Class 12th
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New answer posted
4 months agoContributor-Level 10
P will be the centroid of triangle ABC.
The centroid P is (x? +x? +x? )/3, (y? +y? +y? )/3).
The coordinates of P are given as (17/6, 8/3).
The coordinates of Q are not given, but a calculation is shown.
PQ = √ (24/6)² + (9/3)²) = √ (4² + 3²) = √ (16+9) = √25 = 5.
This implies the coordinates of Q are such that the difference in coordinates with P leads to this result. For example if P= (x? , y? ) and Q= (x? , y? ), then x? -x? =4 and y? -y? =3.
New answer posted
4 months agoContributor-Level 9
dy/dx = 2 (x + 1)
dy = 2 (x + 1)dx.
y = (x + 1)² - c (1)
For y = 0, (x+1)² = c ⇒ x = -1 ± √c.
Area A = ∫? √c? ¹? √c (0 - [ (x+1)² - c])dx = ∫? √c? ¹? √c (c - (x+1)²)dx
A = [cx - (x+1)³/3] from -1-√c to -1+√c
= (c (-1+√c) - (√c)³/3) - (c (-1-√c) - (-√c)³/3)
= -c+c√c - c√c/3 + c+c√c - c√c/3 = 2c√c - 2c√c/3 = (4/3)c√c.
Given A = 4√8 / 3 = 8√2 / 3.
(4/3)c^ (3/2) = 8√2 / 3 ⇒ c^ (3/2) = 2√2 = 2^ (3/2) ⇒ c = 2.
∴ y = (x + 1)² - 2.
∴ y (1) = (1 + 1)² - 2 = 2.
New answer posted
4 months agoContributor-Level 9
f (x) + f (x + 1) = 2 (1)
replace x with x + 1: f (x + 1) + f (x + 2) = 2 (2)
(2) - (1) ⇒ f (x + 2) = f (x)
∴ f (x) is periodic with period 2.
I? = ∫? f (x)dx = 4 ∫? ² f (x)dx.
I? = ∫? ³ f (x)dx = ∫? f (u-1)du. Let u = x+1.
I? = ∫? f (x-1)dx = 2 ∫? ² f (x-1)dx.
From (1), f (x-1) + f (x) = 2.
I? + 2I? = 4∫? ² f (x)dx + 2 (2∫? ² f (x-1)dx) = 4∫? ² f (x)dx + 4∫? ² (2 - f (x)dx
= 4∫? ² (f (x) + 2 - f (x)dx = 4∫? ² 2 dx = 4 [2x] from 0 to 2 = 16.
New answer posted
4 months agoContributor-Level 9
y (x) = ∫? (2t² - 15t + 10)dt
dy/dx = 2x² - 15x + 10.
For tangent at (a, b), slope is m = dx/dy = 1 / (dy/dx) = 1 / (2a² - 15a + 10).
Given slope is -1/3.
2a² - 15a + 10 = -3
2a² - 15a + 13 = 0 (The provided solution has 2a²-15a+7=0, suggesting a different problem or a typo)
Following the image: 2a² - 15a + 7 = 0
(2a - 1) (a - 7) = 0
a = 1/2 or a = 7.
a = 1/2 Rejected as a > 1. So a = 7.
b = ∫? (2t² - 15t + 10)dt = [2t³/3 - 15t²/2 + 10t] from 0 to 7.
6b = [4t³ - 45t² + 60t] from 0 to 7 = 4 (7)³ - 45 (7)² + 60 (7) = 1372 - 2205 + 420 = -413.
|a + 6b| = |7 - 413| = |-406| = 406.
New answer posted
4 months agoContributor-Level 10
The function f (x) is non-differentiable at x=1, 3, 5.
Σ f (f (x) = f (f (1) + f (f (3) + f (f (5).
Assuming f (x) is defined such that f (1)=1, f (3)=1, f (5)=1 (based on context of absolute value functions).
Then Σ f (f (x) = f (1) + f (1) + f (1) = 1 + 1 + 1 = 3.
New answer posted
4 months agoContributor-Level 9
Two drawn cards are spades. There are 50 cards left.
The missing card could be a spade or not a spade.
P (missing card is spade) = 11/50 (since 11 spades remain out of 50 cards).
P (missing card is not spade) = 1 - 11/50 = 39/50
New answer posted
4 months agoContributor-Level 9
A (-3, -6,1), B (2,4, -3). Point P divides AB in ratio k:1.
P = [ (2k-3)/ (k+1), (4k-6)/ (k+1), (-3k+1)/ (k+1)]
P lies on the plane lx + my + nz = 0.
l (2k-3) + m (4k-6) + n (-3k+1) = 0
k (2l + 4m - 3n) = 3l + 6m - n
⇒ k = (3l + 6m - n) / (2l + 4m - 3n)
Plane contains the line (x-1)/-1 = (y+4)/2 = (z+2)/3.
The plane passes through (1, -4, -2) and its normal is perpendicular to the line's direction vector.
-l + 2m + 3n = 0
l (1) + m (-4) + n (-2) = 0 ⇒ l - 4m - 2n = 0
Solving these gives l/-8 = m/-1 = n/-2. Let l=8, m=1, n=2.
k = (3 (8) + 6 (1) - 2) / (2 (8) + 4 (1) - 3 (2) = (24+6-2)/ (16+4-6) = 28/14 = 2.
New answer posted
4 months agoContributor-Level 9
A = [i, -i], [-i, i]
A² = [-2, 2], [2, -2]
A? = [8, -8], [-8, 8]
A? = [-128, 128], [128, -128]
A? [x, y]? =?
-128x + 128y = 8 ⇒ -16x + 16y = 1 ⇒ x - y = -1/16 (I)
128x - 128y = 64 ⇒ 16x - 16y = 8 ⇒ x - y = 1/2 (II)
System is inconsistent hence No solution
New answer posted
4 months ago
Contributor-Level 10
The angle bisector a is parallel to λ ( b? +? ) or μ ( b? -? ).
b? = (i+j)/√2 and? = (i-j+4k)/√ (1+1+16) = (i-j+4k)/ (3√2).
Case 1: a = λ ( (i+j)/√2 + (i-j+4k)/ (3√2) )
a = λ/√2 * (3 (i+j) + (i-j+4k)/3 = λ/ (3√2) * (4i + 2j + 4k).
a is given as αi + 2j + βk.
Comparing the j-component: 2 = λ/ (3√2) * 2 ⇒ λ = 3√2.
So, a = 1 * (4i + 2j + 4k) = 4i + 2j + 4k.
Comparing with αi + 2j + βk, we get α = 4 and β = 4.
(The image has a second case that needs evaluation as well).
Case 2: a = μ ( b? -? )
a = μ/ (3√2) * (3 (i+j) - (i-j+4k) = μ/ (3√2) * (2i + 4j - 4k).
Comparing the j-component: 2 = μ/ (3√2) * 4 ⇒ 4μ
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