Mathematics Previous year Matrices Question for NDA

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Previous year Matrices Question for NDA

Set - 1

Q 2713291149

If ` A = [ (alpha , 2),(2 , alpha) ]` and `det (A^3) = 125,` then `alpha` is equal to
NDA Paper 1 2017

(A)

`±1`

(B)

`±2`

(C)

`±3`

(D)

`±5`

Solution:

`A =[(alpha, 2),(2, alpha)]`

`det (A^3) = 125`

`(det A)^3 = 125`

`(alpha^2 -4)^3 = 125 = 5^3`

`alpha^2 =9`

`alpha= pm 3`

Correct Answer is `=>` (C) `±3`

Q 2743391243

If B is a non-singular matrix and A is a square matrix, then the value of `det `(`B^(-1) AB`) is equal to
NDA Paper 1 2017

(A)

det (B)

(B)

det (A)

(C)

det (`B^(-1)`)

(D)

det (`A^(-1)`)

Solution:

` | B^(-1) AB|`

`= |B^(-1) | * |AB| ` (Since `|AB| = |A| * |B|`)

`= 1/(|B|) * |A| * |B|` (since `|B^(-1)| = 1/(|B|)`)

`=|A|`

`Det (A)`

Correct Answer is `=>` (B) det (A)

Q 2713391249

The equations
`x + 2y + 3z = 1`
`2x + y + 3z = 2`
`5x + 5y + 9z = 4`
NDA Paper 1 2017

(A)

have the unique solution

(B)

have infinitely many solutions

(C)

arc inconsistent

(D)

None of the above

Solution:

`Delta = [ ( 1,2, 3 ), ( 2, 1 , 3 ), ( 5 , 5 , 9 ) ]`

`= 3 `

`Delta _1 = [ ( 1, 2 , 3 ), ( 2, 1 ,3 ), ( 4 , 5 , 9 ) ]`

`= 0`

`Delta _2 = [( 1, 1 , 3 ),(2,2,3),(5,4,9)] = -3`

So `Delta != 3, Delta _1 != 0` Equation have unique solution.

Correct Answer is `=>` (A) have the unique solution

Q 2733491342

`A = [ (x + y , y),(x , x - y)] , B = [ (3),(-2) ] ` and ` C = [ (4),(-2) ] ` If `AB = C`, then what is `A^2` equal to?
NDA Paper 1 2017

(A)

` [ (4 , 8),(-4 , - 16)]`

(B)

` [ (4 , -4),(8 , - 16)]`

(C)

` [ (-4 , 8),(4 , 12)]`

(D)

` [ (-4 , -8),(4 , 12)]`

Solution:

`AB = C`

`[(x + y , y ) , ( x , x -y ) ] [ (3), (-2) ] = [ (4), (-2) ] `

`[ (3x + 3 y - 2 y ) , ( 3 x -2x + 2 y ) ] = [ (4), ( -2 ) ]`

`3x + y = 4`

`x +2 y =-2 `

`x =2 , y = -2`

`A = [ (0, (-2 )) , (2, 4 ) ]`

`A^2 = [ ( -4 , -8 ), ( 4 ,12 ) ]`

Correct Answer is `=>` (D) ` [ (-4 , -8),(4 , 12)]`

Q 2733591442

Consider the set A of all matrices of order `3 xx 3` with entries `0` or `1` only. Let B be the subset of A consisting of all matrices whose determinant is `1`. Let C be the set of A consisting of all matrices whose determinant is `-1`. Then which one of the following is correct ?

NDA Paper 1 2017

(A)

`C` is empty

(B)

`B` has as many elements as `C`

(C)

`A = B cup C`

(D)

`B` has thrice as many elements as `C`

Solution:

(1) `C` cannot be empty as

If `A= [(1, 0,0), (0,0,1),(0,1,0)] det (A)=-1`

(2) For every matrix in B there is a corresponding matrix in C obtained by reversing the first and second columns of

the matrix as value of determinant changes sign , if we replace first and second row and vice versa.

