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Question Number 19812 Answers: 1 Comments: 0

If tangent line of equation y = (x/(3 − x)) at x = a crossed line y = x at (b,b) Find b in terms of a

$$\mathrm{If}\:\mathrm{tangent}\:\mathrm{line}\:\mathrm{of}\:\mathrm{equation}\:{y}\:=\:\frac{{x}}{\mathrm{3}\:−\:{x}}\:\mathrm{at}\: \\ $$$${x}\:=\:{a}\:\mathrm{crossed}\:\mathrm{line}\:{y}\:=\:{x}\:\mathrm{at}\:\left({b},{b}\right) \\ $$$$\mathrm{Find}\:{b}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{a} \\ $$

Question Number 19811 Answers: 0 Comments: 2

If f(x) = (x +1)g(x) − 2 and g(3) = 4 Find the remainder if f(x) divided by (x + 1)(x − 3)

$$\mathrm{If}\:{f}\left({x}\right)\:=\:\left({x}\:+\mathrm{1}\right){g}\left({x}\right)\:−\:\mathrm{2}\:\mathrm{and}\:{g}\left(\mathrm{3}\right)\:=\:\mathrm{4} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{if}\:{f}\left({x}\right)\:\mathrm{divided}\:\mathrm{by}\: \\ $$$$\left({x}\:+\:\mathrm{1}\right)\left({x}\:−\:\mathrm{3}\right) \\ $$

Question Number 19809 Answers: 1 Comments: 0

If sin θ+cosec θ=2, then sin^2 θ+cosec^2 θ is equal to

$$\mathrm{If}\:\mathrm{sin}\:\theta+\mathrm{cosec}\:\theta=\mathrm{2},\:\mathrm{then}\:\mathrm{sin}^{\mathrm{2}} \theta+\mathrm{cosec}^{\mathrm{2}} \theta \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 19978 Answers: 1 Comments: 0

Let f be a one-to-one function from the set of natural numbers to itself such that f(mn) = f(m)f(n) for all natural numbers m and n. What is the least possible value of f(999)?

$$\mathrm{Let}\:{f}\:\mathrm{be}\:\mathrm{a}\:\mathrm{one}-\mathrm{to}-\mathrm{one}\:\mathrm{function}\:\mathrm{from} \\ $$$$\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{numbers}\:\mathrm{to}\:\mathrm{itself} \\ $$$$\mathrm{such}\:\mathrm{that}\:{f}\left({mn}\right)\:=\:{f}\left({m}\right){f}\left({n}\right)\:\mathrm{for}\:\mathrm{all} \\ $$$$\mathrm{natural}\:\mathrm{numbers}\:{m}\:\mathrm{and}\:{n}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{least}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{f}\left(\mathrm{999}\right)? \\ $$

Question Number 19799 Answers: 0 Comments: 2

For a natural number b, let N(b) denote the number of natural numbers a for which the equation x^2 + ax + b = 0 has integer roots. What is the smallest value of b for which N(b) = 20?

$$\mathrm{For}\:\mathrm{a}\:\mathrm{natural}\:\mathrm{number}\:{b},\:\mathrm{let}\:{N}\left({b}\right)\:\mathrm{denote} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{numbers}\:{a}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} \:+\:{ax}\:+\:{b}\:=\:\mathrm{0}\:\mathrm{has} \\ $$$$\mathrm{integer}\:\mathrm{roots}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{smallest} \\ $$$$\mathrm{value}\:\mathrm{of}\:{b}\:\mathrm{for}\:\mathrm{which}\:{N}\left({b}\right)\:=\:\mathrm{20}? \\ $$

Question Number 19796 Answers: 1 Comments: 1

Question Number 19795 Answers: 1 Comments: 0

One morning, each member of Manjul′s family drank an 8-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank 1/7-th of the total amount of milk and 2/17-th of the total amount of coffee. How many people are there in Manjul′s family?

