4/9/22
1) ∫(√x + ³√x²)²/x dx
2) (√x - 1/√x)³ dx
3) ∫dx/(3- 8x)⁵
4) ∫dx/{√(ax + b) - √(ax + c)}.
5) ∫x √(x + a) dx.
6) ∫x/√(x + a) dx
7) ∫(4x+ 3)²⁰²² dx
8) ∫(1+x+ x²)/{x²(1+ x)} dx
9) ∫(2x⁴+ 7x³+ 6x²)/(x²+ 2x) dx
10) ∫(x+ 2)/(x +1)² dx.
11) ∫(8x+ 13)/√(4x +7) dx
12) ∫ x³/(x+ 2) dx
13) ∫(x⁴+ 3)/(x² +1) dx.
14) ∫(2x+ 3)/(x -1)² dx.
15) ∫ 2x/(2x+ 1)²⁰ dx.
25/7/22
Inverse Trigo (R)
Prove
1) 2tan⁻¹1/5 + tan⁻¹1/8=tan⁻¹4/7
2) 1/2 tan⁻¹x = cos⁻¹ √[{1√(1+x²)}/2√(1+x²)]
3) sin⁻¹4/5 +sin⁻¹16/65= π/2
4) sin (π/3 - sin⁻¹(-1/2)) = 1
5) cot (π/4 - 2 cot⁻¹3) =7
6) If cos⁻¹x+ cos⁻¹y+ cos⁻¹z=π then prove x²+y²+z²+2xyz=1
7) If tan⁻¹x+tan⁻¹y+tan⁻¹z=π/2 then prove xy +yz+zx =1
8) If tan⁻¹x+ tan⁻¹y+ tan⁻¹z = 0 then prove x+y+z=xyz
9) prove tan⁻¹(1/2 tan 2A) + tan⁻¹(cot A)+ tan⁻¹(cot³A) =0
10) solve for x: tan⁻¹3x+ tan⁻¹2x =π/4
11) Solve:
tan⁻¹(cos x) =tan⁻¹(2 cosec x)
12) tan⁻¹(x+1) +tan⁻¹(x-1) =tan⁻¹8/31
13) sin⁻¹x/√(1+x²) - sin⁻¹1/√(1+x²) = sin⁻¹{(1+x)/(1+x²)}
14) Sin⁻¹(x/2)+cos⁻¹(x+√3/2) =π/6
15) If Sin⁻¹x +Sin⁻¹y+Sin⁻¹z=3π/2, then prove x²+y²+z²+2xyz=1
16) Simplify:
Cos⁻¹(x+ 1/2)+Cos⁻¹x+Cos⁻¹(x -1/2) = 3π/2.
17) Tan⁻¹{(2sin2θ)/1+2cos2θ)} -1/2 sin⁻¹{(3sin2θ)/(5+4cos2θ)}
18) tan⁻¹1 + tan⁻¹1/2 + tan⁻¹1/3
19) sin(sin⁻¹1/3 + sec⁻¹3) + cos(tan⁻¹1/2 + tan⁻¹2)
Simplify
20) sin{sin⁻¹√5/4 + tan⁻¹√(5/11)}
21) tan⁻¹sin cos⁻¹√(2/3)
22) cos⁻¹3/5+ cos⁻¹12/13 + cos⁻¹63/65 = π/2
23) 1/2 tan⁻¹x = cos⁻¹√{(1+√(1+x²))}/2√(1+x²)
24) prove tan⁻¹x + cot⁻¹(x+1) = tan⁻¹(x²+x+1)
25) tan(2 tan⁻¹m) = 2 tan(tan⁻¹m+ tan⁻¹m³)
26) tan⁻¹(1/2 tan 2A) + tan⁻¹(cotA)
+ tan⁻¹(cot³A) = 0
Solve
27) 3(Cos⁻¹x + 2sin⁻¹x) = 7π
28) tan⁻¹(cot2x) + tan⁻¹(-cot 3x) = x
29) sin⁻¹{2a/(1+a²)} - cos⁻¹{(1-b²)/(1+b²) = 2 tan⁻¹x
30) tan⁻¹x tan⁻¹2x tan⁻¹3x = π
31) 3 tan⁻¹{1/(2+√3)} - tan⁻¹( = 1/x)
= tan⁻¹1/3
32) sin⁻¹x + sin⁻¹(1-x) = cos⁻¹x
33) tan⁻¹(x+1)+tan⁻¹(x-1)=tan⁻¹8/31
34) tan⁻¹{(x-1)/(x-2)} + tan⁻¹{(x+1)/(x+2)} = π/4
35) tan⁻¹{(1/(2x+1)}+ tan⁻¹{1/(4x+1)= tan⁻¹(2/x²)
36) if α+β= tan⁻¹m, α-β=tan⁻¹n Express tan 2α and tan2β in terms of m,n.
