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Question Number 12093    Answers: 1   Comments: 0

log_(abc) b=3 log_(abc) c=4 log_(abc) a=?

$$\mathrm{log}_{\mathrm{abc}} \mathrm{b}=\mathrm{3} \\ $$$$\mathrm{log}_{\mathrm{abc}} \mathrm{c}=\mathrm{4} \\ $$$$\mathrm{log}_{\mathrm{abc}} \mathrm{a}=? \\ $$

Question Number 12092    Answers: 0   Comments: 0

Prove that tan 70°−tan 50°+tan 10° =(√3)

$$\mathrm{Prove}\:\mathrm{that}\:\:\mathrm{tan}\:\mathrm{70}°−\mathrm{tan}\:\mathrm{50}°+\mathrm{tan}\:\mathrm{10}°\:=\sqrt{\mathrm{3}} \\ $$

Question Number 12087    Answers: 1   Comments: 0

In how many ways can 10 objects be split into two groups containing 4 and 6 objects respectively ?

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{10}\:\mathrm{objects}\:\mathrm{be}\:\mathrm{split}\:\mathrm{into}\:\mathrm{two}\:\:\mathrm{groups}\:\mathrm{containing}\: \\ $$$$\mathrm{4}\:\mathrm{and}\:\mathrm{6}\:\mathrm{objects}\:\mathrm{respectively}\:? \\ $$

Question Number 12085    Answers: 2   Comments: 0

Question Number 12082    Answers: 1   Comments: 0

Question Number 12080    Answers: 1   Comments: 0

Question Number 12078    Answers: 1   Comments: 0

If tan^2 α = 1+2tan^2 β then prove that cos 2β =1+2cos 2α .

$$\mathrm{If}\:\mathrm{tan}\:^{\mathrm{2}} \alpha\:=\:\mathrm{1}+\mathrm{2tan}\:^{\mathrm{2}} \beta\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\mathrm{cos}\:\mathrm{2}\beta\:=\mathrm{1}+\mathrm{2cos}\:\mathrm{2}\alpha\:. \\ $$

Question Number 12077    Answers: 1   Comments: 0

A man travels 29 km on an open road at a certain speed. In the city, he reduce his speed by 420 km/hr, and he find that he take him the same time to cover 15 km, Find his average speed. (a) On the open road (b) In the city

$$\mathrm{A}\:\mathrm{man}\:\mathrm{travels}\:\mathrm{29}\:\mathrm{km}\:\mathrm{on}\:\mathrm{an}\:\mathrm{open}\:\mathrm{road}\:\mathrm{at}\:\mathrm{a}\:\mathrm{certain}\:\mathrm{speed}.\:\mathrm{In}\:\mathrm{the}\:\mathrm{city},\:\mathrm{he}\: \\ $$$$\mathrm{reduce}\:\mathrm{his}\:\mathrm{speed}\:\mathrm{by}\:\mathrm{420}\:\mathrm{km}/\mathrm{hr},\:\mathrm{and}\:\mathrm{he}\:\mathrm{find}\:\mathrm{that}\:\mathrm{he}\:\mathrm{take}\:\mathrm{him}\:\mathrm{the}\:\mathrm{same}\:\mathrm{time} \\ $$$$\mathrm{to}\:\mathrm{cover}\:\mathrm{15}\:\mathrm{km},\:\mathrm{Find}\:\mathrm{his}\:\mathrm{average}\:\mathrm{speed}.\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{On}\:\mathrm{the}\:\mathrm{open}\:\mathrm{road} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{In}\:\mathrm{the}\:\mathrm{city} \\ $$

Question Number 12074    Answers: 1   Comments: 0

Question Number 12071    Answers: 0   Comments: 0

Question Number 12064    Answers: 0   Comments: 0

A heat pump gas C.P = 3, while the indoor temperature is 27°C and the outdoor temperature is 8°C. How much work per hour is required to pump 3.07 J of heat per hour into the individual unit of 10^5 J.

$$\mathrm{A}\:\mathrm{heat}\:\mathrm{pump}\:\mathrm{gas}\:\mathrm{C}.\mathrm{P}\:=\:\mathrm{3},\:\mathrm{while}\:\mathrm{the}\:\mathrm{indoor}\:\mathrm{temperature}\:\mathrm{is}\:\mathrm{27}°\mathrm{C}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{outdoor}\:\mathrm{temperature}\:\mathrm{is}\:\mathrm{8}°\mathrm{C}.\:\mathrm{How}\:\mathrm{much}\:\mathrm{work}\:\mathrm{per}\:\mathrm{hour}\:\mathrm{is}\:\mathrm{required}\:\mathrm{to}\:\mathrm{pump} \\ $$$$\mathrm{3}.\mathrm{07}\:\mathrm{J}\:\mathrm{of}\:\mathrm{heat}\:\mathrm{per}\:\mathrm{hour}\:\mathrm{into}\:\mathrm{the}\:\mathrm{individual}\:\mathrm{unit}\:\mathrm{of}\:\mathrm{10}^{\mathrm{5}} \:\mathrm{J}.\: \\ $$

