JEE Mains · Maths · STD 11 - 1. set theory
Let \(A\) and \(B\) be two finite sets with \(m\) and \(n\) elements respectively. The total number of subsets of the set \(A\) is 56 more than the total number of subsets of \(B\). Then the distance of the point \(P ( m , n )\) from the point \(Q (-2,-3)\) is
- A 10
- B 6
- C 4
- D 8
Answer & Solution
Correct Answer
(A) 10
Step-by-step Solution
Detailed explanation
\(2^m-2^n=56\) \(2^n\left(2^{m-n}-1\right)=2^3 \times 7\) \(2^{ n }=2^3\) and \(2^{ m - n }-1=7\) \(\Rightarrow n =3\) and \(2^{ m - n }=8\) \(\Rightarrow n =3\) and \(m - n =3\) \(\Rightarrow n =3\) and \(m =6\) \(P (6,3)\) and \(Q (-2,-3)\) \(P Q=\sqrt{8^2+6^2}=\sqrt{100}=10\)
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
- Let \(A_1\) and \(A_2\) be two arithmetic means and \(G_1, G_2\), \(G _3\) be three geometric means of two distinct positive numbers. The \(G _1^4+ G _2^4+ G _3^4+ G _1^2 G _3^2\) is equal toJEE Mains 2023 Hard
- The value of \(k \in R\), for which the following system of linear equations \(3 x-y+4 z=3\) \(x+2 y-3 x=-2\) \(6 x+5 y+k z=-3\) has infinitely many solutions, is:JEE Mains 2021 Medium
- Let the number \((22)^{2022}+(2022)^{22}\) leave the remainder \(\alpha\) when divided by \(3\) and \(\beta\) when divided by \(7\). Then \(\left(\alpha^2+\beta^2\right)\) is equal toJEE Mains 2023 Hard
- Let the lines \(l_1: \frac{ x +5}{3}=\frac{ y +4}{1}=\frac{ z -\alpha}{-2}\) and \(l_2: 3 x +\) \(2 y+z-2=0=x-3 y+2 z-13\) be coplanar. If the point \(P ( a , b , c )\) on \(l_1\) is nearest to the point \(Q (-\) \(4,-3,2)\), then \(|a|+|b|+|c|\) is equal toJEE Mains 2023 Hard
- The sum of all the elements in the range of \(f(x)=\text{Sgn}(\sin x)+\text{Sgn}(\cos x)+\text{Sgn}(\tan x)+\text{Sgn}(\cot x), x\ne\frac{n\pi}{2}, n\in Z\), where \(\operatorname{Sgn}(t)=\left\{\begin{array}{lll}1, & \text { if } & t>0 \\ -1 & \text { if } & t<0\end{array}\right.\), isJEE Mains 2026 Medium
- A circle passing through the point \(P (\alpha, \beta)\) in the first quadrant touches the two coordinate axes at the points \(A\) and \(B\). The point \(P\) is above the line \(A B\). The point \(Q\) on the line segment \(A B\) is the foot of perpendicular from \(P\) on \(A B\). If \(P Q\) is equal to \(11\) units, then the value of \(\alpha \beta\) is \(.............\).JEE Mains 2023 Hard
More PYQs from JEE Mains
- Consider the system of linear equation \(x+y+z=\) \(4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15\), where \(\lambda, \mu \in R\). Which one of the following statements is \(NOT\) correct?JEE Mains 2024 Hard
- Let \(S = \{\theta \in (-2\pi, 2\pi) : \cos\theta + 1 = \sqrt{3}\sin\theta\}\). Then \(\sum_{\theta \in S}\theta\) is equal to:JEE Mains 2026 Medium
- If the set \(\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in C , \operatorname{Re}(z)=3\right\}\) is equal to the interval \((\alpha, \beta]\), then \(24(\beta-\alpha)\) is equal toJEE Mains 2023 Hard
- Let the volume of a parallelopiped whose coterminous edges are given by \(\overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \overrightarrow{\mathrm{v}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}} \) and \(\overrightarrow{\mathrm{w}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\) be \(1\; cu.\) unit. If \(\theta\) be the angle between the edges \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{w}},\) then \(\cos \theta\) can beJEE Mains 2020 Hard
- Let \(P ( x )= x ^{2}+ bx + c\) be a quadratic polynomial with real coefficients such that \(\int_{0}^{1} P ( x ) dx =1\) and \(P ( x )\) leaves remainder \(5\) when it is divided by \((x-2)\). Then the value of \(9(b+c)\) is equal to:JEE Mains 2021 Hard
- If \(y = y ( x ), x \in\left(0, \frac{\pi}{2}\right)\) be the solution curve of the differential equation \(\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x+2 \sin 4 x\right) y=\)\(2 e ^{-4 x }(2 \sin 2 x +\cos 2 x )\), with \(y \left(\frac{\pi}{4}\right)= e ^{-\pi}\), then \(y \left(\frac{\pi}{6}\right)\) is equal to.JEE Mains 2022 Hard