$$\mathrm{One}\:\mathrm{morning},\:\mathrm{each}\:\mathrm{member}\:\mathrm{of}\:\mathrm{Manjul}'\mathrm{s} \\ $$$$\mathrm{family}\:\mathrm{drank}\:\mathrm{an}\:\mathrm{8}-\mathrm{ounce}\:\mathrm{mixture}\:\mathrm{of} \\ $$$$\mathrm{coffee}\:\mathrm{and}\:\mathrm{milk}.\:\mathrm{The}\:\mathrm{amounts}\:\mathrm{of}\:\mathrm{coffee} \\ $$$$\mathrm{and}\:\mathrm{milk}\:\mathrm{varied}\:\mathrm{from}\:\mathrm{cup}\:\mathrm{to}\:\mathrm{cup},\:\mathrm{but} \\ $$$$\mathrm{were}\:\mathrm{never}\:\mathrm{zero}.\:\mathrm{Manjul}\:\mathrm{drank}\:\mathrm{1}/\mathrm{7}-\mathrm{th} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{total}\:\mathrm{amount}\:\mathrm{of}\:\mathrm{milk}\:\mathrm{and}\:\mathrm{2}/\mathrm{17}-\mathrm{th} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{total}\:\mathrm{amount}\:\mathrm{of}\:\mathrm{coffee}.\:\mathrm{How} \\ $$$$\mathrm{many}\:\mathrm{people}\:\mathrm{are}\:\mathrm{there}\:\mathrm{in}\:\mathrm{Manjul}'\mathrm{s} \\ $$$$\mathrm{family}? \\ $$

Question Number 19794 Answers: 0 Comments: 0

In a triangle ABC with ∠BCA = 90°, the perpendicular bisector of AB intersects segments AB and AC at X and Y, respectively. If the ratio of the area of quadrilateral BXYC to the area of triangle ABC is 13 : 18 and BC = 12 then what is the length of AC?

$$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:{ABC}\:\mathrm{with}\:\angle{BCA}\:=\:\mathrm{90}°, \\ $$$$\mathrm{the}\:\mathrm{perpendicular}\:\mathrm{bisector}\:\mathrm{of}\:{AB} \\ $$$$\mathrm{intersects}\:\mathrm{segments}\:{AB}\:\mathrm{and}\:{AC}\:\mathrm{at}\:{X} \\ $$$$\mathrm{and}\:{Y},\:\mathrm{respectively}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{quadrilateral}\:{BXYC}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{triangle}\:{ABC}\:\mathrm{is}\:\mathrm{13}\::\:\mathrm{18}\:\mathrm{and} \\ $$$${BC}\:=\:\mathrm{12}\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:{AC}? \\ $$

Question Number 19792 Answers: 2 Comments: 0

Let ABCD be a convex quadrilateral with ∠DAB = ∠BDC = 90°. Let the incircles of triangles ABD and BCD touch BD at P and Q, respectively, with P lying in between B and Q. If AD = 999 and PQ = 200 then what is the sum of the radii of the incircles of triangles ABD and BDC?

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{with}\:\angle{DAB}\:=\:\angle{BDC}\:=\:\mathrm{90}°.\:\mathrm{Let}\:\mathrm{the} \\ $$$$\mathrm{incircles}\:\mathrm{of}\:\mathrm{triangles}\:{ABD}\:\mathrm{and}\:{BCD} \\ $$$$\mathrm{touch}\:{BD}\:\mathrm{at}\:{P}\:\mathrm{and}\:{Q},\:\mathrm{respectively}, \\ $$$$\mathrm{with}\:{P}\:\mathrm{lying}\:\mathrm{in}\:\mathrm{between}\:{B}\:\mathrm{and}\:{Q}.\:\mathrm{If} \\ $$$${AD}\:=\:\mathrm{999}\:\mathrm{and}\:{PQ}\:=\:\mathrm{200}\:\mathrm{then}\:\mathrm{what}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{radii}\:\mathrm{of}\:\mathrm{the}\:\mathrm{incircles}\:\mathrm{of} \\ $$$$\mathrm{triangles}\:{ABD}\:\mathrm{and}\:{BDC}? \\ $$

Question Number 19828 Answers: 1 Comments: 1

The speed of a train is reduced from 80km/h to 40km/h after the application of the brake. (i)how much further would the train travel before coming to rest (ii)assuming the acceleration is kept constant,how long will it take to bring the train to rest after the application of the brakes?

$${The}\:{speed}\:{of}\:{a}\:{train}\:{is}\:{reduced} \\ $$$${from}\:\mathrm{80}{km}/{h}\:{to}\:\mathrm{40}{km}/{h}\:{after} \\ $$$${the}\:{application}\:{of}\:{the}\:{brake}. \\ $$$$\left({i}\right){how}\:{much}\:{further}\:{would}\:{the}\: \\ $$$${train}\:{travel}\:{before}\:{coming}\:{to}\:{rest} \\ $$$$\left({ii}\right){assuming}\:{the}\:{acceleration}\:{is} \\ $$$${kept}\:{constant},{how}\:{long}\:{will}\:{it} \\ $$$${take}\:{to}\:{bring}\:{the}\:{train}\:{to}\:{rest} \\ $$$${after}\:{the}\:{application}\:{of}\:{the}\:{brakes}? \\ $$

