CBSE Class 12 Mathematics 2025: question 35

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Step-by-step solution

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1. Let $x,y,z$ be the number of students in the Sports, Music and Drama clubs respectively. The conditions give:
   $x=y+z,y=\frac{x}{2}+20,x+y+z=180$
   That is,
   $x-y-z=0,x-2y=-40,x+y+z=180$

   1½ marks

2. Write the system as $AX=B$, where
   $A=\left[\begin{array}{ccc}1& -1& -1\\ 1& -2& 0\\ 1& 1& 1\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}0\\ -40\\ 180\end{array}\right]$

   ½ mark

3. $|A|=1(-2-0)+1(1-0)-1(1+2)=-4\ne 0$, so $A^{-1}$ exists.

   ½ mark

4. The cofactors are:

   • $C_{11}=-2,C_{12}=-1,C_{13}=3$
   • $C_{21}=0,C_{22}=2,C_{23}=-2$
   • $C_{31}=-2,C_{32}=-1,C_{33}=-1$

   Therefore
   $\text{adj}A=\left[\begin{array}{ccc}-2& 0& -2\\ -1& 2& -1\\ 3& -2& -1\end{array}\right]$

   1 mark

5. $A^{-1}=\frac{1}{|A|}\text{adj}A=\frac{1}{4}\left[\begin{array}{ccc}2& 0& 2\\ 1& -2& 1\\ -3& 2& 1\end{array}\right]$

   ½ mark

6. $X=A^{-1}B=\frac{1}{4}\left[\begin{array}{ccc}2& 0& 2\\ 1& -2& 1\\ -3& 2& 1\end{array}\right]\left[\begin{array}{c}0\\ -40\\ 180\end{array}\right]=\frac{1}{4}\left[\begin{array}{c}360\\ 260\\ 100\end{array}\right]=\left[\begin{array}{c}90\\ 65\\ 25\end{array}\right]$

   So $x=90,y=65,z=25$. The Sports, Music and Drama clubs get 90, 65 and 25 students respectively.

   1 mark

Answer: Sports = 90, Music = 65, Drama = 25

Revise the chapter

• Chapter: Matrices (notes, formulas, where marks are lost and every past question)