21.

A die is rolled twice and the sum of the numbers appearing on them is observed to be 7. What is the conditional probability that the number 2 has appered atleast once ?

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{2}{3}$

• $\frac{2}{5}$

The locus of the mid-points of the focal chord of the parabola y² = 4ax is

• y² = a(x - a)

• y² = 2a(x - a)

• y² = 4a(x - a)

• None of these

Find the value of $\sin \left(12°\right)\sin \left(48°\right)\sin \left(54°\right)$.

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{6}$

• $\frac{1}{8}$

In an equilateral triangle, the inradius, circumradius and one of the exradii are in the ratio

• 2 : 3 : 5

• 1 : 2 : 3

• 1 : 3 : 7

• 3 : 7 : 9

Let p and q be two statements. Then, p v q is false, if

• pis false and q is true

• both p and q are false

• both p and q are true

• None of the above

In how many ways 6 letters be posted in 5 different letter boxes?

• 5⁶

• 6⁵

• 5!

• 6!

If A and B be two sets such that A x B consists of 6 elements. If three elements A x B are (1, 4), (2, 6) and (3, 6), find B x A.

• {(1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6)}

• {(4, 1), (4, 2), (4, 3), (6, 1), (6, 2), (6, 3)}

• {(4, 4), (6, 6)}

• {(4, 1), (6, 2), (6, 3)}

Let f : R $\to$ R be defined as f(x) = x² + 1, find f^-1(- 5).

• $\left\{\mathrm{\varphi }\right\}$

• $\mathrm{\varphi }$

• $\left\{5\right\}$

• $\left\{- 5, 5\right\}$

Which of the following is correct?

• Determinant is a square matrix

• Determinant is a number associated to a matrix

• Determinant is a number associated to a square matrix

• All of the above

If $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ are the roots of x³ + ax² + b = 0, then the value of $\left|\begin{array}{ccc}\mathrm{\alpha }& \mathrm{\beta }& \mathrm{\gamma }\\ \mathrm{\beta }& \mathrm{\gamma }& \mathrm{\alpha }\\ \mathrm{\gamma }& \mathrm{\alpha }& \mathrm{\beta }\end{array}\right|$ is

• - a³

• a³ - 3b

• a³

• a² - 3b