`=> n(B) = n(C) `
(3) If `A= [(1,0,1),(1,0,0),(0,1,1)]` then `det |A| =2`

So, `A ne B cup C`

(4) `n(B) ne 3 n(C) ->` Doesn't make any sense.

Correct Answer is `=>` (B) `B` has as many elements as `C`

Q 2763591445

If `A = [ ( cos theta , sin theta),( - sin theta , cos theta)]` , then what is ` A^3` equal to?
NDA Paper 1 2017

(A)

`[ ( cos 3 theta , sin 3 theta),( - sin 3 theta , cos 3 theta)]`

(B)

`[ ( cos^3 theta , sin^3 theta),( - sin^3 theta , cos^3 theta)]`

(C)

`[ ( cos 3 theta , - sin 3 theta),( sin 3 theta , cos 3 theta)]`

(D)

`[ ( cos^3 theta ,- sin^3 theta),( sin^3 theta , cos^3 theta)]`

Solution:

`A = [ ( cos theta , sin theta),( - sin theta , cos theta)]`

`A^2 = [ (cos theta ,sin theta ), ( - sin theta , cos theta )] [ ( cos theta , sin theta ) , ( - sin theta , cos theta ) ] `

`= [ (cos 2 theta , sin 2 theta ), ( -sin 2 theta , cos 2 theta ) ] `

`A^3 = A .A ^2 = [(cos theta, sin theta),(-sin theta, cos theta)][(cos 2 theta, sin 2 theta),(-sin 2 theta, cos 2 theta)]`

`=> [ (cos theta cos 2 theta+sin theta cos 2 theta, cos theta sin2 theta+ sin theta cos 2 theta), ( - sin theta cos 2 theta- sin 2 theta cos theta , - sin theta sin 2 theta + cos theta cos 2 theta)]`

`=[ ( cos 3 theta , sin 3 theta),( - sin 3 theta, cos 3 theta)]`

Correct Answer is `=>` (A) `[ ( cos 3 theta , sin 3 theta),( - sin 3 theta , cos 3 theta)]`

Q 2783591447

What is the order of
`[ x,y,z] [ (a,h,g),(h,b,f),(g,f,c)] [(x),(y),(z)]`
NDA Paper 1 2017

(A)

`3 xx 1 `

(B)

`1 xx 1`

(C)

`1 xx 3`

(D)

`3 xx 3`

Solution:

`[ x,y,z] [ (a,h,g),(h,b,f),(g,f,c)] [(x),(y),(z)]`

`:. ( 1 xx 3 ) * ( 3 xx 3 ) * ( 3 xx 1 )`

`= ( 1 xx 3 ) * ( 3 xx 1 )`

`= 1 xx 1`

Correct Answer is `=>` (B) `1 xx 1`

Q 2713591449

If `A = [(0,1),(1,0)]` , then the value of `A^4` is
NDA Paper 1 2017

(A)

` [(1,0),(0,1)]`

(B)

` [(1,1),(0,0)]`

(C)

` [(0,0),(1,1)]`

(D)

` [(0,1),(1,0)]`

Solution:

`A = [(0,1),(1,0)]`

`A^2 = [ (1, 0 ) , ( 0, 1 ) ] = I`

`A^4 = A^2 * A^2 = I = [ ( 1, 0), ( 0, 1 ) ]`

Correct Answer is `=>` (A) ` [(1,0),(0,1)]`

Q 2731445322

If A is a square matrix of order `3` and `det A = 5`, then what is `det [(2 A )^-1]` equal to ?
NDA Paper 1 2016

(A)

`1/10`

(B)

`2/5`

(C)

`8/5`

(D)

`1/40`

Solution:

`Det | (2A)^(-1) | = Det |(A^(-1)) * (2^(-1)|`

`= (2^(-1))^3 Det (A^(-1))` [ since of matrix of order `3`]

`= 1/(2^3) |A^(-1)| = 1/(2^3 ) * 1/2^3 * 1/(|A|) = 1/40`

Correct Answer is `=>` (D) `1/40`

Q 2711645520

What is `[ (x, y, z) ] [ (a, h , g ),(h,b,f),(g,f,c)]` equal to ?
NDA Paper 1 2016

