LFM Stats And Pure

jee-main 2023 Q71 View

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then the correct mean and variance are
(1) 14 and 13.5
(2) 14 and 12.5
(3) 15 and 14.5
(4) 14 and 11.5

jee-main 2023 Q73 View

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $\alpha\ (> 0)$, and the mean and standard deviation of marks of class $B$ of $n$ students be respectively 55 and $30 - \alpha$. If the mean and variance of the marks of the combined class of $100 + n$ students are respectively 50 and 350, then the sum of variances of classes $A$ and $B$ is
(1) 500
(2) 450
(3) 650
(4) 900

jee-main 2023 Q73 View

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12 . If the new mean of the marks is 10.2 , then their new variance is equal to:
(1) 4.04
(2) 4.08
(3) 3.96
(4) 3.92

jee-main 2023 Q73 View

Let $S$ be the set of all values of $a_{1}$ for which the mean deviation about the mean of 100 consecutive positive integers $a_{1}, a_{2}, a_{3}, \ldots, a_{100}$ is 25. Then $S$ is
(1) $\phi$
(2) $\{99\}$
(3) $\mathbb{N}$
(4) $\{9\}$

jee-main 2023 Q74 View

If the mean and variance of the frequency distribution

$x_{i}$	2	4	6	8	10	12	14	16
$f_{i}$	4	4	$\alpha$	15	8	$\beta$	4	5

are 9 and 15.08 respectively, then the value of $\alpha^{2} + \beta^{2} - \alpha\beta$ is $\_\_\_\_$.

jee-main 2023 Q74 View

Let the mean and variance of 12 observations be $\frac { 9 } { 2 }$ and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. Find the correct mean and variance.

jee-main 2023 Q74 View

Let $X = \{ 11,12,13 , \ldots , 40,41 \}$ and $Y = \{ 61,62,63 , \ldots , 90,91 \}$ be the two sets of observations. If $\overline { \mathrm { x } }$ and $\overline { \mathrm { y } }$ are their respective means and $\sigma ^ { 2 }$ is the variance of all the observations in $\mathrm { X } \cup \mathrm { Y }$, then $\left| \overline { \mathrm { x } } + \overline { \mathrm { y } } - \sigma ^ { 2 } \right|$ is equal to $\_\_\_\_$

jee-main 2023 Q74 View

Let the mean and variance of 8 numbers $x , y , 10 , 12 , 6 , 12 , 4 , 8$ be 9 and 9.25 respectively. If $x > y$, then $3 x - 2 y$ is equal to $\_\_\_\_$

jee-main 2023 Q86 View

If the variance of the frequency distribution

$x_i$	2	3	4	5	6	7	8
Frequency $f_i$	3	6	16	$\alpha$	9	5	6

is 3, then $\alpha$ is equal to $\underline{\hspace{1cm}}$.

jee-main 2023 Q86 Tangent Lines and Tangent Lengths View

Let a common tangent to the curves $y^2 = 4x$ and $(x-4)^2 + y^2 = 16$ touch the curves at the points $P$ and $Q$. Then $PQ^2$ is equal to $\_\_\_\_$.

jee-main 2023 Q89 View

Two dice $A$ and $B$ are rolled. Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively. If the variance of $\alpha - \beta$ is $\frac { p } { q }$, where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is equal to
(1) 72
(2) 36
(3) 48
(4) 31

jee-main 2024 Q68 View

If the mean and variance of five observations are $\frac { 24 } { 5 }$ and $\frac { 194 } { 25 }$ respectively and the mean of first four observations is $\frac { 7 } { 2 }$, then the variance of the first four observations is equal to
(1) $\frac { 4 } { 5 }$
(2) $\frac { 77 } { 12 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 105 } { 4 }$

jee-main 2024 Q69 View

Consider 10 observations $x_1, x_2, \ldots, x_{10}$, such that $\sum_{i=1}^{10} (x_i - \alpha) = 2$ and $\sum_{i=1}^{10} (x_i - \beta)^2 = 40$, where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. Then $\frac{\beta}{\alpha}$ is equal to:
(1) 2
(2) $\frac{3}{2}$
(3) $\frac{5}{2}$
(4) 1

jee-main 2024 Q69 View

Let $M$ denote the median of the following frequency distribution.

Class	$0 - 4$	$4 - 8$	$8 - 12$	$12 - 16$	$16 - 20$
Frequency	3	9	10	8	6

Then 20 M is equal to :
(1) 416
(2) 104
(3) 52
(4) 208

jee-main 2024 Q69 View

The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12 . The correct standard deviation is
(1) 1.8
(2) 1.94
(3) $\sqrt { 3.96 }$
(4) $\sqrt { 3.86 }$

jee-main 2024 Q69 Probability Distribution Table Completion and Expectation Calculation View

If the mean of the following probability distribution of a random variable $X$ :

