Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

If 1, ω, ω² are the cube roots of unity show that (1+ω²)³ - (1+ω)³ = 0

Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.

Find the principal value of sin^-1(-1).

Prove that \(2{ tan }^{ -1 }\left( \frac { 2 }{ 3 } \right) ={ tan }^{ -1 }\left( \frac { 12 }{ 5 } \right) \)

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

Find the equation of the hyperbola whose vertices are (0, ±7) and e = \(\frac { 4 }{ 3 } \)

Find the Cartesian equation of a line passing through the points A(2, -1, 3) and B(4, 2, 1)

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A man 2 m high walks at a uniform speed of 5 km/ hr away from a lamp post 6 m high. Find the rate at which the length of his shadow increases?

Expand the polynomial f(x)=x²-3x+2 in power of (x-2)

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

If \(w={ e }^{ { x }^{ 2 }+{ y }^{ 2 } }\) ,x=cosθ,y=sinθ, find \(\frac { dw }{ d\theta } \)

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

Find the area bounded by the curve y=sin2x between the ordinates x=0.x=π and x-axis.

Form the differential equation satisfied by are the straight lines in my-plane.

Show that p v (q ∧ r) is a contingency.