JEE 2023 — Mathematics PYQ

Step 1: Define the constants

Let $g^{\prime }(1)=A$ and $g^{\prime \prime }(2)=B$.

Then the function $f(x)$ becomes:

From this, we can find the derivatives of $f(x)$:

• $f^{\prime }(x)=2x+A$

• $f(1)=1+A+B$

Step 2: Substitute into $g(x)$

Now substitute these into the given expression for $g(x)$:

Step 3: Find the derivatives of $g(x)$

To find the values of $A$ and $B$, we differentiate $g(x)$:

• $g^{\prime }(x)=3x^2+2(1+2A+B)x+(B+2)$

• $g^{\prime \prime }(x)=6x+2(1+2A+B)$

Step 4: Solve for $A$ and $B$

Using the definitions $g^{\prime }(1)=A$ and $g^{\prime \prime }(2)=B$:

1. From $g^{\prime }(1)=A$:

   $$3(1{)}^2+2(1+2A+B)(1)+(B+2)=A$$
   $$3+2+4A+2B+B+2=A$$
   $$3A+3B=-7\dots \text{(Eq. 1)}$$

2. From $g^{\prime \prime }(2)=B$:

   $$6(2)+2(1+2A+B)=B$$
   $$12+2+4A+2B=B$$
   $$4A+B=-14⟹B=-14-4A\dots \text{(Eq. 2)}$$

Substitute (Eq. 2) into (Eq. 1):

Step 5: Calculate $f(4)-g(4)$

Instead of calculating $f(4)$ and $g(4)$ separately with fractions, let's find the expression for $f(x)-g(x)$:

At $x=4$:

Substitute $A=-\frac{35}{9}$ and $B=\frac{14}{9}$: