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Question Number 113714 Answers: 1 Comments: 0

Question Number 113710 Answers: 1 Comments: 2

you have 2 identical mathematics books, 2 identical physics books, 2 identical chemistry books, 2 identical biology books and 2 geography books. in how many ways can you compile these books such that same books are not mutually adjacent?

$${you}\:{have}\:\mathrm{2}\:{identical}\:{mathematics} \\ $$$${books},\:\mathrm{2}\:{identical}\:{physics}\:{books},\:\mathrm{2} \\ $$$${identical}\:{chemistry}\:{books},\:\mathrm{2}\:{identical} \\ $$$${biology}\:{books}\:{and}\:\mathrm{2}\:{geography}\:{books}. \\ $$$${in}\:{how}\:{many}\:{ways}\:{can}\:{you}\:{compile} \\ $$$${these}\:{books}\:{such}\:{that}\:{same}\:{books} \\ $$$${are}\:{not}\:{mutually}\:{adjacent}? \\ $$

Question Number 113706 Answers: 4 Comments: 0

∫(√(tanx)) dx =?

$$\:\:\:\int\sqrt{\mathrm{tanx}}\:\mathrm{dx}\:=?\:\:\:\: \\ $$

Question Number 113757 Answers: 2 Comments: 0

∫ (dx/(tan x−sin x)) ?

$$\:\int\:\frac{{dx}}{\mathrm{tan}\:{x}−\mathrm{sin}\:{x}}\:?\: \\ $$

Question Number 113756 Answers: 2 Comments: 1

∫_0 ^1 ((log(x+1))/x)dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{log}\left({x}+\mathrm{1}\right)}{{x}}{dx} \\ $$

Question Number 113708 Answers: 1 Comments: 1

∫x^x dx =?

$$\:\:\:\:\:\int\mathrm{x}^{\mathrm{x}} \:\mathrm{dx}\:=?\:\: \\ $$

Question Number 113689 Answers: 3 Comments: 1

Find p and q such that p^2 +q^2 =101^2 . Where p, q∈Z different from zero.

$$\mathrm{Find}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} =\mathrm{101}^{\mathrm{2}} .\:\mathrm{Where}\:\mathrm{p},\:\mathrm{q}\in\mathbb{Z}\: \\ $$$$\mathrm{different}\:\mathrm{from}\:\mathrm{zero}. \\ $$

Question Number 114192 Answers: 1 Comments: 0

For a positive integer k, we write (1+x)(1+2x)(1+3x)...(1+kx)=a_0 +a_1 x+a_2 x^2 +...a_k x^k Let N=a_0 +a_1 +a_2 +...a_k , if N is divisible by 2019, find the smallest possible value of k.

$$\mathrm{For}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}\:{k},\:\mathrm{we}\:\mathrm{write}\: \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\mathrm{2}{x}\right)\left(\mathrm{1}+\mathrm{3}{x}\right)...\left(\mathrm{1}+{kx}\right)={a}_{\mathrm{0}} +{a}_{\mathrm{1}} {x}+{a}_{\mathrm{2}} {x}^{\mathrm{2}} +...{a}_{{k}} {x}^{{k}} \\ $$$$\mathrm{Let}\:{N}={a}_{\mathrm{0}} +{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +...{a}_{{k}} \:, \\ $$$$\mathrm{if}\:{N}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{2019},\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\mathrm{smallest}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{k}. \\ $$

Question Number 113684 Answers: 1 Comments: 1

A nice question <3 If a quadratic equation (1−q+(p^2 /2))x^2 +p(1+q)x+q(q−1)+(p^2 /2)=0 has equal roots, prove that p^2 =4q

$$\mathrm{A}\:\mathrm{nice}\:\mathrm{question}\:<\mathrm{3} \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{a}\:\mathrm{quadratic}\:\mathrm{equation}\: \\ $$$$\left(\mathrm{1}−{q}+\frac{{p}^{\mathrm{2}} }{\mathrm{2}}\right){x}^{\mathrm{2}} +{p}\left(\mathrm{1}+{q}\right){x}+{q}\left({q}−\mathrm{1}\right)+\frac{{p}^{\mathrm{2}} }{\mathrm{2}}=\mathrm{0} \\ $$$$\mathrm{has}\:\mathrm{equal}\:\mathrm{roots},\:\mathrm{prove}\:\mathrm{that}\:{p}^{\mathrm{2}} =\mathrm{4}{q} \\ $$

