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AllQuestion and Answers: Page 1834

Question Number 17493 Answers: 0 Comments: 0

What is logarithm series?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{logarithm}\:\mathrm{series}? \\ $$

Question Number 16457 Answers: 1 Comments: 0

Find the range of f(x) = (3/(2 − x^2 ))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{f}\left({x}\right)\:=\:\frac{\mathrm{3}}{\mathrm{2}\:−\:{x}^{\mathrm{2}} } \\ $$

Question Number 16455 Answers: 2 Comments: 0

The speed of a projectile when it is at its greatest height is (√(2/5)) times its speed at half the maximum height. What is its angle of projection?

$$\mathrm{The}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{a}\:\mathrm{projectile}\:\mathrm{when}\:\mathrm{it}\:\mathrm{is}\:\mathrm{at} \\ $$$$\mathrm{its}\:\mathrm{greatest}\:\mathrm{height}\:\mathrm{is}\:\sqrt{\frac{\mathrm{2}}{\mathrm{5}}}\:\mathrm{times}\:\mathrm{its} \\ $$$$\mathrm{speed}\:\mathrm{at}\:\mathrm{half}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{height}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{its}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{projection}? \\ $$

Question Number 16451 Answers: 0 Comments: 0

The sixth term of an AP is 2, and its common difference is greater than one. The value of the common difference of the progression so that the product of the first, fourth and fifth terms is greatest is

$$\mathrm{The}\:\mathrm{sixth}\:\mathrm{term}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}\:\mathrm{is}\:\mathrm{2},\:\mathrm{and}\:\mathrm{its} \\ $$$$\mathrm{common}\:\mathrm{difference}\:\mathrm{is}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{one}. \\ $$$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{common}\:\mathrm{difference}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{progression}\:\mathrm{so}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{first},\:\mathrm{fourth}\:\mathrm{and}\:\mathrm{fifth}\:\mathrm{terms}\:\mathrm{is}\:\mathrm{greatest}\:\mathrm{is} \\ $$

Question Number 16430 Answers: 1 Comments: 2

In ΔABC with usual notation (r_1 /(bc)) + (r_2 /(ca)) + (r_3 /(ab)) is (1) (1/r) − (1/R) (2) (1/r) − (1/(2R)) (3) (1/r) + (1/(2R)) (4) (1/r) + (1/R)

$$\mathrm{In}\:\Delta{ABC}\:\mathrm{with}\:\mathrm{usual}\:\mathrm{notation} \\ $$$$\frac{{r}_{\mathrm{1}} }{{bc}}\:+\:\frac{{r}_{\mathrm{2}} }{{ca}}\:+\:\frac{{r}_{\mathrm{3}} }{{ab}}\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{1}}{{r}}\:−\:\frac{\mathrm{1}}{{R}} \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}}{{r}}\:−\:\frac{\mathrm{1}}{\mathrm{2}{R}} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{1}}{{r}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{R}} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{1}}{{r}}\:+\:\frac{\mathrm{1}}{{R}} \\ $$

Question Number 17647 Answers: 0 Comments: 5

ABC is a triangular park with AB = AC = 100 m. A clock tower is situated at the midpoint of BC. The angles of elevation of top of the tower at A and B are cot^(−1) (3.2) and cosec^(−1) (2.6) respectively. The height of tower is

$${ABC}\:\mathrm{is}\:\mathrm{a}\:\mathrm{triangular}\:\mathrm{park}\:\mathrm{with}\:{AB}\:= \\ $$$${AC}\:=\:\mathrm{100}\:\mathrm{m}.\:\mathrm{A}\:\mathrm{clock}\:\mathrm{tower}\:\mathrm{is}\:\mathrm{situated} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of}\:{BC}.\:\mathrm{The}\:\mathrm{angles}\:\mathrm{of} \\ $$$$\mathrm{elevation}\:\mathrm{of}\:\mathrm{top}\:\mathrm{of}\:\mathrm{the}\:\mathrm{tower}\:\mathrm{at}\:{A}\:\mathrm{and} \\ $$$${B}\:\mathrm{are}\:\mathrm{cot}^{−\mathrm{1}} \left(\mathrm{3}.\mathrm{2}\right)\:\mathrm{and}\:\mathrm{cosec}^{−\mathrm{1}} \left(\mathrm{2}.\mathrm{6}\right) \\ $$$$\mathrm{respectively}.\:\mathrm{The}\:\mathrm{height}\:\mathrm{of}\:\mathrm{tower}\:\mathrm{is} \\ $$

