JEE Advanced 2018 Paper 2 Offline | Matrices and Determinants Question 21 | Mathematics

JEE Advanced 2018 Paper 2 Offline

MCQ (More than One Correct Answer)

-1

Let S be the set of all column matrices $$\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]$$ such that $${b_1},{b_2},{b_3} \in R$$ and the system of equations (in real variables)

$$\eqalign{ & - x + 2y + 5z = {b_1} \cr & 2x - 4y + 3z = {b_2} \cr & x - 2y + 2z = {b_3} \cr} $$

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $$\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]$$$$ \in $$S?

$$x + 2y + 3z = {b_1}$$, $$\,4y + 5z = {b_2}$$ and $$x + 2y + 6z = {b_3}$$

$$x + y + 3z = {b_1}$$, $$5x + 2y + 6z = {b_2}$$ and $$ - 2x - y - 3z = {b_3}$$

$$ - x + 2y - 5z = {b_1}$$, $$\,2x - 4y + 10z = {b_2}$$ and $$x - 2y + 5z = {b_3}$$

$$x + 2y + 5z = {b_1}$$, $$2x + 3z = {b_2}$$ and $$x + 4y - 5z = {b_3}$$

JEE Advanced 2017 Paper 1 Offline

MCQ (More than One Correct Answer)

-1

Which of the following is(are) NOT the square of a 3 $$ \times $$ 3 matrix with real entries?

$$\left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & { - 1} \cr } } \right]$$

$$\left[ {\matrix{ 1 & 0 & 0 \cr 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr } } \right]$$

$$\left[ {\matrix{ { - 1} & 0 & 0 \cr 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr } } \right]$$

$$\left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right]$$

JEE Advanced 2016 Paper 2 Offline

MCQ (More than One Correct Answer)

-2

Let a, $$\lambda$$, m $$\in$$ R. Consider the system of linear equations

ax + 2y = $$\lambda$$

3x $$-$$ 2y = $$\mu$$

Which of the following statements is(are) correct?

If a = $$-$$3, then the system has infinitely many solutions for all values of $$\lambda$$ and $$\mu$$.

If a $$\ne$$ $$-$$3, then the system has a unique solution for all values of $$\lambda$$ and $$\mu$$.

If $$\lambda$$ + $$\mu$$ = 0, then the system has infinitely many solutions for a = $$-$$3.

If $$\lambda$$ + $$\mu$$ $$\ne$$ 0, then the system has no solution for a = -3.

JEE Advanced 2016 Paper 1 Offline

MCQ (More than One Correct Answer)

-2

Let $$P = \left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$$, where $$\alpha$$ $$\in$$ R. Suppose $$Q = [{q_{ij}}]$$ is a matrix such that PQ = kl, where k $$\in$$ R, k $$\ne$$ 0 and I is the identity matrix of order 3. If $${q_{23}} = - {k \over 8}$$ and $$\det (Q) = {{{k^2}} \over 2}$$, then

$$\alpha$$ = 0, k = 8

$$4\alpha - k + 8 = 0$$

$$\det (Padj(Q)) = {2^9}$$

$$\det (Qadj(P)) = {2^{13}}$$

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