Question Number 113682 Answers: 8 Comments: 1

Question Number 113675 Answers: 1 Comments: 3

Question Number 113668 Answers: 1 Comments: 0

Question Number 113667 Answers: 1 Comments: 0

In △ABC, BC=5cm AC=4cm cos(A−B)=((31)/(32)) Find the area of △ABC.

$$\mathrm{In}\:\bigtriangleup\mathrm{ABC},\:\mathrm{BC}=\mathrm{5cm}\:\mathrm{AC}=\mathrm{4cm} \\ $$$$\mathrm{cos}\left(\mathrm{A}−\mathrm{B}\right)=\frac{\mathrm{31}}{\mathrm{32}}\:\: \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\bigtriangleup\mathrm{ABC}. \\ $$

Question Number 113657 Answers: 2 Comments: 4

Two guns situated at the top of a hill of height 10m, fire one shot each with the same speed 5(√3)ms^(−1) at some interval of time. One gun fires horizontally and the other fires upwards at an angle of 60° with the horizontal. The shots collide in air at a point P. Find (i) the time interval between the firings and (ii) the coordinates of the hill right below the muzzle and trajectories in x-y plane.

$$\mathrm{Two}\:\mathrm{guns}\:\mathrm{situated}\:\mathrm{at}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{a}\:\mathrm{hill}\:\mathrm{of}\:\mathrm{height}\:\mathrm{10m},\:\mathrm{fire}\:\mathrm{one}\:\mathrm{shot}\:\mathrm{each}\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{speed}\:\mathrm{5}\sqrt{\mathrm{3}}\mathrm{ms}^{−\mathrm{1}} \:\mathrm{at}\:\mathrm{some}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{time}.\:\mathrm{One}\:\mathrm{gun}\:\mathrm{fires}\:\mathrm{horizontally} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{other}\:\mathrm{fires}\:\mathrm{upwards}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{60}°\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{The}\:\mathrm{shots} \\ $$$$\mathrm{collide}\:\mathrm{in}\:\mathrm{air}\:\mathrm{at}\:\mathrm{a}\:\mathrm{point}\:\mathrm{P}.\:\mathrm{Find}\:\left(\mathrm{i}\right)\:\mathrm{the}\:\mathrm{time}\:\mathrm{interval}\:\mathrm{between}\:\mathrm{the}\:\mathrm{firings}\:\mathrm{and} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{the}\:\mathrm{coordinates}\:\mathrm{of}\:\mathrm{the}\:\mathrm{hill}\:\mathrm{right}\:\mathrm{below}\:\mathrm{the}\:\mathrm{muzzle}\:\mathrm{and}\:\mathrm{trajectories}\:\mathrm{in}\:\mathrm{x}-\mathrm{y} \\ $$$$\mathrm{plane}. \\ $$

Question Number 113656 Answers: 1 Comments: 0

∫ (((1+tan (((3x)/2)))^2 )/(1+sin 3x)) dx ?

$$\:\:\int\:\frac{\left(\mathrm{1}+\mathrm{tan}\:\left(\frac{\mathrm{3x}}{\mathrm{2}}\right)\right)^{\mathrm{2}} }{\mathrm{1}+\mathrm{sin}\:\mathrm{3x}}\:\mathrm{dx}\:? \\ $$

Question Number 113651 Answers: 3 Comments: 0

lim_(x→0) ((1/2)−(1/(1+e^(−x) ))).(1/(3x)) = ?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{1}+{e}^{−{x}} }\right).\frac{\mathrm{1}}{\mathrm{3}{x}}\:=\:? \\ $$

Question Number 113857 Answers: 2 Comments: 0

find the values of k for which the line y=kx−3 does not meet the curve y=2x^2 −3x+k.

$${find}\:{the}\:{values}\:{of}\:{k}\:{for}\:{which}\:{the} \\ $$$${line}\:{y}={kx}−\mathrm{3}\:{does}\:{not}\:{meet}\:{the}\:{curve} \\ $$$${y}=\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+{k}. \\ $$

Question Number 113641 Answers: 0 Comments: 1

Prove that there exists M>0 such that for any positive integers n, we have (√(1+(√(2+(√(...+(√(n+1))))))))≤M