X	0	2	4	6	8
$\mathrm { P } ( \mathrm { X } )$	$a$	$2a$	$a + b$	$2b$	$3b$

is $\frac { 46 } { 9 }$, then the variance of the distribution is
(1) $\frac { 173 } { 27 }$
(2) $\frac { 566 } { 81 }$
(3) $\frac { 151 } { 27 }$
(4) $\frac { 581 } { 81 }$

jee-main 2024 Q69 View

If the variance of the frequency distribution

$x$	$c$	$2c$	$3c$	$4c$	$5c$	$6c$
$f$	2	1	1	1	1	1

is 160, then the value of $c \in N$ is
(1) 7
(2) 8
(3) 5
(4) 6

jee-main 2024 Q70 View

Let the mean and the variance of 6 observations $a, b, 68, 44, 48, 60$ be 55 and 194, respectively. If $a > b$, then $a + 3b$ is
(1) 200
(2) 190
(3) 180
(4) 210

jee-main 2024 Q70 View

Let $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 } , \ldots , \mathrm { a } _ { 10 }$ be 10 observations such that $\sum _ { \mathrm { k } = 1 } ^ { 10 } \mathrm { a } _ { \mathrm { k } } = 50$ and $\sum _ { \forall \mathrm { k } < \mathrm { j } } \mathrm { a } _ { \mathrm { k } } \cdot \mathrm { a } _ { \mathrm { j } } = 1100$. Then the standard deviation of $a _ { 1 } , a _ { 2 } , \ldots , a _ { 10 }$ is equal to:
(1) 5
(2) $\sqrt { 5 }$
(3) 10
(4) $\sqrt { 115 }$

jee-main 2024 Q84 Circles Tangent to Each Other or to Axes View

Consider a circle $x - \alpha ^ { 2 } + y - \beta ^ { 2 } = 50$, where $\alpha , \beta > 0$. If the circle touches the line $y + x = 0$ at the point P , whose distance from the origin is $4 \sqrt { 2 }$, then $( \alpha + \beta ) ^ { 2 }$ is equal to $\_\_\_\_$ .

jee-main 2024 Q85 Chord Length and Chord Properties View

Consider two circles $C_1: x^2 + y^2 = 25$ and $C_2: (x - \alpha)^2 + y^2 = 16$, where $\alpha \in (5, 9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\frac{\sqrt{63}}{8}$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha\beta)^2$ equals $\underline{\hspace{1cm}}$.

jee-main 2024 Q86 View

If the mean and variance of the data $65,68,58,44,48,45,60 , \alpha , \beta , 60$ where $\alpha > \beta$ are 56 and 66.2 respectively, then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to

jee-main 2024 Q86 View

If the variance $\sigma^2$ of the data $$\begin{array}{cccccccc} x_i & 0 & 1 & 5 & 6 & 10 & 12 & 17 \\ f_i & 3 & 2 & 3 & 2 & 6 & 3 & 3 \end{array}$$ is $k$, then the value of $\lfloor k \rfloor$ is $\underline{\hspace{1cm}}$ (where $\lfloor \cdot \rfloor$ denotes the greatest integer function).

jee-main 2024 Q86 View

Let the mean and the standard deviation of the probability distribution

X	$\alpha$	1	0	- 3
$\mathrm { P } ( \mathrm { X } )$	$\frac { 1 } { 3 }$	K	$\frac { 1 } { 6 }$	$\frac { 1 } { 4 }$

be $\mu$ and $\sigma$, respectively. If $\sigma - \mu = 2$, then $\sigma + \mu$ is equal to $\_\_\_\_$

jee-main 2024 Q86 View

Let $\mathrm { a } , \mathrm { b } , \mathrm { c } \in \mathrm { N }$ and $\mathrm { a } < \mathrm { b } < \mathrm { c }$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $9,25 , \mathrm { a } , \mathrm { b } , \mathrm { c }$ be 18,4 and $\frac { 136 } { 5 }$, respectively. Then $2 \mathrm { a } + \mathrm { b } - \mathrm { c }$ is equal to $\_\_\_\_$

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LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

Solving quadratics and applications 106

Curve Sketching 295

Factor & Remainder Theorem 90

Polynomial Division & Manipulation 44

Binomial Theorem (positive integer n) 244

Function Transformations 65

Composite & Inverse Functions 249

Partial Fractions 22

Generalised Binomial Theorem 16

Generalised Binomial Theorem and Partial Fractions 3

Complex Numbers Arithmetic 283

Complex Numbers Argand & Loci 135

Permutations & Arrangements 276

Combinations & Selection 268

Measures of Location and Spread 233

Probability Definitions 413

Tree Diagrams 5

Principle of Inclusion/Exclusion 23

Independent Events 51

Conditional Probability 127

Discrete Probability Distributions 210

Binomial Distribution 107

Geometric Distribution 11

Hypergeometric Distribution 10

Negative Binomial Distribution 2

Modelling and Hypothesis Testing 38

Hypothesis test of binomial distributions 12

Data representation 29

Continuous Probability Distributions and Random Variables 30

Continuous Uniform Random Variables 15

Geometric Probability 28

Normal Distribution 96

Approximating Binomial to Normal Distribution 7

Sign Change & Interval Methods 74

Fixed Point Iteration 12