Question 1 :
Prove that
Solution :
First let us factor "a" from the 1st row, "b" from the 2nd row and c from the 3rd row.
Now we have to multiply column 1, 2 and by a, b and c respectively.
Let us subtract 2nd row from 1st row and subtract 3rd row from the 2nd row.
= x2(c2x2 + x4 + b2x2) + x2(0 + a2 x2)
= x2(c2x2 + x4 + b2x2) + x2(0 + a2 x2)
= x4 c2 + x6 + b2x4 + a2x4
= x4(c2 + x2 + b2+ a2)
Hence it is divisible by x4.
Question 2 :
If a, b, c are all positive, and are pth, qth and rth terms of a G.P., show that
Solution :
nth term of G.P
an = arn-1
a = pth term of G.P = arp-1 --(1)
b = qth term of G.P = arq-1 --(2)
c = rth term of G.P = arr-1 --(3)
By using properties of determinants, let us write them as sum of two determinants.
In the second determinant, let us add 1st and 3rd column.
In the first determinant column 1 and are identical. In the second determinant column 1 and 2 are identical.
= log a (0) + log r (0)
= 0
Hence it is proved.
Question 3 :
Find the value of
if x, y and z ≠ 1
Solution :
By expanding the above determinant, we get
= 1[1 - logzy logyz] - logxy[logyx - logzx logyz] + logxz[logyxlogzy - logzx]
By using the properties of logarithms
= [1 - logyy] - logxylogyx + logxylogzx logyz + logxzlogyxlogzy - logxzlogzx
= [1 - logyy] - logyy + logzylogyz + logyzlogzy - logzz
= [1 - 1] - 1 + logyy + logzz - 1
= - 1 + 1 + 1 - 1
= 0
Hence the answer is 0.
Question 4 :
If A =
Solution :
|A| = 1/4
det (Ak) = (1/4)k
if k = 1 det (A1) = (1/4)1 |
if k = 2 det (A2) = (1/4)2 |
if k = 3 det (A3)=(1/4)3 |
By finding the sum, we get
= (1/4) + (1/4)2 + (1/4)3 + ..................n terms
Sum of geometric series
Sn = a(rn - 1) / (r - 1)
a = 1/4, r = 1/4
Sn = (1/4)((1/4)n - 1) / ((1/4) - 1)
= (1/4)((1/4)n - 1) / (-3/4)
= (-1/3) ((1/4)n - 1)
= (1/3)(1 - (1/4)n)
Hence it is proved.
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