37) If sin⁻¹x = tan⁻¹y then find the value of 1/x² - 1/y²
38) If r²= x²+y²+z², prove that tan⁻¹yz/xr + tan⁻¹zx/yr + tan⁻¹xy/zr = π/2
39) Solve:
cos(2tan⁻¹1/7) = sin(4tan⁻¹x)
40) cos(tan⁻¹x) = sin(cot⁻¹3/4).
prove :
41) cos⁻¹(63/65)+2tan⁻¹(1/5) = sin⁻¹(3/5).
42) 4 tan⁻¹1/5 - tan⁻¹1/70 + tan⁻¹1/99 =π/4
43) sin⁻¹{x/√(1+x²)} + cos⁻¹{(x+1)/√(x²+2x+2)} = tan⁻¹(x²+x+1)
44) cot(π/4 - 2 cot⁻¹3) = 7
45) sin cot⁻¹ cos(tan⁻¹x) = √{(1+x²)/(2+x²)
46) sin⁻¹√3/2 + 2 tan⁻¹1/√3 = 2π/3
47) tan⁻¹1/3 +tan⁻¹1/5 + tan⁻¹1/7
+tan⁻¹1/8 = π/4
48) tan⁻¹(1/2 tan 2A) +tan⁻¹(cot A)
+ tan⁻¹(cot³A) = 0
Solve:
49) tan⁻¹(2+x)+ tan⁻¹(2-x) =tan⁻¹2/3
50) sin⁻¹5/x + sin⁻¹12/x = π/2
51) sin⁻¹6x + sin⁻¹(6√3 x)= - π/2
52) sin⁻¹{2a/(1+a²)} + sin⁻¹{2b/(1+b²)} = 2 tan⁻¹x
53) sin{2 cos⁻¹ cot(2 tan⁻¹x)}=0
Find
54) If tan⁻¹a + tan⁻¹b + tan⁻¹c = π then Prove that a+ b + c = abc
55) If tan⁻¹a + tan⁻¹b + tan⁻¹c = π/2 prove ab + bc + ca = 2
56) If cos⁻¹a/2 + cos⁻¹b/3 = K prove 9x² - 12 xy cos K + 4y² = 36 sin²K
prove
57) tan⁻¹(1/2) + tan⁻¹(1/3) = π/4
58) 4tan⁻¹1/5 + tan⁻¹1/70 + tan⁻¹1/89 =π/4
59) tan⁻¹(1/4)+tan⁻¹2/9= 1/2 cos⁻¹3/5
60) sin⁻¹4/5 + sin⁻¹5/13 + sin⁻¹16/65 =π/2
61) 1/2 tan⁻¹x = cos⁻¹√[{(1+√(1+x²)}/2√(1+x²)]
62) tan⁻¹x + cot⁻¹(x+1) = tan⁻¹(x²+x+1)
63) tan⁻¹1+tan⁻¹2+tan⁻¹3 = 0
64) 2(tan⁻¹1+tan⁻¹1/2 +tan⁻¹1/3) =π
65) tan(2tan⁻¹a) = 2tan(tan⁻¹a+ tan⁻¹a³)
66) tan⁻¹(1/2 tan 2A)+ tan⁻¹(cot A)
+ tan⁻¹(cot³A) =0
Solve
67) tan⁻¹(x+1)+tan⁻¹(x-1) = tan⁻¹8/31
68) tan⁻¹(x-1)+tan⁻¹x+tan⁻¹(x+1)= tan⁻¹3x
69) tan⁻¹{(x-1)/(x-2)} + tan⁻¹{(x+1)/(x+2)} = π/4
70) If cos⁻¹x + cos⁻¹y + cos⁻¹z =π prove x²+y²+z²+2xyz =1
71) prove sin⁻¹(12/13) + cos⁻¹(4/5) + tan⁻¹(63/16) = π .
