Q.By using the properties of definite integrals, evaluate the integral $\int_{-\pi/2}^{\pi/2}\sin^2 x\,dx$

Concept. A function $f:\mathbb{R}\to\mathbb{R}$ is onto iff its range is all of $\mathbb{R}$. For a piecewise function, take the union of the ranges of the branches over their own domains.

Step 1 — range of the linear branch. For $x \le \frac43$: $f(x) = 2x + 3$ is increasing, so its values run over $\left(-\infty,\; 2\cdot\tfrac43 + 3\right] = \left(-\infty, \tfrac{17}{3}\right]$.

Step 2 — range of the quadratic branch. For $x > \frac43$: $f(x) = -3x^2 + 8x$ is a downward parabola with vertex at $x = \dfrac{8}{2\cdot3} = \dfrac43$. So on $x > \frac43$ it is strictly decreasing from the (unattained) value $-3\cdot\tfrac{16}{9} + \tfrac{32}{3} = \tfrac{16}{3}$ down to $-\infty$: range $= \left(-\infty, \tfrac{16}{3}\right)$.

Step 3 — total range and conclusions. …

The key insight is that every power of $x$ in the polynomial is even. That means the equation depends only on $x^2$, not on $x$ itself. So we can treat it as a polynomial in $x^2$, which immediately reduces the degree and makes the problem tractable.

1. Substitute $t = x^2$. Since $x^{2k} = (x^2)^k = t^k$, the equation becomes

This is a quintic in $t$. Every real $t \ge 0$ gives $x = \pm \sqrt{t}$ (two real $x$ if $t > 0$, one if $t = 0$). Negative $t$ gives purely imaginary $x$, i.e., non-real roots. So the number of real $x$ is twice the number of positive real roots of this quintic (plus possibly one from $t=0$).

2. Check for obvious roots of the quintic. Try $t = 1$:

So $t = 1$ is a root. Factor out $(t-1)$ using synthetic division or polynomial division.

3. Factor the quintic.

   Divide $t^5 - 3t^4 + 5t^3 - 5t^2 + 3t - 1$ by $(t-1)$:

   • Coefficients: $1, -3, 5, -5, 3, -1$
   • Bring down $1$, multiply by $1$ → $1$, add to $-3$ → $-2$
   • Multiply $-2$ by $1$ → $-2$, add to $5$ → $3$
   • Multiply $3$ by $1$ → $3$, add to $-5$ → $-2$
   • Multiply $-2$ by $1$ → $-2$, add to $3$ → $1$
   • Multiply $1$ by $1$ → $1$, add to $-1$ → $0$ So the quotient is $t^4 - 2t^3 + 3t^2 - 2t + 1$.

4. Factor the quartic.

   Notice $t^4 - 2t^3 + 3t^2 - 2t + 1$ looks symmetric. Write it as

Expand $(t^2 + at + 1)^2 = t^4 + 2a t^3 + (a^2 + 2) t^2 + 2a t + 1$.

Compare: $2a = -2 \Rightarrow a = -1$, then $a^2 + 2 = 1 + 2 = 3$ matches, and $2a = -2$ matches. Perfect.

So $t^4 - 2t^3 + 3t^2 - 2t + 1 = (t^2 - t + 1)^2$.

5. Solve for $t$. The quintic factors as

So $t = 1$ or $t^2 - t + 1 = 0$.

The quadratic $t^2 - t + 1 = 0$ has discriminant $\Delta = 1 - 4 = -3 < 0$, so its roots are complex conjugates:

These are non-real (and not positive real).

6. Back to $x$. …

We are given four functions (List-I) and five characteristics (List-II). The key is to examine each function’s fundamental properties: its domain (where it is defined), its range (the set of output values), whether it is periodic, and whether it is even or odd. Let’s define the lists clearly (the problem statement omits them, but typical JEE/competitive exam problems use these):

List-I (Functions):

• A: $f(x) = \sin^{-1} x$
• B: $f(x) = \cos^{-1} x$
• C: $f(x) = \tan^{-1} x$
• D: $f(x) = \cot^{-1} x$

List-II (Characteristics):

• I: Domain is $[-1,1]$ and range is $[0,\pi]$
• II: Domain is $\mathbb{R}$ and range is $(0,\pi)$
• III: Domain is $[-1,1]$ and range is $[-\pi/2, \pi/2]$
• IV: Domain is $\mathbb{R}$ and range is $(-\pi/2, \pi/2)$
• V: Domain is $\mathbb{R}$ and range is $[0,\pi]$

Now, let’s match step by step.

1. Function A: $f(x) = \sin^{-1} x$

   • The inverse sine function is defined only for inputs between $-1$ and $1$, so its domain is $[-1,1]$.
   • Its principal value range is $[-\pi/2, \pi/2]$ (angles whose sine is $x$).
   • This matches characteristic III (Domain $[-1,1]$, range $[-\pi/2, \pi/2]$).
   • So A → III.

2. Function B: $f(x) = \cos^{-1} x$

   • Domain is also $[-1,1]$ because cosine outputs only between $-1$ and $1$.
   • Its principal value range is $[0,\pi]$ (angles whose cosine is $x$).
   • This matches characteristic I (Domain $[-1,1]$, range $[0,\pi]$).
   • So B → I.

3. Function C: $f(x) = \tan^{-1} x$

   • Tangent can take any real number, so the domain of its inverse is all real numbers $\mathbb{R}$.
   • Its principal value range is $(-\pi/2, \pi/2)$ (open interval, since $\pm \pi/2$ are asymptotes).
   • This matches characteristic IV (Domain $\mathbb{R}$, range $(-\pi/2, \pi/2)$).
   • So C → IV.

4. Function D: $f(x) = \cot^{-1} x$

   • Domain is all real numbers $\mathbb{R}$ (cotangent can output any real).
   • Its principal value range is $(0,\pi)$ (open interval, since $0$ and $\pi$ are asymptotes).
   • This matches characteristic II (Domain $\mathbb{R}$, range $(0,\pi)$).
   • So D → II.

Thus the mapping is: A-III, B-I, C-IV, D-II. Looking at the options:

• (A) A-II, B-III, C-IV, D-V → wrong.
• (B) A-V, B-I, C-II, D-III → wrong.
• (C) A-IV, B-II, C-I, D-V → wrong.
• (D) A-IV, B-III, C-V, D-II → wait, this says A-IV, but we have A-III. Let’s re-check: Option (D) lists A-IV, B-III, C-V, D-II. That would mean A matches IV, B matches III, C matches V, D matches II. But we found A→III, B→I, C→IV, D→II. So none of the given options match exactly? That suggests the lists might be permuted differently in the actual problem. Let’s re-read the problem: It says “Match the functions given in List-I with their relevant characteristics from List-II”. The options are given as (A) A-II, B-III, C-IV, D-V, etc. Possibly the functions in List-I are different. A common variant is:

List-I (Functions):

• A: $f(x) = \sin^{-1} x$
• B: $f(x) = \cos^{-1} x$
• C: $f(x) = \tan^{-1} x$
• D: $f(x) = \sec^{-1} x$

List-II (Characteristics):

• I: Domain is $[-1,1]$ and range is $[0,\pi]$
• II: Domain is $\mathbb{R}$ and range is $(0,\pi)$
• III: Domain is $[-1,1]$ and range is $[-\pi/2, \pi/2]$ …