(A)

`[ (ax + hy +gz , h + b +f , g +f + c )]`

(B)

`[(a,h,g),(hx, by, fz ), (g,f, c ) ]`

(C)

`[(ax + by + gz), (hx + by + fz ), (gx + fy + cz )]`

(D)

`[ (ax + hy + gz , hx + by + fz , gx + fy + cz ) ]`

Solution:

`[ (x, y, z) ] [ (a, h , g ),(h,b,f),(g,f,c)]= [(ax+by +g^2),(hx+by+f^2),(gx+fy +c^2)]`

Correct Answer is `=>` (D) `[ (ax + hy + gz , hx + by + fz , gx + fy + cz ) ]`

Q 2731656522

Let `ax^3 + bx^2 + cx +d = | ( x+1 , 2x, 3x ) , ( 2x +3 , x +1 , x ) , ( 2-x , 3x + 4 , 5 x -1 ) | ` then
What is the value of `a + b + c +d ?
NDA Paper 1 2016

(A)

(B)

(C)

(D)

Solution:

`[ (x, y, z) ] [ (a, h , g ),(h,b,f),(g,f,c)]= [(ax+by +g^2),(hx+by+f^2),(gx+fy +c^2)]`

Correct Answer is `=>` (B) 63

Q 2721856721

If `m = [ (1, 0), (0,1) ] ` and `n = [ (0,1), ( -1 , 0 ) ]` , then what is the value of the determinant of `m cos theta - n sin theta`?
NDA Paper 1 2016

(A)

-1

(B)

(C)

(D)

Solution:

Det ` (m cos theta - n sin theta )`

Det `([ (cos theta , 0 ), ( 0 , cos theta ) ] - [ ( 0, sin theta ), ( -sin theta , 0) ] )`

`1 cos^2 theta - (0- sin^2 theta) 1`

`=1`

Correct Answer is `=>` (C) 1

Q 2741056823

If `f (x ) = [ (cos x , -sin x , 0 ), ( sin x , cos x , 0 ) , ( 0, 0 , 1) ]` , then which of the following are correct ?

1. `f (theta) xx f(phi) = f( theta + phi )`
2. The value of the determinant of the matrix `f (theta ) xx f (phi ) ` is 1
3. The determinant of f(x) is an even function.

Select the correct answer using the code given below :
NDA Paper 1 2016

(A)

1 and 2 only

(B)

2 and 3 only

(C)

1 and 3 only

(D)

1, 2 and 3

Solution:

`f(x) = [ ( cos x , - sin x , 0 ), ( sin x, cos x ,0 ), ( 0,0, 1) ]`

(i) `f(theta) xx f (phi) = [ (cos theta , - sin theta , 0 ), ( sin theta , cos theta , 0 ), ( 0,0, 1) ] [ (cos phi , - sin phi , 0 ), (sin phi , cos phi , 0 ), ( 0,0,1) ]`

`= [ (cos theta cos phi - sin theta sin phi , - cos theta sin phi - sin theta cos phi , 0 ), ( sin theta cos phi + cos theta sin phi , cos theta cos phi- sin theta sin phi,0), (0,0,1) ]`

`= [ (cos (theta + phi ), - sin (theta + phi) , 0 ), (sin (theta + phi ) , cos ( theta + phi ),0 ), ( 0,0,1) ]`

`= f (theta + phi )`

(ii) ` | f (theta) xx f (phi) | = | f (theta + phi ) |`

`= | (cos (theta + phi ) , - sin (theta + phi) , 0 ), (sin (theta + phi ) , cos (theta + phi ) , 0 ), ( 0,0,1) | `

` = cos^2 (theta + phi ) + sin^2 (theta + phi )`

`=1`

(iii) Det `f(x)= | f(x) | =1`

`f(x) = f (-x) =1` so `fx`is even function

Correct Answer is `=>` (D) 1, 2 and 3

Q 2731167022

If `A = [ (1, -1 ) , (2,3 ) ]` and `B = [ (2,3),(-1,-2) ]` ,then which of the following is/are correct ?