Question Number 16419 Answers: 1 Comments: 0

3 cubes of metal whose edges are 3,4 and 5 respectively are melted and formed into a single cube. If there be no loss of metal in the process find the side of the new cube.

$$\mathrm{3}\:\mathrm{cubes}\:\mathrm{of}\:\mathrm{metal}\:\mathrm{whose}\:\mathrm{edges}\:\mathrm{are}\:\mathrm{3},\mathrm{4} \\ $$$$\mathrm{and}\:\mathrm{5}\:\mathrm{respectively}\:\mathrm{are}\:\mathrm{melted}\:\mathrm{and} \\ $$$$\mathrm{formed}\:\mathrm{into}\:\mathrm{a}\:\mathrm{single}\:\mathrm{cube}.\:\mathrm{If}\:\mathrm{there}\:\mathrm{be} \\ $$$$\mathrm{no}\:\mathrm{loss}\:\mathrm{of}\:\mathrm{metal}\:\mathrm{in}\:\mathrm{the}\:\mathrm{process}\:\mathrm{find} \\ $$$$\mathrm{the}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\:\mathrm{new}\:\mathrm{cube}. \\ $$

Question Number 16418 Answers: 1 Comments: 0

A solid sphere of radius 4 cm is melted and recast into ′n′ solid hemispheres of radius 2 cm each. Find n.

$$\mathrm{A}\:\mathrm{solid}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{4}\:\mathrm{cm}\:\mathrm{is}\:\mathrm{melted} \\ $$$$\mathrm{and}\:\mathrm{recast}\:\mathrm{into}\:'{n}'\:\mathrm{solid}\:\mathrm{hemispheres}\:\mathrm{of} \\ $$$$\mathrm{radius}\:\mathrm{2}\:\mathrm{cm}\:\mathrm{each}.\:\mathrm{Find}\:{n}. \\ $$

Question Number 16409 Answers: 3 Comments: 9

Question Number 16406 Answers: 0 Comments: 0

D={(x,y)∣∣x∣+∣y∣≤1,∣x∣+∣y∣≥0.5} ∫∫_(D) ln (x^2 +y^2 )dxdy a)≥0; b)≤0; c)=0; d)non-existent.

$$\mathrm{D}=\left\{\left({x},\mathrm{y}\right)\mid\mid{x}\mid+\mid\mathrm{y}\mid\leqslant\mathrm{1},\mid{x}\mid+\mid{y}\mid\geqslant\mathrm{0}.\mathrm{5}\right\} \\ $$$$\underset{\mathrm{D}} {\int\int}\mathrm{ln}\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right){dxdy} \\ $$$$\left.\mathrm{a}\right)\geqslant\mathrm{0}; \\ $$$$\left.\mathrm{b}\right)\leqslant\mathrm{0}; \\ $$$$\left.\mathrm{c}\right)=\mathrm{0}; \\ $$$$\left.\mathrm{d}\right)\mathrm{non}-\mathrm{existent}. \\ $$

Question Number 16401 Answers: 1 Comments: 0

v=2i+2j+5k r=i+9j−8k Find 𝛚 I can do ((r×v)/r^2 )=𝛚 and i get 𝛚= ((61i−21j−16k)/(146)) but i dont get w×r=v. why?

