Maths

Normals of parabola y² = 4x at P and Q meets at R(x₂, 0) and tangents at P and Q meets at T(x₁, 0). If x₂ = 3, then find the area of quadrilateral PTQR.

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Contributor-Level 10

PQ is focal chord.
Quadrilateral PTQR is square.

Area = (PQ * TR) / 2 = (4 * 4) / 2 = 8

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2 months ago

Let g(x) = x+2|-3|. If a denotes the number of relative minima, b denotes the number of relative maxima and c denotes the product of the zeros. Then the value of (a+2b-c) is

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Maths Continuity and Differentiability

Contributor-Level 10

c = -5, a = 2, b = 1

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2 months ago

If 1+x⁴-x⁵ = Σ_i=0⁵ a?(1+x)ⁱ, for all x ∈ R, then a² is equal to ____________

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Contributor-Level 10

1 + x? - x? = a? (1+x)? + a? (1+x) + a? (1+x)² . + a? (1+x)?
Differentiate
4x³ - 5x? = a? + 2a? (1+x) + 3a? (1+x)².
12x² - 20x³ = 2a? + 6a? (1+x).
Put x = -1
12 + 20 = 2a? ⇒ a? = 16

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2 months ago

Maths Matrices

Let A be a square matrix of order 2 such that A² – 4A + 4I = 0, where I is an identity matrix of order 2. If B = A? + 4A? + 6A³ + 4A² + A – 162 I, then det(B) is equal to

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Contributor-Level 10

A² - 4A + 4I = 0
(A - 2I)² = 0 ⇒ A-2I ≠ 0
B = 297A - 594I
= 297 (A - 2I)

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2 months ago

The value of Σ_i?0⁸ ⁸C_i(⁸C_r+1 - ⁸C_r) equals ____________

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Contributor-Level 10

? C? +? C? =? C?
Σ (from r=0 to 3)? C? =? C? +? C? +? C? +? C?
=? C? +? C? +? C? +? C?
= ¹²C? = (12*11*10)/6 = 220

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2 months ago

lim (x→0) (ae^2x - bcosx + c)/(xsinx) = 1, then a + b + c is equal to ____________

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Contributor-Level 10

lim (x→0) (ae²? - bcosx + c) / (x sinx)
= lim (x→0) (ae²? - bcosx + c) / x² * lim (x→0) x/sinx
For limit to exist
a - b + c = 0
2a + b = 0
(4a-b)/2 = 1
c = 2, b = 2, a = -1
a+b+c = 3

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2 months ago

The value of ∫₀^(π/2) |xsin²x - 1/2|dx is equal to aπ/b where a, b are co-prime numbers, then a.b is ____________

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Contributor-Level 10

∫ (from 0 to π/2) (x/2)|cos (2x)|dx
∫ (from 0 to π/4) (x/2)cos (2x)dx - ∫ (from π/4 to π/2) (x/2)cos (2x)dx
= [x sin (2x)/4 - ∫sin (2x)/4 dx] (from 0 to π/4) - [x sin (2x)/4 - ∫sin (2x)/4 dx] (from π/4 to π/2)
= [x sin (2x)/4 + cos (2x)/8] (from 0 to π/4) - [x sin (2x)/4 + cos (2x)/8] (from π/4 to π/2)
= (π/16 - 1/8) - (-1/8 - π/16) = π/8

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2 months ago

If a,b are any two perpendicular vectors of equal magnitude and |3a+4b| + |4a-3b| = 20, then |a| equals: ____________

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Contributor-Level 10

|3a+4b|² = 9|a|² + 16|b|² + 24a·b
But a·b = 0, |a|=|b|=k
|3a+4b| = 5k
|4a-3b|
10k = 20? k = 2 = |a| = |b|

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2 months ago

A circle passes through the points (2, 2) and (9, 9) and touches the x-axis. The absolute value of the difference of possible x-coordinate of the point of contact is ______

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Contributor-Level 10

Family of circle through (2, 2) and (9, 9), (x-2) (x-9) + (y-2) (y-9) + λ (y-x) = 0
Touches y=0 (x-axis)
(x-2) (x-9) + 18 - λx = 0
x² - 11x - λx + 36 = 0
x² - (11+λ)x + 36 = 0 has repeated roots
D = 0 ⇒ λ = 1, λ = -23
x = 6 or x = -6
Difference = 12

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2 months ago

Number of ways in which three distinct numbers can be selected between 1 and 20 both inclusive, whose sum is even is ____________

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