The system of linear equations x+λy−z=0; λx−y−z=0; x+y−λz=0 has a non-trivial solution for

• infinitely many values of λ.

• exactly one value of λ.

• exactly two values of λ.

• exactly two values of λ.

For x ε R, f (x)=|log2−sinx| and g(x)=f(f(x)), then :

• g is not differentiable at x=0

• g'(0)=cos(log2)

• g'(0)=-cos(log2)

• g'(0)=-cos(log2)

13.

Consider f(x) = tan^-1 . A normal to y = f (x) at x = π/6 also passes through the point

• (0,0)

• (0, 2π/3)

• (π/6 ,0)

• (π/6 ,0)

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side=x units and a circle of radius=r units. If the sum of the areas of the square and the circle so formed is minimum, then:

• 2x=(π+4)r

• (4−π)x=πr

• x=2r

• x=2r

• 18/e⁴

• 27/e²

• 9/e²

• 9/e²

The area (in sq. units) of the region is

If a curve y=f(x) passes through the point (1, −1) and satisfies the differential equation, y(1+xy) dx=x dy, then f(-1/2) is equal to

• -2/5

• -4/5

• 2/5

• 2/5

Let two fair six-faced dice A and B be thrown simultaneously. If E₁ is the event that die A shows up four, E₂ is the event that die B shows up two and E₃ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true?

• E₁ and E₂ are independent

• E₂ and E₃ are independent.

• E₁ and E₃ are independent.

• E₁ and E₃ are independent.

If 0≤x<2π, then the number of real values of x, which satisfy the equation cosx+cos2x+cos3x+cos4x=0, is :

• 3

• 5

• 7

• 7

The integral  is equal to