(i) The next pattern is 
(ii)

(iii)

First number is 2
2^nd number is 2 + 3 = 5
3^rd number is 5 + 4 = 9
4^th number is 9 + 5 = 14
5^th number is 14 + 6 = 20
(iv) Number of dots in the 25^th pattern is

\(\therefore \) Number of dots in the 25^th pattern = 350.

(i) Total chairs in a stadium = 7,689
Chairs in one row = 90
\(\frac{7689}{90}=85\ \frac{39}{90}\)
There will be 85 rows
(ii) Yes, There are 39 chairs left over.

(i) February
(ii) April
(iii) March
(iv) 1250
(v) Brand A has sold 200 mobiles more than brand B.

Run rate = Ratio of runs to overs
Run rate of Karmegam = \(\frac { 54 }{ 9 } \) = \(\frac { 6 }{ 1 } \) = 6 :1
Run rate of Asif = \(\frac { 77 }{ 11 } \) = \(\frac { 7 }{ 1 } \) = 7 :1
∴ Asif's run rate is better.

i) \(\angle L M N, \angle N M L, \angle M\)
ii) \(\angle P Q R, \angle R Q P, \angle Q\)
iii) \(\angle \mathrm{MNO}, \angle \mathrm{ONM}, \angle \mathrm{N}\)
iv) \(\angle \mathrm{TAS}, \angle \mathrm{SAT}, \angle \mathrm{A} \)

Star = 8, Face = 7, Triangle = 10 

(i) all pairs of parallel lines.
l₃ and l₄ ,l₄ and l₅ ,l₃ and l₅
(ii) all pairs of intersecting lines.
l₁ and l₂, l₁ and l₃ , l₁ and l₄ , l₁ and l₅ , l₂ and l₃ , l₂ and l₄, l₂ and l₅
(iii) pair of lines whose point of intersection is ‘V’.
l₁ and l₂
(iv) point of intersection of the lines ‘ l₂’ and ‘l₃’. l₁ and l2
Q
(v) point of intersection of the lines ‘l₁’ and ‘l₅’.
U

According to the given statement,

This can be written as, x – 6 gives 10, where ‘x’ denotes the unknown number. Now, we will find out for what value of ‘x’, x − 6 gives 10.

Hence, the unknown number (variable) ‘x’ takes the value 16.

A, B, C, D are four triangles.
Combining A & D, B & D and D & C do not form any triangles.
Combining A, B & D; A, C & D and B, D & C also do not form any triangles.
Combining A, B, C & D = 1 triangle
\(\therefore\)Total number of triangles = 4 +1 = 5

Teacher to Student ratio of 6^th Std = 6 : 12 = \(\frac{6}{12}=\frac{1}{2}\) = 1 : 2
Teacher to Student ratio of 7^th Std = 9 : 27 = \(\frac{9}{27}=\frac{1}{3}\) = 1 : 3
Teacher to Student ratio of 8^th Std = 4 : 16 = \(\frac{4}{16}=\frac{1}{4}\) = 1 : 4
In standard 8th the ratio of teacher to Student is 1 : 4 and which is the least.

Here the extremes are 3 and 20 and the means are 2 and 30.
Product of extremes, ad = 3 × 20 = 60.
Product of means, bc = 2 × 30 = 60.
Thus by proportionality law, we find ad = bc and hence 3 : 2 and 30 : 20 are in proportion.

\(1:2=\frac{1}{2}\)
\(2:1=\frac{2}{1}\times\frac{2}{2}=\frac{4}{2}\)
Here \(\frac{4}{2}\) larger than \(\frac{1}{2}\)
\(\therefore\) 2 : 1 larger than 1: 2

Step (i): Preparing place value chart with 8 digits 6 in ten lakh place and 9 in Ten thousand place
Step (ii): Fill the other places with any of the numbers

\(\therefore\) The number may be 56897432. Similarly we can write many numbers

11 added to 10 times x

(i) Obtuse angle
(ii) Zero angle
(iii) Straight angle
(iv) Acute angle
(v) Right angle

(i) 3, 9, 4, 12 .
Here product of extremes = 3 \(\times\) 12 = 36
Product of means = 9 \(\times\) 4 = 36
∴ 3 : 9 : : 4: 12
(ii) Also if we take 9, 3, 12, 4
Product of extremes = 9 \(\times\) 4 = 36
Product of means = 3 \(\times\) 12 = 36
∴ 9 : 3 : : 12: 4.

Place value chart is given by

\(\Rightarrow\) The number with more digits is the greater number
Step 1: \(\therefore\) 128435 is the larger number and 6348 is the least number
Step 2:  For the remaining 5 digit numbers we can compare the left most digits and find 25840 > 21354 > 10835
\(\therefore\)  The descending order:
128435> 25840 > 21354 > 10835> 6348