NIMCET 2007 — Mathematics PYQ

Step 1: Analyze the Given Relationships

We are given two system relationships linking functions $f(x)$ and $g(x)$:

1. $f^{\prime }(x)=g(x)$

2. $g^{\prime }(x)=-f(x)$

We need to evaluate the expression:

$$E=f^{\prime \prime }(19)+g^{\prime \prime }(19)$$

Step 2: Find the Second Derivatives

Let's find $f^{\prime \prime }(x)$ by differentiating both sides of equation (1):

$$f^{\prime \prime }(x)=\frac{d}{dx}[g(x)]=g^{\prime }(x)$$

Substituting the value of $g^{\prime }(x)$ from equation (2) into this:

$$f^{\prime \prime }(x)=-f(x)$$

Now, let's find $g^{\prime \prime }(x)$ by differentiating both sides of equation (2):

$$g^{\prime \prime }(x)=\frac{d}{dx}[-f(x)]=-f^{\prime }(x)$$

Substituting the value of $f^{\prime }(x)$ from equation (1) into this:

$$g^{\prime \prime }(x)=-g(x)$$

Step 3: Simplify the Target Expression

Now substitute our expressions for $f^{\prime \prime }(x)$ and $g^{\prime \prime }(x)$ into our target formula:

$$f^{\prime \prime }(x)+g^{\prime \prime }(x)=-f(x)+(-g(x))$$

$$f^{\prime \prime }(x)+g^{\prime \prime }(x)=-[f(x)+g(x)]$$

For $x=19$, this gives:

$$f^{\prime \prime }(19)+g^{\prime \prime }(19)=-[f(19)+g(19)]$$

Step 4: Formulate a Constant Value Function

To determine the behavior of $f(x)$ and $g(x)$, let's consider a test function $H(x)=[f(x){]}^2+[g(x){]}^2$. Differentiating it with respect to $x$:

$$H^{\prime }(x)=2f(x)f^{\prime }(x)+2g(x)g^{\prime }(x)$$

$$H^{\prime }(x)=2f(x)(g(x))+2g(x)(-f(x))=2f(x)g(x)-2f(x)g(x)=0$$

Since the derivative is $0$, $[f(x){]}^2+[g(x){]}^2$ is a constant. This indicates standard sinusoidal behavior (like $\sin x$ and $\cos x$). Let's solve the differential equations to find the exact functional behavior.

Given:

$$f^{\prime \prime }(x)+f(x)=0$$

The general solution for this differential equation is:

$$f(x)=C_1\cos x+C_2\sin x$$

Since $f^{\prime }(x)=g(x)$, differentiating $f(x)$ gives:

$$g(x)=-C_1\sin x+C_2\cos x$$

Now let's compute the value of $f(x)+g(x)$:

$$f(x)+g(x)=(C_1+C_2)\cos x+(C_2-C_1)\sin x$$

This sum varies periodically depending on $x$, meaning $f(19)+g(19)$ depends on the specific angle values of $19$ radians and is not universally constrained to a simple integer independent of $x$ unless the sum happens to evaluate identically to zero, which we can check:

Let's find the constants using the given condition at $x=3$:

• $f(3)=5$

• $g(3)=f^{\prime }(3)=5$

This means at $x=3$:

$$f(3)+g(3)=5+5=10$$

Since $f(x)+g(x)=\sqrt{2}\cdot A\sin (x+\varphi )$, its value changes with $x$ and will not stay a constant $10$ or look like any fixed integer at an arbitrary point like $x=19$.

Therefore, $-[f(19)+g(19)]$ does not match any of the constant choices $16$, $32$, or $64$.

Conclusion

The expression evaluations do not yield a fixed matching constant integer from choices (a), (b), or (c).

Correct Option: (d) none