For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.

Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p. 

Simplify \({ sin }^{ -1 }\left( \frac { sinx+cosx }{ \sqrt { 2 } } \right) ,\frac { \pi }{ 4 }\) 

A kho-kho player In a practice session while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.

Find the shortest distance between the following pairs of lines \(\frac { x-3 }{ 3 } =\frac { y-8 }{ -1 } =\frac { z-3 }{ 1 } \)and \(\frac { x+3 }{ -3 } =\frac { y+7 }{ 2 } =\frac { z-6 }{ 4 } \) 

Show that the curves 4x = y² and 4xy = k cut at right angles if k² = 512.

A water tank has a shape of an inverted cone with its axis vertical and vertex lower most. Its semi vertical angle is tan⁻¹(0.5). Water is poured into it at a constant rate of 5 cm3/hr. Find the rate at which the level of the water is rising at that instant when the depth of the water is 4 m.

A manufacturer can sell x items at a price of rupees \(\left( 5-\frac { x }{ 100 } \right) \) each. The cost price of x items is Rs.\(\left( \frac { x }{ 5 } +500 \right) \) .Find the numbers of items he should sell to earn maximum profit.

If V = log r and r² = x² +y² + z², then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 } }{ \partial { z }^{ 2 } } =\frac { 1 }{ { r }^{ 2 } } \)

Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = \(\frac { \pi }{ 3 } \) and x axis are as 2:3

Find the area of the region common to the circle x²+y²=16 and the parabola y²=6x.

Show that the ratio of the area under the curve y=sinx and y=sin2x between x=0 and \(x=\frac { \pi }{ 3 } \) and x- axis are as 2 : 3.

A population grows at the rate of 2% per year. How long does it take for the population to double?

Solve : (1+y²)(1 + log x)dx + x dy = 0, given that x = 1,y = 1.

It is given that the rate at which some bacteria multiply is proportional to the instantaneous number present. If the original number of bacteria doubles in two hours, in how many hours will it be five times.

Construct the truth table for (p ∧ q) v r.