JEE Mains · Maths · STD 11 - 9. straight line
Let a rectangle \(A B C D\) of sides \(2\) and \(4\) be inscribed in another rectangle \(P Q R S\) such that the vertices of the rectangle \(A B C D\) lie on the sides of the rectangle \(P Q R S\). Let \(a\) and \(b\) be the sides of the rectangle \(P Q R S\) when its area is maximum. Then \((a+b)^2\) is equal to :
- A \(72\)
- B \(60\)
- C \(80\)
- D \(64\)
Answer & Solution
Correct Answer
(A) \(72\)
Step-by-step Solution
Detailed explanation
\( \text { Area }=(4 \cos \theta+2 \sin \theta)(2 \cos \theta+4 \sin \theta) \) \( =8 \cos ^2 \theta+16 \sin \theta \cos \theta+4 \sin \theta \cos \theta+8 \sin ^2 \theta \) \( =8+20 \sin \theta \cos \theta \) \( =8+10 \sin 2 \theta \)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If the distances of the point (1, 2, a) from the line \(\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}\) along the lines
\(L_{1}:\frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b}\) and
\(L_{2}:\frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c}\) are equal,
then \(a+b+c\) is equal toJEE Mains 2026 Easy - In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is \(\mathbb{R}-(a,b)\), then \(a^{2}+b^{2}\) is equal to ___ .JEE Mains 2026 Easy
- If the system of equations
\(\begin{aligned} & (\lambda-1) x+(\lambda-4) y+\lambda z=5 \\ & \lambda x+(\lambda-1) y+(\lambda-4) z=7 \\ & (\lambda+1) x+(\lambda+2) y-(\lambda+2) z=9\end{aligned}\)
has infinitely many solutions, then \(\lambda^2+\lambda\) is equal toJEE Mains 2025 Easy - Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be such that \(f(xy) = f(x)f(y)\), for all \(x, y \in \mathbb{R}\) and \(f(0) \neq 0\). Let \(g: [1, \infty) \rightarrow \mathbb{R}\) be a differentiable function such that
\(x^2 g(x) = \int\limits_1^x (t^2 f(t) - tg(t))\,dt\).
Then \(g(2)\) is equal to :JEE Mains 2026 Hard - Let \(a , b , c\) and \(d\) be positive real numbers such that \(a+b+c+d=11\). If the maximum value of \(a^5 b^3 c^2 d\) is \(3750 \beta\), then the value of \(\beta\) isJEE Mains 2023 Hard
- Let the centre of the circle \(x^2 + y^2 + 2gx + 2fy + 25 = 0\) be in the first quadrant and lie on the line \(2x - y = 4\). Let the area of an equilateral triangle inscribed in the circle be \(27\sqrt{3}\). Then the square of the length of the chord of the circle on the line \(x = 1\) is _______.JEE Mains 2026 Hard
More PYQs from JEE Mains
- If non-zero real numbers \(b\) and \(c\) are such that \(min \,f\left( x \right) > \max \,g\left( x \right)\), where \(f\left( x \right) = {x^2} + 2bx + 2{c^2}\) and \(g\left( x \right) = {-x^2} - 2cx + {b^2}\)\(\left( {x \in R} \right)\); then \(\left| {\frac{c}{b}} \right|\) lies in the intervalJEE Mains 2014 Hard
- Let the locus of the mid points of the chords of circle \(x^2+(y-1)^2=1\) drawn from the origin intersect the line \(x+y=1\) at \(P\) and \(Q\). Then, the length of \(P Q\) is :JEE Mains 2024 Hard
- Let \(f(x)\) be a quadratic polynomial such that \(f(-2)\) \(+f(3)=0\). If one of the roots of \(f(x)=0\) is \(-1\), then the sum of the roots of \(f(x)=0\) is equal toJEE Mains 2022 Hard
- The sum, \(\sum\limits_{n=1}^{7} \frac{n(n+1)(2 n+1)}{4}\) is equal toJEE Mains 2020 Medium
- If \(\mathrm{b}\) is very small as compared to the value of \(\mathrm{a}\), so that the cube and other higher powers of \(\frac{b}{a}\) can be neglected in the identity \(\frac{1}{a-b}+\frac{1}{a-2 b}+\frac{1}{a-3 b} \ldots .+\frac{1}{a-n b}=\alpha n+\beta n^{2}+\gamma n^{3}\), then the value of \(\gamma\) is:JEE Mains 2021 Hard
- The angle of elevation of a jet plane from a point \(A\) on the ground is \(60^{\circ}\). After a flight of \(20\, seconds\) at the speed of \(432\, km / hour\), the angle of elevation changes to \(30^{\circ}\). If the jet plane is flying at a constant height, then its height is ..... \(m.\)JEE Mains 2021 Hard