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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?

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

Question Number 19783 Answers: 1 Comments: 3

The sides of a triangle are of lengths (√((m^2 −n^2 ))) ,m^2 +n^2 , 2mn. Show that it is a right angle Δ.

$${The}\:{sides}\:{of}\:{a}\:{triangle}\:{are}\:{of} \\ $$$${lengths}\:\sqrt{\left({m}^{\mathrm{2}} −{n}^{\mathrm{2}} \right)}\:,{m}^{\mathrm{2}} +{n}^{\mathrm{2}} ,\:\mathrm{2}{mn}. \\ $$$${Show}\:{that}\:{it}\:{is}\:{a}\:{right}\:{angle}\:\Delta. \\ $$

Question Number 19700 Answers: 1 Comments: 0

What is the maximum possible value of k for which 2013 can be written as a sum of k consecutive positive integers?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of} \\ $$$${k}\:\mathrm{for}\:\mathrm{which}\:\mathrm{2013}\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{a} \\ $$$$\mathrm{sum}\:\mathrm{of}\:{k}\:\mathrm{consecutive}\:\mathrm{positive}\:\mathrm{integers}? \\ $$

Question Number 19699 Answers: 0 Comments: 0

Let S be a circle with centre O. A chord AB, not a diameter, divides S into two regions R_1 and R_2 such that O belongs to R_2 . Let S_1 be a circle with centre in R_1 , touching AB at X and S internally. Let S_2 be a circle with centre in R_2 , touching AB at Y, the circle S internally and passing through the centre of S. The point X lies on the diameter passing through the centre of S_2 and ∠YXO = 30°. If the radius of S_2 is 100 then what is the radius of S_1 ?

$$\mathrm{Let}\:{S}\:\mathrm{be}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{centre}\:{O}.\:\mathrm{A}\:\mathrm{chord} \\ $$$${AB},\:\mathrm{not}\:\mathrm{a}\:\mathrm{diameter},\:\mathrm{divides}\:{S}\:\mathrm{into}\:\mathrm{two} \\ $$$$\mathrm{regions}\:{R}_{\mathrm{1}} \:\mathrm{and}\:{R}_{\mathrm{2}} \:\mathrm{such}\:\mathrm{that}\:{O}\:\mathrm{belongs} \\ $$$$\mathrm{to}\:{R}_{\mathrm{2}} .\:\mathrm{Let}\:{S}_{\mathrm{1}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{centre}\:\mathrm{in} \\ $$$${R}_{\mathrm{1}} ,\:\mathrm{touching}\:{AB}\:\mathrm{at}\:{X}\:\mathrm{and}\:{S}\:\mathrm{internally}. \\ $$$$\mathrm{Let}\:{S}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{centre}\:\mathrm{in}\:{R}_{\mathrm{2}} , \\ $$$$\mathrm{touching}\:{AB}\:\mathrm{at}\:{Y},\:\mathrm{the}\:\mathrm{circle}\:{S}\:\mathrm{internally} \\ $$$$\mathrm{and}\:\mathrm{passing}\:\mathrm{through}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:{S}. \\ $$$$\mathrm{The}\:\mathrm{point}\:{X}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{the}\:\mathrm{diameter} \\ $$$$\mathrm{passing}\:\mathrm{through}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:{S}_{\mathrm{2}} \:\mathrm{and} \\ $$$$\angle{YXO}\:=\:\mathrm{30}°.\:\mathrm{If}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:{S}_{\mathrm{2}} \:\mathrm{is}\:\mathrm{100} \\ $$$$\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:{S}_{\mathrm{1}} ? \\ $$

Question Number 19698 Answers: 1 Comments: 0

Let f(x) = x^3 − 3x + b and g(x) = x^2 + bx − 3, where b is a real number. What is the sum of all possible values of b for which the equations f(x) = 0 and g(x) = 0 have a common root?

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:−\:\mathrm{3}{x}\:+\:{b}\:\mathrm{and}\:{g}\left({x}\right)\:=\:{x}^{\mathrm{2}} \:+ \\ $$$${bx}\:−\:\mathrm{3},\:\mathrm{where}\:{b}\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{What} \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:{b}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{equations}\:{f}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{and}\:{g}\left({x}\right) \\ $$$$=\:\mathrm{0}\:\mathrm{have}\:\mathrm{a}\:\mathrm{common}\:\mathrm{root}? \\ $$

Question Number 22315 Answers: 0 Comments: 0

Prove that the greatest coefficient in the expansion of (x_1 +x_2 +x_3 +...+x_k )^n = ((n!)/((q!)^(k−r) [(q+1)!]^r )) , where n = qk + r, 0 ≤ r ≤ k − 1

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{coefficient}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +...+{x}_{{k}} \right)^{{n}} \\ $$$$=\:\frac{{n}!}{\left({q}!\right)^{{k}−{r}} \left[\left({q}+\mathrm{1}\right)!\right]^{{r}} }\:,\:\mathrm{where}\:{n}\:=\:{qk}\:+\:{r}, \\ $$$$\mathrm{0}\:\leqslant\:{r}\:\leqslant\:{k}\:−\:\mathrm{1} \\ $$

Question Number 19696 Answers: 1 Comments: 0

Let m be the smallest odd positive integer for which 1 + 2 + ... + m is a square of an integer and let n be the smallest even positive integer for which 1 + 2 + ... + n is a square of an integer. What is the value of m + n?

$$\mathrm{Let}\:{m}\:\mathrm{be}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{odd}\:\mathrm{positive} \\ $$$$\mathrm{integer}\:\mathrm{for}\:\mathrm{which}\:\mathrm{1}\:+\:\mathrm{2}\:+\:...\:+\:{m}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{square}\:\mathrm{of}\:\mathrm{an}\:\mathrm{integer}\:\mathrm{and}\:\mathrm{let}\:{n}\:\mathrm{be}\:\mathrm{the} \\ $$$$\mathrm{smallest}\:\mathrm{even}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{1}\:+\:\mathrm{2}\:+\:...\:+\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{square}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{integer}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{m}\:+\:{n}? \\ $$

Question Number 22313 Answers: 1 Comments: 2

Question Number 19690 Answers: 1 Comments: 0

If ∣z − (4/z)∣ = 2, then find the maximum value of ∣z∣.

$$\mathrm{If}\:\mid{z}\:−\:\frac{\mathrm{4}}{{z}}\mid\:=\:\mathrm{2},\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mid{z}\mid. \\ $$

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