Question Number 19774 Answers: 1 Comments: 0

A balloon is ascending vertically with an acceleration of 0.2 ms^(−2) . Two stones are dropped from it at an interval of 2 s. The distance between them when the second stone dropped is (take g = 9.8 ms^(−2) )

$$\mathrm{A}\:\mathrm{balloon}\:\mathrm{is}\:\mathrm{ascending}\:\mathrm{vertically}\:\mathrm{with} \\ $$$$\mathrm{an}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{0}.\mathrm{2}\:\mathrm{ms}^{−\mathrm{2}} .\:\mathrm{Two}\:\mathrm{stones} \\ $$$$\mathrm{are}\:\mathrm{dropped}\:\mathrm{from}\:\mathrm{it}\:\mathrm{at}\:\mathrm{an}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{2}\:\mathrm{s}. \\ $$$$\mathrm{The}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{them}\:\mathrm{when}\:\mathrm{the} \\ $$$$\mathrm{second}\:\mathrm{stone}\:\mathrm{dropped}\:\mathrm{is}\:\left(\mathrm{take}\:{g}\:=\:\mathrm{9}.\mathrm{8}\right. \\ $$$$\left.\mathrm{ms}^{−\mathrm{2}} \right) \\ $$

Question Number 19760 Answers: 1 Comments: 0

If sin x + cos x + tan x + cosec x + cot x + sec x = 7, then find the value of sin 2x.

$$\mathrm{If}\:\mathrm{sin}\:{x}\:+\:\mathrm{cos}\:{x}\:+\:\mathrm{tan}\:{x}\:+\:\mathrm{cosec}\:{x}\:+ \\ $$$$\mathrm{cot}\:{x}\:+\:\mathrm{sec}\:{x}\:=\:\mathrm{7},\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{sin}\:\mathrm{2}{x}. \\ $$

Question Number 19758 Answers: 1 Comments: 1

Question Number 19741 Answers: 0 Comments: 1

If z = x + iy and arg(((z − 2)/(z + 2))) = (π/6), then find the locus of z.

$$\mathrm{If}\:{z}\:=\:{x}\:+\:{iy}\:\mathrm{and}\:\mathrm{arg}\left(\frac{{z}\:−\:\mathrm{2}}{{z}\:+\:\mathrm{2}}\right)\:=\:\frac{\pi}{\mathrm{6}},\:\mathrm{then} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}. \\ $$

Question Number 19740 Answers: 1 Comments: 0

If ∣z^2 − 1∣ = ∣z∣^2 + 1, then z lies on

$$\mathrm{If}\:\mid{z}^{\mathrm{2}} \:−\:\mathrm{1}\mid\:=\:\mid{z}\mid^{\mathrm{2}} \:+\:\mathrm{1},\:\mathrm{then}\:{z}\:\mathrm{lies}\:\mathrm{on} \\ $$

Question Number 19739 Answers: 1 Comments: 0

If z = x + iy is a complex number satisfying ∣z + (i/2)∣^2 = ∣z − (i/2)∣^2 , then the locus of z is

$$\mathrm{If}\:{z}\:=\:{x}\:+\:{iy}\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number} \\ $$$$\mathrm{satisfying}\:\mid{z}\:+\:\frac{{i}}{\mathrm{2}}\mid^{\mathrm{2}} \:=\:\mid{z}\:−\:\frac{{i}}{\mathrm{2}}\mid^{\mathrm{2}} ,\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{is} \\ $$

Question Number 19738 Answers: 1 Comments: 1

Locus of the point z satisfying the equation ∣iz − 1∣ + ∣z − i∣ = 2 is

$$\mathrm{Locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:{z}\:\mathrm{satisfying}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mid{iz}\:−\:\mathrm{1}\mid\:+\:\mid{z}\:−\:{i}\mid\:=\:\mathrm{2}\:\mathrm{is} \\ $$

Question Number 19736 Answers: 1 Comments: 1

Question Number 19735 Answers: 1 Comments: 0

If z = λ + 3 + i(√(5 − λ^2 )), then the locus of z is a

$$\mathrm{If}\:{z}\:=\:\lambda\:+\:\mathrm{3}\:+\:{i}\sqrt{\mathrm{5}\:−\:\lambda^{\mathrm{2}} },\:\mathrm{then}\:\mathrm{the}\:\mathrm{locus} \\ $$$$\mathrm{of}\:{z}\:\mathrm{is}\:\mathrm{a} \\ $$

