23/9/22
Trigonometrical ratios of angles
g) sin 135° cos 210° tan 240° cot 300° sec 330°. 1/√2
h) cos 24°+ cos 55° + cos 125° + cos 204° + cos 300°. 1/2
i) tan π/12 tan 5π/12 tan 7π/12 tan 11π/12. 1
j) tan 1° tan 2° tan 3°....... tan 87° tan 88° tan 89°. 1
k) sin²120+ cos²120+ tan²120+ cos180 - tan 135. 9/2
a) If A, B, C, D are the successive angles of a cyclic quadrilateral then prove
i)cosA+ cos B + cos C + cos D = 0.
ii) tan(A+ B)+ tan(C + D)= 0.
b) If cos x - sin x =√2 sin x then show cosx + sinx =√2 cosx.
c) If x= r cos k cos m, y= r cos k sin m and z= r sin k then show x² + y² + z² = r².
d) If tan⁴x + tan²x = 1, then show cos⁴x + cos²x = 1.
e) If x= a sec k cos m, y= b sec k sin m and z= c tan k, then show x²/a² + y²/b² - z²/c² = 1.
f) if tan x= (siny - cos y)/(sin y + cos y) then show sin y + cos y= ± √2 cos k.
g) If x be an angle of fourth quadrant and sec x= 5/3 then find the value of (6tan x + 5 cosx)/(5 cotx + cosecx). 1
h) If tanA + sin A= m and tanA- sin A= n, then show m² - n² = 4√(mn).
i) If 3 sinx + 4 cosx = 5, then show sin x= 3/5.
j) If tan x + sinx = m, tan x - sinx = n, then show mn= tan²x. sin²x and 4√(mn)= m² - n².
k) show 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)] - 2[sin⁶(π/2 + x) + sin⁶(5π - x)] independent of x.
l) If sinx + cosecx = 2, then show sin⁷x + cosec⁷x = 2.
j) If 1 + sin²A = 3 sinA cosA then find the value of tan A. 1, 1/2
k) A, B, C are the three angles of an acute angled triangle and cos(B+ C - A)= 0, sin(C+ A- B)= √3/2. Find the value of A, B, C. 45, 60, 75
l) If cos²x - sin²x = tan²y, then show cos²y - sin²y = tan²x.
m) Show: 4(sin⁶x + cos⁶x) - 6(sin⁴x + cos⁴x)= - 2.
n) If x sin³a + y cos³a= sina and x sina - y cosa = 0, then prove x² + y² = 1.
o) If a sinx = b cos x = (2c tanx)/(1- tan²x), prove (a² - b²)²= 4c²(a² + b²).
p) If a sinx + b cos x = c then show a cosx + b sinx = ±√(a²+ b²+c²)
q) If 4x secA = 1+ 4x² then show secA + tanA = 2x or 1/2x.
r) If cos⁴x + cos²x = 1 then prove tan⁴x + tan²x = 1.
s) If p tanx = tan px then show sin²px/sin²x = p²/{1+ (p² -1)sin²x}
t) Eliminate x: tanx - cotx = a, cosx + sinx = b.
u) If tanA= n tan B, sinA= m sin B, then show cos²A = (m² -1)/(n² -1).
8/6/22
prove:
a) {1+ cot x - sec(π/2+ x)}{1+ cot x + sec(π/2+ x)}= 2 cot x.
b) sin 420 cos 390 + cos(-300) sin(-330)= 1
c) cot π/20 . cot 3π/20 . cot 5π/20. cot 7π/20 . cot 9π/20 = 1
d) cos 24+ cos 55+ cos 125+ cos 204 + cos 270+ cos 300 = 1/2
e) (sin 150 - 5 cos 300+ 7 tan 225)/tan 135 + 3 sin 210) = -2
6/6/22
1) Show that:
a) sin 105°+ cos°105= 1/√2.
b) sin(40+x) cos(10+x) - cos(40+ x) sin(10+ x)= 1/2.
c) cos(3π/2+ x) cos(2π + x)[cot(3π/2 - x) + cot(2π+ x)]= 1.
d) {cos(π+x) cos(-x)}/{cos (π- x) cos(π/2 + x) = - cot x.
e) cos x/sin(90+ x) + sin(-x)/sin(180+ x) - tan(90+x)/cot x = 3.
Day-1
1) Find the value of 405°.
2) Find the value of cos(-2220°).
3) 4 sin(π/6) sin²(π/3)+ 3cos(π/3) tan(π/4) + cosec²(π/2).
4) Prove:
{Cos (90+x)sec (270+x) sin (180+x)}/ {Cosec(-x)cos (270-x) tan (180+x)} = cosx.
5) Prove Cosx/sin (90+x) + sin (-x)/sin(180+x) - tan(90+x)/cotx= 3
******
Complex numbers
1) If z is a complex number such that |z|=1, prove that {(z -2)/(z+1)} is purely imaginary. What will be your conclusion if z= 1?
2) If a+ ib = (c+i)/(c - i), where c is real, show that a²+ b²=1 and b/a = 2c/(c²-1).
3) Find real values of x and y for which the following equalities hold:
a) (1+ i)y²+ (6+ i)= (2+ i)x. 5,±2
b) (x⁴+2xi) - (3x²+ iy)= (3- 5i)+ (1+2iy). ±2,(3,1/3)
4) Express (1-2 i)⁻³ in the standard form a + ib. -11/125 + -2i/125
5) Find real values of x and y for which the complex numbers - 3+ ix²y and x²+ y+ 4i are conjugate of each other. ±1, -4
Quadratic equations
1) Solve:
a) 25x²- 30x +11=0. 3/5 ± √2i/5
b) x² - (7- i)x + (18- i)= 0 over C. 4 - 3i, 3+ 2i
c) x²-4x +13=0. 2 ± 3i
d) 2x²+ 3ix +2 =0. i/2, -2i
Linear Inequalities
1) If |2x -3|< |x+5|, then find the interval in which x lies. x ∈(-2/3,8)
2) The solution set of (x +3)/(x-2) < 2. x∈ (-∞,2) U[7,∞)
3) (2x +3)/5 < (4x-1)/2 then find the interval in which x lies. x∈ (11/6, ∞)
4) 7x -2 < 4 - 3x and 3x -1 < 2+ 5x, then find the interval in which x lies. x∈ (-3/2, 3/5)
5) The set of all real numbers x for which x²- |x +2|+ x > 0, is ______. x∈ (-∞, - √2) U (√2, ∞).
No comments:
Post a Comment