72) Solve cos⁻¹(sin cos⁻¹x) = π/6
73) solve) cos (sin⁻¹x)= 1/7
74) value of tan(2 tan⁻¹ 1/5)
Solve:
75) tan⁻¹{1/(2x+1)} +tan⁻¹{1/(4x+1)} = tan⁻¹2/x²
76) tan⁻¹{(x+1)/(x-2)} + tan⁻¹{(x-1)/x} = tan⁻¹(-7)
77) If tan⁻¹x +tan⁻¹y+tan⁻¹z=π prove x+y+z=xyz
78) if tan⁻¹x+tan⁻¹y+tan⁻¹z=π/2 show that xy+yz+zx=1
79) prove tan(π/4 + 1/2cos-¹ 1/3) + Tan(π/4 - 1/2 cos-¹ 1/3) =6
80) solve cos⁻¹(sin (cos⁻¹x))=π/3.
81) Prove
Sin(2tan⁻¹3/5-sin⁻¹7/25)=304/425
82) If tan⁻¹x + tan⁻¹y + tan⁻¹z=0, prove that x+ y +z = xyz.
83) tan⁻¹1/4 + tan⁻¹2/9 = 1/2 cos⁻¹3/5
19/7/22
-------:::
1) If A= 2 -1
-1 2 and I is the unit matrix of order 2, then A² is
A) 4A - 3I. B) 4A - 3I C) A - I D) A+I
2) The multiplicative inverse matrix of 2 1
7 4
A) 4 -1 B) 4 -1 C) 4 -7 D) -4 -1
-7 -2 -7 2. 7 2 7 -2
3) A is a square matrix such that A³ = I; then inverse of A is
A) A². B) A C) A³ D) none
4) Assuming that the sums and products given below are defined which of the following is not true matrices ?
A) AB= AC does not imply B= C
B) A + B = B+ A C) (AB)'= B'A'
D) AB= O implies A= O or B= O.
5) If A=1 0 2 and Adj A=5 a -2
-1 1 -2 1 1 0
0 2 1 -2 -2. b then the value of a and b are
A) -4,1 B) -4,-1 C) 4,1. D) 4,-1
6) If A= -1 0
0 2 then value of A³- A²
A) I B) A C) 2A. D) 2J
7) If A= - x -y
z t then the transpose of adj A is
A) t z B) t y C) t -z D) none
-y -x. -z -x y - x
8) If A=3 5 & B= 1 17
2 0 0 -10 then |AB|=
A) 80 B) 100 C) -110 D) 92
9) A= 5 -2
3 1 find inverse of A
A) -2/13 5/13 B) 1 2
1/13 3/13 -3 5
C) 1/11 2/11 D) 1 3
-3/11 5/11. -2 5
10) If A is a singular matrix of order n then A.(adj A) is
A) a null matrix.
B) a row matrix
C) a column matrix D) none
11) If A and B are two matrices and if A⁻¹ and B⁻¹ exist, then (AB)⁻¹ is
A) A ⁻¹B⁻¹ B) AB⁻¹ C) A⁻¹B D) B⁻¹A⁻¹.
