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Prove that ∣z_1 + z_2 ∣ = ∣z_1 − z_2 ∣ ⇔ arg(z_1 ) − arg(z_2 ) = (π/2) |
Prove that ∣z_1 + z_2 + z_3 + .... + z_n ∣ ≤ ∣z_1 ∣ + ∣z_2 ∣ + ∣z_3 ∣ + .... + ∣z_n ∣ |
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A thin bi − convex lens rest on a plane mirror . it is found that a point objects placed 20cm above the object coincide with it own image. Determine the position and nature of the image when the object is placed (i) 8cm and (ii) 12 from the lens mirror combinatiom |
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STATEMENT-1 : The graph between kinetic energy and vertical displacement is a straight line for a projectile. STATEMENT-2 : The graph between kinetic energy and horizontal displacement is a straight line for a projectile. STATEMENT-3 : The graph between kinetic energy and time is a parabola for a projectile. |
Let S_n = n^2 + 20n + 12, n a positive integer. What is the sum of all possible values of n for which S_n is a perfect square? |
Convert i(√((2(√2)−1)/2)) into polarform. |
A triangle with perimeter 7 has integer side lengths. What is the maximum possible area of such a triangle? |
Solve the equation y^3 = x^3 + 8x^2 − 6x + 8 for positive integers x and y. |
y=tan^(−1) 3a^2 x−x^3 /a(a^2 −3x^2 ) |
y=sin (2tan^(−1) (√(1−x/1+x))) |
Prove that ∣z_1 ± z_2 ∣^2 = ∣z_2 ∣^2 + ∣z_1 ∣^2 ± 2Re(z_1 z_2 ^ ) = ∣z_1 ∣^2 + ∣z_2 ∣^2 ± 2Re(z_1 ^ .z_2 ) |
Product of n, n^(th) roots of unity = 1.α.α^2 .α^3 ..... α^(n−1) = (−1)^(n−1) Why? How to get RHS? |
Parallel tangents to a circle at A and B are cut in the points C and D by a tangent to the circle at E. Prove that AD, BC and the line joining the middle points of AE and BE are concurrent. |
e^(iπ) +1=0 |
xcos^(−1) x/(√(1−x^2 )) |
tan^(−1) ((√x)−x/1+x^(3/2) ) |
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Let AC be a line segment in the plane and B a point between A and C. Construct isosceles triangles PAB and QBC on one side of the segment AC such that ∠APB = ∠BQC = 120° and an isosceles triangle RAC on the other side of AC such that ∠ARC = 120°. Show that PQR is an equilateral triangle. |
Prove the equality sin (π/(2n)) sin ((2π)/(2n)) ... sin (((n − 1)π)/(2n)) = ((√n)/2^(n−1) ) . |
Prove that (1/(cos 6°)) + (1/(sin 24°)) + (1/(sin 48°)) = (1/(sin 12°)) . |
lim_(x→π) (((2x)/(cot(1/x)))) |
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lim_(x→π) (2 − cos^2 x)^((2(√(2(1 + cos x))))/((x − π)^3 )) |
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