Question(s) from Search: IIT

If x = a + b, y = aα + bβ, z = aβ + bα where α, β are cube roots of unity show that .

If the system of equations

x – ky – z = 0

kx – y –z = 0

x + y –z = 0

has a non zero solution then possible values of k are

a) −1, 2

b) 1, 2

c) 0, 1

d) −1, 1

Given 2x – y – z = 2, x – 2y + z = − 4, x + y + λz = 4 then the value of λ such that the given system of equations has no solution is

a) 3

b) −2

c) 0

d) −3

Find all non zero complex numbers satisfying .

(Multiple choices)
The determinant
  is equal to zero if

a) a, b, c are in arithmetic progression

b) a, b, c are in geometric progression

c) a, b, c are in harmonic progression

d) α is a root of the equation ax² + bx + c = 0

e) x – α is a factor of ax² + 2bx + c

The cube roots of unity when represented on argand diagram form the vertices of an equilateral triangle.

a) True

b) False

The value of the integral
 is

a) sin⁻¹ x – 6tan⁻¹(sin⁻¹ x) + c

b) sin⁻¹x – 2(sinx)⁻¹ + c

c) sin⁻¹x – 2(sinx)⁻¹ − 6tan⁻¹(sin⁻¹x) + c

d) sin⁻¹x – 2(sinx)⁻¹ + 5tan⁻¹(sin⁻¹x) + c

Integrate

a)

b)

c)

d)

If f(x) be the interval of  find

a) ½

b) 1

c) 2

d) 4

 =

a)

b)

c)

d)

Show that
 =   where y =

For any natural number m, show that
 

Let f : ℝ → ℝ and g : ℝ → ℝ be continuous functions. Then the value of integral  is

a) π

b) 1

c) – 1

d) 0

Let f(x) =  where p is a constant

Then  at x = 0 is

a) p

b)

c)

d) Independent of p

a)

b)

c)

d)

 equals

a)

b)

c)

d)

If  f(x) = x – [x] for every real number x, where [x] is the integral part of x, then  is

a) 1

b) 2

c) 0

d)

(Assertion and reason)

The question contains statement – 1 (assertion) and statement 2 (reason). Of these statements mark correct choice if

a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation for statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true

Statement 1 – The curve  is symmetric with respect to the line x = 1

Statement 2 – The parabola is symmetric about its axis.

Let P = (x, y) be any point on  with focii  and  equals

a) 8

b) 6

c) 10

d) 12

a) True

b) False

An ellipse has OB as a semi-minor axis. F, F’ are its focii and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is . . . . .

If tan θ =  then sin θ is

a)  but not  

b)  or

c)  but not −

d) None of these

Find the sum of the series
 

If x = 9 is the chord of contact of the hyperbola  then the equation of the corresponding pair of tangents is

a)

b)

c)

d)

The general solution of
 is

a)

b)

c)

d)