Q.Evaluate $\cos\left[\cos^{-1}\left(\frac{-\sqrt3}{2}\right)+\frac{\pi}{6}\right]$.

Showing the 12 most recent of 20 on this concept.

1. TG EAPCET 2026Set eng-2026-05-09-AN1 markMCQ
   Q.If $\sqrt[3]{i} = \text{cis} \alpha$, $\alpha$ belongs to second quadrant and $\sqrt[3]{-i} = \text{cis} \beta$, $\beta$ belongs to third quadrant then $\text{cis} \alpha + \text{cis} \beta =$ (A) $\sqrt{3}$ (B) $i$ (C) $-i$ (D) $-\sqrt{3}$

   ›Reveal solutionSolution

   The cube roots of $i$ and $-i$ are found by De Moivre’s theorem; using the quadrant conditions picks the correct angles, and their sum simplifies to $-\sqrt{3}$.

   The problem asks for the sum of two complex numbers, each given in cis form (cis $\theta = \cos\theta + i\sin\theta$). The cube root of a complex number is not unique — there are three distinct cube roots. The quadrant condition for $\alpha$ and $\beta$ picks exactly one of those three for each. Once we have the correct angles, adding the two cis values is straightforward trigonometry.

   ---

   1. Find the cube roots of $i$. Write $i$ in polar form: $i = \text{cis}\, \frac{\pi}{2}$. By De Moivre’s theorem, the cube roots are

   $\text{cis}\left(\frac{\pi/2 + 2k\pi}{3}\right),\quad k = 0,1,2.$

   That gives the three angles:

   $\frac{\pi}{6},\quad \frac{\pi}{6} + \frac{2\pi}{3} = \frac{5\pi}{6},\quad \frac{\pi}{6} + \frac{4\pi}{3} = \frac{3\pi}{2}.$

   The second quadrant contains angles between $\frac{\pi}{2}$ and $\pi$. Among these, only $\frac{5\pi}{6}$ lies there.

   Hence $\alpha = \frac{5\pi}{6}$.

   2. Find the cube roots of $-i$. Write $-i = \text{cis}\left(-\frac{\pi}{2}\right)$ or equivalently $\text{cis}\,\frac{3\pi}{2}$. The cube roots are

   $\text{cis}\left(\frac{-\pi/2 + 2k\pi}{3}\right),\quad k = 0,1,2.$

   The three angles are:

   $-\frac{\pi}{6},\quad -\frac{\pi}{6} + \frac{2\pi}{3} = \frac{\pi}{2},\quad -\frac{\pi}{6} + \frac{4\pi}{3} = \frac{7\pi}{6}.$

   The third quadrant covers angles from $\pi$ to $\frac{3\pi}{2}$. Only $\frac{7\pi}{6}$ falls there.

   Hence $\beta = \frac{7\pi}{6}$.

   Watch out

   A common mistake is to take the principal cube root (the one with the smallest positive angle) without checking the quadrant condition. That would give $\alpha = \pi/6$ and $\beta = -\pi/6$, leading to a wrong sum.

   3. Compute $\text{cis}\,\alpha + \text{cis}\,\beta$.

   $\text{cis}\,\frac{5\pi}{6} = \cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6} = -\frac{\sqrt{3}}{2} + i\cdot\frac{1}{2}.$

   $\text{cis}\,\frac{7\pi}{6} = \cos\frac{7\pi}{6} + i\sin\frac{7\pi}{6} = -\frac{\sqrt{3}}{2} + i\cdot\left(-\frac{1}{2}\right).$

   Adding:

   $\left(-\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}\right) + i\left(\frac{1}{2} - \frac{1}{2}\right) = -\sqrt{3} + 0i.$

   Tip

   Notice that the imaginary parts cancel exactly because $\sin(5\pi/6) = 1/2$ and $\sin(7\pi/6) = -1/2$. The sum is purely real and negative.

   ✓Final answer

   The value is $\boxed{-\sqrt{3}}$, which corresponds to option (D).

