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If ((cos θ)/(1 − sin θ)) = a a ≠ (π/2) + 2kπ So, tan (θ/2) = ... (A) (a/(a + 1)) (D) ((a + 1)/(a − 1)) (B) (1/(a + 1)) (E) (a/(a − 1)) (C) ((a − 1)/(a + 1))

$$\mathrm{If}\:\:\frac{\mathrm{cos}\:\theta}{\mathrm{1}\:−\:\mathrm{sin}\:\theta}\:=\:{a}\:\:\:\:\:\:\:\:\:\:\:{a}\:\neq\:\frac{\pi}{\mathrm{2}}\:+\:\mathrm{2}{k}\pi \\ $$$$\mathrm{So},\:\:\mathrm{tan}\:\frac{\theta}{\mathrm{2}}\:=\:... \\ $$$$\left(\mathrm{A}\right)\:\:\frac{{a}}{{a}\:+\:\mathrm{1}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\frac{{a}\:+\:\mathrm{1}}{{a}\:−\:\mathrm{1}} \\ $$$$\left(\mathrm{B}\right)\:\:\frac{\mathrm{1}}{{a}\:+\:\mathrm{1}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{E}\right)\:\:\frac{{a}}{{a}\:−\:\mathrm{1}} \\ $$$$\left(\mathrm{C}\right)\:\:\frac{{a}\:−\:\mathrm{1}}{{a}\:+\:\mathrm{1}} \\ $$

Question Number 10948 Answers: 1 Comments: 0

If cos^(−1) (x/a)+cos^(−1) (y/b)=α prove (x^2 /a^2 )−((2xy)/(ab))cos α+(y^2 /b^2 )=sin^2 α

$$\mathrm{If}\:\mathrm{cos}^{−\mathrm{1}} \frac{{x}}{{a}}+\mathrm{cos}^{−\mathrm{1}} \frac{{y}}{{b}}=\alpha \\ $$$${prove}\: \\ $$$$\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }−\frac{\mathrm{2}{xy}}{{ab}}\mathrm{cos}\:\alpha+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{sin}^{\mathrm{2}} \alpha \\ $$

Question Number 10947 Answers: 0 Comments: 0

In a triangle ABC prove the following (((a+b+c)^2 )/(a^2 +b^2 +c^2 )) = ((cot (A/2)+cot (B/2)+cot (C/2))/(cot A+cot B+cot C))

$$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{ABC}\:\mathrm{prove}\:\mathrm{the}\:\mathrm{following} \\ $$$$\frac{\left({a}+{b}+{c}\right)^{\mathrm{2}} }{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }\:=\:\frac{\mathrm{cot}\:\frac{{A}}{\mathrm{2}}+\mathrm{cot}\:\frac{{B}}{\mathrm{2}}+\mathrm{cot}\:\frac{{C}}{\mathrm{2}}}{\mathrm{cot}\:{A}+\mathrm{cot}\:{B}+\mathrm{cot}\:{C}} \\ $$

Question Number 10944 Answers: 1 Comments: 2

Find all ordered pairs (a,b) so that ((ab)/(a+b)) is an integer. (a and b are integers).

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{ordered}\:\mathrm{pairs}\:\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{so}\:\mathrm{that}\:\frac{\mathrm{ab}}{\mathrm{a}+\mathrm{b}}\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer}. \\ $$$$\left(\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{integers}\right). \\ $$

Question Number 10933 Answers: 2 Comments: 0

If p and q are the roots for the x^2 − (a + 1)x + (−a − (5/2)) = 0 The minimum value of p^(2 ) + q^2 is ...

$$\mathrm{If}\:\:{p}\:\:\mathrm{and}\:\:{q}\:\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{for}\:\mathrm{the} \\ $$$${x}^{\mathrm{2}} \:−\:\left({a}\:+\:\mathrm{1}\right){x}\:+\:\left(−{a}\:−\:\frac{\mathrm{5}}{\mathrm{2}}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{The}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\:{p}^{\mathrm{2}\:} +\:{q}^{\mathrm{2}} \:\:\mathrm{is}\:... \\ $$

Question Number 10917 Answers: 1 Comments: 0

hence show that i)((1−cos4θ)/(sin4θ))=tan2θ ii)((1−cos6θ)/(sin6θ))=tan3θ

$${hence}\:{show}\:{that} \\ $$$$\left.{i}\right)\frac{\mathrm{1}−{cos}\mathrm{4}\theta}{{sin}\mathrm{4}\theta}={tan}\mathrm{2}\theta \\ $$$$\left.{ii}\right)\frac{\mathrm{1}−{cos}\mathrm{6}\theta}{{sin}\mathrm{6}\theta}={tan}\mathrm{3}\theta \\ $$

Question Number 10916 Answers: 1 Comments: 0

find all possible values of cosθ such that 2cot^2 θ+cosθ=0

$${find}\:{all}\:{possible}\:{values}\:{of}\:{cos}\theta\:{such} \\ $$$${that}\:\mathrm{2}{cot}^{\mathrm{2}} \theta+{cos}\theta=\mathrm{0} \\ $$

Question Number 10914 Answers: 1 Comments: 0

A 200 N force inclined at 40° above the horizontal , drag load along the horizontal floor. coefficient of the kinetic friction between the load is 0.30 and the load experiences an acceleration of 1.2 m/s^2 , What is the mass of the load.

$$\mathrm{A}\:\mathrm{200}\:\mathrm{N}\:\mathrm{force}\:\mathrm{inclined}\:\mathrm{at}\:\mathrm{40}°\:\mathrm{above}\:\mathrm{the}\:\mathrm{horizontal}\:,\:\mathrm{drag}\:\mathrm{load}\:\mathrm{along}\:\mathrm{the} \\ $$$$\mathrm{horizontal}\:\mathrm{floor}.\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{the}\:\mathrm{kinetic}\:\mathrm{friction}\:\mathrm{between}\:\mathrm{the}\:\mathrm{load}\:\mathrm{is}\:\mathrm{0}.\mathrm{30}\: \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{load}\:\mathrm{experiences}\:\mathrm{an}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{1}.\mathrm{2}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} , \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{the}\:\mathrm{load}. \\ $$

Question Number 10913 Answers: 1 Comments: 0

∫_( 1) ^( 3) x^x dx

$$\int_{\:\mathrm{1}} ^{\:\mathrm{3}} \:\mathrm{x}^{\mathrm{x}} \:\:\mathrm{dx} \\ $$

Question Number 10912 Answers: 0 Comments: 1

5^(log_2 3) is transcendental? General: Let a,b and c algebraic and log_b c transcendental. If a^(log_b c) is algebraic, so b = a^q , with q rational?

$$\mathrm{5}^{\mathrm{log}_{\mathrm{2}} \mathrm{3}} \:\mathrm{is}\:\mathrm{transcendental}? \\ $$$$\mathrm{General}: \\ $$$$\mathrm{Let}\:{a},{b}\:\mathrm{and}\:{c}\:\mathrm{algebraic}\:\mathrm{and}\:\mathrm{log}_{{b}} {c}\: \\ $$$$\mathrm{transcendental}.\:\mathrm{If}\:{a}^{\mathrm{log}_{{b}} {c}} \:\mathrm{is}\:\mathrm{algebraic},\:\mathrm{so} \\ $$$${b}\:=\:{a}^{{q}} ,\:\mathrm{with}\:{q}\:\mathrm{rational}? \\ $$

Question Number 10908 Answers: 1 Comments: 0