Q.Find $\frac{dy}{dx}$ in the following: $x \cdot \cos x$

Showing the 12 most recent of 50 on this concept.

1. TG EAPCET 2026Set eng-2026-05-09-AN1 markMCQ
   Q.$e^{\sinh^{-1}(2\sqrt{2})} + e^{\cosh^{-1}(3)} =$ (A) $2e^{\tanh^{-1}\left(\dfrac{1}{2\sqrt{2}}\right)}$ (B) $\dfrac{2}{3} e^{\cosech^{-1}(3)}$ (C) $2e^{\sech^{-1}\left(\dfrac{1}{3}\right)}$ (D) $\dfrac{1}{3} e^{\coth^{-1}(2\sqrt{2})}$

   ›Reveal solutionSolution

   The key idea is to simplify the exponentials of inverse hyperbolic functions by converting them into algebraic expressions using the definitions of $\sinh^{-1}$ and $\cosh^{-1}$, then matching the result to one of the given options. The final value is $2e^{\tanh^{-1}(1/(2\sqrt{2}))}$, which corresponds to option (A).

   The problem asks for the value of $e^{\sinh^{-1}(2\sqrt{2})} + e^{\cosh^{-1}(3)}$. Instead of trying to compute the inverse functions directly, we can use the fundamental definitions: for any real $x$, $\sinh^{-1}(x) = \log(x + \sqrt{x^2+1})$ and $\cosh^{-1}(x) = \log(x + \sqrt{x^2-1})$ for $x \ge 1$. Then $e^{\sinh^{-1}(x)} = x + \sqrt{x^2+1}$ and $e^{\cosh^{-1}(x)} = x + \sqrt{x^2-1}$. This turns the problem into simple arithmetic.

   1. Simplify $e^{\sinh^{-1}(2\sqrt{2})}$. Using $e^{\sinh^{-1}(x)} = x + \sqrt{x^2+1}$, with $x = 2\sqrt{2}$:

   $e^{\sinh^{-1}(2\sqrt{2})} = 2\sqrt{2} + \sqrt{(2\sqrt{2})^2 + 1} = 2\sqrt{2} + \sqrt{8 + 1} = 2\sqrt{2} + 3.$

   2. Simplify $e^{\cosh^{-1}(3)}$. Using $e^{\cosh^{-1}(x)} = x + \sqrt{x^2-1}$, with $x = 3$:

   $e^{\cosh^{-1}(3)} = 3 + \sqrt{9 - 1} = 3 + \sqrt{8} = 3 + 2\sqrt{2}.$

   3. Add the two results.

   $e^{\sinh^{-1}(2\sqrt{2})} + e^{\cosh^{-1}(3)} = (2\sqrt{2} + 3) + (3 + 2\sqrt{2}) = 6 + 4\sqrt{2}.$

   So the sum is $6 + 4\sqrt{2}$.

   4. Match this to one of the given options. Each option is of the form $k \cdot e^{\text{some inverse hyperbolic function}}$. We need to see which simplifies to $6 + 4\sqrt{2}$. Let’s test option (A): $2e^{\tanh^{-1}(1/(2\sqrt{2}))}$. Recall that $e^{\tanh^{-1}(x)} = \sqrt{\frac{1+x}{1-x}}$ for $|x| < 1$. Here $x = \frac{1}{2\sqrt{2}}$, so:

   $e^{\tanh^{-1}\left(\frac{1}{2\sqrt{2}}\right)} = \sqrt{\frac{1 + \frac{1}{2\sqrt{2}}}{1 - \frac{1}{2\sqrt{2}}}}.$

   Simplify the fraction inside:

   $\frac{1 + \frac{1}{2\sqrt{2}}}{1 - \frac{1}{2\sqrt{2}}} = \frac{\frac{2\sqrt{2} + 1}{2\sqrt{2}}}{\frac{2\sqrt{2} - 1}{2\sqrt{2}}} = \frac{2\sqrt{2} + 1}{2\sqrt{2} - 1}.$

   Rationalize the denominator:

   $\frac{2\sqrt{2} + 1}{2\sqrt{2} - 1} \times \frac{2\sqrt{2} + 1}{2\sqrt{2} + 1} = \frac{(2\sqrt{2} + 1)^2}{(2\sqrt{2})^2 - 1^2} = \frac{8 + 4\sqrt{2} + 1}{8 - 1} = \frac{9 + 4\sqrt{2}}{7}.$

   So $e^{\tanh^{-1}(1/(2\sqrt{2}))} = \sqrt{\frac{9 + 4\sqrt{2}}{7}}$.

   Then $2e^{\tanh^{-1}(1/(2\sqrt{2}))} = 2\sqrt{\frac{9 + 4\sqrt{2}}{7}}$. This does not look like $6 + 4\sqrt{2}$ at first glance. But wait — we might have misapplied the formula. Actually, $e^{\tanh^{-1}(x)} = \sqrt{\frac{1+x}{1-x}}$ is correct, but let’s check if this simplifies to $6 + 4\sqrt{2}$ numerically: $2\sqrt{(9+4\sqrt{2})/7} \approx 2\sqrt{(9+5.656)/7} = 2\sqrt{14.656/7} = 2\sqrt{2.0937} \approx 2 \times 1.447 = 2.894$, which is far from $6+4\sqrt{2} \approx 6+5.656=11.656$. So option (A) seems wrong? Let’s re-evaluate.

   Watch out

   The formula $e^{\tanh^{-1}(x)} = \sqrt{\frac{1+x}{1-x}}$ is correct only if we interpret $\tanh^{-1}(x)$ as the inverse hyperbolic tangent. But here the exponent is $\tanh^{-1}(1/(2\sqrt{2}))$, and $e^{\tanh^{-1}(x)} = \frac{1+x}{\sqrt{1-x^2}}$? Actually, let’s derive properly: $\tanh^{-1}(x) = \frac{1}{2}\log\frac{1+x}{1-x}$, so $e^{\tanh^{-1}(x)} = \sqrt{\frac{1+x}{1-x}}$. That is correct. So option (A) gives about 2.894, not 11.656. Something is off — perhaps we miscomputed the sum? Let’s double-check the sum: $2\sqrt{2}+3$ plus $3+2\sqrt{2}$ is indeed $6+4\sqrt{2} \approx 11.656$. So option (A) is not matching. Let’s test the other options quickly.

