JEE Mains · Maths · STD 12 - 13. probability
The probability that a randomly selected \(2\) digit number belongs to the set \(\left(n \in N:\left(2^{n}-2\right)\right.\) is a multiple of \(3\, )\) is equal to:
- A \(\frac{1}{2}\)
- B \(\frac{1}{3}\)
- C \(\frac{2}{3}\)
- D \(\frac{1}{6}\)
Answer & Solution
Correct Answer
(A) \(\frac{1}{2}\)
Step-by-step Solution
Detailed explanation
Total number of cases \(={ }^{90} \mathrm{C}_{1}=90\) Now, \(2^{n}-2=(3-1)^{n}-2\) \({ }^{n} C_{0} 3^{n}-{ }^{n} C_{1} \cdot 3^{n-1}+\ldots+(-1)^{n-1} \cdot{ }^{n} C_{n-1} 3+(-1)^{n} \cdot{ }^{n} C_{n}-2\) \(3\left(3^{n-1}-n 3^{n-2}+\ldots+(-1)^{n-1} \cdot n\right)+(-1)^{n}-2\)…
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 S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :JEE Mains 2025 Medium
- The mean deviation about the mean for the data
is equal to:\(x_i\) \(5\) \(7\) \(9\) \(10\) \(12\) \(15\) \(f_i\) \(8\) \(6\) \(2\) \(2\) \(2\) \(6\) JEE Mains 2026 Medium - Let \(f:[-1,2] \rightarrow \mathrm{R}\) be given by \(f(x)=2 x^2+x+\left[x^2\right]-[x]\), where \([t]\) denotes the greatest integer less than or equal to \(t\). The number of points, where \(f\) is not continuous, is :JEE Mains 2024 Hard
- Marks obtains by all the students of class 12 are presented in a freqency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students isJEE Mains 2025 Medium
- A line passing through the point \(A(9,0)\) makes an angle of \(30^{\circ}\) with the positive direction of \(\mathrm{x}\)-axis. If this line is rotated about \(A\) through an angle of \(15^{\circ}\) in the clockwise direction, then its equation in the new position isJEE Mains 2024 Medium
- \(\lim _{x \rightarrow \infty} \frac{\left(2 x^2-3 x+5\right)(3 x-1)^{\frac{x}{2}}}{\left(3 x^2+5 x+4\right) \sqrt{(3 x+2)^x}}\) is equal to :JEE Mains 2025 Medium
More PYQs from JEE Mains
- Let \(B=\left[\begin{array}{ccc}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha > 2\) be the adjoint of \(a\) matrix \(A\) and \(| A |=2\), then \([\alpha\,\,-2 \alpha \,\, \alpha \,\,] B \left[\begin{array}{c}\alpha \\ -2 \alpha \\ \alpha\end{array}\right]\) is equal to :-JEE Mains 2023 Hard
- Let \(I(x)=\int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x\). If \(I(0)=0\) the \(I\) \(\left(\frac{\pi}{4}\right)\) is equal toJEE Mains 2023 Hard
- If the sum of the coefficients of all the positive even powers of \(x\) in the binomial expansion of \(\left(2 x^{3}+\frac{3}{x}\right)^{10}\) is \(5^{10}-\beta \cdot 3^{9}\), then \(\beta\) is equal toJEE Mains 2022 Hard
- Let \(\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}\) and \(\vec{c}\) be vectors such that \(\vec{a} \times \vec{c}=\vec{a} \times \vec{b}\). If \(\vec{a} \cdot \vec{c}=-12\), \(\vec{c} .(\hat{i}-2 \hat{j}+\hat{k})=5\), then \(\vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})\) is equal to \(.............\).JEE Mains 2023 Hard
- Let in a Binomial distribution, consisting of \(5\) independent trials, probabilities of exactly \(1\) and \(2\) successes be \(0.4096\) and \(0.2048\) respectively. Then the probability of getting exactly \(3\) successes is equal to ....... .JEE Mains 2021 Hard
- Let \(p, q\) and \(r\) be real numbers \((p \ne q,r \ne 0),\) such that the roots of the equation \(\frac{1}{{x + p}} + \frac{1}{{x + q}} = \frac{1}{r}\) are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to .JEE Mains 2018 Hard