2. TG EAPCET 2026Set eng-2026-05-10-AN1 markMCQ
   Q.If $z = (\sqrt{3} - i)^{2025} + (-1 - \sqrt{3}i)^{2026}$ then the point corresponding to $z$ in Argand plane lies in (A) $1^{\text{st}}$ quadrant (B) $2^{\text{nd}}$ quadrant (C) $3^{\text{rd}}$ quadrant (D) $4^{\text{th}}$ quadrant

   ›Reveal solutionSolution

   $z = -2^{2025} + i\,2^{2025}(1-\sqrt3)$: real part $<0$ and imaginary part $<0$, so $z$ lies in the third quadrant.

   First term $(\sqrt3 - i)^{2025}$: here $\sqrt3 - i = 2\big(\cos(-\tfrac{\pi}{6}) + i\sin(-\tfrac{\pi}{6})\big)$, so

   $(\sqrt3 - i)^{2025} = 2^{2025}\Big(\cos\tfrac{-2025\pi}{6} + i\sin\tfrac{-2025\pi}{6}\Big).$

   Reducing the angle: $-\tfrac{2025\pi}{6} = -337.5\pi \equiv \tfrac{\pi}{2} \pmod{2\pi}$, so this term $= 2^{2025}\,i$.

   Second term $(-1 - \sqrt3\,i)^{2026}$: here $-1-\sqrt3\,i = 2\big(\cos\tfrac{4\pi}{3} + i\sin\tfrac{4\pi}{3}\big)$, so

   $(-1 - \sqrt3\,i)^{2026} = 2^{2026}\Big(\cos\tfrac{2026\cdot 4\pi}{3} + i\sin\tfrac{2026\cdot 4\pi}{3}\Big).$

   Reducing: $\tfrac{2026\cdot 4\pi}{3} = \tfrac{8104\pi}{3} \equiv \tfrac{4\pi}{3}\pmod{2\pi}$, giving

   $2^{2026}\Big(-\tfrac12 - i\tfrac{\sqrt3}{2}\Big) = -2^{2025} - i\,2^{2025}\sqrt3.$

   Sum:

   $z = 2^{2025}i + \big(-2^{2025} - i\,2^{2025}\sqrt3\big) = -2^{2025} + i\,2^{2025}\big(1-\sqrt3\big).$

   Since $\text{Re}(z) = -2^{2025} < 0$ and $\text{Im}(z) = 2^{2025}(1-\sqrt3) < 0$ (because $\sqrt3 > 1$), the point lies in the third quadrant.

   ✓Final answer

   $z$ lies in the $3^{\text{rd}}$ quadrant — option (C).

3. TG EAPCET 2026Set eng-2026-05-10-AN1 markMCQ
   Q.If one of the values of $\sqrt{-1-\sqrt{3}i}$ is $a+i\beta$, $\alpha < 0$ and $\beta > 0$, then $\alpha =$ (A) $-\dfrac{1}{\sqrt{2}}$ (B) $-\dfrac{\sqrt{3}}{\sqrt{2}}$ (C) $-\sqrt{3}$ (D) $-\sqrt{2}$

   ›Reveal solutionSolution

   $-1-\sqrt3\,i = 2\,\text{cis}\tfrac{4\pi}{3}$; its square roots are $\sqrt2\,\text{cis}\tfrac{2\pi}{3}$ and $\sqrt2\,\text{cis}\tfrac{5\pi}{3}$. The root with $\alpha<0,\beta>0$ gives $\alpha = -\tfrac{1}{\sqrt2}$.

   Write the radicand in polar form. For $-1 - \sqrt3\,i$: modulus $= \sqrt{1+3} = 2$, and the point $(-1,-\sqrt3)$ is in the third quadrant, so its argument is $\tfrac{4\pi}{3}$ (equivalently $-\tfrac{2\pi}{3}$):

   $-1 - \sqrt3\,i = 2\left(\cos\tfrac{4\pi}{3} + i\sin\tfrac{4\pi}{3}\right).$

   The square roots have modulus $\sqrt2$ and arguments $\tfrac{1}{2}\cdot\tfrac{4\pi}{3} = \tfrac{2\pi}{3}$ and $\tfrac{2\pi}{3} + \pi = \tfrac{5\pi}{3}$:

   $\sqrt2\left(\cos\tfrac{2\pi}{3} + i\sin\tfrac{2\pi}{3}\right) = \sqrt2\left(-\tfrac12 + i\tfrac{\sqrt3}{2}\right) = -\frac{1}{\sqrt2} + i\,\frac{\sqrt6}{2},$

   $\sqrt2\left(\cos\tfrac{5\pi}{3} + i\sin\tfrac{5\pi}{3}\right) = \frac{1}{\sqrt2} - i\,\frac{\sqrt6}{2}.$

   The root with real part $\alpha < 0$ and imaginary part $\beta > 0$ is the first one, for which

   $\alpha = -\frac{1}{\sqrt2}.$

   ✓Final answer

   $\alpha = -\dfrac{1}{\sqrt2}$ — option (A).