1. `AB(A^-1B^-1)` is a unit matrix.
2. `(AB)^-1 = A^-1 B^-1`

Select the correct answer using the code given below :
NDA Paper 1 2016

(A)

1 only

(B)

2 only

(C)

Both 1 and 2

(D)

Neither 1 nor 2

Solution:

`A= [ ( 1,-1 ), (2,3) ] B = [ (2,3), (-1,-2) ]`

1. `AB (A^(-1) B^(-1) )`

`= [ (1,-1), (2,3) ][ ( 2,3), (-1,-2) ]( 1/5 [ (3,1), (-2,1) ] ) 1/1 [ (-2,-3), (1,2) ] `

`= 1/(-5) [ ( -5,0), (0,-5) ] = [ (1,0), (0,1) ] =I`

2. `AB= [ (1,-1),(2,3) ] [ (2,3), (-1,-2) ] = [ (3,5), (1,0) ]`

`(AB)^(-1) = -1/5 [ ( 0,-5), (-1,3) ]`

`A^(-1) * B^(-1) = 1/5 [ (3,1), (-2,1) ] * 1/(-1) [ (-2,3), (1,2) ] = 1/(-5) [ (-5,11),(5,-4)]`

clearly `(AB)^(-1) ne A^(-1) B^(-1)`

Correct Answer is `=>` (A) 1 only

Q 2701667528

For the system of linear equation `2 x + 3y + 5 z = 9 , 7x + 3y - 2 z = 8` and `2 x + 3 y + lamda z = mu`
Under what condition does the above system of equations have infinitely many solution ?
NDA Paper 1 2016

(A)

`lamda = 5 ` and `mu != 9`

(B)

`lamda != 5 ` and ` mu = 7` only

(C)

`lamda != 5`` and ` mu ` has any real value

(D)

`lamda ` has any real value and `mu != 9`

Correct Answer is `=>` (C) `lamda != 5`` and ` mu ` has any real value

Q 2127201181

If A is a square matrix, then what is adj `(A^(-1)) - (adj A)^(-1)`
equal to?
NDA Paper 1 2016

(A)

`2 | A|`

(B)

Null matrix

(C)

Unit matrix

(D)

None of these

Solution:

`adj (A^(-1)) - (adj A)^(-1) = (adj A)^(-1) - (adj A)^(-1)`

`= 0` ` quad [∵ adj (A^(-1) ) = (adj A)^(-1)]`

Hence, it is a null matrix.

Correct Answer is `=>` (B) Null matrix

Q 2177312286

Consider the following in respect of the matrix

` A = [ ( -1 , 1) , (1 , -1) ]`
`1. A^(2) = - A`
`2. A^(3) = 4A`

Which of the above is/are correct ?
NDA Paper 1 2016

(A)

Only `1`

(B)

Only `2`

(C)

Both `1` and `2`

(D)

Neither `1` nor `2`

Solution:

Given, `A = [ ( -1 , 1) , (1 , -1) ] `

1. Now, ` A^(2) = A xx A = [ ( -1 , 1) , (1 , -1) ] [ ( -1 , 1) , (1 , -1) ]`

` = [ ( 1 + 1 ,- 1 -1 ) , (-1 -1 ,1 +1 ) ]`

` A^(2) = [ ( 2 , -2) , (-2 , 2 ) ] = 2 [ ( 1 , -1) , (-1 , 1) ]`

` A^(3) = A^(2) . A = [ ( 2 , -2) , (-2 , 2 ) ] [ ( -1 , 1) , (1 , -1) ]`

` = [ ( -2 -2 ,2 +2 ) , (2 +2 ,-2 -2 ) ] = [ ( -4 , 4) , (4 , -4) ]`

` = 4 [ ( -1 , 1) , (1 , -1) ]`

`A^(3) = 4A`

Hence, only 2 is correct.

