AIEEE 2008 | Matrices and Determinants Question 292 | Mathematics

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 $${a^2} + {b^2} + {c^2} + 2abc$$ is equal to :

$$2$$

$$-1$$

$$0$$

$$1$$

AIEEE 2008

MCQ (Single Correct Answer)

-1

Out of Syllabus

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

If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ exists but all its entries are not necessarily integers

If det $$A \ne \pm 1,$$ then $${A^{ - 1}}$$ exists and all its entries are non integers

If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ exists but all its entries are integers

If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ need not exists

AIEEE 2007

MCQ (Single Correct Answer)

-1

Out of Syllabus

Let $$A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 & 0 & 5 \cr } } \right|.$$ If $$\,\,\left| {{A^2}} \right| = 25,$$ then $$\,\left| \alpha \right|$$ equals

$$1/5$$

$$5$$

$${5^2}$$

$$1$$

AIEEE 2007

MCQ (Single Correct Answer)

-1

If $$D = \left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + x} & 1 \cr 1 & 1 & {1 + y} \cr } } \right|$$ for $$x \ne 0,y \ne 0,$$ then $$D$$ is :

divisible by $$x$$ but not $$y$$

divisible by $$y$$ but not $$x$$

divisible by neither $$x$$ nor $$y$$

divisible by both $$x$$ and $$y$$