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Question Number 21390 Answers: 0 Comments: 0

∫_0 ^1 ((x^7 − 1)/(ln x)) dx

$\underset{0}{\int^{1}} \frac{x^{7} - 1}{\ln x} d x$

Question Number 21388 Answers: 0 Comments: 4

A block of mass m is connected with another block of mass 2m by a light spring. 2m is connected with a hanging mass 3m by an inextensible light string. At the time of release of block 3m, find tension in the string and acceleration of all the masses.

$A block of mass m is connected with$ $another block of mass 2 m by a light$ $spring . 2 m is connected with a hanging$ $mass 3 m by an inextensible light string .$ $At the time of release of block 3 m, find$ $tension in the string and acceleration$ $of all the masses .$

Question Number 21377 Answers: 0 Comments: 0

Balls are dropped from the roof of a tower at a fixed interval of time. At the moment when 9th ball reaches the ground the nth ball is (3/4)th height of the tower. What is the value of n?

$Balls are dropped from the roof of a$ $tower at a fixed interval of time . At the$ $moment when 9 th ball reaches the$ $ground the n th ball is (3 / 4) th height$ $of the tower . What is the value of n ?$

Question Number 21366 Answers: 3 Comments: 0

Question Number 21357 Answers: 1 Comments: 0

Solve : log_(2x+3) x^2 < 1

$Solve : \log_{2 x + 3} x^{2} < 1$

Question Number 21356 Answers: 1 Comments: 0

Solve : (2^((3x−1)/(x−1)) )^(1/3) < 8^((x−3)/(3x−7))

$Solve : \sqrt[3]{2^{\frac{3 x - 1}{x - 1}}} < 8^{\frac{x - 3}{3 x - 7}}$

Question Number 21355 Answers: 1 Comments: 0

Solve : ∣x^2 + 3x∣ + x^2 − 2 ≥ 0

$Solve : ∣ x^{2} + 3 x ∣ + x^{2} - 2 ⩾ 0$

Question Number 21354 Answers: 0 Comments: 4

Solve : (√(2x + 5)) + (√(x − 1)) > 8

$Solve : \sqrt{2 x + 5} + \sqrt{x - 1} > 8$

Question Number 21374 Answers: 0 Comments: 2

In how many ways can the letters of the word PATLIPUTRA be arranged, so that the relative order of vowels are consonants do not alter?

$In how many ways can the letters of the$ $word PATLIPUTRA be arranged, so$ $that the relative order of vowels are$ $consonants do not alter ?$

Question Number 21350 Answers: 0 Comments: 0

prove (√2) < log_8 19 < (3)^(1/3)

$p r o v e$ $\sqrt{2} < {l o g}_{8} 19 < \sqrt[3]{3}$

Question Number 21342 Answers: 1 Comments: 0

Question Number 21341 Answers: 2 Comments: 0

the angle between the straight lines x^2 +4xy+3y^2 =0 is

$t h e a n g l e b e t w e e n t h e s t r a i g h t l i n e s x^{2} + 4 x y + 3 y^{2} = 0 i s$

Question Number 21321 Answers: 1 Comments: 0

The number of real solutions of the equation 4x^(99) + 5x^(98) + 4x^(97) + 5x^(96) + ..... + 4x + 5 = 0 is

$The number of real solutions of the$ $equation 4 x^{99} + 5 x^{98} + 4 x^{97} + 5 x^{96} +$ $. . . . . + 4 x + 5 = 0 is$

Question Number 21319 Answers: 0 Comments: 0

If x, y, z are three real numbers such that x + y + z = 4 and x^2 + y^2 + z^2 = 6, then (1) (2/3) ≤ x, y, z ≤ 2 (2) 0 ≤ x, y, z ≤ 2 (3) 1 ≤ x, y, z ≤ 3 (4) 2 ≤ x, y, z ≤ 3

$If x, y, z are three real numbers such$ $that x + y + z = 4 and x^{2} + y^{2} + z^{2} = 6,$ $then$ $(1) \frac{2}{3} ⩽ x, y, z ⩽ 2$ $(2) 0 ⩽ x, y, z ⩽ 2$ $(3) 1 ⩽ x, y, z ⩽ 3$ $(4) 2 ⩽ x, y, z ⩽ 3$

Question Number 21316 Answers: 0 Comments: 0

Let p = (x_1 − x_2 )^2 + (x_1 − x_3 )^2 + .... + (x_1 − x_6 )^2 + (x_2 − x_3 )^2 + (x_2 − x_4 )^2 + .... + (x_2 − x_6 )^2 + .... + (x_5 − x_6 )^2 = Σ_(1≤i<j≤6) ^6 (x_i − x_j )^2 . Then the maximum value of p if each x_i (i = 1, 2, ....., 6) has the value 0 and 1 is

$Let p = {(x_{1} - x_{2})}^{2} + {(x_{1} - x_{3})}^{2} + . . . . +$ ${(x_{1} - x_{6})}^{2} + {(x_{2} - x_{3})}^{2} + {(x_{2} - x_{4})}^{2} +$ $. . . . + {(x_{2} - x_{6})}^{2} + . . . . + {(x_{5} - x_{6})}^{2} =$ $\underset{1 ⩽ i < j ⩽ 6}{\sum^{6}} {(x_{i} - x_{j})}^{2} .$ $Then the maximum value of p if each$ $x_{i} (i = 1, 2, . . . . ., 6) has the value 0 and$ $1 is$

