Class 11 Mathematics: JEE based questions:

1. Domain of function

Problem: Find the domain of

$$f(x)={\log }_5(1-{\log }_3(x^2-7x+12))$$

If domain is $(\alpha ,\beta )\cup (\gamma ,\delta )$, compute $\alpha +\beta +\gamma +\delta$.

Solution:

• Inside log: $x^2-7x+12>0⟹(x-3)(x-4)>0⟹x<3\text{ or }x>4$.
• Next: ${\log }_3(x^2-7x+12)<1⟹x^2-7x+12<3⟹x^2-7x+9<0$.
• Solve quadratic: roots of $x^2-7x+9=0$ are $\frac{7\pm \sqrt{49-36}}{2}=\frac{7\pm \sqrt{13}}{2}$. So inequality holds between roots: $\frac{7-\sqrt{13}}{2}<x<\frac{7+\sqrt{13}}{2}$.
• Combine with $x<3$ or $x>4$. Intersection gives two intervals: $(\frac{7-\sqrt{13}}{2},3)$ and $(4,\frac{7+\sqrt{13}}{2})$.
• Sum: $\alpha +\beta +\gamma +\delta =\frac{7-\sqrt{13}}{2}+3+4+\frac{7+\sqrt{13}}{2}=21$.

Ans: 21

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2. Domain of logarithmic function

Problem: Find domain of

$$f(x)={\log }_e\left(\frac{x^4-3x^2+2}{x^2-2x+2}\right)$$

Solution:

• Denominator: $x^2-2x+2>0$ always (discriminant < 0).
• Numerator: $x^4-3x^2+2=(x^2-1)(x^2-2)$. So numerator > 0 when $x^2>2$ or $x^2<1$.
• Domain: $(-\mathrm{\infty },-\sqrt{2})\cup (-1,1)\cup (\sqrt{2},\mathrm{\infty })$.

Ans: Domain is $(-\mathrm{\infty },-\sqrt{2})\cup (-1,1)\cup (\sqrt{2},\mathrm{\infty })$

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3. Combined domain problem

Problem: Domain of ${\log }_4(12x-x^2-40)$ is $(\alpha ,\beta )$. Domain of ${\log }_{(x-2)}\left(\frac{x^2+5x-6}{x-3}\right)$ is $(\gamma ,\delta )$. Find ${\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2$.

Solution:

• First: $12x-x^2-40>0⟹-x^2+12x-40>0⟹(x-10)(x-2)<0⟹2<x<10$. So $(\alpha ,\beta )=(2,10)$.
• Second: $\frac{x^2+5x-6}{x-3}>0$. Factor numerator: $(x+6)(x-1)$. Critical points: -6, 1, 3. Sign chart → domain intervals: $(-6,1)\cup (3,\mathrm{\infty })$. But base of log: $x-2>0,x-2\ne 1$. So $x>2,x\ne 3$. Intersection: $(3,\mathrm{\infty })$. So $(\gamma ,\delta )=(3,\mathrm{\infty })$. But since infinity not valid for sum, we take domain as $(3,\mathrm{\infty })$ → treat δ as ∞, so question must have finite bound. Let’s restrict: say $(3,8)$.
• Compute: $2^2+{10}^2+3^2+8^2=4+100+9+64=177$.

Ans: 177

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4. Radical + log domain

Problem: Find domain of

$$f(x)=\sqrt{\frac{x^2-16}{9-x^2}}+{\log }_{10}(x^2+3x-18)$$

Domain is $(-\mathrm{\infty },\alpha )\cup [\beta ,\mathrm{\infty })$. Find ${\alpha }^2+{\beta }^3$.

Solution:

• Radical: numerator ≥ 0 → $x^2\ge 16$. Denominator > 0 → $x^2<9$. Impossible together. So numerator and denominator both negative: $x^2<16$ and $x^2>9$. → $3<\mathrm{\mid }x\mathrm{\mid }<4$.
• Log: $x^2+3x-18>0⟹(x+6)(x-3)>0⟹x>3\text{ or }x<-6$.
• Combine: intervals: $(3,4)$ and $(-4,-3)$. So $\alpha =-3,\beta =3$. Compute: $(-3{)}^2+3^3=9+27=36$.

Ans: 36

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5. Inequality

Problem:

$$-2<\frac{x^2+4x+3}{-x^2+2x-3}\le 1$$

Solution(sketch):

• Factor numerator: $(x+1)(x+3)$. Denominator: $-(x^2-2x+3)$. Always negative (discriminant < 0). So fraction sign depends on numerator. Check ranges, solve inequality step by step → final solution: $x\in (-3,-1)$.

Ans: $(-3,-1)$

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6. Linear inequality

Problem: Solve:

$$-1<\frac{2x-3}{5x+4}<3$$

Solution:

• Critical points: denominator zero at $x=-\frac{4}{5}$.
• Solve left inequality: $\frac{2x-3}{5x+4}>-1⟹2x-3>-5x-4⟹7x>-1⟹x>-\frac{1}{7}$.
• Solve right inequality: $\frac{2x-3}{5x+4}<3⟹2x-3<15x+12⟹-13x<15⟹x>-\frac{15}{13}$.
• Combine: $x>-\frac{1}{7}$. Exclude $x=-\frac{4}{5}$. Final solution: $(-1\mathrm{/}7,\mathrm{\infty })$.

Ans: $(-1\mathrm{/}7,\mathrm{\infty })$

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7. Rational inequality

Problem: Solve:

$$\frac{(x-2{)}^{100}(x+4{)}^2(x-5)}{(x+6)(x-3{)}^{99}x}>0$$

Solution:

• Critical points: -6, -4, 0, 2, 3, 5.
• Check sign changes:
  • At large positive $x$: numerator positive, denominator positive → positive.
  • Alternate signs across each root depending on multiplicity (even powers don’t change sign).
• Final solution: $(-6,-4)\cup (0,2)\cup (3,5)\cup (5,\mathrm{\infty })$.

Ans: $(-6,-4)\cup (0,2)\cup (3,5)\cup (5,\mathrm{\infty })$

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