   Option (B): $\frac{2}{3} e^{\cosech^{-1}(3)}$. Recall $\cosech^{-1}(x) = \log(1/x + \sqrt{1/x^2+1})$ for $x>0$, so $e^{\cosech^{-1}(3)} = 1/3 + \sqrt{1/9+1} = 1/3 + \sqrt{10/9} = 1/3 + \sqrt{10}/3 = (1+\sqrt{10})/3$. Then $\frac{2}{3} \times (1+\sqrt{10})/3 = 2(1+\sqrt{10})/9 \approx 2(1+3.162)/9 = 8.324/9 \approx 0.925$, not 11.656.

   Option (C): $2e^{\sech^{-1}(1/3)}$. $\sech^{-1}(x) = \log(1/x + \sqrt{1/x^2-1})$ for $0<x\le1$, so $e^{\sech^{-1}(1/3)} = 3 + \sqrt{9-1} = 3 + \sqrt{8} = 3+2\sqrt{2}$. Then $2(3+2\sqrt{2}) = 6+4\sqrt{2}$. That matches exactly!

   Option (D): $\frac{1}{3} e^{\coth^{-1}(2\sqrt{2})}$. $\coth^{-1}(x) = \frac{1}{2}\log\frac{x+1}{x-1}$ for $|x|>1$, so $e^{\coth^{-1}(2\sqrt{2})} = \sqrt{\frac{2\sqrt{2}+1}{2\sqrt{2}-1}}$. Rationalizing gives $\sqrt{\frac{9+4\sqrt{2}}{7}}$, then $\frac{1}{3}$ of that is about $0.333 \times 1.447 \approx 0.482$, not 11.656.

   So the correct match is option (C). But wait — we earlier thought option (A) might be correct? Let’s re-check the problem statement: the options are given as (A), (B), (C), (D). Option (C) is $2e^{\sech^{-1}(1/3)}$, and we just computed $e^{\sech^{-1}(1/3)} = 3+2\sqrt{2}$, so $2(3+2\sqrt{2}) = 6+4\sqrt{2}$, which is exactly our sum. So the answer is (C).

   Tip

   A quick way to see this: $\sech^{-1}(x) = \cosh^{-1}(1/x)$, so $e^{\sech^{-1}(1/3)} = e^{\cosh^{-1}(3)} = 3+2\sqrt{2}$. Then $2e^{\sech^{-1}(1/3)} = 2(3+2\sqrt{2}) = 6+4\sqrt{2}$, which is the sum. So option (C) is directly the sum itself, not just a match.

   Thus the sum simplifies elegantly to option (C).

   ✓Final answer

   The correct option is (C).

2. TG EAPCET 2026Set eng-2026-05-09-FN1 markMCQ
   Q.If the angle between the curves $y^2 = 4x$ and $y = ax^2 - 5$ at the point $(1,2)$ is $\alpha$, then $(a-2)|\tan\alpha|=$ (A) $13$ (B) $\dfrac{13}{3}$ (C) $\dfrac{5}{13}$ (D) $\dfrac{13}{5}$

   ›Reveal solutionSolution

   The key idea is to compute the slopes of the two curves at the intersection point (1,2), then use the tangent-of-angle formula between curves. The result simplifies to $(a-2)|\tan\alpha| = \frac{13}{3}$, so the correct option is (B).

   ---

   Concept & Intuition

   When two curves intersect, the angle between them at that point is defined as the angle between their tangent lines. So we find the derivative of each curve at the point, get the slopes $m_1$ and $m_2$, then use

   $\tan\alpha = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|.$

   We are given that the point $(1,2)$ lies on both curves — this determines the unknown parameter $a$. Then we compute $m_1$, $m_2$, and finally the expression $(a-2)|\tan\alpha|$.

   ---

   Step-by-step solution

   1. Find $a$ using the intersection point The second curve is $y = a x^2 - 5$. Since $(1,2)$ lies on it:

   $2 = a(1)^2 - 5 \quad\Rightarrow\quad a = 7.$

   So $a-2 = 5$.

   2. Slope of the first curve $y^2 = 4x$. Differentiate implicitly:

   $2y \frac{dy}{dx} = 4 \quad\Rightarrow\quad \frac{dy}{dx} = \frac{2}{y}.$

   At $(1,2)$: $m_1 = \frac{2}{2} = 1$.

   3. Slope of the second curve $y = 7x^2 - 5$. Differentiate:

   $\frac{dy}{dx} = 14x.$

   At $x=1$: $m_2 = 14$.

   4. Angle between the curves Using the formula:

   $\tan\alpha = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| = \left|\frac{1 - 14}{1 + 1 \cdot 14}\right| = \left|\frac{-13}{15}\right| = \frac{13}{15}.$

   5. Compute the required expression

   $(a-2)|\tan\alpha| = 5 \cdot \frac{13}{15} = \frac{13}{3}.$

   ---

   Watch out

   A common mistake is to forget that the point must satisfy both curves — without finding $a$ first, the slopes are wrong. Also, note that the angle formula uses absolute value, so sign doesn’t matter.

   Tip

   The slopes turned out to be nice numbers (1 and 14), making the arithmetic clean. Always check if the point satisfies the curve equation before differentiating.

   ---

   ✓Final answer

   The correct option is (B).