4. TG EAPCET 2026Set eng-2026-05-10-FN1 markMCQ
   Q.Number of values of a complex number $z$ satisfying the condition $z = z^2 + i \operatorname{Im}(\overline{z})$ is (A) $0$ (B) $2$ (C) $1$ (D) $3$

   ›Reveal solutionSolution

   By substituting $z = x + iy$ into the given equation and equating the real and imaginary parts, we find two distinct values for $z$. The number of such values is $\boxed{2}$.

   The problem asks us to find the number of complex numbers $z$ that satisfy the given equation $z = z^2 + i \operatorname{Im}(\overline{z})$. When an equation involves both the complex number $z$ and its real or imaginary parts, or its conjugate, the most straightforward approach is often to express $z$ in its rectangular form, $z = x + iy$, where $x$ and $y$ are real numbers. This allows us to separate the equation into two real equations by equating the real and imaginary parts on both sides.

   Here's how we solve it:

   1. Represent $z$ in rectangular form:

      Let $z = x + iy$, where $x$ and $y$ are real numbers.

   2. Express $\operatorname{Im}(\overline{z})$ in terms of $x$ and $y$:

      If $z = x + iy$, then its conjugate is $\overline{z} = x - iy$.

      The imaginary part of $\overline{z}$ is $\operatorname{Im}(\overline{z}) = -y$.

   3. Substitute these into the given equation:

      The given equation is $z = z^2 + i \operatorname{Im}(\overline{z})$.

      Substitute $z = x + iy$ and $\operatorname{Im}(\overline{z}) = -y$:

      $x + iy = (x + iy)^2 + i(-y)$

      Expand the term $(x + iy)^2$:

      $(x + iy)^2 = x^2 + 2ixy + (iy)^2 = x^2 + 2ixy - y^2$.

      So the equation becomes:

      $x + iy = (x^2 - y^2 + 2ixy) - iy$

      Group the real and imaginary terms on the right side:

      $x + iy = (x^2 - y^2) + i(2xy - y)$

   4. Equate the real and imaginary parts:

      For two complex numbers to be equal, their real parts must be equal, and their imaginary parts must be equal.

      Equating the real parts:

      $x = x^2 - y^2 \quad \text{(Equation 1)}$

      Equating the imaginary parts:

      $y = 2xy - y \quad \text{(Equation 2)}$

   5. Solve the system of real equations:

      Let's start with Equation 2, as it often simplifies nicely:

      $y = 2xy - y$

      Add $y$ to both sides:

      $2y = 2xy$

      Move all terms to one side:

      $2y - 2xy = 0$

      Factor out $2y$:

      $2y(1 - x) = 0$

      This equation implies that either $2y = 0$ or $1 - x = 0$.

      So, we have two cases:

      • Case A: $y = 0$
      • Case B: $x = 1$

      Let's analyze each case:

      Case A: $y = 0$

      Substitute $y = 0$ into Equation 1:

      $x = x^2 - (0)^2$

      $x = x^2$

      Rearrange into a quadratic equation:

      $x^2 - x = 0$

      Factor out $x$:

      $x(x - 1) = 0$

      This gives two possible values for $x$: $x = 0$ or $x = 1$.

      • If $x = 0$ and $y = 0$, then $z = 0 + i(0) = 0$.
      • If $x = 1$ and $y = 0$, then $z = 1 + i(0) = 1$. Both $z=0$ and $z=1$ are distinct solutions.

      Case B: $x = 1$

      Substitute $x = 1$ into Equation 1:

      $1 = (1)^2 - y^2$

      $1 = 1 - y^2$

      Subtract 1 from both sides:

      $0 = -y^2$

      This implies $y^2 = 0$, so $y = 0$.

      This case gives $x = 1$ and $y = 0$, which corresponds to $z = 1 + i(0) = 1$. This is the same solution we found in Case A.

