm{electron}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{in}\:\mathrm{3}^{\mathrm{rd}} \:\mathrm{orbit}\:\mathrm{of} \\ $$$$\mathrm{Hydrogen}\:\mathrm{atom}.\:\mathrm{The}\:\mathrm{frequency}\:\mathrm{of} \\ $$$$\mathrm{moving}\:\mathrm{electron}\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{2}.\mathrm{19}\:×\:\mathrm{10}^{\mathrm{14}} \:\mathrm{rps} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{7}.\mathrm{3}\:×\:\mathrm{10}^{\mathrm{14}} \:\mathrm{rps} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{2}.\mathrm{44}\:×\:\mathrm{10}^{\mathrm{14}} \:\mathrm{rps} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{7}.\mathrm{3}\:×\:\mathrm{10}^{\mathrm{10}} \:\mathrm{rps} \\ $$

Question Number 16134 Answers: 0 Comments: 0

The mathematical expression which is true for the uncertainty principle is (1) (Δx) (Δv) ≥ (h/(4π)) (2) (ΔE) (Δx) ≥ (h/(4π)) (3) (Δθ) (Δφ) ≥ (h/(4π)) (4) (Δx) (Δm) ≥ (h/(4π))

Question Number 16133 Answers: 1 Comments: 0

H_α line of Balmer series is 6500 A^o . The wave length of Hγ is (1) 4815 A^o (2) 4298 A^o (3) 7800 A^o (4) 3800 A^o

$$\mathrm{H}_{\alpha} \:\mathrm{line}\:\mathrm{of}\:\mathrm{Balmer}\:\mathrm{series}\:\mathrm{is}\:\mathrm{6500}\:\overset{\mathrm{o}} {\mathrm{A}}.\:\mathrm{The} \\ $$$$\mathrm{wave}\:\mathrm{length}\:\mathrm{of}\:\mathrm{H}\gamma\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{4815}\:\overset{\mathrm{o}} {\mathrm{A}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{4298}\:\overset{\mathrm{o}} {\mathrm{A}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{7800}\:\overset{\mathrm{o}} {\mathrm{A}} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{3800}\:\overset{\mathrm{o}} {\mathrm{A}} \\ $$

Question Number 16116 Answers: 0 Comments: 0

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