   ANSWER: B

3. TG EAPCET 2026Set eng-2026-05-09-FN1 markMCQ
   Q.

   [!FORMULA] $\int \frac{dx}{\sqrt{4x^2 + 11x + 6}} = \frac{1}{2} \cosh^{-1} \left( \frac{f(x)}{5} \right) + c \text{ and } f(1) = 19, \text{ then } f(2) =$

   (A) 11 (B) 15 (C) 23 (D) 27

   ›Reveal solutionSolution

   The key idea is to complete the square inside the square root, then use the standard integral form for $\cosh^{-1}$. Matching the given result yields $f(x)=8x+11$, so $f(2)=27$, option (D).

   We are given that

   $\int \frac{dx}{\sqrt{4x^2 + 11x + 6}} = \frac{1}{2} \cosh^{-1}\left( \frac{f(x)}{5} \right) + c$

   and $f(1)=19$. We need $f(2)$.

   Concept and intuition:

   The integral $\int \frac{dx}{\sqrt{ax^2+bx+c}}$ is a standard form that leads to an inverse hyperbolic cosine (or sine) when the quadratic is positive. The trick is to rewrite the quadratic as a perfect square plus a constant — completing the square — so that the integrand becomes $\frac{dx}{\sqrt{(something)^2 - A^2}}$ or $\frac{dx}{\sqrt{(something)^2 + A^2}}$. Here, the presence of $\cosh^{-1}$ tells us the expression inside the square root will be of the form $(linear)^2 - (constant)^2$, because $\frac{d}{du}\cosh^{-1}(u) = \frac{1}{\sqrt{u^2-1}}$.

   Let’s work through it.

   1. Complete the square for $4x^2+11x+6$. Factor out the leading coefficient from the $x^2$ and $x$ terms:

   $4x^2+11x+6 = 4\left(x^2 + \frac{11}{4}x\right) + 6.$

   Complete the square inside:

   $x^2 + \frac{11}{4}x = \left(x + \frac{11}{8}\right)^2 - \left(\frac{11}{8}\right)^2 = \left(x + \frac{11}{8}\right)^2 - \frac{121}{64}.$

   So

   $4x^2+11x+6 = 4\left[\left(x + \frac{11}{8}\right)^2 - \frac{121}{64}\right] + 6 = 4\left(x + \frac{11}{8}\right)^2 - \frac{121}{16} + 6.$

   Now $6 = \frac{96}{16}$, so

   $4x^2+11x+6 = 4\left(x + \frac{11}{8}\right)^2 - \frac{25}{16}.$

   Factor the constant:

   $= 4\left(x + \frac{11}{8}\right)^2 - \left(\frac{5}{4}\right)^2.$

   2. Rewrite the integral in standard form. The square root becomes

   $\sqrt{4x^2+11x+6} = \sqrt{4\left(x + \frac{11}{8}\right)^2 - \left(\frac{5}{4}\right)^2}.$

   Factor out $4$ from inside the square root:

   $= \sqrt{4} \cdot \sqrt{\left(x + \frac{11}{8}\right)^2 - \left(\frac{5}{8}\right)^2} = 2 \sqrt{\left(x + \frac{11}{8}\right)^2 - \left(\frac{5}{8}\right)^2}.$

   So the integral is

   $\int \frac{dx}{2 \sqrt{\left(x + \frac{11}{8}\right)^2 - \left(\frac{5}{8}\right)^2}} = \frac{1}{2} \int \frac{dx}{\sqrt{\left(x + \frac{11}{8}\right)^2 - \left(\frac{5}{8}\right)^2}}.$

   3. Apply the standard integral formula. Recall:

   $\int \frac{du}{\sqrt{u^2 - a^2}} = \cosh^{-1}\left(\frac{u}{a}\right) + c \quad (u > a > 0).$

   Here $u = x + \frac{11}{8}$ and $a = \frac{5}{8}$. Thus

   $\frac{1}{2} \int \frac{dx}{\sqrt{\left(x + \frac{11}{8}\right)^2 - \left(\frac{5}{8}\right)^2}} = \frac{1}{2} \cosh^{-1}\left( \frac{x + \frac{11}{8}}{\frac{5}{8}} \right) + c.$

   Simplify the argument:

   $\frac{x + \frac{11}{8}}{\frac{5}{8}} = \frac{8x + 11}{5}.$

   So the integral equals

   $\frac{1}{2} \cosh^{-1}\left( \frac{8x+11}{5} \right) + c.$

   4. Match with the given form. The problem states the integral equals $\frac{1}{2} \cosh^{-1}\left( \frac{f(x)}{5} \right) + c$. Comparing, we see

   $\frac{f(x)}{5} = \frac{8x+11}{5} \quad \Rightarrow \quad f(x) = 8x + 11.$

   Check: $f(1) = 8(1)+11 = 19$, which matches the given condition. Perfect.

   5. Find $f(2)$.

   $f(2) = 8(2) + 11 = 16 + 11 = 27.$

   Tip

   A quick sanity check: the derivative of $\frac{1}{2}\cosh^{-1}\left(\frac{8x+11}{5}\right)$ is $\frac{1}{2} \cdot \frac{1}{\sqrt{\left(\frac{8x+11}{5}\right)^2 - 1}} \cdot \frac{8}{5} = \frac{4}{5\sqrt{\frac{(8x+11)^2}{25} - 1}} = \frac{4}{\sqrt{(8x+11)^2 - 25}}$. But $(8x+11)^2 - 25 = 64x^2 + 176x + 121 - 25 = 64x^2 + 176x + 96 = 16(4x^2+11x+6)$, so the derivative becomes $\frac{4}{\sqrt{16(4x^2+11x+6)}} = \frac{1}{\sqrt{4x^2+11x+6}}$, confirming the integral.

   Watch out

   A common mistake is to forget the factor of $4$ when completing the square, leading to an incorrect linear function. Always check that the derivative of your candidate $f(x)$ reproduces the original integrand.

   ✓Final answer

   The correct option is (D).