Question Number 19733 Answers: 1 Comments: 0

If ∣z + 1∣ = (√2)∣z − 1∣, then the locus described by the point z in the argand diagram is a

$$\mathrm{If}\:\mid{z}\:+\:\mathrm{1}\mid\:=\:\sqrt{\mathrm{2}}\mid{z}\:−\:\mathrm{1}\mid,\:\mathrm{then}\:\mathrm{the}\:\mathrm{locus} \\ $$$$\mathrm{described}\:\mathrm{by}\:\mathrm{the}\:\mathrm{point}\:{z}\:\mathrm{in}\:\mathrm{the}\:\mathrm{argand} \\ $$$$\mathrm{diagram}\:\mathrm{is}\:\mathrm{a} \\ $$

Question Number 19734 Answers: 1 Comments: 0

If the imaginary part of ((2z + 1)/(iz + 1)) is −2, then the locus of the point representing z in the complex plane is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{imaginary}\:\mathrm{part}\:\mathrm{of}\:\frac{\mathrm{2}{z}\:+\:\mathrm{1}}{{iz}\:+\:\mathrm{1}}\:\mathrm{is}\:−\mathrm{2}, \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:\mathrm{representing} \\ $$$${z}\:\mathrm{in}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{plane}\:\mathrm{is} \\ $$

Question Number 19732 Answers: 1 Comments: 0

The locus of z given by ∣((z − 1)/(z − i))∣ = 1 is

$$\mathrm{The}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{given}\:\mathrm{by}\:\mid\frac{{z}\:−\:\mathrm{1}}{{z}\:−\:{i}}\mid\:=\:\mathrm{1}\:\mathrm{is} \\ $$

Question Number 19730 Answers: 1 Comments: 0

If z = x + iy and ∣z − 2i∣ = 1, then (1) z lies on x-axis (2) z lies on y-axis (3) z lies on a circle (4) None of these

$$\mathrm{If}\:{z}\:=\:{x}\:+\:{iy}\:\mathrm{and}\:\mid{z}\:−\:\mathrm{2}{i}\mid\:=\:\mathrm{1},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{z}\:\mathrm{lies}\:\mathrm{on}\:{x}-\mathrm{axis} \\ $$$$\left(\mathrm{2}\right)\:{z}\:\mathrm{lies}\:\mathrm{on}\:{y}-\mathrm{axis} \\ $$$$\left(\mathrm{3}\right)\:{z}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{a}\:\mathrm{circle} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{None}\:\mathrm{of}\:\mathrm{these} \\ $$

Question Number 19729 Answers: 1 Comments: 0

2x + 9y^2 = 4 2x^2 − 45y^2 + xy = 0 Find the value of xy

$$\mathrm{2}{x}\:+\:\mathrm{9}{y}^{\mathrm{2}} \:=\:\mathrm{4} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} \:−\:\mathrm{45}{y}^{\mathrm{2}} \:+\:{xy}\:=\:\mathrm{0} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{xy} \\ $$

Question Number 19709 Answers: 1 Comments: 0

In the cyclic quadrilateral ABCD AB=7,BC=8,CD=8,DA=15. Calculate the angle ADC and the length ofAC.

$${In}\:{the}\:{cyclic}\:{quadrilateral}\:{ABCD} \\ $$$${AB}=\mathrm{7},{BC}=\mathrm{8},{CD}=\mathrm{8},{DA}=\mathrm{15}. \\ $$$${Calculate}\:{the}\:{angle}\:{ADC}\:{and} \\ $$$${the}\:{length}\:{ofAC}. \\ $$

Question Number 19704 Answers: 1 Comments: 0

What is the sum (in base 10) of all the natural numbers less than 64 which have exactly three ones in their base 2 representation?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\left(\mathrm{in}\:\mathrm{base}\:\mathrm{10}\right)\:\mathrm{of}\:\mathrm{all}\:\mathrm{the} \\ $$$$\mathrm{natural}\:\mathrm{numbers}\:\mathrm{less}\:\mathrm{than}\:\mathrm{64}\:\mathrm{which} \\ $$$$\mathrm{have}\:\mathrm{exactly}\:\mathrm{three}\:\mathrm{ones}\:\mathrm{in}\:\mathrm{their}\:\mathrm{base}\:\mathrm{2} \\ $$$$\mathrm{representation}? \\ $$

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