M(x_K + x_L) ; (y_K + y_L)
         2              2
M(1 +(-3)  ; 7 + (-1) 
         2            2 
M(-1 ; 3)

OR
x_M =x₁ + x₂ , y_M y₁ + y₂ 
            2                 2
x_M =1 + (3)  ,7 + (-1) 
          2            2 
M(-1;3)

y = ³/₂x + ¹⁷/₂
LHS = - 2
RHS= ³/₂ (-4) + ¹⁷/₂ = ⁵/2
LHS ≠ RHS
the point does NOT lie on the line

OR
m_KL = 2
n_New = y1 - y2 = -2 - 1 = - 3    
            x1 - x2    -4 + 5
m_New ≠ m_KL
the point does NOT lie on the line

OR
y + 2 = 2( x + 4)
y = 2x + 6
(- 4 ; - 2)
does NOT lie ony = 2x +11

sin ² B                        A
cosec B                      
A secB                          
A - cosB                        
A                          
AsinB
I                  
AcosB
S                               
CA
OR        
Asin (p - B)     
Acos( 2p - B)
sin ² B                       
A sin B                           
A cos B                          
A - cosB                         
A  
                               
CA (7)

x ∈ (90^ο ; 135°]
x =180° 
OR
90° < x ≤ 135° CA
x =180°  CA(2)
[12]

O3 = 44°
∠at centre = 2 x ∠at circum
O₃ ≠ R₁
OT is not parallel to SR (alt. ∠s are not equal)

OR
O3 = 44°
∠at centre = 2 x ∠at circum
O2 = 68° 
∠at centre = 2 x ∠at circum
SOT + OSR = 44°+ 68°+ 56°=168°¹180°
OT is not parallel to SR (co-int ∠s ≠180°)

OR
S₂ = T₂ = 34° [∠s opp. = sides]∠ in the semi-circle/
S3 = 90° - 68° = 22
T_2≠ S₃
OT is not parallel to SR  (alt. ∠s are not equal)

ST
RE A 

OR
ST    A
ST  A
RE  A 

OR
ST  A
ST  A
RE  A  (3)

M = 98° ≠ 90°
LN is not a diameter (∠ subtended by LN ≠ 90°)

OR
P2 + 98°=180° (Opp. ∠s of cyclic quad.)
P2 =82°¹ 90°
LN is not a diameter (∠ subtended by LN ≠ 90°)

ST M = 98° ≠ 90° A
RE              A 

OR
ST P2 =82° ≠ 90° A 
RE      A(2)

P1 +82°=180°  ( ∠s on straight line)
P1 =98° 

OR
P1 =98°   (Ext. ∠ of a cyclic quad.)

ST / RE    A
P1 = 98°    CA

OR
ST / RE        A
P1 = 98°    CA(2)

K is common
L1 = N1   (both = 27° / tan- chord)
P1 = KLN ( 3rd ∠ of D )
ΔKLP|||ΔKNL (∠∠∠)

OR Equiangular

K + 27°+ 98°=180°  (∠s of/van D)
K = 55°
K + M = 55°+ 98°¹ 180°
KLMN is not a cyclic quad.

OR
K + L1 + = 86° ext∠ = sum of opp.in t ∠s 
K = 55°
K + M = 55°+ 98°¹ 180°
KLMN is not a cyclic quad.

ST/RE        CA
value of  K   A 
Conclusion    A 

OR
ST/RE          CA
value of  K    A 
Conclusion

CF = 1 =1  (BF = FC; F is the midpoint of BC)
FB    1
CE = 2 = 2
EA    1
CF ≠ CE
FB    EA
EF is NOT parallel to AB (sides are not prop)

OR
BF = FC; F is the midpoint of BC
AE≠ EC; ; E is NOT the midpoint of AC
EF is NOT parallel to AB(FE not joining midpoints of two sides of a triangle)