   ANSWER: D

4. TG EAPCET 2026Set eng-2026-05-09-FN1 markMCQ
   Q.If $\dfrac{dy}{dx} = (x^3 - x) - (1 - 3x^2)\tan x + (x^3 - x)\tan^2 x$ and $y(1) = 0$, then $\dfrac{64}{\pi} y\left(\dfrac{\pi}{4}\right) =$ (A) $1$ (B) $\pi^2 + 16$ (C) $\pi^2 - 16$ (D) $16\pi^2$

   ›Reveal solutionSolution

   The key is to notice that the derivative simplifies to a perfect derivative of a product involving $\tan x$ and a polynomial, so we can integrate directly and then evaluate at $x = \pi/4$. The final value is $\pi^2 - 16$, which corresponds to option (C).

   ---

   We are given:

   $\frac{dy}{dx} = (x^3 - x) - (1 - 3x^2)\tan x + (x^3 - x)\tan^2 x$

   with $y(1) = 0$, and we need $\frac{64}{\pi} y(\pi/4)$.

   Concept and intuition:

   The expression looks messy, but the presence of $\tan x$ and $\tan^2 x$ alongside a polynomial suggests it might be the derivative of something like $(\text{polynomial}) \cdot \tan x$ plus something simpler. When we differentiate a product $u(x) \tan x$, we get $u'(x) \tan x + u(x) \sec^2 x$. Since $\sec^2 x = 1 + \tan^2 x$, this can produce terms like $u(x) + u(x) \tan^2 x$ plus a $u'(x) \tan x$ term. That matches our structure perfectly.

   Let’s try to match it.

   ---

   1. Guess the form Suppose $y(x) = (x^3 - x) \tan x + \text{something}$. Differentiate:

   $\frac{d}{dx}\big[(x^3 - x)\tan x\big] = (3x^2 - 1)\tan x + (x^3 - x)\sec^2 x.$

   Since $\sec^2 x = 1 + \tan^2 x$, this becomes:

   $(3x^2 - 1)\tan x + (x^3 - x) + (x^3 - x)\tan^2 x.$

   2. Compare with given $\frac{dy}{dx}$ The given derivative is:

   $(x^3 - x) - (1 - 3x^2)\tan x + (x^3 - x)\tan^2 x.$

   Notice that $-(1 - 3x^2) = 3x^2 - 1$. So the given derivative is exactly:

   $(x^3 - x) + (3x^2 - 1)\tan x + (x^3 - x)\tan^2 x.$

   This matches the derivative we computed for $(x^3 - x)\tan x$ term by term.

   3. Conclusion about $y(x)$ Hence,

   $\frac{dy}{dx} = \frac{d}{dx}\big[(x^3 - x)\tan x\big].$

   Integrating both sides:

   $y(x) = (x^3 - x)\tan x + C.$

   4. Use the initial condition $y(1) = 0$ gives:

   $0 = (1^3 - 1)\tan 1 + C = 0 \cdot \tan 1 + C = C.$

   So $C = 0$, and

   $y(x) = (x^3 - x)\tan x.$

   5. Evaluate at $x = \pi/4$

   $y\left(\frac{\pi}{4}\right) = \left(\left(\frac{\pi}{4}\right)^3 - \frac{\pi}{4}\right) \tan\left(\frac{\pi}{4}\right).$

   Since $\tan(\pi/4) = 1$,

   $y\left(\frac{\pi}{4}\right) = \frac{\pi^3}{64} - \frac{\pi}{4}.$

   6. Compute the required expression

   $\frac{64}{\pi} y\left(\frac{\pi}{4}\right) = \frac{64}{\pi} \left( \frac{\pi^3}{64} - \frac{\pi}{4} \right) = \frac{64}{\pi} \cdot \frac{\pi^3}{64} - \frac{64}{\pi} \cdot \frac{\pi}{4} = \pi^2 - 16.$

   Tip

   Spotting the product rule pattern early saves you from having to integrate a messy trigonometric expression. The key was rewriting $-(1-3x^2)$ as $3x^2-1$ to see the match.

   Watch out

   A common mistake is to try to integrate term-by-term directly, which leads to complicated integrals. Always look for derivative patterns first.

   ✓Final answer

   The correct option is (C).

   ANSWER: C

5. TG EAPCET 2026Set eng-2026-05-10-FN1 markMCQ
   Q.Approximate volume of a cone whose semi vertical angle is $\tan^{-1}(3)$ and base radius is $15.001$ is (A) $(375.025)\pi$ (B) $(325.025)\pi$ (C) $(375.075)\pi$ (D) $(325.075)\pi$

   ›Reveal solutionSolution

   With $\tan\alpha=r/h=3$, $V=\dfrac{\pi r^3}{9}$; using $dV=\dfrac{\pi r^2}{3}\,dr$ at $r=15,\ dr=0.001$ gives $V\approx(375.075)\pi$.

   Semi-vertical angle $\alpha=\tan^{-1}(3)$, so $\tan\alpha=\dfrac{r}{h}=3\Rightarrow h=\dfrac{r}{3}$.

   $V=\frac13\pi r^2 h=\frac13\pi r^2\cdot\frac{r}{3}=\frac{\pi r^3}{9}.$

   At $r=15$: $V=\dfrac{\pi(3375)}{9}=375\pi$.

   Differential for a small change $dr=0.001$ about $r=15$:

   $dV=\frac{dV}{dr}\,dr=\frac{\pi\cdot 3r^2}{9}\,dr=\frac{\pi r^2}{3}\,dr=\frac{\pi(225)}{3}(0.001)=0.075\pi.$

   Hence $V\approx 375\pi+0.075\pi=(375.075)\pi$.

   ✓Final answer

   Approximate volume $=(375.075)\pi$ — option (C).

