Ncert Solutions Maths class 12th
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The equation of the plane which contains the y-axis and passes through the point (1,2,3) is:
The equation of the plane which contains the y-axis and passes through the point (1,2,3) is:
Let the equation of the plane passing through (1, 2, 3) be:
a (x - 1) + b (y - 2) + c (z - 3) = 0
The plane contains the y-axis, which has direction ratios (0, 1, 0).
Therefore, the normal to the plane must be perpendicular to the y-axis.
a (0) + b (1) + c (0) = 0 ⇒ b = 0
The equation becomes: a (x -
Upvote 
Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, a
Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, a
P (at least 2 show 3 or 5) =? C? (2/6)² (4/6)² +? C? (2/6)³ (4/6)¹ +? C? (2/6)?
= (384+128+16)/6? = 11/27
n=27
∴ expectation of number of times = np
= 27 ⋅ (11/27) = 11
Upvote The probability of a man hitting a target is 1/10. The least number of shots required, so that the probability of his hitting the target at least once is greater than 1/4, is
(Probability)
The probability of a man hitting a target is 1/10. The least number of shots required, so that the probability of his hitting the target at least once is greater than 1/4, is
(Probability)
We have, 1 - (probability of all shots result in failure) > 1/4
? 1 - (9/10)? > 1/4
? 3/4 > (9/10)? ? n? 3 > (9/10)? ? n? 3
Upvote Consider the data on x taking the values 0,2,4,8, ., 2? with frequencies ?C?, ?C?, ?C?, ., ?C? respectively. If the mean of this data is 728/2?, then n is equal to
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Consider the data on x taking the values 0,2,4,8, ., 2? with frequencies ?C?, ?C?, ?C?, ., ?C? respectively. If the mean of this data is 728/2?, then n is equal to
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x? = Σf? x? / Σf?
Σf? =? C? +? C? +? C? + . +? C? = 2?
Σf? x? = (0 ×? C? ) + (2 ×? C? ) + (2² ×? C? ) + . + (2? ×? C? )
This sum is Σ? ? C? 2? = (Σ? ? C? 2? ) -? C?2? = (1+2)? - 1 = 3? - 1.
x? = (3? - 1)/2?
Given x? = 728 / (something that resolves to 2? ). Assuming it is 728/2?
(3? - 1)/2? = 728/2
Upvote 
The probabilities of three events A, B and C are given by P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5. If P(A ∪ B) = 0.8, P(A ∩ C) = 0.3, P(A ∩ B ∩ C) = 0.2, P(B ∩ C) = β and P(A ∪ B ∪ C) = α, where 0.85 ≤ α ≤ 0.95, then β lies in the interval:
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The probabilities of three events A, B and C are given by P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5. If P(A ∪ B) = 0.8, P(A ∩ C) = 0.3, P(A ∩ B ∩ C) = 0.2, P(B ∩ C) = β and P(A ∪ B ∪ C) = α, where 0.85 ≤ α ≤ 0.95, then β lies in the interval:
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P (A∪B∪C) = P (A) + P (B) + P (C) – P (A∩B) – P (B∩C) – P (C∩A) + P (A∩B∩C)
Given relations lead to: α = 1.4 – P (A∩B) – β ⇒ α + β = 1.4 - P (A∩B)
Again, from P (A∪B) = P (A) + P (B) – P (A∩B), and given values, it is found that P (A∩B) = 0.2.
From (1) and (2), α = 1.2 – β.
Now given 0.85 ≤ α ≤ 0.95
⇒ 0.
Upvote Let λεR. The system of linear equations.
2x₁ – 4x₂ + λx₃ = 1
x₁ - 6x₂ + x₃ = 2
λx₁ – 10x₂ + 4x₃ = 3
Is inconsistent for:
Let λεR. The system of linear equations.
2x₁ – 4x₂ + λx₃ = 1
x₁ - 6x₂ + x₃ = 2
λx₁ – 10x₂ + 4x₃ = 3
Is inconsistent for:
Here D = | 2 -4 λ |
| 1 -6 1 | = (λ-3) (3λ+2)
| λ -10 4 |
D = 0 ⇒ λ = 3, -2/3
Upvote A plane P meets the coordinate axes at A, B and C respectively. The centroid of ? ABC is given to be (1,1,2). Then the equation of the line through this centroid and perpendicular to the plane P is:
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A plane P meets the coordinate axes at A, B and C respectively. The centroid of ? ABC is given to be (1,1,2). Then the equation of the line through this centroid and perpendicular to the plane P is:
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Let A (α, 0,0), B (0, β, 0), C (0,0, γ), then the centroid is G (α/3, β/3, γ/3) = (1,1,2).
α = 3, β = 3, γ = 6
∴ Equation of plane is x/α + y/β + z/γ = 1
⇒ x/3 + y/3 + z/6 = 1
⇒ 2x + 2y + z = 6
∴ Required line passing through G (1,1,2) and normal to the plane is (x-1)/2 = (y-1)/2 = (z-2)/1.
Upvote If the tangent to the curve, y = f(x) = xlog? x, (x > 0) at a point (c, f(c)) is parallel to the line - segment joining the points (1,0) and (e, e) then c is equal to:
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If the tangent to the curve, y = f(x) = xlog? x, (x > 0) at a point (c, f(c)) is parallel to the line - segment joining the points (1,0) and (e, e) then c is equal to:
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f' (c) = 1 + lnc = e/ (e-1)
lnc = e/ (e-1) - 1 = (e - (e-1)/ (e-1) = 1/ (e-1)
c = e^ (1/ (e-1)
Upvote The set of all real values of λ for which the function f(x) = (1 - cos²x) ⋅ (λ + sinx), x ∈ (-π/2, π/2), has exactly one maxima and exactly one minima, is
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The set of all real values of λ for which the function f(x) = (1 - cos²x) ⋅ (λ + sinx), x ∈ (-π/2, π/2), has exactly one maxima and exactly one minima, is
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f (x) = sin²x (λ + sinx)
f' (x) = 2sinxcosx (λ + sinx) + sin²x (cosx) = sinxcosx (2λ + 3sinx)
For extrema, f' (x) = 0
sinx = 0, cosx = 0, or sinx = -2λ/3
For more than 2 points of extrema in the interval, sinx = -2λ/3 must have solutions other than where sinx=0 or cosx=0.
-1 < -2λ/3 < 1 and -2λ/3 ≠ 0
This gives λ ∈ (-3/2, 3/2) -
Upvote