12) If A= 3 -5
-4 2 then the value of A² - 5A is
A) I B) 14I. C) O d) none
13) If A= 5 6 -3
-4 3 2
-4 -7 3 then the cofactors of the elements of second row are
A) 3,3,11. B) 1-3,11 C) -39,3,-11 D) 39, -3,11
14) If A= 1 2 & B= 1 2
2 3 2 1
3 4
Then
A) both AB and BA exist
B) neither AB nor BA exist
C) AB exist but BA dies not exist.
D) AB does not exist but BA exist
15) If A= 2 -1 & B= 1 0
0 1 -1 -1 then (A+ B)² is not equal to
A) A²+ AB+ BA + B²
B) A²+ AB+ BA + B²I
C) A²I + AB+ BA + B²
D) A²+ 2AB + B².
16) If A be an n x n matrix and k any scalar, then det kA is equal
A) k det A B) nᵏdet A C) k ⁿdet A D) kn det A
17) If A= 1 2
3 -5 find inverse of A
A) -5 -2 B) -5/11 -2/11
-3 1 C) -3/11 1/11
C) 5/11 2/11 D) 5 2
3/11 -1/11 3 -1
18) If A= -1 2 & B= 5
2 -1 7 and AX= B, then X is equal to
A) 19 17 B)19/3. C) 19/3 17/3 D) 19
17/3 17
19) If A= 0 1 2
1 2 3
3 1 1 and it's inverse B=[bᵢⱼ], then the element b₂₃ of matrix B is
A) -1. B) 1 C) -2 D) 2
20) If A= a b c &B= 1 2 3
d e f 2 3 4
g h i 3 4 5
C= -1 -2 & D= -4 -5 -6
-2 0 0 0 1
0 -4
And the relation A= BCD, then value of e
A) 40. B) -40 C) -20 D) 20
21) A= 1 2 & B= 3 8
3 4 7 2 and the relation 2X+ A = B then metrix B is
A) 2 6 B) 1 -3 C) 1 3 D) 2 -6
4 -2 2 -1 2 -1. 4 -2
22) If A= a 2
2 a and|A³|= 125, then the value of a is
A) ±2 B) ±3. C) ±5 D) 0
-----------------------------------------------
17/7/22
-----------
2- marks
1) y= sin(x²+1) find dy/dx
2) y= tan⁻¹x find d²y/dx²
3) y= sin 3x cos 5x find dy/dx
5- marks (any three)
4) If 3 sin⁻¹{2x/(1+ x²)} - 4 cos⁻¹{(1- x²)/(1+ x²)+ 2 tan⁻¹{2x/(1- x²)=π/3 then find x.
5) If xᵐ yⁿ = (x+ y)ᵐ⁺ⁿ, show dy/dx= y/x
6) If y= x log{x/(a+ bx)} then show that x³ d²y/dx² = (x dy/dx - y)²
7) Find dy/dx: y= xˢᶦⁿ ˣ + (sin x)ᶜᵒˢ ˣ
6 marks . (Any two)
8) If x= a[cos t + log|tan (t/2)|] and y= a sin t then find dy/dx at t =π/4
9) If yˣ + xʸ + xˣ = aᵇ, find dy/dx
10) If (x - a)² + (y - b)²= c², Prove that [1+ (dy/dx)²]³⁾²/d²y/dx² is a constant independent of a and b.
5/7/22
Question 1. 6x2=12
i) If A= 3 1
7 5 find x and y do that A² + xI= yA.
ii) Evaluate: tan⁻¹a/b - tan⁻¹{a -b)/(a+b)}
iii) Prove that the relation R on Z defined by (a,b) ∈ R <=> a-b is divisible by 5 is an equivalence relation on Z.
iv) Differentiate sin(x². eˣ) w.r.t.x
v) find the derivative of the function sec⁻¹{1/(2x²-1)} w.r.t.x
vi) Find the interval in which the function f(x)=10 - 6x - 2x² are increasing or decreasing.