6. TG EAPCET 2026Set eng-2026-05-11-FN1 markMCQ
   Q.If $1^\circ \approx 0.01745$ then the approximate value of $\sec 29^\circ$ is (A) 1.1530 (B) 1.1430 (C) 1.1525 (D) 1.1493

   ›Reveal solutionSolution

   Linearise $\sec x$ about $30^\circ$ with $\Delta x=-1^\circ$: $\sec 29^\circ \approx \sec 30^\circ - \sec 30^\circ\tan 30^\circ\cdot(0.01745) \approx 1.1430$ — option (B).

   Method — differential approximation. For a small angle change, $f(x+\Delta x)\approx f(x)+f'(x)\,\Delta x$ with $\Delta x$ in radians. Take $f(x)=\sec x$, $x=30^\circ$ (a known angle) and $\Delta x=-1^\circ=-0.01745$ rad.

   Known values.

   $\sec 30^\circ=\frac{2}{\sqrt{3}}\approx 1.15470,\qquad \tan 30^\circ=\frac{1}{\sqrt{3}}.$

   Derivative.

   $f'(x)=\sec x\tan x,\qquad f'(30^\circ)=\frac{2}{\sqrt{3}}\cdot\frac{1}{\sqrt{3}}=\frac{2}{3}.$

   Apply the approximation.

   $\sec 29^\circ \approx \sec 30^\circ + f'(30^\circ)\,\Delta x = 1.15470 + \frac{2}{3}\,(-0.01745).$

   $\frac{2}{3}\times 0.01745 = 0.011633,$

   $\sec 29^\circ \approx 1.15470 - 0.011633 = 1.14307 \approx 1.1430.$

   ✓Final answer

   $\sec 29^\circ \approx 1.1430$ — option (B).

7. TG EAPCET 2026Set eng-2026-05-11-AN1 markMCQ
   Q.If a normal drawn at the point $t=1$ on the parabola $x^2 + 4y + 2x - 8 = 0$ intersects the parabola again at a point $A(\alpha, \beta)$, then $4\alpha\beta=$ (A) 189 (B) $-24$ (C) 152 (D) $-38$

   ›Reveal solutionSolution

   The key idea is to rewrite the parabola in standard form, find the normal at the given point, solve for its second intersection with the parabola, and compute $4\alpha\beta$. The result is $-24$, so the correct option is (B).

   ---

   We start with the equation

   $x^2 + 4y + 2x - 8 = 0.$

   This is a parabola, but it’s not in vertex form. The first step is always to complete the square in $x$ to see its orientation and find its axis.

   1. Rewrite the parabola in standard form

   Group the $x$-terms:

   $x^2 + 2x + 4y - 8 = 0.$

   Complete the square:

   $(x^2 + 2x + 1) + 4y - 8 - 1 = 0 \quad\Rightarrow\quad (x+1)^2 + 4y - 9 = 0.$

   So

   $(x+1)^2 = -4y + 9 = -4\left(y - \frac{9}{4}\right).$

   Thus the parabola is

   $(x+1)^2 = -4\left(y - \frac{9}{4}\right).$

   This is a downward-opening parabola with vertex at $(-1, \frac{9}{4})$ and focal length $1$ (since $4a = 4$ gives $a=1$).

   2. Find the point at $t=1$

   The problem says “at the point $t=1$”. For a parabola of the form $(x+1)^2 = -4(y - 9/4)$, a common parametric form is:

   Let $x+1 = 2t$ (or sometimes $x+1 = -2t$; we’ll check consistency). Then

   $(2t)^2 = -4(y - 9/4) \quad\Rightarrow\quad 4t^2 = -4y + 9 \quad\Rightarrow\quad y = \frac{9}{4} - t^2.$

   So a convenient parametrization is

   $x = 2t - 1, \quad y = \frac{9}{4} - t^2.$

   At $t=1$:

   $x = 2(1) - 1 = 1, \quad y = \frac{9}{4} - 1 = \frac{5}{4}.$

   So the point is $P(1, \frac{5}{4})$.

   3. Find the equation of the normal at $P$

   Differentiate implicitly:

   $2x + 2 + 4\frac{dy}{dx} = 0 \quad\Rightarrow\quad \frac{dy}{dx} = -\frac{x+1}{2}.$

   At $x=1$, slope of tangent is

   $m_t = -\frac{1+1}{2} = -1.$

   Thus slope of normal is the negative reciprocal: $m_n = 1$.

   Equation of normal through $P(1, \frac{5}{4})$:

   $y - \frac{5}{4} = 1(x - 1) \quad\Rightarrow\quad y = x - 1 + \frac{5}{4} = x - \frac{1}{4}.$

   4. Find the second intersection $A(\alpha, \beta)$

   Solve the system:

   $y = x - \frac{1}{4}, \quad x^2 + 4y + 2x - 8 = 0.$

   Substitute $y$:

   $x^2 + 4\left(x - \frac{1}{4}\right) + 2x - 8 = 0.$

   Simplify:

   $x^2 + 4x - 1 + 2x - 8 = 0 \quad\Rightarrow\quad x^2 + 6x - 9 = 0.$

   Solve:

   $x = \frac{-6 \pm \sqrt{36 + 36}}{2} = \frac{-6 \pm 6\sqrt{2}}{2} = -3 \pm 3\sqrt{2}.$

   One root is $x=1$ (the original point), the other is $\alpha = -3 - 3\sqrt{2}$ (since $-3 + 3\sqrt{2} \approx 1.242$ is not 1? Wait check: $-3 + 3\sqrt{2} \approx -3 + 4.242 = 1.242$, not exactly 1. But we know $x=1$ must satisfy the quadratic because $P$ lies on both curves. Let’s verify: plug $x=1$ into $x^2+6x-9$: $1+6-9=-2 \neq 0$. Something’s off — we made an algebra slip.)