Correct Answer is `=>` (B) Only `2`

Q 1608323208

Which one of the following matrices is an

elementary matrix?
NDA Paper 1 2015

(A)

` [(1,0,0),(0,0,0),(0,0,1)]`

(B)

` [(1,5,0),(0,1,0),(0,0,1)]`

(C)

` [(0,2,0),(1,0,0),(0,0,1)]`

(D)

` [(1,0,0),(0,1,0),(0,5,2)]`

Solution:

` [(1,5,0),(0,1,0),(0,0,1)]` is an elementary matrix. Since, the 'value

of determinant of the given matrix is `1` .

Correct Answer is `=>` (B) ` [(1,5,0),(0,1,0),(0,0,1)]`

Q 1638423302

If `A = [(2,7),(1,5)]` then what is `A + 3A ^(- 1)` equal to?

where, `I` is the identity matrix of order `2`.
NDA Paper 1 2015

(A)

`3I`

(B)

`5I`

(C)

`7I`

(D)

None of these

Solution:

We have, `A = [(2,7),(1,5)]`

`:. | A| = 10 - 7 = 3`

Now `A^(-1) = 1/( | A| ) adj A`

`:. | A| = 1/3 [(5,-1),(-7,2)] = 1/3 [(5,-7),(-1,2)]`

`:. A+ 3A^(-1) = [(2,7),(1,5)] + 3 xx 1/3 [(5,-7),(-1,2)] = [(7,0),(0,7)] = 7I .`

Correct Answer is `=>` (C) `7I`

Q 2240367213

If `A = [(1,0,-2) , (2 ,-3, 4) ]` then the matrix `X` for which
`2X + 3A = 0` holds true is
NDA Paper 1 2015

(A)

` [ (- 3/2 , 0, -3),(-3, - 9/2 , -6) ]`

(B)

` [ ( 3/2 , 0, -3),(3, - 9/2 , -6) ]`

(C)

` [ ( 3/2 , 0, 3),(3, 9/2 , 6) ]`

(D)

` [ (- 3/2 , 0, 3),(-3, 9/2 , -6) ]`

Solution:

Given, `A = [(1,0,-2) , (2 ,-3, 4) ]`

We have, `2X + 3A= 0`

`=> X=(-3)/2 A`

` => X =(-3)/2 [(1,0,-2) , (2 ,-3, 4) ] => [ (- 3/2 , 0, 3),(-3, 9/2 , -6) ]`

Correct Answer is `=>` (D) ` [ (- 3/2 , 0, 3),(-3, 9/2 , -6) ]`

Q 2210367219

If `A= [ ( 1,1,-1), (2,-3,4),(3,-2,3) ]` and ` B = [ (-1,-2,-1),(6,12,6),(5,10,5)]`
then which of the following is/ are correct?
1. `A` and `B` commute.
2. `AB` is a null matrix.

Select the correct answer using the code given below
NDA Paper 1 2015

(A)

Only `1`

(B)

Only `2`

(C)

Both `1` and `2`

(D)

Neither `1` nor `2`

Solution:

We have `A= [ ( 1,1,-1), (2,-3,4),(3,-2,3) ]` and ` B = [ (-1,-2,-1),(6,12,6),(5,10,5)]`

` AB = [ (-1+6-5 , -2+12 -10 , -1+6-5),( -2 -18+20 , -4 -36 +40, -2 -18+20), (-3 -12 +15, -6 -24+30 , -3 -12 +15)]`

` = [(0,0,0),(0,0,0),(0,0,0)]`

Hence, `AB ` is a null matix.

Correct Answer is `=>` (B) Only `2`

Q 1668423305

The matrix ` [(0, -4 + i) ,( 4 +i ,0)]` is
NDA Paper 1 2015

(A)

symmetric

(B)

skew-symmetric

(C)

hermitian

(D)

skew-hermitian

Solution:

A square matrix `A` is said to be skew-hermitian, if

`A' = -A` or `a_(ij) = a_(ji) AA i ` and `j`.

Here, `a_(12) = -4 + i` and `a_(21) = 4 + i`

Now, `a_(21) = - (-4 + i) = - (- i - 4) = 4 + i`

Hence, the given matrix is skew-hermitian matrix.

