If z₁ and z₂ are two complex numbers, such that |z₁| = Iz₂|, then is it necessary that z₁ = z₂?

If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1

If z =\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\), then show that Im (z) = 0

If sin ∝, cos ∝ are the roots of the equation ax² + bx + c-0 (c ≠ 0), then prove that (n + c)² - b² + c²

If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)

Prove that \({ tan }^{ -1 }\left( \frac { 1 }{ 7 } \right) +{ tan }^{ -1 }\left( \frac { 1 }{ 13 } \right) ={ tan }^{ -1 }\left( \frac { 2 }{ 9 } \right) \)

Evaluate \(sin\left( \frac { 1 }{ 2 } { cos }^{ -1 }\frac { 4 }{ 5 } \right) \)

If a parabolic reflector is 24 cm in diameter and 6 cm deep, find its locus.

Find the locus of a point which divides so that the sum of its distances from (-4, 0) and (4, 0) is 10 units.

A force of magnitude 6 units acting parallel to \(\overset { \wedge }{ 2i } -\overset { \wedge }{ 2j } +\overset { \wedge }{ k } \) displaces the point of application from (1, 2, 3) to (5, 3, 7). Find the work done.

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b

Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line \({ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right) \)

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p=-1

If the planes \({ \overset { \rightarrow }{ r } }.\left( \overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \right) =7\) and \({ \overset { \rightarrow }{ r } }.\left( \lambda \overset { \wedge }{ i } +2\overset { \wedge }{ j } -7\overset { \wedge }{ k } \right) =26\) are perpendicular. Find the value of λ.

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∆=4

Find the maximum and minimum values of f(x) = |x+3| ∀ \(x\in R\).

Find x if the rate of decrease of \(\frac { { x }^{ 2 } }{ 2 } -2x+5\) is twice the decrease of x.

Using Rolle’s theorem find the value of c for f(x) = sin x in[0,2π]

Obtain Maclaurin’s Series expansion for e^2x.

Determine the domain of concavity of the curve y=2-x²

A circular metal plate expands under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate if the radius of the plate before heating is 10cm.

If f (x, y) = 2x³ - 11x²y + 3y³, prove that \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } =3f\)

IF u(x, y) = x² + 3xy + y², x, y, ∈ R, find tha linear appraoximation for u at (2, 1) 

If w=xye^xy find \(\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \)

Find a linear approximation to f(x)=3xe^2x-10 at x=5

Prove that \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log(tan \ x)dx } \)

Evaluate \(\int _{ 1 }^{ 2 }{ \frac { 3x }{ { 9x }^{ 2 }-1 } dx } \)