   Correction:

   Substitute $y = x - \frac{1}{4}$ into the original parabola equation:

   $x^2 + 4(x - \frac{1}{4}) + 2x - 8 = x^2 + 4x - 1 + 2x - 8 = x^2 + 6x - 9.$

   But $x=1$ gives $1+6-9=-2$, so indeed $x=1$ is not a root. That means our parametrization at $t=1$ gave a point that does not satisfy the normal? Let’s re-check the parametrization.

   Re-parametrization:

   Standard form: $(x+1)^2 = -4(y - 9/4)$. A common parametric choice is $x+1 = -2t$, then

   $(-2t)^2 = 4t^2 = -4(y - 9/4) \Rightarrow y = 9/4 - t^2.$

   So $x = -2t - 1$. At $t=1$: $x = -3$, $y = 9/4 - 1 = 5/4$. That point is $(-3, 5/4)$. Check: $(-3)^2 + 4(5/4) + 2(-3) - 8 = 9 + 5 - 6 - 8 = 0$. Good.

   So the correct point at $t=1$ is $P(-3, \frac{5}{4})$.

   5. Normal at correct point

   Slope of tangent: $\frac{dy}{dx} = -\frac{x+1}{2}$. At $x=-3$:

   $m_t = -\frac{-3+1}{2} = -\frac{-2}{2} = 1.$

   So tangent slope = 1, normal slope = $-1$.

   Normal equation:

   $y - \frac{5}{4} = -1(x + 3) \quad\Rightarrow\quad y = -x - 3 + \frac{5}{4} = -x - \frac{7}{4}.$

   6. Second intersection

   Solve

   $y = -x - \frac{7}{4}, \quad x^2 + 4y + 2x - 8 = 0.$

   Substitute:

   $x^2 + 4(-x - \frac{7}{4}) + 2x - 8 = x^2 - 4x - 7 + 2x - 8 = x^2 - 2x - 15 = 0.$

   Factor:

   $(x - 5)(x + 3) = 0.$

   Roots: $x = -3$ (the original point) and $x = 5$. So $\alpha = 5$.

   Then $\beta = -5 - \frac{7}{4} = -\frac{20}{4} - \frac{7}{4} = -\frac{27}{4}$.

   7. Compute $4\alpha\beta$

   $4\alpha\beta = 4 \cdot 5 \cdot \left(-\frac{27}{4}\right) = 5 \cdot (-27) = -135.$

   That’s not among the options. Something is still off — let’s check the parametrization convention.

   Watch out

   The parameter $t$ is often defined differently for parabolas. For a parabola $(x+1)^2 = -4(y - 9/4)$, a standard parametric form is $x+1 = 2t$, $y = 9/4 - t^2$. At $t=1$, that gave $(1, 5/4)$, which didn’t satisfy the normal correctly because the normal slope was 1, not -1. But the quadratic gave $x^2+6x-9=0$ with roots not including 1 — meaning the normal from that point didn’t pass through it? That’s impossible. So the parametrization must be $x+1 = -2t$ to have $t=1$ correspond to a point on the parabola. That gave $(-3, 5/4)$, and the normal slope was -1, leading to $x^2 - 2x - 15 = 0$ and $\alpha=5$, $\beta=-27/4$, product -135. Still not an option.

   Let’s re-check the original equation: $x^2 + 4y + 2x - 8 = 0$. At $(-3, 5/4)$: $9 + 5 - 6 - 8 = 0$, correct. At $(5, -27/4)$: $25 - 27 + 10 - 8 = 0$, correct. So the normal from $(-3, 5/4)$ indeed meets the parabola again at $(5, -27/4)$. Then $4\alpha\beta = 4 \cdot 5 \cdot (-27/4) = -135$. But -135 is not listed.

   Tip

   Perhaps the parameter $t$ is defined differently: sometimes for a parabola $x^2 = 4ay$, the parametric form is $(2at, at^2)$. Here our parabola is shifted. Let’s shift origin: let $X = x+1$, $Y = y - 9/4$. Then $X^2 = -4Y$. Parametric: $X = 2t$, $Y = -t^2$. Then $x = 2t - 1$, $y = 9/4 - t^2$. At $t=1$: $(1, 5/4)$. That’s the point we first tried. The normal there had slope 1, equation $y = x - 1/4$. Intersection gave $x^2+6x-9=0$, roots $x = -3 \pm 3\sqrt{2}$. One root is $x=1$? No, $1$ is not a root. So the normal from $(1, 5/4)$ does not pass through $(1, 5/4)$? That’s impossible — we built it to. Let’s check: at $x=1$, $y = 1 - 1/4 = 3/4$, not $5/4$. So we made an arithmetic error in the normal equation earlier.

   Re-do for $P(1, 5/4)$:

   Slope of tangent: $dy/dx = -(x+1)/2$. At $x=1$: $-(2)/2 = -1$. So tangent slope = -1, normal slope = 1.

   Normal: $y - 5/4 = 1(x - 1) \Rightarrow y = x - 1 + 5/4 = x + 1/4$.

   Now substitute: $x^2 + 4(x + 1/4) + 2x - 8 = x^2 + 4x + 1 + 2x - 8 = x^2 + 6x - 7 = 0$.

   Roots: $x = \frac{-6 \pm \sqrt{36 + 28}}{2} = \frac{-6 \pm \sqrt{64}}{2} = \frac{-6 \pm 8}{2}$. So $x = 1$ or $x = -7$.

   Thus $\alpha = -7$, $\beta = -7 + 1/4 = -27/4$.

   Then $4\alpha\beta = 4 \cdot (-7) \cdot (-27/4) = 7 \cdot 27 = 189$.

   That matches option (A).

   Tip

   The pitfall was misidentifying which point corresponds to $t=1$. Using the standard parametric $x = 2t - 1$, $y = 9/4 - t^2$ gives $P(1, 5/4)$ at $t=1$, and the normal there indeed yields the second intersection at $(-7, -27/4)$, giving $4\alpha\beta = 189$.

   ✓Final answer

   The correct option is (A).

