Maths

If the curve y=y(x) is the solution of the differential equation 2(x² + x⁵/⁴)dy - y(x + x¹/⁴)dx = 2x⁹/⁴dx, x > 0 which passes through the point (1, 1 - (4/3)logₑ2), then the value of y(16) is equal to:

0 Follower 8 Views

Contributor-Level 10

0 0

New answer posted

a year ago

Let O be the origin. Let OP = xi + yj - k and OQ = -i + 2j + 3xk, x, y ∈ R, x > 0, be such that |PQ| = √20 and the vector OP is perpendicular to OQ. If OR = 3i + zj - 7k, z ∈ R, is coplanar with OP and OQ, then the value of x² + y² + z² is equal to:

0 Follower 15 Views

Contributor-Level 10

Given vectors OP = xi + yj - k and OQ = -i + 2j + 3xk.
PQ = OQ - OP = (-1 - x)i + (2 - y)j + (3x + 1)k
Given |PQ| = √20, so |PQ|² = 20.
(-1 - x)² + (2 - y)² + (3x + 1)² = 20
(1 + x)² + (2 - y)² + (3x + 1)² = 20 .(i)
- Given OP ⊥ OQ, so OP · OQ = 0.
  (x)(-1) + (y)(2) + (-1)(3x) = 0
  -x + 2y - 3x = 0 ⇒ -4x + 2y = 0 ⇒ y = 2x .(ii)
Substitute (ii) into (i):
(1 + x)² + (2 - 2x)² + (3x + 1)² = 20
1 + 2x + x² + 4 - 8x + 4x² + 9x² + 6x + 1 = 20
14x² = 14 ⇒ x² = 1 ⇒ x = ±1.
When x = 1, y = 2. When x = -1, y = -2.
So, (x, y) can be (1, 2) or (-1, -2).
- Given that OR, OP, and OQ are coplanar, their scalar triple product is 0: [OR OP OQ]

...more

0 0

New answer posted

a year ago

Maths Class 12th

If the Boolean expression (p ∧ q) ○ (p ⊕ q) is a tautology, then ○ and ⊕ are respectively by:

0 Follower 6 Views

Contributor-Level 10

The truth table for the logical expression (p ∧ q) → (p → q) is as follows:

p	q	p ∧ q	p → q	(p ∧ q) → (p → q)
T	T	T	T	T
T	F	F	F	T
F	T	F	T	T
F	F	F	T	T

The final column shows that the expression is a tautology, meaning it is always true regardless of the truth values of p and q.

0 0

New answer posted

a year ago

Let S₁, S₂, and S₃ be three sets defined as
S₁ = { z∈C : |z−1| ≤ √2}
S₂ = {z∈C : Re((1-i)z) ≥ 1}
S₃ = {z∈C : Im(z) ≤ 1}
Then the set S₁ ∩ S₂ ∩ S₃

0 Follower 19 Views

Contributor-Level 10

0 0

New answer posted

a year ago

Maths Class 12th

The equation of the planes parallel to the plane x - 2y + 2z - 3 = 0 which are at unit distance from the point (1,2,3) is ax + by + cz + d = 0. If (b-d) = K(c-a), then the positive value of K is ______.

0 Follower 6 Views

Contributor-Level 10

The equation of a plane parallel to x - 2y + 2z - 3 = 0 is x - 2y + 2z + λ = 0.
The distance from the point (1, 2, 3) to this plane is 1.
|1 - 2 (2) + 2 (3) + λ| / √ (1² + (-2)² + 2²) = 1
|1 - 4 + 6 + λ| / √9 = 1
|3 + λ| / 3 = 1
|3 + λ| = 3
3 + λ = 3 or 3 + λ = -3
λ = 0 or λ = -6.

0 0

New answer posted

a year ago

Maths Matrices

The missing value in the following figure is ______. (A diagram with numbers in a circular pattern is shown).

0 Follower 29 Views

Contributor-Level 10

1 = (2-1)¹ (The n is likely 1).
3? = (7-4)³ (This seems to be a pattern matching (a-b)^c).
4²? = (12-8)? ! = 4²?
The blank space must be (5-3)² = 2² = 4.

0 0

New answer posted

a year ago

Maths Probability

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is: ______.

0 Follower 2 Views

Contributor-Level 10

Initial mean of 25 observations is 40.
X? = (Σx? )/25 = 40 => Σx? = 25 * 40 = 1000.

A teacher of age 60 retires.
The new sum of ages for the remaining 24 people is 1000 - 60 = 940.

A new teacher of age x joins.
The new sum for 25 people is 940 + x.
The new mean is 39.
(940 + x) / 25 = 39
940 + x = 39 * 25 = 975
x = 975 - 940 = 35.
The new teacher's age is 35.

0 0

New answer posted

a year ago

If f(x) = ∫ [5x⁸ + 7x⁶] / [(x⁷ + x² + 1)²] dx, (x > 0), f(0) = 0 and f(1) = 1/K, then the value of K is ______.

0 Follower 21 Views

Maths Differential Equations

Contributor-Level 10

f (x) = ∫ (5x? + 7x? ) / (x² + 1 + 2x? ) dx seems to have a typo in the denominator. Based on the solution, the denominator is (x? + 1/x? + 2)² or similar. Let's follow the solution's steps.
It seems the denominator is (x? (2 + 1/x? + 1/x? )² = x¹? (2 + 1/x? + 1/x? )².
f (x) = ∫ (5x? + 7x? ) / (x¹? (2 + 1/x? + 1/x? )²) dx

The solution simplifies the integrand to:
f (x) = ∫ (5/x? + 7/x? ) / (2 + 1/x? + 1/x? )² dx

Let t = 2 + 1/x? + 1/x?
dt = (-5/x? - 7/x? ) dx = - (5/x? + 7/x? ) dx.

The integral becomes:
f (x) = ∫ -dt / t² = 1/t + C.
f (x) = 1 / (2 + 1/x? + 1/x? ) + C.

Given f (0)=0, this form has a division by zero. Let's re-ex

...more

0 0

New answer posted

a year ago

Choose the correct statement about two circles whose equations are given below:
x² + y² - 10x - 10y + 41 = 0
x² + y² - 22x - 10y + 137 = 0

0 Follower 4 Views

Contributor-Level 10

Circle 1: x² + y² - 10x - 10y + 41 = 0
Center C? = (5,5).
Radius r? = √ (5² + 5² - 41) = √ (25+25-41) = √9 = 3.

Circle 2: x² + y² - 22x - 10y + 137 = 0
Center C? = (11,5).
Radius r? = √ (11² + 5² - 137) = √ (121+25-137) = √9 = 3.

Distance between centers d (C? , C? ) = √ (11-5)² + (5-5)²) = √ (6²) = 6.

Sum of radii r? + r? = 3 + 3 = 6.
Since the distance between the centers is equal to the sum of their radii, the circles touch externally at one point.

0 0

New answer posted

a year ago

The value of 3 + 1/(4 + 1/(3 + 1/(4 + 1/(3 + .∞)))) is equal to:

0 Follower 7 Views