   6. Count the distinct values of $z$:

      From our analysis, the distinct values of $z$ that satisfy the given condition are $z = 0$ and $z = 1$.

      There are two such values.

   ✓Final answer

   The number of values of a complex number $z$ satisfying the condition is $\boxed{2}$.

5. TG EAPCET 2025Set eng-2025-05-02-FN1 markMCQ
   Q.If $f(x) = \tan\left(\dfrac{\pi}{\sqrt{x+1}+4}\right)$ is a real valued function then the range of $f$ is (A) $[-1,1]$ (B) $(0,1]$ (C) $[-1,\infty)$ (D) $\mathbb{R}$

   ›Reveal solutionSolution

   The function is a composition of a decreasing inner square-root expression and the tangent function over a restricted domain; the range is $(0,1]$, so option (B) is correct.

   We need the range of $f(x) = \tan\left(\dfrac{\pi}{\sqrt{x+1}+4}\right)$ for real $x$ where the function is defined.

   The key is to find what values the argument of $\tan$ can take, then see what outputs $\tan$ gives on that interval.

   ---

   Concept & Intuition

   The tangent function is periodic and unbounded, but here its argument is squeezed into a small interval because the denominator $\sqrt{x+1}+4$ is always at least $4$. That means the fraction $\frac{\pi}{\text{big number}}$ stays small — specifically between $0$ and $\frac{\pi}{4}$. On $(0, \frac{\pi}{4})$, $\tan$ is positive, increasing, and bounded between $0$ and $1$. So the range is $(0,1]$.

   ---

   Step-by-step reasoning

   1. Domain of $f$

      We need $\sqrt{x+1}$ defined, so $x+1 \ge 0 \Rightarrow x \ge -1$.

      Also, the denominator $\sqrt{x+1}+4$ is never zero (it’s always $\ge 4$), so no further restriction.

      Domain: $[-1, \infty)$.

   2. Behavior of the inner expression

      Let $t = \sqrt{x+1}$. For $x \ge -1$, $t \ge 0$.

      Then the argument of $\tan$ is

   $\theta = \frac{\pi}{t+4}.$

   As $x$ increases from $-1$ to $\infty$, $t$ increases from $0$ to $\infty$.

   3. Range of $\theta$

      • When $x = -1$: $t = 0$, so $\theta = \frac{\pi}{4}$.
      • As $x \to \infty$: $t \to \infty$, so $\theta \to 0^+$. Since $t+4$ increases, $\theta$ decreases continuously. Hence $\theta \in (0, \frac{\pi}{4}]$.

   4. Apply $\tan$

      On the interval $(0, \frac{\pi}{4}]$, $\tan$ is:

      • Continuous and strictly increasing.
      • $\tan(0^+) = 0^+$ (approaches 0 from above).
      • $\tan(\frac{\pi}{4}) = 1$.

      Therefore, the outputs cover $(0, 1]$.

   5. Check endpoints

      • At $x = -1$, $f(-1) = \tan(\frac{\pi}{4}) = 1$, so $1$ is included.
      • $0$ is never attained because $\theta$ never equals $0$ (only approaches it). So range is $(0, 1]$.

   ---

   Watch out

   A common mistake is to think $\tan$ can give negative values here — but the argument is always positive and less than $\frac{\pi}{2}$, so $\tan$ stays positive. Also, $\tan$ does not blow up because the argument never reaches $\frac{\pi}{2}$.

   Tip

   Notice that the denominator $\sqrt{x+1}+4$ is always at least 4, so the fraction $\frac{\pi}{\text{denominator}}$ is at most $\frac{\pi}{4}$. That immediately bounds the tangent between 0 and 1.

   ---

   ✓Final answer

   The correct option is (B).

   ANSWER: B

6. TG EAPCET 2025Set eng-2025-05-02-FN1 markMCQ
   Q.Let $z = x + iy$ and $P(x, y)$ be a point on the Argand plane. If $z$ satisfies the condition

   [!FORMULA] $\operatorname{Arg} \left( \frac{z - 3i}{z + 2i} \right) = \frac{\pi}{4}$

   then the locus of $P$ is (A) $x^2 + y^2 - y - 6 = 0, (x, y) \neq (0, -2)$ (B) $x^2 + y^2 - x - y - 6 = 0, (x, y) \neq (0, -2)$ (C) $x^2 + y^2 + 5x - y - 6 = 0, (x, y) \neq (0, -2)$ (D) $x^2 + y^2 + x - y - 6 = 0, (x, y) \neq (0, -2)$

   ›Reveal solutionSolution

   Rationalising $\dfrac{z-3i}{z+2i}$ and setting its argument to $\tfrac{\pi}{4}$ gives $x^2+y^2+5x-y-6=0$ (with $(0,-2)$ excluded) — option (C).

