INVERSE TRIGONOMETRY
MATRIX
MARKS -2
1) If M= 1 2
2 3 Find the value of k and M² - kM - I= 0
2) If A= 3 1
7 5 , find x and y so that A²+ xI= yA.
3) If the matrix A= 6 x 2
2 -1 2
-10 5 2 is a singular matrix, find the value of x.
4) If A= 3 -2
4 -2 , find x such that A²= xA - 2. Hence, find inverse of A.
5) If A = x² B= 2x C= 7
y² 3y -3 and A+ 2B = 3C, then find x and y
6) If (B - 2I)(B - 3I)= 0, where.
B =4 2
-1 y and I is the unit matrix, find the value of y.
7) If A= 5 4. And B= 1 -2
1 1 1 3 with the relation AX= B. Then find the matrix X
8) If A = 2 3
1 2 , find inverse of A
9) If A= cost sint
- sint cost , show that A. A' = I
10) For what value of x is the given matrix 2x+4 4
x+5 3 a singular matrix.
MARKS -4
1) If A = [x 4 1]
B= 2 1 2
1 0 2
0 2 -4
C= x
4
-1 with the relation ABC= 0, Find x.
2) If A= 3
1
-2
B= [1 -5 7]
Verify (AB)' = B' A'
3) Solve: 2x - 3y =1; x+5y=7
4) If A = 3 1
7 2 , find inverse of A, hence Solve the following Equation: 3x+7y= 4; x+2y= 1
Marks - 6
1) Evaluate:
3 -2 3 -1 -5 -1
2 1 -1 x -8 -6 9
4 -3 2 -10 1 7
Hence, Solve system of equation, 3x-2y+3z=8; 2x+y-z=1; 4x-3y+2z=4.
2) Solve: x - 2y - 2z - 5= 0; -x + 3y +4= 0; - 2x + z - 4=0
3) Solve: 2/x + 3/y + 10/z= 4;
4/x - 6/y + 5/z= 1;
6/x + 9/y - 20/z= 2
4) If A= 4 2 3
1 1 1
3 1 -2 , find inverse of A. Hence, solve the following system of linear equations: 4x+2y+3z= 2, x+y+z= 1, 3x + y - 2z= 5.
5) If A= 2 0 -1
5 1 0
0 1 3 find inverse of A.
6) If A= 1 2 5
2 3 1
-1 1 1 , Find A⁻¹ and verify A⁻¹A = I= A A⁻¹
ANSWER: 2 Marks
1) k= 4. 2) x=8, y=8. 3) -3
4) x=1 , inverse of A= -1 1
-2 3/2
5) x= -7, 3 and y= -3,-3. 6) 1
7) -3 -14
4 17
8) - 1 2
2 -3
10) x= 4
Marks 4:
1) x= -2 or -1. 3) x=2, y= 1
4) -2 1
7 -3 , x= -1, y= 1
Marks 6:
1) -17; x=1, y=2, z= 3
2) -31/11, -25/11, -18/11
3) x=2, y=3, z=5
4) -3 7 -1
1/8. 5 -17 -1
-2 2 2
x= 1/2, y= 3/2, z= - 1
5) 3 -1 1
-15 6 -5
5 -2 2
6) 2 3 -13
1/21x. -3 6 9
5 -3 -1
2 Marks
-----------
A) Solve:
.*1) sin⁻¹cos(sin⁻¹x) = π/3. ±1/2
.2) tan(cos⁻¹x) = 2/√5. ± √5/3
.3) sin⁻¹(6x)+sin⁻¹(6√3x)=-π/2. ±1/12
.4) cos(tan⁻¹x)= sin(cot⁻¹3/4). ±3/4
5) tan⁻¹{(1-x)/(1+x)}= 1/2 tan⁻¹ x, x> 0. 1/√3
6) Cos (tan⁻¹x) = sin(cot⁻¹3/4) 3/4
7) cos⁻¹x + sin⁻¹x/2= π/6. ±1
B) Prove:
.1) tan(2 tan⁻¹1/5) - π/4) = -7/17
.2) sec²(tan⁻¹2) +cosec²(cot⁻¹3)= 15
.3) tan⁻¹1/4 +tan⁻¹2/9=1/2 sin⁻¹4/5
.*4) 1/2 cos⁻¹{(1-x)/(1+x)}=tan⁻¹√x
*.5) tan⁻¹{1/(1+2x)}+tan⁻¹{1/(4x+1)} = tan⁻¹(2/x²)
6) 4(2tan⁻¹1/3 + tan⁻¹1/7)= π
7) tan⁻¹1/2 + tan⁻¹1/5+ tan⁻¹1/8 = π/4
8) 2 sin⁻¹x = sin⁻¹{2x√(1-x²)}
9) (1/2)tan⁻¹x = cos⁻¹[√{(1+√(1+x²)}/{2√(1+x²)} ]
10) If sin⁻¹x +sin⁻¹y= 2π/3 then find the value of cos⁻¹x + cos⁻¹y. π/3
4 marks
------------
A) Solve:
.1) sin⁻¹{x/√(1+x²)} - sin⁻¹{1/√(1+x²)} = sin⁻¹{(1+x)/(1+x²)}. 2
.2) Sin⁻¹5/x + sin⁻¹12/x= π/2, x≠ 0 ±13
.3) sin⁻¹x + sin⁻¹(1-x)= cos⁻¹x, x≠0.
1/2
4) tan(cos⁻¹x)= sin(tan⁻¹2). ±√5/3
B) PROVE:
1) 2tan⁻¹1/2 + tan⁻¹1/7= tan⁻¹31/17
.2) tan⁻¹1/2 = π/4 -1/2 cos⁻¹4/5
.3) tan[sin⁻¹1/√17] + cos⁻¹9/√85 =1/12
.4) sin[2tan⁻¹3/5 - sin⁻¹7/25]= 304/425
.5) 2 tan⁻¹1/5 + cos⁻¹7/5√2+ 2 tan⁻¹1/8= π/4
.6) cos⁻¹63/65 +2 tan⁻¹1/5= sin⁻¹ 3/5
7) Sin⁻¹{x/√(1+x²)} +cos⁻¹{(x+1)/√(x²+x+2)}= tan⁻¹(x²+x+1)
8) sin⁻¹12/13 + cos⁻¹4/5 + tan⁻¹63/16 =π
.9) If sin⁻¹x + tan⁻¹x= π/2 then prove 2x² + 1= √5
.10) If cos⁻¹x+ cos⁻¹y+ cos⁻¹z= π then show x²+y²+z²+2xyz= 1
11) If tan⁻¹x + tan⁻¹y+ tan⁻¹z = 0 then show x+y+z= xyz
12) cos⁻¹x/a + cos⁻¹y/b= K then show x²/a² - (2xy cos K)/ab + y²/b² = sin² K
13) 1/2 tan⁻¹x = cos⁻¹√[{1+√(1+x²)}/2√(1+x²)]
No comments:
Post a Comment