ST             A 
ST            A 
Conclusion      CA

OR
F is the midpoint of BC          A
E is NOT the midpoint of  AC       A
Conclusion   CA(3)

h = 2, 95 m and d =13, 9 m
4h² - 4dh + x² = 0
4(2, 95)² - 4(13, 9) (2, 95) + x² = 0
-129, 21+ x² = 0
x² = 129, 21
x = 11, 36ED ≈ 11, 37 

OR
Using the half chord of PQ 
½ ED = √(6,95)²  - (4)²
½ED = √32,3025
½ED = 5,68
ED ≈ 11,37

formula            A 
SF         CA
S           CA  
length    CA 

OR
Pythagoras       A
SF         CA
S               CA  
length   CA 
(4)
NPR

angle of sector, FOG = 20% x 2π
= ²/₅π = 1, 26 rad

OR
angle of sector, FOG = 360°x ²⁰/₁₀₀ = 72°
72° = 72° x ^π/_180°
= ²/₅π

OR 1, 26 rad

OR
Circmf. = 2π
= 2π(6, 95)
= 43, 67m
20% x 43, 67m = 8,73
s = rθ
8, 73 = 6, 95θ
θ = 1, 26 rad

A
radian         CA

OR
angle size   A
M                A
radian   A

OR
Circumf.   A
M           CA
radian  CA
NPR       NPU         (3)

A = r ²θ 
        2
= (6,95)²(1,26)
             2
≈30,43 m ²

OR
A = rs 
      2
= (6,95)(8,73)
          2
= 30,43 m ²

Formula  A
SF              A
area    CA
NPU

OR
Formula    A
SF            A
area   CA
NPUNPR 
 (3)

n =   18   
      3600
n (in rev/sec) = 0, 005 rev / sec 
n(in rad/sec) = 0, 005 rev / sec x 2π
= 0, 01p rad / sec or
0, 03141..rad 

D = 2 x 10 m = 20 m
v = π D n                           

OR
v = 2πr n
=π x 20 x (  18  )   = 2π x 10 x (  18  )
                 3600                         3600 
= 0,1p m/s   

OR
≈ 0, 31 m/s

w = 2πn
= 2π (  18  )
          3600 
= 0, 01π rad

OR
≈3,14 x 10 ^-2 rad

Area = length´breadth
187,5 =length x 7,5

OR
length = ^A/_b = ^187,5/_7.5
length = 25m

A_T = a(o₁ + o_n+ o₂ + o₃ + ....+ o_n-1)
                2
= 1, 5 (15 + 7 + 12 +18 +15 +11) m²
                2
=1, 5 (11+ 12 +18 +15 +11) m²
=100, 50 m²
Damaged area =187, 5 -100, 50 = 87m²
It will take 87 x 0, 25 hours = 21, 75 hours to repair the damaged area

OR
A_T = a(m₁ + m₂ + m₃ +....+ m_n)
=1.5(15 + 12 + 12 + 18 + 18 + 115 + 15 + 11 + 11 + 7)m²
              2              2              2               2            2
= 1, 5 (13, 5 + 15 +16, 5 +13 + 9) m²
=100, 50 m²
Damaged area = 187, 5 -100, 50 = 87m²
It will take 87 x 0, 25 hours = 21, 75 hours to repair the damaged area

OR
A_T = a(o₁ + o_n+ o₂ + o₃ + ....+ o_n-1)
                2
= 1, 5 (10 + 18 + 13 +7 +10 +14) m²
                2
= 1.5(14 + 13 + 7 + 10 + 14)m²
= 87m ² 
It will take 87 x 0, 25 hours = 21, 75 hours to repair the damaged area

formula  A
value of a   A
SF    CA 
value of A_TCA
87m²   CA 
time  CA

OR
F         A
value of a  
A
S     CA
value of A_T  CA
87m²    CA
Time    CA 

OR
F         A
value of a  
NEW ordinates 
S     CA
value of A_T  CA
87m²    CA
Time    CA