$$\boldsymbol{{v}}=\mathrm{2}\boldsymbol{{i}}+\mathrm{2}\boldsymbol{{j}}+\mathrm{5}\boldsymbol{{k}} \\ $$$$\boldsymbol{{r}}=\boldsymbol{{i}}+\mathrm{9}\boldsymbol{{j}}−\mathrm{8}\boldsymbol{{k}} \\ $$$$\mathrm{Find}\:\boldsymbol{\omega} \\ $$$$\mathrm{I}\:\mathrm{can}\:\mathrm{do}\:\frac{\boldsymbol{{r}}×\boldsymbol{{v}}}{{r}^{\mathrm{2}} }=\boldsymbol{\omega} \\ $$$$\mathrm{and}\:\mathrm{i}\:\mathrm{get}\:\boldsymbol{\omega}=\:\frac{\mathrm{61}\boldsymbol{{i}}−\mathrm{21}\boldsymbol{{j}}−\mathrm{16}\boldsymbol{{k}}}{\mathrm{146}} \\ $$$$\mathrm{but}\:\mathrm{i}\:\mathrm{dont}\:\mathrm{get}\:\boldsymbol{{w}}×\boldsymbol{{r}}=\boldsymbol{{v}}. \\ $$$${why}? \\ $$

Question Number 16392 Answers: 3 Comments: 0

Question Number 16387 Answers: 0 Comments: 0

Why sinx is a power series?

$${Why}\:{sinx}\:{is}\:{a}\:{power}\:{series}? \\ $$

Question Number 16383 Answers: 2 Comments: 0

If P : Q : R = 2 : 3 : 4 and P^2 +Q^2 +R^2 =11600, then find (P+Q−R).

$$\mathrm{If}\:\mathrm{P}\::\:\mathrm{Q}\::\:\mathrm{R}\:=\:\mathrm{2}\::\:\mathrm{3}\::\:\mathrm{4}\:\mathrm{and}\:\mathrm{P}^{\mathrm{2}} +\mathrm{Q}^{\mathrm{2}} +\mathrm{R}^{\mathrm{2}} =\mathrm{11600}, \\ $$$$\mathrm{then}\:\mathrm{find}\:\left(\mathrm{P}+\mathrm{Q}−\mathrm{R}\right). \\ $$

Question Number 16373 Answers: 1 Comments: 1

if Σ_(k=0) ^(200) i^k +Π_(p=1) ^(50) i^p =x+iy then..(x,y)is... a. (0,1) b. (1,−1) c. (2,3) d. (4,8)

$${if}\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{200}} {\sum}}{i}^{{k}} +\underset{{p}=\mathrm{1}} {\overset{\mathrm{50}} {\prod}}{i}^{{p}} ={x}+{iy}\:{then}..\left({x},{y}\right){is}... \\ $$$${a}.\:\left(\mathrm{0},\mathrm{1}\right) \\ $$$${b}.\:\left(\mathrm{1},−\mathrm{1}\right) \\ $$$${c}.\:\left(\mathrm{2},\mathrm{3}\right) \\ $$$${d}.\:\left(\mathrm{4},\mathrm{8}\right) \\ $$

Question Number 16364 Answers: 1 Comments: 0

Question Number 16363 Answers: 1 Comments: 3

A car drives due north at 50 km/hr. Wind blows due North-West at 50(√2) km/hr. In what direction, a flag hoisted on the roof of the car points?

$$\mathrm{A}\:\mathrm{car}\:\mathrm{drives}\:\mathrm{due}\:\mathrm{north}\:\mathrm{at}\:\mathrm{50}\:\mathrm{km}/\mathrm{hr}. \\ $$$$\mathrm{Wind}\:\mathrm{blows}\:\mathrm{due}\:\mathrm{North}-\mathrm{West}\:\mathrm{at}\:\mathrm{50}\sqrt{\mathrm{2}} \\ $$$$\mathrm{km}/\mathrm{hr}.\:\mathrm{In}\:\mathrm{what}\:\mathrm{direction},\:\mathrm{a}\:\mathrm{flag} \\ $$$$\mathrm{hoisted}\:\mathrm{on}\:\mathrm{the}\:\mathrm{roof}\:\mathrm{of}\:\mathrm{the}\:\mathrm{car}\:\mathrm{points}? \\ $$