   ANSWER: A

8. TG EAPCET 2026Set eng-2026-05-11-AN1 markMCQ
   Q.If the local maximum value of the real valued function $f(x) = 2(x^3 - 1) - 3a x(x + 4a)$, $a > 0$ is 54, then $a =$ (A) 4 (B) 2 (C) 9 (D) 27

   ›Reveal solutionSolution

   The key is to find the critical point of the cubic, enforce that it gives a local maximum of 54, and solve for $a$. The answer is $a = 2$.

   We are given

   $f(x) = 2(x^3 - 1) - 3a x(x + 4a), \quad a > 0$

   and told that its local maximum value is 54. We need to find $a$.

   ---

   Concept and Intuition

   A local maximum of a differentiable function occurs at a critical point where the derivative is zero and the second derivative is negative. Here $f$ is a cubic (after expanding), so it will have exactly one local maximum and one local minimum (since the leading coefficient is positive). The problem gives us the value of the local maximum, so we can set up equations:

   1. Find $x$ where $f'(x)=0$.
   2. Check which critical point gives a local maximum (second derivative test).
   3. Set $f$ at that point equal to 54 and solve for $a$.

   ---

   Step-by-step solution

   1. Expand and simplify $f(x)$.

   $f(x) = 2x^3 - 2 - 3a x^2 - 12a^2 x$

   So

   $f(x) = 2x^3 - 3a x^2 - 12a^2 x - 2$

   2. Find the derivative.

   $f'(x) = 6x^2 - 6a x - 12a^2$

   Factor out 6:

   $f'(x) = 6(x^2 - a x - 2a^2)$

   3. Set $f'(x)=0$ to find critical points.

   $x^2 - a x - 2a^2 = 0$

   This factors nicely:

   $(x - 2a)(x + a) = 0$

   So the critical points are

   $x = 2a \quad \text{and} \quad x = -a$

   4. Determine which is the local maximum.

   Take the second derivative:

   $f''(x) = 12x - 6a$

   • At $x = 2a$: $f''(2a) = 12(2a) - 6a = 24a - 6a = 18a > 0$ (since $a>0$), so this is a local minimum.
   • At $x = -a$: $f''(-a) = 12(-a) - 6a = -12a - 6a = -18a < 0$, so this is a local maximum.

   Thus the local maximum occurs at $x = -a$.

   Tip

   For a cubic with positive leading coefficient, the leftmost critical point is the local maximum and the rightmost is the local minimum. Here $-a < 2a$ for $a>0$, so indeed $-a$ gives the max.

   5. Evaluate $f$ at the local maximum and set equal to 54.

   $f(-a) = 2(-a)^3 - 3a(-a)^2 - 12a^2(-a) - 2$

   Compute term by term:

   • $2(-a)^3 = 2(-a^3) = -2a^3$
   • $-3a(-a)^2 = -3a(a^2) = -3a^3$
   • $-12a^2(-a) = +12a^3$
   • Then $-2$

   Sum:

   $-2a^3 - 3a^3 + 12a^3 - 2 = 7a^3 - 2$

   So

   $f(-a) = 7a^3 - 2$

   Set equal to 54:

   $7a^3 - 2 = 54 \quad \Rightarrow \quad 7a^3 = 56 \quad \Rightarrow \quad a^3 = 8$

   Since $a > 0$, we get

   $a = 2$

   Watch out

   A common mistake is to forget that the local maximum occurs at $x = -a$, not at $x = 2a$. Always check the sign of the second derivative.

   ---

   ✓Final answer

   The correct option is (B).

   ANSWER: B

9. TG EAPCET 2025Set eng-2025-05-02-FN1 markMCQ
   Q.Let ‘a’ be a positive real number. If a real valued function

   [!FORMULA] $f(x) = \begin{cases} \frac{6^{x} - 3^{x} - 2^{x} + 1}{1 - \cos\left(\frac{x}{a}\right)} & \text{if } x \neq 0 \\ \log 3 \log 4 & \text{if } x = 0 \end{cases}$

   is continuous at $x = 0$, then $a =$ (A) 1 (B) 2 (C) 3 (D) 4

   ›Reveal solutionSolution

   Factoring the numerator as $(2^x-1)(3^x-1)$ and matching the limit to $f(0)$ gives $a=1$ — option (A).

   Continuity condition. $\displaystyle\lim_{x\to0}f(x)=f(0)=\log 3\,\log 4$.

   Numerator.

   $6^x-3^x-2^x+1=3^x(2^x-1)-(2^x-1)=(2^x-1)(3^x-1).$

   As $x\to0$, $2^x-1\sim x\ln2$ and $3^x-1\sim x\ln3$, so the numerator $\sim x^2\ln2\,\ln3$.

   Denominator. $1-\cos\dfrac{x}{a}\sim\dfrac12\Big(\dfrac{x}{a}\Big)^2=\dfrac{x^2}{2a^2}$.

   Limit.

   $\lim_{x\to0}f(x)=\frac{x^2\ln2\,\ln3}{\dfrac{x^2}{2a^2}}=2a^2\ln2\,\ln3.$

   Match $f(0)$. Since $\log 4=2\log 2$, we have $f(0)=2\ln2\,\ln3$. Therefore

   $2a^2\ln2\,\ln3=2\ln2\,\ln3\ \Longrightarrow\ a^2=1\ \Longrightarrow\ a=1\ (a>0).$

   ✓Final answer

   $a=1$ — option (A).

10. TG EAPCET 2025Set eng-2025-05-02-FN1 markMCQ
    Q.If $f(x) = \sqrt{\cos^{-1} \sqrt{1-x^2}}$, then $f'\left(\dfrac{1}{2}\right) =$ (A) $\sqrt{\dfrac{2}{\pi}}$ (B) $\sqrt{\dfrac{\pi}{2}}$ (C) $-\sqrt{\dfrac{2}{\pi}}$ (D) $-\sqrt{\dfrac{\pi}{2}}$

    ›Reveal solutionSolution

    For $x\ge0$, $\cos^{-1}\sqrt{1-x^2}=\sin^{-1}x$, so $f(x)=\sqrt{\sin^{-1}x}$ and $f'\!\left(\tfrac12\right)=\sqrt{\tfrac{2}{\pi}}$ — option (A).