Question Number 12063    Answers: 1   Comments: 0

Question Number 12062    Answers: 2   Comments: 0

lim_(x→y) ((x^n − y^n )/(x − y))

$$\underset{{x}\rightarrow\mathrm{y}} {\mathrm{lim}}\:\:\frac{\mathrm{x}^{\mathrm{n}} \:−\:\mathrm{y}^{\mathrm{n}} }{\mathrm{x}\:−\:\mathrm{y}} \\ $$

Question Number 12047    Answers: 1   Comments: 1

∫4x^3 (3x^2 + 2)^5 dx

$$\int\mathrm{4}{x}^{\mathrm{3}} \left(\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{2}\right)^{\mathrm{5}} \:{dx} \\ $$

Question Number 12046    Answers: 1   Comments: 0

how much matrices of integers number A= [(a,b),(c,d) ]if A^2 +A=2I, c=0, det(A)=4

$${how}\:{much}\:{matrices}\:{of}\:{integers}\:{number} \\ $$$${A}=\begin{bmatrix}{{a}}&{{b}}\\{{c}}&{{d}}\end{bmatrix}{if}\:{A}^{\mathrm{2}} +{A}=\mathrm{2}{I},\:{c}=\mathrm{0},\:{det}\left({A}\right)=\mathrm{4} \\ $$

Question Number 12045    Answers: 1   Comments: 0

how much matrices of integers number A= [(a,b),(c,d) ]if A^2 =I and b=c

$${how}\:{much}\:{matrices}\:{of}\:{integers}\:{number} \\ $$$${A}=\begin{bmatrix}{{a}}&{{b}}\\{{c}}&{{d}}\end{bmatrix}{if}\:{A}^{\mathrm{2}} ={I}\:{and}\:{b}={c} \\ $$

Question Number 12044    Answers: 1   Comments: 0

A∈M_(n×n) A^2 =A (I+A)^(−1) =....???

$${A}\in{M}_{{n}×{n}} \\ $$$${A}^{\mathrm{2}} ={A} \\ $$$$\left({I}+{A}\right)^{−\mathrm{1}} =....??? \\ $$

Question Number 12326    Answers: 1   Comments: 0

Π_(n=1) ^x a_n =9^(x!) a_4 =?

$$\underset{{n}=\mathrm{1}} {\overset{{x}} {\prod}}{a}_{{n}} =\mathrm{9}^{{x}!} \\ $$$${a}_{\mathrm{4}} =? \\ $$

Question Number 12040    Answers: 2   Comments: 0

Prove that ∀x,y∈R ⇒ 1+ ((x^2 +y^2 +xy)/3) ≥ x+y

$$\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\:\forall\boldsymbol{{x}},\boldsymbol{{y}}\in\boldsymbol{{R}} \\ $$$$\Rightarrow\:\mathrm{1}+\:\frac{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{xy}}}{\mathrm{3}}\:\geqslant\:\boldsymbol{{x}}+\boldsymbol{{y}} \\ $$

Question Number 12030    Answers: 1   Comments: 0

Question Number 12028    Answers: 1   Comments: 0

Express : sin(33) in surd form.

$$\mathrm{Express}\::\:\mathrm{sin}\left(\mathrm{33}\right)\:\mathrm{in}\:\mathrm{surd}\:\mathrm{form}. \\ $$

Question Number 12023    Answers: 3   Comments: 0

Solve simultaneously x + y − (√(xy)) = 3 .......... (i) (√(x + 1)) + (√(y + 1)) = 4 .......... (ii)

$$\mathrm{Solve}\:\mathrm{simultaneously} \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:−\:\sqrt{\mathrm{xy}}\:=\:\mathrm{3}\:\:\:\:\:..........\:\left(\mathrm{i}\right) \\ $$$$\sqrt{\mathrm{x}\:+\:\mathrm{1}}\:+\:\sqrt{\mathrm{y}\:+\:\mathrm{1}}\:=\:\mathrm{4}\:\:\:\:\:..........\:\left(\mathrm{ii}\right) \\ $$

Question Number 12019    Answers: 1   Comments: 0

Question Number 12018    Answers: 0   Comments: 0

Question Number 12017    Answers: 0   Comments: 0

How Can we expand (a+b)^(1/2) and (a+b)^(−n) ?

$${How}\:{Can}\:{we}\:{expand}\:\left({a}+{b}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:{and} \\ $$$$\left({a}+{b}\right)^{−{n}} \:? \\ $$

Question Number 12015    Answers: 1   Comments: 0

prove that for all x ∈R, e^x ≥x^e

$${prove}\:{that}\:{for}\:{all}\:{x}\:\in{R}, \\ $$$${e}^{{x}} \geqslant{x}^{{e}} \\ $$

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