   Let $z = x+iy$, so $z-3i = x+i(y-3)$ and $z+2i = x+i(y+2)$. Multiply by the conjugate of the denominator:

   $\frac{z-3i}{z+2i} = \frac{[x+i(y-3)][x-i(y+2)]}{x^2+(y+2)^2}.$

   Expand the numerator:

   • Real part: $x^2 + (y-3)(y+2) = x^2 + y^2 - y - 6.$
   • Imaginary part: $x(y-3) - x(y+2) = -5x.$

   So the number is

   $\frac{(x^2+y^2-y-6) - 5x\,i}{x^2+(y+2)^2}.$

   Since $\operatorname{Arg} = \dfrac{\pi}{4}$ and $\tan\dfrac{\pi}{4}=1$, the imaginary and real parts are equal (both positive):

   $-5x = x^2 + y^2 - y - 6 \quad\Rightarrow\quad x^2 + y^2 + 5x - y - 6 = 0.$

   The point $z=-2i$, i.e. $(0,-2)$, makes the denominator zero and is excluded. (The positivity requirement $-5x>0$ further restricts the locus to the arc with $x<0$.)

   ✓Final answer

   $x^2 + y^2 + 5x - y - 6 = 0,\ (x,y)\neq(0,-2)$ — option (C).

7. TG EAPCET 2025Set eng-2025-05-03-AN1 markMCQ
   Q.The amplitude of the complex number $\dfrac{(\sqrt{3}+i)(1-\sqrt{3}i)}{(-1+i)(-1-i)}$ is (A) $\dfrac{\pi}{2}$ (B) $\dfrac{\pi}{3}$ (C) $-\dfrac{5\pi}{12}$ (D) $-\dfrac{\pi}{6}$

   ›Reveal solutionSolution

   The amplitude (argument) is $-\dfrac{\pi}{6}$, so the answer is (D).

   Write each factor in polar form and combine arguments (add for products, subtract for the quotient).

   Numerator

   • $\sqrt{3}+i$: modulus $2$, argument $\dfrac{\pi}{6}$.
   • $1-\sqrt{3}\,i$: modulus $2$, argument $-\dfrac{\pi}{3}$.

   Numerator argument $=\dfrac{\pi}{6}-\dfrac{\pi}{3}=-\dfrac{\pi}{6}$.

   Denominator

   $(-1+i)(-1-i)=(-1)^2-(i)^2=1-(-1)=2$, a positive real number, so its argument is $0$.

   Combine

   $\arg(z)=-\dfrac{\pi}{6}-0=-\dfrac{\pi}{6},$

   which lies in $(-\pi,\pi]$ and is therefore the principal amplitude.

   ✓Final answer

   Amplitude $=-\dfrac{\pi}{6}$, option (D).

8. TG EAPCET 2025Set eng-2025-05-03-FN1 markMCQ
   Q.The amplitude of the complex number $\dfrac{(\sqrt{3}+i)(1-\sqrt{3}i)}{(-1+i)(-1-i)}$ is (A) $\dfrac{\pi}{2}$ (B) $\dfrac{\pi}{3}$ (C) $-\dfrac{5\pi}{12}$ (D) $-\dfrac{\pi}{6}$

   ›Reveal solutionSolution

   The expression reduces to $\sqrt{3}-i$, whose amplitude is $-\dfrac{\pi}{6}$. Option (D).

   Solution

   Simplify numerator and denominator directly.

