JEE Mains · Maths · STD 11 - 13. statistics
The number of values of \(a \in N\) such that the variance of \(3,7,12 a, 43-a\) is a natural number is (Mean \(=13\))
- A \(0\)
- B \(2\)
- C \(5\)
- D infinite
Answer & Solution
Correct Answer
(A) \(0\)
Step-by-step Solution
Detailed explanation
Mean \(=13\) Variance \(=\frac{9+49+144+ a ^{2}+(43- a )^{2}}{5}-13^{2} \in N\) \(\Rightarrow \frac{2 a^{2}-a+1}{5} \in N\) \(\Rightarrow 2 a^{2}-a+1-5 n=0\) must have solution as natural numbers its \(D=40 n-7\) always has \(3\) at unit place \(\Rightarrow D\) can't be perfect…
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 system of equations:
\(x+y+z=5\)
\(x+2y+3z=9\)
\(x+3y+\lambda z=\mu\)
has infinitely many solutions, then the value of \(\lambda+\mu\) is:JEE Mains 2026 Medium - \(\int\limits_{0}^{5} \cos \left(\pi\left(x-\left[\frac{x}{2}\right]\right)\right) d x\) Where \([t]\) denotes greatest integer less than or equal to \(t\), is equal toJEE Mains 2022 Hard
- Suppose a class has \(7\) students. The average marks of these students in the mathematics examination is \(62\), and their variance is \(20\) . A student fails in the examination if \(he/she\) gets less than \(50\) marks, then in worst case, the number of students can fail isJEE Mains 2022 Medium
- Let \(\mathrm{a} \in \mathbf{R}\) and A be a matrix of order \(3 \times 3\) such that \(\operatorname{det}(A)=-4\) and \(A+I=\left[\begin{array}{lll}1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2\end{array}\right]\), where \(I\) is the identity matrix of order \(3 \times 3\).
If \(\operatorname{det}((a+1) \operatorname{adj}((a-1) A))\) is \(2^m 3^n, m, n \in\) \(\{0,1,2, \ldots .20\}\), then \(\mathrm{m}+\mathrm{n}\) is equal to :JEE Mains 2025 Hard - Let \(\left\{a_{n}\right\}_{n-1}^{\infty}\) be a sequence such that \(a_{1}=1, a_{2}=1\) and \(a_{n+2}=2 a_{n+1}+a_{n}\) for all \(n \geq 1 .\) Then tha value of \(47 \sum_{n=1}^{\infty} \frac{a_{n}}{2^{3 n}}\) is equal to \(.....\)JEE Mains 2021 Hard
- Let \(y=y(x)\) be the solution curve of the differential equation
\(x\left(x^2+e^x\right) d y+\left(e^x(x-2) y-x^3\right) d x=0, x \gt 0\) passing through the point \((1,0)\). Then \(y(2)\) is equal to :JEE Mains 2025 Medium
More PYQs from JEE Mains
- \(\lim \limits_{x \rightarrow \frac{\pi}{2}}(\tan ^{2} x((2 \sin ^{2} x+3 \sin x+4)^{\frac{1}{2}}\) \(-(\sin ^{2} x+6 \sin x+2)^{\frac{1}{2}}))\) is equal toJEE Mains 2022 Hard
- The differential equation of the family of curves, \(x^{2}=4 b(y+b), b \in R,\) isJEE Mains 2020 Hard
- The mean age of \(25\) teachers in a school is \(40\) years. A teacher retires at the age of \(60\) years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is \(39\) years, then the age (in years) of the newly appointed teacher is..........JEE Mains 2021 Medium
- Let \(\displaystyle\int_{-2}^{2} (|\sin x| + [x \sin x])\,dx = 2(3 - \cos 2) + \beta\), where \([\cdot]\) is the greatest integer function. Then \(\beta \sin\left(\dfrac{\beta}{2}\right)\) equals:JEE Mains 2026 Medium
- Let the hyperbola \(H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1\) pass through the point \((2 \sqrt{2},-2 \sqrt{2})\). A parabola is drawn whose focus is same as the focus of \(H\) with positive abscissa and the directrix of the parabola passes through the other focus of \(H\). If the length of the latus rectum of the parabola is e times the length of the latus rectum of \(H\), where \(e\) is the eccentricity of \(H\), then which of the following points lies on the parabola?JEE Mains 2022 Hard
- Let \(\mathrm{C}\) be the set of all complex numbers. Let \(\mathrm{S}_{1} =\left\{\mathrm{z} \in \mathrm{C}|| \mathrm{z}-3-\left.2 \mathrm{i}\right|^{2}=8\right\}\) \(\mathrm{S}_{2} =\{\mathrm{z} \in \mathrm{C} \mid \operatorname{Re}(\mathrm{z}) \geq 5\} \text { and }\) \(\mathrm{S}_{3} =\{\mathrm{z} \in \mathrm{C} \| \mathrm{z}-\bar{z} \mid \geq 8\}\) Then the number of elements in \(\mathrm{S}_{1} \cap \mathrm{S}_{2} \cap \mathrm{S}_{3}\) is equal to:JEE Mains 2021 Hard