    Simplify. For $0\le x\le 1$, let $\theta=\cos^{-1}\sqrt{1-x^2}\in\big[0,\tfrac{\pi}{2}\big]$. Then $\cos^2\theta=1-x^2$, so $\sin\theta=x$ and $\theta=\sin^{-1}x$. Hence

    $f(x)=\sqrt{\sin^{-1}x}.$

    Differentiate.

    $f'(x)=\frac{1}{2\sqrt{\sin^{-1}x}}\cdot\frac{1}{\sqrt{1-x^2}}.$

    Evaluate at $x=\tfrac12$. $\sin^{-1}\tfrac12=\dfrac{\pi}{6}$ and $\sqrt{1-\tfrac14}=\dfrac{\sqrt3}{2}$:

    $f'\!\left(\tfrac12\right)=\frac{1}{2\sqrt{\pi/6}}\cdot\frac{2}{\sqrt3}=\frac{1}{\sqrt{\pi/6}\,\sqrt3}=\frac{1}{\sqrt{\pi/2}}=\sqrt{\frac{2}{\pi}}.$

    Since $f$ is increasing on $[0,1]$, the sign is positive.

    ✓Final answer

    $f'\!\left(\tfrac12\right)=\sqrt{\dfrac{2}{\pi}}$ — option (A).

11. TG EAPCET 2025Set eng-2025-05-02-AN1 markMCQ
    Q.If $f(x) = \begin{cases} \dfrac{a \sin x - b x + c x^2 + x^3}{2 \log(1+x) - 2x^3 + x^4}, & x \neq 0 \\ 0, & x = 0 \end{cases}$ is continuous at $x = 0$, then (A) $a = 2b$ (B) $a = b$ (C) $a = b = c$ (D) $b = c$

    ›Reveal solutionSolution

    Continuity at $x=0$ needs $\lim_{x\to0}f(x)=0$; since the denominator is of order $x$, the numerator's $x$-term must vanish, forcing $a=b$ — option (B).

    Continuity requirement. Because $f(0)=0$, we need

    $\lim_{x\to0}\frac{a\sin x-bx+cx^2+x^3}{2\log(1+x)-2x^3+x^4}=0.$

    Series expansions. Using $\sin x=x-\tfrac{x^3}{6}+\cdots$ and $\log(1+x)=x-\tfrac{x^2}{2}+\tfrac{x^3}{3}-\cdots$:

    $\text{Numerator}=(a-b)x+cx^2+\left(1-\tfrac{a}{6}\right)x^3+\cdots,$

    $\text{Denominator}=2\log(1+x)-2x^3+x^4=2x-x^2-\tfrac{4}{3}x^3+\cdots.$

    Order condition. The denominator's lowest-order term is $2x$ (order $1$). For the ratio to tend to $0$, the numerator must be of order higher than $1$, so its $x$-coefficient must vanish:

    $a-b=0\quad\Rightarrow\quad a=b.$

    Then the numerator begins at order $x^2$ and the quotient behaves like $\tfrac{c}{2}x\to0$, giving the required limit for any $c$. Thus the only forced relation is $a=b$.

    ✓Final answer

    Continuity forces $a=b$ — option (B).

12. TG EAPCET 2025Set eng-2025-05-02-AN1 markMCQ
    Q.If a particle is moving in a straight line so that after $t$ seconds its distance $S$ (in cms) from a fixed point on the line is given by $S = f(t) = t^3 - 5t^2 + 8t$ then the acceleration of the particle at $t = 5$ sec is (in cm/sec$^2$) (A) 10 (B) 30 (C) 20 (D) 40

    ›Reveal solutionSolution

    Acceleration is the second derivative of position with respect to time. For $S = t^3 - 5t^2 + 8t$, the acceleration at $t=5$ is $20\ \text{cm/s}^2$, so the correct option is (C).

    ---

    The key idea is that acceleration is the rate of change of velocity, and velocity is the rate of change of position. So if we have position $S(t)$, we differentiate once to get velocity $v(t)$, and differentiate again to get acceleration $a(t)$. Then we just plug in $t = 5$.

    Let’s walk through it step by step.

    1. Find the velocity function. Velocity is the first derivative of position with respect to time:

    $v(t) = \frac{dS}{dt} = \frac{d}{dt}(t^3 - 5t^2 + 8t)$

    Using the power rule:

    $v(t) = 3t^2 - 10t + 8$

    2. Find the acceleration function. Acceleration is the derivative of velocity (or the second derivative of position):

    $a(t) = \frac{dv}{dt} = \frac{d}{dt}(3t^2 - 10t + 8)$

    Again using the power rule:

    $a(t) = 6t - 10$

    3. Evaluate at $t = 5$ seconds. Substitute $t = 5$ into the acceleration function:

    $a(5) = 6(5) - 10 = 30 - 10 = 20$

    So the acceleration is $20\ \text{cm/s}^2$.

    Tip

    A common shortcut: since acceleration is the second derivative, you can directly differentiate $S$ twice in one step:

    $a(t) = \frac{d^2S}{dt^2} = \frac{d}{dt}(3t^2 - 10t + 8) = 6t - 10$

    Then evaluate. This avoids writing the intermediate velocity step separately, though it’s the same process.

    Watch out

    A classic mistake is to confuse acceleration with velocity and plug $t=5$ into the first derivative. That gives $v(5) = 3(25) - 50 + 8 = 33$, which isn’t even one of the options — but it’s a trap if you forget to differentiate twice.

    ✓Final answer

    The correct option is (C).

    ANSWER: C