1
MCQ (Single Correct Answer)

### JEE Main 2018 (Online) 15th April Evening Slot

Suppose A is any 3$\times$ 3 nonsingular matrx and ( A $-$ 3I) (A $-$ 5I) = O where I = I3 and O = O3. If $\alpha$A + $\beta$A-1 = 4I, then $\alpha$ + $\beta$ is equal to :
A
8
B
7
C
13
D
12

## Explanation

Given,

( A $-$ 3I) (A $-$ 5I) = O

$\Rightarrow$ A2 - 8A + 15I = O

Multiplying both sides by A- 1, we get,

A- 1A.A - 8A- 1A + 15A- 1I = A- 1O

$\Rightarrow$ A - 8I + 15A- 1 = O

$\Rightarrow$ A + 15A- 1 = 8I

$\Rightarrow$${A \over 2} + {{15{A^{ - 1}}} \over 2} = 4I$

Comparing with the equation $\alpha$A + $\beta$A-1 = 4I, we get

$\alpha$ = ${1 \over 2}$ and $\beta$ = ${15 \over 2}$

$\therefore$ $\alpha$ + $\beta$ = ${1 \over 2}$ + ${15 \over 2}$ = ${16 \over 2}$ = 8
2
MCQ (Single Correct Answer)

### JEE Main 2018 (Online) 15th April Evening Slot

If the system of linear equations
x + ay + z = 3
x + 2y + 2z = 6
x + 5y + 3z = b
has no solution, then :
A
a = $-$ 1,    b = 9
B
a = $-$ 1,    b $\ne$ 9
C
a $\ne$ $-$ 1,    b = 9
D
a = 1,    b $\ne$ 9

## Explanation

As the given system of equations has no solution then

$\Delta$ = 0 and at least one of $\Delta$1, $\Delta$2 and $\Delta$2 should not be zero.

$\therefore$ $\Delta$ = $\left| {\matrix{ 1 & a & 1 \cr 1 & 2 & 2 \cr 1 & 5 & 3 \cr } } \right| = 0$

$\Rightarrow$ - $a$ - 1 = 0

$\Rightarrow$ a = - 1

$\Delta$2 = $\left| {\matrix{ 1 & 3 & 1 \cr 1 & 6 & 2 \cr 1 & b & 3 \cr } } \right| \ne 0$

$\Rightarrow$ b $\ne$ 0
3
MCQ (Single Correct Answer)

### JEE Main 2018 (Online) 16th April Morning Slot

The number of values of k for which the system of linear equations,
(k + 2)x + 10y = k
kx + (k +3)y = k -1
has no solution, is :
A
1
B
2
C
3
D
infinitely many

## Explanation

System of linear equation have no solution,

$\therefore\,\,\,$ determinant of coefficient = 0

$\left| {\matrix{ {k + 2} & {10} \cr k & {k + 3} \cr } } \right| = 0$

$\Rightarrow$ $\,\,\,\,$ (k + 2) (k + 3) $-$ 10 K = 0

$\Rightarrow$ $\,\,\,\,$ k2 $-$ 5k + 6 = 0

$\therefore\,\,\,\,$ k = 2, 3

When, k = 2 then equations become,

4x + 10y = 2

and 2x + 5y = 1

It has in finite number of solutions.

When k = 3, equations becomes

5x + 10y = 3

3x + 6y = 2

Those equation has no solutions.

$\therefore\,\,\,\,$ When k = 3, then system of equations have no solutions.
4
MCQ (Single Correct Answer)

### JEE Main 2018 (Online) 16th April Morning Slot

Let A = $\left[ {\matrix{ 1 & 0 & 0 \cr 1 & 1 & 0 \cr 1 & 1 & 1 \cr } } \right]$ and B = A20. Then the sum of the elements of the first column of B is :
A
210
B
211
C
231
D
251

=

## Explanation

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

A2 = A.A = $\left[ {\matrix{ 1 & 0 & 0 \cr 1 & 1 & 0 \cr 1 & 1 & 1 \cr } } \right] \times \left[ {\matrix{ 1 & 0 & 0 \cr 1 & 1 & 0 \cr 1 & 1 & 1 \cr } } \right]$

=   $\left[ {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right]$

A3 = A2.A =  $\left[ {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right] \times \left[ {\matrix{ 1 & 0 & 0 \cr 1 & 1 & 0 \cr 1 & 1 & 1 \cr } } \right]$

=   $\left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 0 \cr 6 & 3 & 1 \cr } } \right]$

Similarly

A4 =   $\left[ {\matrix{ 1 & 0 & 0 \cr 4 & 1 & 0 \cr {10} & 4 & 1 \cr } } \right]$

From this we can say,

An =   $\left[ {\matrix{ 1 & 0 & 0 \cr n & 1 & 0 \cr {{{n\left( {n + 1} \right)} \over 2}} & n & 1 \cr } } \right]$

$\therefore\,\,\,$ A20 =   $\left[ {\matrix{ 1 & 0 & 0 \cr {20} & 1 & 0 \cr {210} & {20} & 1 \cr } } \right]$

$\therefore\,\,\,$ Sum of the first column

= 1 + 20 + 210

= 231

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