Day -3
1) Without expanding at any stage , find the value of the determinant:
∆= 2 x y+ z
2 y z + x
2 z x + y
2) If M= 1 2
2 3 and M² - kM - I₂= 0, then find the value of k.
3) Find dy/dx , if x= at² and y= 2at.
4) Using properties of determinants, prove
x x² 1+ px³
y y² 1+ py³
z z² 1+ pz³ =
(1+ pxyz)(x - y)(y - z) (z - x) Where p is any scalar.
5) Prove that the function f(x)= |x -1|, x∈R, is continuous at x= 1.
6) If A= 3 -2 3 & B= -1 -5 -1
2 1 -1 -8 -6 9
4 -3 2 -10 1 7
Evaluate. Hence, solve the system of equations, 3x - 2y + 3z= 8; 2x + y - z = 1; 4x - 3y + 2z = 4.
7) Using elementary transformation, find the inverse of the matrix
1 3 -2
-3 0 -1
2 1 0
Day- 2
1) If y= (cosx - since)/(cosx + since), then find dy/dx
2) If x = a sin2t, y= a(cos2t + log tan t) then find dy/dx.
3) If A= 1. 4 x
z 2 y
-3 -1 3 is a symmetric Matrix, then the value of x+ y+ z is
4) If A= 3 2
5 -7, then find inverse of A.
5) Evaluate the determinant:
x -1 1
x³ x²+ x +1
6) If 2 2 = 3x 1
2 3 4x 2 then find x
7) f(x= (since)/x + cosx, if x≠ 0
2 If x= 0
Show that it continuous at x= 2.
8) If A= 2 -3 5
3 2 -4
1 1 -2 then find inverse of A.
Using the inverse of A solve the following system of equations:
2x - 3y + 5z= 11
3x + 2y - 4z= -5
x + y -2z= -3.
Day-1
A) Using properties of determinants. Prove that:
a) x² y² z²
x³ y³ z³
xyz xyz xyz
= xyz(x - y)(y - z)(z - x)(xy+ yz+ zx)
b) y+ z x+ y x
z + x y+ x y = x³+ y³+ z³- 3xyz
x+ y z + x z
c) a b - c c- b
a- c b c- a
a - b b - a c
= (a+ b -c)(b+ c -a)(c +a -b)
d) (b+ c)² a² a²
b² (c +a)² b²
c² c² (a +b)²
= 2abc(a+ b+ c)³
B) If X is a 2x2 matrix given that
1) A= 1 3 & B= 1 -1
0 1 0 1 with the relation AX= B, find the matrix X.
2) If A= 1 2 3 B= 1 0 2 & C= 4 4 10
-1 -3 2 3 4 5 4 2 14 with the relation 2A + kB= C find the value of k (given k≠ 0).
3) If A= 4 2
1 1 find (A - 2I)(A - 3I), where I is a unit matrix.
4) Solve by matrix method: 5x - 12y = -9; 7x - 6y = -8.
C)1) f(x)= (1- cosx)/(x sinx), when x≠ 0
= 2, When x=0
Find the value of m such that f(x) is continuous at x= 0.
2) Given f(x)= x²+4, x≤2 & €(x)= 2x, x ≤2
= x+6, x>2 = 4, x> 2.
Show that f(x) x €(x) is continuous at x= 2.
3) f(x)= {log(1+ax) - log(1- bx)}/x is not defined at x= 0; find the value of f(0) so that f(x) is continuous at x= 0.
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