Correct Answer is `=>` (D) skew-hermitian

Q 2220291111

If `A` is an invertible matrix of order `n` and `k` is any positive
real number, then the value of `[det(kA)]^(-1) det (A)` is
NDA Paper 1 2015

(A)

`k^(-n)`

(B)

`k^(-1)`

(C)

`k^(n)`

(D)

`nk`

Solution:

`[ det (kA)]^(-1) det (A)`

` = 1/(det (kA)) xx det (A) quadquadquadquadquad [ :. a^(-m) = 1/a^m]`

` = 1/ (k^(n) det (A)) xx det ( A) = 1/ k^(n) = k^(-n)`

Correct Answer is `=>` (A) `k^(-n)`

Q 1648523403

Consider the following in respect of two
non-singular matrices `A` and `B` of same order

I. det `(A+ B) = det A + det B`

II. `(A+ B)^(-1) =A ^(-1) + B^(-1)`

Which of the above is/are correct?
NDA Paper 1 2015

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

If `A = B + C`, then it is not necessary that

`det (A) = det (B) + det (C)`

Also, `(A+ B)^(-1) =A^(-1) + B^(-1)` is false.

Correct Answer is `=>` (D) Neither I nor II

Q 1628623501

If `X = [(3,-4),(1,-1) ], B=[ (5,2),(-2,1)]` and `A = [(p,q),(r,s)]` satisfy

the equation `AX = B`, then the matrix `A` is equal to
NDA Paper 1 2015

(A)

` [(-7,26),(1,-5) ]`

(B)

` [(7,26),(4,17) ]`

(C)

` [(-7,-4),(26,13) ]`

(D)

` [(-7,26),(-6,23) ]`

Solution:

`:. AX = - B`

`:. [(p,q),(r,s)] [(3,-4),(1,-1) ] = [ (5,2),(-2,1)]`

`=> [(3p+ q, -4p-q),(3r + s , -4r - s)] = [ (5,2),(-2,1)]`

`=> 3p + q = 5` and `-4p - q = 2`

`=> -p = 7 => p = -7`

Now, `q = 5 + 21 = 26, 3r + s = - 2`

Also, `-4r- s = 1`

`=> -r = - 1 => r = 1` and `s = -2 - 3 = - 5`

`:. A= [(-7,26),(1,-5) ]`

Correct Answer is `=>` (A) ` [(-7,26),(1,-5) ]`

Q 1618823700

Let `A= [ (x+y,y) ,(2x,x-y)] , B [(2),(-1)] ` and ` c= [3/2]` if

`AB = C`, then what is `A^2` equal to?
NDA Paper 1 2015

(A)

` [ (6 , -10), (4 , 26)]`

(B)

` [ (-10 , 5), (4 , 24)]`

(C)

` [ (-5 , -6), (-4 , -20)]`

(D)

` [ (-5 , -7), (-5 , 20)]`

Solution:

We have `AB = C`

`:. [(x +y ,y) ,(2x , x-y)] [(2),(-1)] = [3/2]`

` => [( 2x+2y , -y) ,( 4x,-x-y)] =[3/2]`

` => [ (2x+y),(3x-y)] = [3/2]`

` => 2x + y = 3` and `3x + y = 2 => x = 2- 3 = -1`

`:. y = 5`

` :. A^2 = [(x +y ,y) ,(2x , x-y)]^2`

` = [ (4 , 5), (-2 , -6)] [ (4 , 5), (-2 , -6)]`

` = [ (16 - 10 , 20 - 30), (-8 +12 ,-10 + 36 )] = [ (6 , -10), (4 , 26)]`

Correct Answer is `=>` (A) ` [ (6 , -10), (4 , 26)]`

Q 2261401325

If `A` is an orthogonal matrix of order `3` and `B = [(1,2,3),(-3,0,2),(2,5,0)]`
then which following is/are correct?
1.`|AB| =pm 47`
2. `AB=BA`

Select the correct answer using the code given below
NDA Paper 1 2015

(A)

Only `1`

(B)

Only `2`

(C)

Both `1` and `2`

(D)

Neither `1` nor `2`

Solution:

`A` is an orthogonal matrix.