Question Number 21315 Answers: 0 Comments: 0

The number of real solutions of the equation ((97 − x))^(1/4) + (x)^(1/4) = 5

$The number of real solutions of the$ $equation \sqrt[4]{97 - x} + \sqrt[4]{x} = 5$

Question Number 21314 Answers: 1 Comments: 0

Let α and β be the root of x^2 + px − (1/(2p^2 )) = 0, p ∈ R. The minimum value of α^4 + β^4 is

$Let α and β be the root of x^{2} + p x - \frac{1}{2 p^{2}} = 0,$ $p \in R . The minimum value of α^{4} + β^{4} is$

Question Number 21313 Answers: 0 Comments: 4

Let k be a real number such that the inequality (√(x − 3)) + (√(6 − x)) ≥ k has a solution then the maximum value of k is

$Let k be a real number such that the$ $inequality \sqrt{x - 3} + \sqrt{6 - x} ⩾ k has a$ $solution then the maximum value of k$ $is$

Question Number 21311 Answers: 0 Comments: 0

Let a and b be positive real numbers with a^3 + b^3 = a − b, and k = a^2 + 4b^2 , then (1) k < 1 (2) k >1 (3) k = 1 (4) k > 2

$Let a and b be positive real numbers$ $with a^{3} + b^{3} = a - b, and k = a^{2} + 4 b^{2},$ $then$ $(1) k < 1$ $(2) k > 1$ $(3) k = 1$ $(4) k > 2$

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 21309 Answers: 0 Comments: 0

Suppose p is a polynomial with complex coefficients and an even degree. If all the roots of p are complex non-real numbers with modulus 1, prove that p(1) ∈ R iff p(−1) ∈ R.

$Suppose p is a polynomial with complex$ $coefficients and an even degree . If all$ $the roots of p are complex non - real$ $numbers with modulus 1, prove that$ $p (1) \in R iff p (- 1) \in R .$

Question Number 21308 Answers: 0 Comments: 0

Find all complex numbers z such that ∣z − ∣z + 1∣∣ = ∣z + ∣z − 1∣∣

$Find all complex numbers z such that$ $∣ z - ∣ z + 1 ∣∣ = ∣ z + ∣ z - 1 ∣∣$

Question Number 21307 Answers: 0 Comments: 5

Let z_1 , z_2 , z_3 be complex numbers such that (i) ∣z_1 ∣ = ∣z_2 ∣ = ∣z_3 ∣ = 1 (ii) z_1 + z_2 + z_3 ≠ 0 (iii) z_1 ^2 + z_2 ^2 + z_3 ^2 = 0 Prove that for all n ≥ 2, ∣z_1 ^n + z_2 ^n + z_3 ^n ∣ ∈ {0, 1, 2, 3}.

$Let z_{1}, z_{2}, z_{3} be complex numbers such$ $that$ $(i) ∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = 1$ $(ii) z_{1} + z_{2} + z_{3} \neq 0$ $(iii) z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 0$ $Prove that for all n ⩾ 2,$ $∣ z_{1}^{n} + z_{2}^{n} + z_{3}^{n} ∣ \in {0, 1, 2, 3} .$

Question Number 21295 Answers: 1 Comments: 0

For a particle performing uniform circular motion, angular momentum is constant in magnitude but direction keeps changing. Am I right or wrong?

$For a particle performing uniform$ $circular motion, angular momentum is$ $constant in magnitude but direction$ $keeps changing .$ $Am I right or wrong ?$

Question Number 21294 Answers: 1 Comments: 0

Let z_1 , z_2 , z_3 be complex numbers, not all real, such that ∣z_1 ∣ = ∣z_2 ∣ = ∣z_3 ∣ = 1 and 2(z_1 + z_2 + z_3 ) − 3z_1 z_2 z_3 ∈ R. Prove that max(arg z_1 , arg z_2 , arg z_3 ) ≥ (π/6) . Where 0 < arg(z_1 ), arg(z_2 ), arg(z_3 ) < 2π.

$Let z_{1}, z_{2}, z_{3} be complex numbers, not$ $all real, such that ∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = 1$ $and 2 (z_{1} + z_{2} + z_{3}) - 3 z_{1} z_{2} z_{3} \in R .$ $Prove that \max (\arg z_{1}, \arg z_{2}, \arg z_{3}) ⩾$ $\frac{π}{6} . Where 0 < \arg (z_{1}), \arg (z_{2}), \arg (z_{3})$ $< 2 π .$

Question Number 21293 Answers: 1 Comments: 0

Let n be an even positive integer such that (n/2) is odd and let α_0 , α_1 , ...., α_(n−1) be the complex roots of unity of order n. Prove that Π_(k=0) ^(n−1) (a + bα_k ^2 ) = (a^(n/2) + b^(n/2) )^2 for any complex numbers a and b.

$Let n be an even positive integer such$ $that \frac{n}{2} is odd and let α_{0}, α_{1}, . . . ., α_{n - 1} be$ $the complex roots of unity of order n .$ $Prove that \underset{k = 0}{\prod^{n - 1}} (a + b α_{k}^{2}) = {(a^{\frac{n}{2}} + b^{\frac{n}{2}})}^{2}$ $for any complex numbers a and b .$

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