Suppose the cube x³– px + q has three distinct real roots where p > 0 and q > 0. Then which one of the following holds?

• The cubic has minima at  and maxima at –

• The cubic has minima at – and maxima at

• The cubic has minima at both  and- 

• The cubic has minima at both  and- 

22.

The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is 

• (x – 2)y′² = 25 – (y – 2)² 

• (y – 2)y′² = 25 – (y – 2)²

• (y – 2)²y′²= 25 – (y – 2)² 

• (y – 2)²y′²= 25 – (y – 2)² 

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

• If det A = ± 1, then A^–1 exists but all its entries are not necessarily integers

• If detA ≠ ± 1, then A^–1 exists and all its entries are non-integers

• If detA = ± 1, then A^–1 exists and all its entries are integers

• If detA = ± 1, then A^–1 exists and all its entries are integers

Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx and z = bx + ay. Then a² + b² + c² + 2abc is equal to

• 2

• -1

• 0

• 0

It is given that the events A and B are such that  then P(B) is 

• 1/6

• 1/3

• 2/3

• 2/3

The vector  lies in the plane of the vectors  and bisects the angle between .Then which one of the following gives possible values of α and β?

• α = 2, β = 2

• α = 1, β = 2

• α = 2, β = 1

• α = 2, β = 1

The non−zero verctors are related by .Then the angle between  is

• 0

• π/4

• π/2

• π/2

The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz−plane at the point . Then

• a = 2, b = 8

• a = 4, b = 6

• a = 6, b = 4

• a = 6, b = 4

If the straight lines  intersect at a point, then the integer k is equal to

• -5

• 2

• 5

• 5

In a shop, there are five types of ice-creams available. A child buys six ice-creams.
Statement -1: The number of different ways the child can buy the six ice-creams is ¹⁰C₅.
Statement -2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A’s and 4 B’s in a row.

• Statement −1 is false, Statement −2 is true

• Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

• Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

• Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.