Question Number 16360 Answers: 1 Comments: 0

Question Number 16359 Answers: 1 Comments: 0

In a ΔABC if ((s − a)/(a − b)) = ((s − c)/(b − c)) , then prove that r_1 , r_2 , r_3 are in A.P. Here r_1 , r_2 and r_3 are the exradii opposite to angles A, B and C respectively.

$$\mathrm{In}\:\mathrm{a}\:\Delta{ABC}\:\mathrm{if}\:\frac{{s}\:−\:{a}}{{a}\:−\:{b}}\:=\:\frac{{s}\:−\:{c}}{{b}\:−\:{c}}\:,\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:{r}_{\mathrm{1}} ,\:{r}_{\mathrm{2}} ,\:{r}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{in}\:\mathrm{A}.\mathrm{P}. \\ $$$$\mathrm{Here}\:{r}_{\mathrm{1}} ,\:{r}_{\mathrm{2}} \:\mathrm{and}\:{r}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{exradii} \\ $$$$\mathrm{opposite}\:\mathrm{to}\:\mathrm{angles}\:{A},\:{B}\:\mathrm{and}\:{C}\:\mathrm{respectively}. \\ $$

Question Number 16358 Answers: 1 Comments: 1

In any triangle ABC, a cot A + b cot B + c cot C is equal to (1) r + R (2) r − R (3) 2(r + R) (4) 2(r − R)

$$\mathrm{In}\:\mathrm{any}\:\mathrm{triangle}\:{ABC},\:{a}\:\mathrm{cot}\:{A}\:+\:{b}\:\mathrm{cot}\:{B} \\ $$$$+\:{c}\:\mathrm{cot}\:{C}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{1}\right)\:{r}\:+\:{R} \\ $$$$\left(\mathrm{2}\right)\:{r}\:−\:{R} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{2}\left({r}\:+\:{R}\right) \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}\left({r}\:−\:{R}\right) \\ $$

Question Number 16450 Answers: 0 Comments: 3

Ten balls were manufactured, nine of them have the same mass, while just one of them has a slightly higher or slightly lower mass. Given is just a beam balance and no weights. comparing the masses of balls only, with the help of the balance , and in just 3 weighings explain how to judge which is the defective one and whether it is heavier or lighter than the rest (as the case may be).

$${Ten}\:{balls}\:{were}\:{manufactured}, \\ $$$$\:{nine}\:{of}\:{them}\:{have}\:{the}\:{same} \\ $$$${mass},\:{while}\:{just}\:{one}\:{of}\:{them} \\ $$$${has}\:{a}\:{slightly}\:{higher}\:{or}\:{slightly} \\ $$$${lower}\:{mass}.\:{Given}\:{is}\:{just}\:{a}\:{beam} \\ $$$${balance}\:{and}\:{no}\:{weights}.\:{comparing} \\ $$$${the}\:{masses}\:{of}\:{balls}\:{only},\:{with}\:{the} \\ $$$${help}\:{of}\:{the}\:{balance}\:,\:{and}\:{in}\:{just} \\ $$$$\mathrm{3}\:{weighings}\:{explain}\:{how}\:{to}\:{judge} \\ $$$${which}\:{is}\:{the}\:{defective}\:{one}\:{and} \\ $$$${whether}\:{it}\:{is}\:{heavier}\:{or}\:{lighter} \\ $$$${than}\:{the}\:{rest}\:\left({as}\:{the}\:{case}\:{may}\:{be}\right). \\ $$

Question Number 16441 Answers: 1 Comments: 2

Question Number 16338 Answers: 1 Comments: 2

Question Number 16330 Answers: 0 Comments: 0

Question Number 16310 Answers: 1 Comments: 3

Question Number 16302 Answers: 1 Comments: 5

Related to Q16140 What if the three lines d_1 ,d_2 ,d_3 are not parallel, but concurrent?

Pg 1829 Pg 1830 Pg 1831 Pg 1832 Pg 1833 Pg 1834 Pg 1835 Pg 1836 Pg 1837 Pg 1838

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