The number of positive integral solutions of  where n is positive integer, is

f𝐨𝐮𝐫 𝐛𝐨𝐲𝐬 𝐩𝐢𝐜𝐤𝐞𝐝 𝐮𝐩 𝟑𝟎 𝐦𝐚𝐧𝐠𝐨𝐞𝐬. 𝐈𝐧 𝐡𝐨𝐰 𝐦𝐚𝐧𝐲 𝐰𝐚𝐲𝐬 𝐜𝐚𝐧 𝐭𝐡𝐞𝐲 𝐝𝐢𝐯𝐢𝐝𝐞 𝐭𝐡𝐞𝐦, 𝐢𝐟 𝐚𝐥𝐥 𝐦𝐚𝐧𝐠𝐨𝐞𝐬 𝐛𝐞 𝐢𝐝𝐞𝐧𝐭𝐢𝐜𝐚𝐥

A non zero vector is parallel to the line of intersection of the plane determined by the vectors and the plane determined by the vectors then the angle between and is

For such questions, we should know how to find the normal to the plane. We should know how to find the line of intersection of two planes. And, we should know how to take a cross product.

Let and . Then, the line of intersection of planes one determined by and other determined by is perpendicular to

For such questions, we should know how to find the line intersecting two planes. We should also know how to find the normal to the plane using two vectors.

𝐀 𝐠𝐮𝐚𝐫𝐝 𝐨𝐟 𝟏𝟐 𝐦𝐞𝐧 𝐢𝐬 𝐟𝐨𝐫𝐦𝐞𝐝 𝐟𝐫𝐨𝐦 𝐧 𝐬𝐨𝐥𝐝𝐢𝐞𝐫𝐬.𝐈𝐟 𝐀 𝐚𝐧𝐝 𝐁 𝐚𝐫𝐞 𝐭𝐡𝐫𝐞𝐞 𝐭𝐢𝐦𝐞𝐬 𝐚𝐬 𝐨𝐟𝐭𝐞𝐧 𝐭𝐨𝐠𝐞𝐭𝐡𝐞𝐫 𝐨𝐧 𝐠𝐮𝐚𝐫𝐝 𝐚𝐬 𝐂, 𝐃, 𝐄 𝐭𝐡𝐞𝐧 𝐧 =

Number of ways of selecting 3 consecutive objects out of 15 objects (distinct) placed around a circle is

If , then 

The number of ways in which 6 gentlemen and 3 ladies be seated round a table so that every gentleman may have a lady by his side is

If 
then the values of a and b are

A circle having radius 7 touches both the axes and lies in 1^st quadrant. If the circle is rolled along the positive x-axis and completes on rotation. Then equation of circle in final position is

If 

For such questions, we should know different formulas.

Let α and β be the distinct roots of