`|AB|= pm |B|`

` :. |B| = |(1,2,3),(-3,0,2),(2,5,0)| = 47`

Now, ` |AB| =pm 47`

Correct Answer is `=>` (A) Only `1`

Q 1618123900

If `E( theta ) = [ (cos theta , sin theta),(- sin theta , cos theta) ]` then `E( alpha) E (beta)` is equal to
NDA Paper 1 2015

(A)

`E(alpha beta)`

(B)

`E(alpha - beta)`

(C)

`E(alpha + beta)`

(D)

`-E(alpha + beta)`

Solution:

Given, ` E( theta ) = [ (cos theta , sin theta),(- sin theta , cos theta) ]`

` :. E (alpha) = [ (cos alpha , sin alpha ),(- sin alpha , cos alpha ) ]`

and ` E (beta) = [ (cos beta , sin beta ),(- sin beta , cos beta ) ]`

` :. E(alpha)E(beta) [ (cos alpha , sin alpha ),(- sin alpha , cos alpha ) ] [ (cos beta , sin beta ),(- sin beta , cos beta ) ]`

` = [( cos2 . cos beta - sin2 . sin beta , cos2 . sin beta - sin2 . cos beta),(-sin 2 . cos beta - sin beta . cos 2 ,- sin 2 . sin beta - cos2 cos beta )]`

` = [ (cos (alpha +beta) , sin(alpha + beta)) ,(-sin (alpha +beta) , cos (alpha + beta))]`

` = E (alpha + beta)`

Correct Answer is `=>` (C) `E(alpha + beta)`

Q 1608180008

The area of a triangle, whose vertices are `(3, 4),
(5, 2)` and the point of intersection of the lines `x = a`
and `y = 5`, is `3` square units. When is the value of `a`?
NDA Paper 1 2015

(A)

`2`

(B)

`3`

(C)

`4`

(D)

`5`

Solution:

We have, ` Delta = 3,` sq units

` :. 1/2 | (5, 2, 1) ,(3,4,1) ,(a,5,1)| = 3`

` 1/2 [5(4- 5)- 2(3- a)+ 1(15- 4a)] = 3`

`=> 1/2 [-5- 2(3- a)+ (15- 4a)] = 3`

`=> 1/2 [-5-6+ 2a + 15- 4a] = 3`

` => 2 - a = pm 3`

` => a= 5` or `-1`

`:. a=5`

Correct Answer is `=>` (B) `3`

Q 1629691511

If `A` and `B` are square matrices of second order such that
`| A | = -1` and `| B | = 3`, then what is `| 3 AB |` equal to?
NDA Paper 1 2014

(A)

`3`

(B)

`-9`

(C)

`-27`

(D)

None of these

Solution:

`A` and `B` are square matrices of order `2`.

We know that, `| kA | = k^n | A |`, where `n` is order of matrix `A`.

`:. | 3AB | = 3^2 | A | | B | quad ( ∵ | AB | = | A || B |)`

` = 9 (-1) (3)`

` = -27 quad ( ∵ | A | = -1, | B | = 3)`

Correct Answer is `=>` (C) `-27`

Q 1782334237

If, `A` and `B` be two matrices such that `AB =A` and `BA =B`.
Then, which of the following statements are correct?
1. `A^2 =A`
2. `B^2 =B`
3. `(AB)^2 = AB`
Select the correct answer using the code given below.
NDA Paper 1 2014

(A)

`1` and `2`

(B)

`2` and `3`

(C)

`1` and `3`

(D)

`1, 2` and `3`

Solution:

1. We have, `AB = A`

`:. A^2 = (AB). (AB) =A. (BA) B`

`= ABB quad (∵ BA = B)`

` = AB`

` = A quad (∵ AB = A)`

Also, `B^2 = (BA). (BA)`

`= B. (AB). A`

`= B. A. A quad (∵ AB = A)`

` = B. A`

` = B quad (∵ BA = B)`

Again `(AB)^2 = (AB).(AB)`

` = A. (BA)B`

` = A. B. B quad (∵ BA = B)`

` = AB`

` = A quad (∵ AB = A)`

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