Maths

Let A(-1,1), B(3, 4) and C(2,0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A? and A? be the area of ?ABC and ?PQC respectively, such that A? = 3A?, then the value of m is equal to:

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8 months ago

Maths Vector Algebra

Let a = i + 2j - 3k and b = 2i - 3j + 5k. If r x a = b x r, r.(αi+2j+k) = 3 and r.(2i + 5j - αk) = -1, α ∈ R, then the value of α + |r|² is equal to :

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Maths Ncert Solutions Maths class 11th

Contributor-Level 10

Given r x a = b x r, which means r x a + r x b = 0, so r x (a+b) = 0.
This implies r is parallel to (a+b). So, r = λ (a+b).
a = I + 2j - 3k, b = 2i - 3j + 5k
a+b = 3i - j + 2k.
r = λ (3i - j + 2k).
Given r ⋅ (αi + 2j + k) = 3. The OCR is unclear, but the equation appears to be r ⋅ (αi + 2j + k) = 3. The solution works with r ⋅ (αi + 2j + k) = 3, but theOCR says r. (ai+2j+k). Let's assume it's α.
λ (3i - j + 2k) ⋅ (αi + 2j + k) = 3
λ (3α - 2 + 2) = 3 => λα = 1.
Given r ⋅ (2i + 5j - αk) = -1.
λ (3i - j + 2k) ⋅ (2i + 5j - αk) = -1
λ (6 - 5 - 2α) = -1 => λ (1 - 2α) = -1.
From λα = 1, α = 1/λ.
λ (1 - 2/λ) = -1 => λ

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If for x ∈ (0, π/2), log?sinx + log?cosx = -1 and log?(sinx + cosx) = ½(log?n - 1), n > 0, then the value of n is equal to :

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Contributor-Level 9

Kindly consider the following Image

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8 months ago

Maths Class 12th

The range of a ∈ R for which the function f(x) = (4a - 3)(x + log?5) + 2(a - 7)cot(x/2)sin²(x/2), x ≠ 2nπ, n ∈ N has critical points is:

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Contributor-Level 9

f (x) = (4a - 3) (x + ln5) + 2 (a - 7)cot (x/2)sin² (x/2)
= (4a - 3) (x + ln5) + (a - 7)sin (x)
f' (x) = (4a - 3) + (a - 7)cos (x)
For critical points f' (x) = 0
cos (x) = - (4a - 3) / (a - 7) = (3 - 4a) / (a - 7)
⇒ -1 ≤ (3 - 4a) / (a - 7) ≤ 1
Solving this inequality leads to:
⇒ a ∈ [4/3, 2]

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The number of elements in the set {x ∈ R : (|x| - 3)|x + 4| = 6} is equal to :

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Contributor-Level 9

(|x| - 3)|x + 4| = 6

Case (i) x < -4
(-x - 3) (- (x + 4) = 6
(x + 3) (x + 4) = 6 ⇒ x² + 7x + 12 = 6 ⇒ x² + 7x + 6 = 0
(x + 1) (x + 6) = 0 ⇒ x = -6 (since x < -4)

Case (ii) -4 ≤ x < 0
(-x - 3) (x + 4) = 6
⇒ -x² - 7x - 12 = 6
⇒ x² + 7x + 18 = 0
The discriminant is D = 7² - 4 (1) (18) = 49 - 72 < 0, so no real solution.

Case (iii) x ≥ 0
(x - 3) (x + 4) = 6
⇒ x² + x - 12 = 6
⇒ x² + x - 18 = 0
x = [-1 ± √ (1² - 4 (1) (-18)] / 2 = [-1 ± √73] / 2
Since x ≥ 0 ⇒ x = (√73 - 1) / 2
Only two solutions.

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Consider three observations a, b and c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

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Contributor-Level 9

Variance of a, b, c & a+2, b+2, c+2, are same.
Given: b = a + c (i)
d² = (1/3) (a² + b² + c²) - [ (a+b+c)/3]²
as a + c = b
d² = (1/3) (a² + b² + c²) - (2b/3)²
9d² = 3 (a² + b² + c²) - 4b²
⇒ b² = 3 (a² + c²) - 9d²

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Let a complex number z, |z| ≥ 1, satisfy log?/√² (|z|+11)/(|z|-1)² ≤ 2. Then, the largest value of |z| is equal to…

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Maths Differential Equations

Contributor-Level 9

log? /? [ (|z|+11)/ (|z|-1)²] < 2
(|z|+11)/ (|z|-1)² > (1/2)²
(|z|+11)/ (|z|-1)² > 1/4
⇒ 4|z| + 44 > |z|² - 2|z| + 1
⇒ |z|² - 6|z| - 43 < 0

⇒ |z| - 7 ≤ 0
∴ |z|max = 7

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Let C₁ be the curve obtained by the solution of differential equation 2xy dy/dx = y² - x², x > 0. Let the curve C₂ be the solution of 2xy/(x²-y²) = dy/dx. If both the curves pass through (1, 1), then the area enclosed by the curves C₁ and C₂ is equal to :

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Maths Class 12th

Let f:S → S where S = (0,∞) be a twice differentiable function such that f(x+1) = xf(x). If g: S → R be defined as g(x) = logₑf(x), then the value of |g''(5)-g''(1)| is equal to :

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Maths Class 12th

Let P(x) = x² + bx + c be a quadratic polynomial with real coefficients such that ∫₀¹ P(x)dx = 1 and P(x) leaves remainder 5 when it is divided by (x - 2). Then the value of 9(b + c) is equal to:

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