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Question Number 21401 Answers: 0 Comments: 0

∫_0 ^∞ ((xdx)/(e^x +1))=?

$\underset{0}{\int^{\infty}} \frac{x d x}{e^{x} + 1} = ?$

Question Number 21310 Answers: 1 Comments: 8

What do you guys think of creating a Telegram group to discuss theory and more descriptive questions?

$What do you guys think of$ $creating a Telegram group to$ $discuss theory and more descriptive$ $questions ?$

Question Number 21231 Answers: 0 Comments: 0

Consider the areas of the four triangles obtained by drawing the diagonals AC and BD of a trapezium ABCD. The product of these areas, taken two at time, are computed. If among the six products so obtained, two products are 1296 and 576, determine the square root of the maximum possible area of the trapezium to the nearest integer.

$Consider the areas of the four triangles$ $obtained by drawing the diagonals A C$ $and B D of a trapezium A B C D . The$ $product of these areas, taken two at$ $time, are computed . If among the six$ $products so obtained, two products are$ $1296 and 576, determine the square$ $root of the maximum possible area of$ $the trapezium to the nearest integer .$

Question Number 21228 Answers: 0 Comments: 0

Let P be an interior point of a triangle ABC whose sidelengths are 26, 65, 78. The line through P parallel to BC meets AB in K and AC in L. The line through P parallel to CA meets BC in M and BA in N. The line through P parallel to AB meets CA in S and CB in T. If KL, MN, ST, are of equal lengths, find this common length.

$Let P be an interior point of a triangle$ $A B C whose sidelengths are 26, 65, 78 .$ $The line through P parallel to B C meets$ $A B in K and A C in L . The line through$ $P parallel to C A meets B C in M and B A$ $in N . The line through P parallel to A B$ $meets C A in S and C B in T . If K L, M N,$ $S T, are of equal lengths, find this$ $common length .$

Question Number 21001 Answers: 0 Comments: 0

determinant (((a 1 1)),((1 b 1)),((1 1 c)))>0 then showthat abc>−8−99

$| \begin{matrix} a 1 1 \\ 1 b 1 \\ 1 1 c \end{matrix} | > 0 t h e n s h o w t h a t a b c > - 8 - 99$

Question Number 20953 Answers: 0 Comments: 2

Question Number 20944 Answers: 1 Comments: 0

Question Number 20599 Answers: 1 Comments: 0

In a rectangle ABCD, E is the midpoint of AB; F is a point on AC such that BF is perpendicular to AC; and FE perpendicular to BD. Suppose BC = 8(√3). Find AB.

$In a rectangle A B C D, E is the midpoint$ $of A B; F is a point on A C such that B F$ $is perpendicular to A C; and F E$ $perpendicular to B D . Suppose B C = 8 \sqrt{3} .$ $Find A B .$

Question Number 20545 Answers: 0 Comments: 1

Let ABC be an acute-angled triangle with AC ≠ BC and let O be the circumcenter and F be the foot of altitude through C. Further, let X and Y be the feet of perpendiculars dropped from A and B respectively to (the extension of) CO. Prove that FY ⊥ CA using that ∠CFY = ∠CBY = ∠CAF.

$Let ABC be an acute - angled triangle$ $with AC \neq BC and let O be the$ $circumcenter and F be the foot of$ $altitude through C . Further, let X and$ $Y be the feet of perpendiculars dropped$ $from A and B respectively to (the$ $extension of) CO . Prove that FY ⊥ CA$ $using that ∠ CFY = ∠ CBY = ∠ CAF .$

Question Number 20187 Answers: 0 Comments: 1

t_n =(t_(n−1) /n^2 ), t_1 =3;t_2 ,t_3 ,(n≥2)

$t_{n} = \frac{t_{n - 1}}{n^{2}}, t_{1} = 3; t_{2}, t_{3}, (n ⩾ 2)$

Question Number 20156 Answers: 1 Comments: 0

Solve: inverse laplace. L^(−1) ((s/(s^(2 ) + 6s + 25)))

$Solve : inverse laplace . L^{- 1} (\frac{s}{s^{2} + 6 s + 25})$

Question Number 20131 Answers: 1 Comments: 4

In rectangle ABCD,AB=8, BC=20.P is a point on AD so that ∠BPC=90°.If r_1 ,r_2 ,r_3 are the radii of the incircles of APB, BPC, and CPD. find r_1 +r_2 +r_3

$I n r e c t a n g l e A B C D, A B = 8,$ $B C = 20 . P i s a p o i n t o n A D s o$ $t h a t ∠ B P C = 90 ° . I f r_{1}, r_{2}, r_{3} a r e t h e$ $r a d i i o f t h e i n c i r c l e s o f A P B,$ $B P C, a n d C P D . f i n d r_{1} + r_{2} + r_{3}$

Question Number 19939 Answers: 0 Comments: 0

f(x)=lnx

$f (x) = l n x$

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?