   Numerator:

   $(\sqrt{3}+i)(1-\sqrt{3}\,i)=\sqrt{3}-3i+i-\sqrt{3}\,i^{2}=\sqrt{3}-2i+\sqrt{3}=2\sqrt{3}-2i.$

   Denominator: the factors are conjugates, so

   $(-1+i)(-1-i)=(-1)^{2}-(i)^{2}=1+1=2.$

   Therefore

   $z=\frac{2\sqrt{3}-2i}{2}=\sqrt{3}-i.$

   Its argument: real part $\sqrt{3}>0$, imaginary part $-1<0$ (fourth quadrant), so

   $\arg z=\arctan\!\left(\frac{-1}{\sqrt{3}}\right)=-\frac{\pi}{6}.$

   ✓Final answer

   Option (D): $\;-\dfrac{\pi}{6}$.

9. TG EAPCET 2025Set eng-2025-05-04-FN1 markMCQ
   Q.If $\omega \neq 1$ is a cube root of unity, then one root among the $7^{\text{th}}$ roots of $(1+\omega)$ is (A) $1 + \omega$ (B) $\dfrac{\omega}{\omega - \omega^2}$ (C) $1 - \omega$ (D) $\omega - \omega^2$

   ›Reveal solutionSolution

   Since $1+\omega=-\omega^{2}$ and $(1+\omega)^{6}=\omega^{12}=1$, the number $1+\omega$ is itself a $7^{\text{th}}$ root of $(1+\omega)$ — option (A).

   Using $1+\omega+\omega^{2}=0$ gives $1+\omega=-\omega^{2}$. A number $z$ is a $7^{\text{th}}$ root of $(1+\omega)$ iff $z^{7}=1+\omega$.

   Test $z=1+\omega=-\omega^{2}$:

   $z^{7}=(-\omega^{2})^{7}=-\omega^{14}=-\omega^{12}\cdot\omega^{2}=-\omega^{2}=1+\omega,$

   because $\omega^{12}=(\omega^{3})^{4}=1$. Hence $z^{7}=1+\omega$, so $1+\omega$ is a $7^{\text{th}}$ root of $(1+\omega)$ — equivalently $(1+\omega)^{6}=\omega^{12}=1$.

   The other options fail the modulus test: a $7^{\text{th}}$ root of $-\omega^{2}$ (modulus $1$) must have modulus $1$, but $\omega-\omega^{2}=i\sqrt3$ and $1-\omega$ have modulus $\sqrt3$, and $\dfrac{\omega}{\omega-\omega^{2}}=\dfrac{1}{1-\omega}$ has modulus $\tfrac{1}{\sqrt3}$.

   ✓Final answer

   One $7^{\text{th}}$ root of $(1+\omega)$ is $1+\omega$ — option (A).

   ANSWER: A

10. TG EAPCET 2024Set eng-2024-05-09-FN1 markMCQ
    Q.The domain of the real valued function $f(x) = \sqrt{\cos(\sin x)} + \cos^{-1} \left( \frac{1 + x^2}{2x} \right)$ is (A) $(-1, 1)$ (B) $[-1, 1]$ (C) $\mathbb{R} - (-1, 1)$ (D) $\{-1, 1\}$

    ›Reveal solutionSolution

    The domain is the set of real numbers where both terms are defined: the square root requires $\cos(\sin x) \ge 0$ (true for all real $x$), and the inverse cosine requires $\left|\frac{1+x^2}{2x}\right| \le 1$ and $x \neq 0$, which forces $x = \pm 1$. So the domain is $\{-1, 1\}$, option (D).

    ---

    The key idea: a function defined by a sum is only valid where every piece is defined. Here we have a square root and an inverse cosine. The square root’s domain is where its inside is non‑negative; the inverse cosine’s domain is where its argument lies in $[-1, 1]$. We find the intersection of these conditions.

    Why this approach works:

    Instead of guessing, we treat each term separately. The square root term looks scary but simplifies because $\sin x$ is bounded and cosine is always non‑negative on the range of $\sin x$. The inverse cosine term is the restrictive one — it forces a very narrow set of $x$ values.

    ---

    1. First term: $\sqrt{\cos(\sin x)}$

       For the square root to be real, we need $\cos(\sin x) \ge 0$.

       Since $\sin x \in [-1, 1]$, we check: for any $y \in [-1, 1]$, is $\cos y \ge 0$?

       The cosine function is non‑negative on $[-\pi/2, \pi/2]$, and $[-1, 1]$ is a subset of that interval (because $1 < \pi/2 \approx 1.57$).

       Therefore $\cos(\sin x) > 0$ for all real $x$.

       So the first term imposes no restriction — its domain is $\mathbb{R}$.

