USING PROPERTY PROVE
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1) y+x x y
z+x z x = (x+y+z)(x - z)²
x+y y z
2) 1 x+y x²+y²
1 y+z y²+ z² = (x -y)(y -z)(z - x)
1 z+x z²+ x²
3) a+ x y z
x a + y z = a²(a+x+y+z)
x y a+z
4) 1 1 1
x y z = (x-y)(y-z)(z-x)
yz zx xy
5) b+c c+a a+b
c+a a+b b+c
a+b b+c c+a
= 2(a+b+c)(ab+bc+ca-a²-b²-c²)
6) a- b-c 2a 2a
2a b- c-a 2b = (a+b+c)³
2c 2c c -a-b
7) 3a -a+b -a+c
a-b 3b c-b
a-c b-c 3c
= 3(a+b+c)(ab+ bc+ ca)
8) a+b b+c c+a a b c
b+c c+a a+b = 2. b c a
c+a a+b b+c c a b
9) x+4 2x 2x
2x x+4 2x = (5x+4)(4-x)²
2x 2x x+4
10) x x² 1+x³
y y² 1+y³ = 0,
z z² 1+z³
Then show that xyz = - 1
11) a² +1 ab ac
ab b²+1 bc = 1+a²+b²+ c²
ca cb c²+1
12) 1 +a 1 1
1 1+b 1 = abc+bc+ca+ab
1 1 1+c
13) 1 1+p 1+ p+q
2 3+2p 1+3p+2q = 1
3 6+3p 1+6p+3q
14) x x² 1+ px³
y y² 1+ py³
z z² 1+pz³
= (1+pxyz)(x -y)(y-z)(z-x)
15) If a, b, c are positive and unequal, show that the following determinant is negative:
a b c
A = b c a
c a b
Solve for x:
1) a+x a-x a- x
a-x a+x a-x = 0
a- x a-x a+x
2) x 4 = 0
2 2x
Evaluate
1) sin 30° cos 30°
- sin 60° cos 30°
2) a - b b - c c - a
b - c c - a a - b
c - a a - b b - c
MATRIX
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1) If A = 4 1
5 8
Show that A + A' is a symmetric matrix, where A' denotes the transpose of matrix A
2) If A = 3 4
5 1
Show that A - A' is a skew symmetric matrix, where A' denotes the transpose of matrix A
3) If A = 1 - 3 2 B= 2 - 1 - 1
2 0 2 1 0 - 1
Find the matrix such that A+B+C is a zero matrix.
4) Construct a 2x3 matrix A, whose elements are given (I+j)²/2
5) Compute the adjoint of the matrix A= 1 2
3 - 5
and verify that A. (Adj A) = | A| I.
6) If A= 3 - 2
4 - 2
find k If A²= kA - 2I₂
7) If A= 1 2 2
2 1 2
2 2 1
prove thatA² - 4A - 5A = 0. Hence find A⁻¹.
8) If A = 1 2
2 1
and f(x)= x² - 2x - 3, show that
f(A)= O
9) Express the metrix
A = 1 3 5
-6 8 3
-4 6 5
as the sum of a symmetric and a skew-symmetric matrix.
10) If A= 6 5
7 6
Show that A² - 12A + I= 0. Hence find A⁻¹.
11) If A= 3 2
1 1
find the value of a and such that
A² + A a+ b I=0. Hence find A⁻¹.
12) If A= 3 1
-1 2
Show that A² - 5A + 7I= 0. Hence find A⁻¹.
13) If A = 3 -2
4 -2
Find the value of λ, if A² +λA+ 2 I= 0. Hence find A⁻¹.
14) For what value of x, is following matrix is singular ?
3 -2x x +1
2 4
15) Find x, if 3x+y -y 1 2
2y-x 3 = -5 3
16) Find the inverse of
3 0 -1
2 3 0
0 4 1
17) write the adjoint of 2 - 1
4 3
18) Express the matrix as the sum of a symmetric and skew symmetric matrix, and verify result.
3 - 2 - 4
3 - 2 - 5
-1 1 2
Solve
1) 2x + y +2z =3; x +y + 2z= 2 ;
2x + 3y - z = - 2
2) x + 2y - 3z = - 4; 2x +3y + 2z= 2 ;
3x - 3y - 4z = 11
3) 3x+y+z= 3, 2x - y - z= 2,
- x - y + z = 1
4) x+y+z= 6; 2x - y+z= 3;
x - 2y + 3z = 6
5) x+2y+z=7, x+3z=11,2x-3y=1
6) x+y-z=1, 3x+y-2z=3,x-y-z=1
7) 3x-y+z=5,2x-2y+3z=7, x+y-z=-1
8) x+2y -3z=6, 3x+2y-2z=3,
2x-y+z=2.
9) 2x - y+z=0, x+y-z=6, 3x-y+4z=7.
10) x+y+z=6, x+2z=7, 3x+y+z=12
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