$In a triangle A B C with ∠ B C A = 90 °,$ $the perpendicular bisector of A B$ $intersects segments A B and A C at X$ $and Y, respectively . If the ratio of the$ $area of quadrilateral B X Y C to the$ $area of triangle A B C is 13 : 18 and$ $B C = 12 then what is the length of A C ?$

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?

$Let A B C D be a convex quadrilateral$ $with ∠ D A B = ∠ B D C = 90 ° . Let the$ $incircles of triangles A B D and B C D$ $touch B D at P and Q, respectively,$ $with P lying in between B and Q . If$ $A D = 999 and P Q = 200 then what is$ $the sum of the radii of the incircles of$ $triangles A B D and B D C ?$

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.

$I n t h e c y c l i c q u a d r i l a t e r a l A B C D$ $A B = 7, B C = 8, C D = 8, D A = 15 .$ $C a l c u l a t e t h e a n g l e A D C a n d$ $t h e l e n g t h o f A C .$

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.

$T h e s i d e s o f a t r i a n g l e a r e o f$ $l e n g t h s \sqrt{(m^{2} - n^{2})}, m^{2} + n^{2}, 2 m n .$ $S h o w t h a t i t i s a r i g h t a n g l e Δ .$

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 ?

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

Question Number 19668 Answers: 0 Comments: 2

Question Number 19659 Answers: 1 Comments: 0

((1+secθ)/(secθ))=((sin^(2 ) θ)/(1−cosθ))

$\frac{1 + s e c θ}{s e c θ} = \frac{{s i n}^{2} θ}{1 - c o s θ}$

Question Number 19415 Answers: 1 Comments: 0

PS is a line segment of length 4 and O is the midpoint of PS. A semicircular arc is drawn with PS as diameter. Let X be the midpoint of this arc. Q and R are points on the arc PXS such that QR is parallel to PS and the semicircular arc drawn with QR as diameter is tangent to PS. What is the area of the region QXROQ bounded by the two semicircular arcs?

$P S is a line segment of length 4 and O$ $is the midpoint of P S . A semicircular$ $arc is drawn with P S as diameter . Let$ $X be the midpoint of this arc . Q and R$ $are points on the arc P X S such that Q R$ $is parallel to P S and the semicircular$ $arc drawn with Q R as diameter is$ $tangent to P S . What is the area of the$ $region Q X R O Q bounded by the two$ $semicircular arcs ?$

Question Number 19394 Answers: 1 Comments: 0

Question Number 19388 Answers: 0 Comments: 1

related to Q.19333 the side lengthes of a triangle are integer. if the perimeter of the triangle is 100, how many different triangles exist? what is the maximum area of them?

Question Number 19301 Answers: 1 Comments: 0

e^(iπ) +1=0

$e^{i π} + 1 = 0$

Question Number 19293 Answers: 1 Comments: 3

Let AC be a line segment in the plane and B a point between A and C. Construct isosceles triangles PAB and QBC on one side of the segment AC such that ∠APB = ∠BQC = 120° and an isosceles triangle RAC on the other side of AC such that ∠ARC = 120°. Show that PQR is an equilateral triangle.

$Let A C be a line segment in the plane$ $and B a point between A and C .$ $Construct isosceles triangles P A B and$ $Q B C on one side of the segment A C$ $such that ∠ A P B = ∠ B Q C = 120 ° and$ $an isosceles triangle R A C on the other$ $side of A C such that ∠ A R C = 120 ° .$ $Show that P Q R is an equilateral$ $triangle .$

Question Number 19279 Answers: 0 Comments: 0

lim_(x→π) (((2x)/(cot(1/x))))

$\lim_{x \to π} (\frac{2 x}{c o t \frac{1}{x}})$

Pg 95 Pg 96 Pg 97 Pg 98 Pg 99 Pg 100 Pg 101 Pg 102 Pg 103 Pg 104

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