    2. Second term: $\cos^{-1}\left( \frac{1 + x^2}{2x} \right)$

       The inverse cosine function $\cos^{-1}(t)$ is defined only for $t \in [-1, 1]$.

       So we require

    $-1 \le \frac{1 + x^2}{2x} \le 1.$

    Also, the denominator $2x$ cannot be zero, so $x \neq 0$.

    3. Solve the inequality Consider the right inequality first:

    $\frac{1 + x^2}{2x} \le 1.$

    Bring to one side:

    $\frac{1 + x^2}{2x} - 1 \le 0 \quad\Rightarrow\quad \frac{1 + x^2 - 2x}{2x} \le 0 \quad\Rightarrow\quad \frac{(x-1)^2}{2x} \le 0.$

    The numerator $(x-1)^2$ is always $\ge 0$ and equals $0$ only at $x=1$.

    So the fraction is $\le 0$ only when the denominator is negative (since numerator is non‑negative).

    That means $2x < 0$, i.e. $x < 0$, or numerator = 0 (i.e. $x=1$).

    So from this inequality: $x < 0$ or $x = 1$.

    Now the left inequality:

    $-1 \le \frac{1 + x^2}{2x} \quad\Rightarrow\quad \frac{1 + x^2}{2x} + 1 \ge 0 \quad\Rightarrow\quad \frac{1 + x^2 + 2x}{2x} \ge 0 \quad\Rightarrow\quad \frac{(x+1)^2}{2x} \ge 0.$

    The numerator $(x+1)^2 \ge 0$, zero only at $x=-1$.

    The fraction is $\ge 0$ when denominator $2x > 0$ (i.e. $x > 0$) or numerator = 0 (i.e. $x = -1$).

    So from this inequality: $x > 0$ or $x = -1$.

    4. Combine both conditions

       From the right inequality: $x < 0$ or $x = 1$.

       From the left inequality: $x > 0$ or $x = -1$.

       The intersection (and remembering $x \neq 0$) gives:

       • $x < 0$ and $x > 0$ is impossible.
       • $x < 0$ and $x = -1$ gives $x = -1$.
       • $x = 1$ and $x > 0$ gives $x = 1$.
       • $x = 1$ and $x = -1$ is impossible.

       So the only solutions are $x = -1$ and $x = 1$.

    5. Final domain

       The first term accepts all real $x$, the second term accepts only $x = \pm 1$.

       Hence the domain of $f$ is $\{-1, 1\}$.

    Watch out

    A common mistake is to forget that $\cos^{-1}(t)$ requires $t$ to be in $[-1,1]$, not just $t \le 1$. Also, dividing by $x$ without considering sign can lead to missing the $x = \pm 1$ cases.

    Tip

    Notice that $\frac{1+x^2}{2x}$ is exactly $\frac{1}{2}\left(x + \frac{1}{x}\right)$, which by AM–GM has magnitude at least $1$ for $x>0$ and at most $-1$ for $x<0$, with equality only at $x = \pm 1$. That instantly gives the domain.

    ✓Final answer

    The correct option is (D).

    ANSWER: D

11. TG EAPCET 2024Set eng-2024-05-09-FN1 markMCQ
    Q.$x$ and $y$ are two complex numbers such that $|x| = |y| = 1$. If $\operatorname{Arg}(x) = 2\alpha$, $\operatorname{Arg}(y) = 3\beta$ and $\alpha + \beta = \dfrac{\pi}{36}$, then $x^6 y^4 + \dfrac{1}{x^6 y^4} =$ (A) $0$ (B) $-1$ (C) $1$ (D) $\dfrac{1}{2}$

    ›Reveal solutionSolution

    The key is to express $x^6 y^4$ in polar form using the given arguments, simplify the exponent using $\alpha+\beta = \pi/36$, and then evaluate $z + 1/z$ for a complex number on the unit circle — the result is $1$.

    We are given two complex numbers $x$ and $y$ on the unit circle ($|x|=|y|=1$), with arguments $\operatorname{Arg}(x)=2\alpha$ and $\operatorname{Arg}(y)=3\beta$, and the condition $\alpha+\beta = \pi/36$.

    We need $x^6 y^4 + \frac{1}{x^6 y^4}$.