Question 2) Solve the following set of equation using cramer's rule: 2x-z=1; 2x + 4y -z=1; x - 8y -3 z= -2;. (5)
Question 3) Find dy/dx if (eˣ+1)y = (eˣ-1). (5)
Question 4) given two matrices A and B as
A= 1 2 1 & B= 1 4 0
1 -1 1 -1 2 2
2 3 -1 0 0 2 . Find the value of the expression AB - 2B. (5)
Question 5) From the following data, find the best fit regression equation.
X. Y
56 147
42 125
72 160
36 118
63 149
47 128
55 150
49 145
38 115. (5)
15/5/22
21) y= log(tan x/2)
22) x² w r.t. log x
23) ₘ(ax²+bx+c)
24) log(ax+ b)³
25) {f(x)}ⁿ
26) ₑf(x)
27) log{√f(x)}
28) ₃f(x)
29) sin {f(x)}
30) sec {f(x)}
11/5/22
11) y= logₓ2
12) x² + y² = a²
13) x= a cos t, y= a sin t
14) cos(logx)
15) log(sinx)
16) cot x°
17) ₁₀10ˣ
18) xˣ
19) If f(x)= √(x²+1), then f'(-1) is
20) y= 1/(3x -1) at x= 0
10/5/22
1) y= eˢᶦⁿ ˣ
2) y= sin³x.
3) y= ₂3x²
4) If y= log(x² -5) and dy/dx = ky/(x² -5) then find k
5) if y= log₁₀x= m/x then m is..
6) If d/dx(2x³ +5)¹⁰= 60x² f(x) then f(x) is .
7) if d/dx √(2x² +9)= f(x)/√(2x²+9) then f(x) is
8) y= √(x² + a²)
9) y= log f(x)
10) y= 10ᵐˣ
22/4/22
1) If A= 2 -1
-1 2 and I is the unit matrix of order 2, then A² is
A) 4A - 3I B) 3A - 4I
C) A - I D) A + I
2) The multiplicative inverse of matrix 2 1
7 4
3) if A= 1 0 2 5 a -2
-1 1 -2 & adj of A=1 1 0
0 2 1 -2 -2 b then find the values of a and b.
4) If A = -1 0
0 2 then the value of A³ - A² is
5) If A= - x - y
z t then find the transpose of adj A.
20/4/22
1) Find Adjoint of 2 3
-4 -6
2) Find inverse of 2 -2
4 3
3) If A= 3 1
-1 2 then show A² - 5A + 7I = 0, hence find inverse of A.
19/4/22
1) Find Adjoint of 2 3
-4 -6
2) Find inverse of 2 -2
4 3
3) If A= 3 1
-1 2 then show A² - 5A + 7I = 0, hence find inverse of A.
4) If 1 1 1
A= 1 2 -3
2 -1 3 then show that A³ - 6A² + 5A + 11I= 0, hence find inverse of A.
18/4/22
1) Construct a 2x2 matrix whose elements are (2i + j)²/2.
2) x+3 z+4 2y-7 0 6 3y -2
4x+6 a-1 0 =2x -3 2c -2
b-3 3b z+ 2c 2b+4 -21 0 Obtain the values of a, b, c, x, y , z
3) Let A= 2 3
-1 2 and
f(x)= x² - 4x + 7. Show that f(A)= O. Use this result to find A⁵.
4) Show that AB= BA where
A= 5 -1 and B= 2 1
6 7 3 4
1) If A= 1 2
3 4 then A⁴ - 5A³ - A² - 4A - 2I = ?. (I is unit metrix).
A) 0 B) A C) I D) 2I
2) Two matrices A and B are said to be conformable for multiplication in the same order iff the number of rows of A is equal to number of columns of B. T/F
3) If A= 2 0 0 B= x & C= 4
0 3 0 y -1
0 0 4 z 8 then find the value of 3xyz.
4) Find the inverse of -1 4
7 -20
5) If A= a 2
2 a and | A³|= 125, then find a.
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