    Concept & Intuition

    When a complex number lies on the unit circle, its reciprocal equals its conjugate. So $z + 1/z = z + \overline{z} = 2\operatorname{Re}(z)$.

    Thus the problem reduces to finding the real part of $x^6 y^4$. Since $x$ and $y$ are on the unit circle, their powers are also on the unit circle, and their arguments add. The given relation $\alpha+\beta = \pi/36$ will simplify the total argument to a nice angle whose cosine we know exactly.

    Step-by-step

    1. Write $x$ and $y$ in polar form Since $|x|=|y|=1$, we have

    $x = e^{i\cdot 2\alpha}, \qquad y = e^{i\cdot 3\beta}.$

    2. Compute $x^6 y^4$

    $x^6 = (e^{i\cdot 2\alpha})^6 = e^{i\cdot 12\alpha}, \qquad y^4 = (e^{i\cdot 3\beta})^4 = e^{i\cdot 12\beta}.$

    Hence

    $x^6 y^4 = e^{i(12\alpha + 12\beta)} = e^{i\cdot 12(\alpha+\beta)}.$

    3. Use the given $\alpha+\beta = \pi/36$

    $12(\alpha+\beta) = 12\cdot\frac{\pi}{36} = \frac{\pi}{3}.$

    So

    $x^6 y^4 = e^{i\pi/3}.$

    4. Evaluate $z + 1/z$ for $z = e^{i\pi/3}$ Since $|z|=1$, we have $1/z = \overline{z} = e^{-i\pi/3}$. Therefore

    $z + \frac{1}{z} = e^{i\pi/3} + e^{-i\pi/3} = 2\cos\left(\frac{\pi}{3}\right) = 2\cdot\frac{1}{2} = 1.$

    Tip

    A common pitfall is forgetting that $1/z = \overline{z}$ only when $|z|=1$ — here it holds because $x$ and $y$ are on the unit circle, so their product is also on the unit circle.

    ✓Final answer

    The correct option is (C).

    ANSWER: C

12. TG EAPCET 2024Set eng-2024-05-10-AN1 markMCQ
    Q.If $Z_1 = \sqrt{3} + i\sqrt{3}$ and $Z_2 = \sqrt{3} + i$, and $\left(\dfrac{Z_1}{Z_2}\right)^{50} = x + iy$, then the point $(x, y)$ lies in (A) first quadrant (B) second quadrant (C) third quadrant (D) fourth quadrant

    ›Reveal solutionSolution

    Work in polar form: $\arg(Z_1/Z_2)=\tfrac{\pi}{4}-\tfrac{\pi}{6}=\tfrac{\pi}{12}$, so the 50th power has argument $\tfrac{50\pi}{12}=\tfrac{25\pi}{6}\equiv\tfrac{\pi}{6}$. Both $x$ and $y$ are positive, so $(x,y)$ is in the first quadrant — option (A).

    Step 1 — argument of $Z_1=\sqrt3+i\sqrt3$.

    $\theta_1=\arctan\!\frac{\sqrt3}{\sqrt3}=\arctan 1=\frac{\pi}{4}.$

    Step 2 — argument of $Z_2=\sqrt3+i$.

    $\theta_2=\arctan\!\frac{1}{\sqrt3}=\frac{\pi}{6}.$

    Step 3 — argument of the quotient.

    $\arg\!\left(\frac{Z_1}{Z_2}\right)=\theta_1-\theta_2=\frac{\pi}{4}-\frac{\pi}{6}=\frac{3\pi-2\pi}{12}=\frac{\pi}{12}.$

    Step 4 — raise to the 50th power (De Moivre).

    $\arg\!\left(\frac{Z_1}{Z_2}\right)^{50}=50\cdot\frac{\pi}{12}=\frac{25\pi}{6}.$

    Reduce modulo $2\pi$: $\dfrac{25\pi}{6}-4\pi=\dfrac{25\pi-24\pi}{6}=\dfrac{\pi}{6}.$

    Step 5 — locate the point.

    The modulus $\left(\tfrac{\sqrt6}{2}\right)^{50}$ is positive and the reduced angle $\tfrac{\pi}{6}$ lies in the first quadrant, so $x=r\cos\tfrac{\pi}{6}>0$ and $y=r\sin\tfrac{\pi}{6}>0$.

    ✓Final answer

    The point $(x,y)$ lies in the first quadrant — option (A).