LFM Stats And Pure

jee-main 2021 Q69 View

Consider three observations $a , b$ and $c$ such that $b = a + c$. If the standard deviation of $a + 2 , b + 2 , c + 2$ is $d$, then which of the following is true?
(1) $b ^ { 2 } = 3 \left( a ^ { 2 } + c ^ { 2 } \right) + 9 d ^ { 2 }$
(2) $b ^ { 2 } = a ^ { 2 } + c ^ { 2 } + 3 d ^ { 2 }$
(3) $b ^ { 2 } = 3 \left( a ^ { 2 } + c ^ { 2 } + d ^ { 2 } \right)$
(4) $b ^ { 2 } = 3 \left( a ^ { 2 } + c ^ { 2 } \right) - 9 d ^ { 2 }$

jee-main 2021 Q71 View

The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation $\sqrt { 13.44 }$, then the standard deviation of the second sample is:
(1) 8
(2) 6
(3) 4
(4) 5

jee-main 2021 Q84 View

Consider a set of $3 n$ numbers having variance 4 . In this set, the mean of first $2 n$ numbers is 6 and the mean of the remaining $n$ numbers is 3 . A new set is constructed by adding 1 into each of the first $2 n$ numbers, and subtracting 1 from each of the remaining $n$ numbers. If the variance of the new set is $k$, then $9 k$ is equal to $\_\_\_\_$ .

jee-main 2021 Q85 View

The mean of 10 numbers $7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14 , \ldots$ is

jee-main 2021 Q87 View

An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2 . The variance of marks obtained by 30 girls is also 2 . The average marks of all 50 candidates is 15 . If $\mu$ is the average marks of girls and $\sigma ^ { 2 }$ is the variance of marks of 50 candidates, then $\mu + \sigma ^ { 2 }$ is equal to

jee-main 2022 Q68 View

Let the mean of 50 observations is 15 and the standard deviation is 2. However, one observation was wrongly recorded. The sum of the correct and incorrect observations is 70. If the mean of the correct set of observations is 16, then the variance of the correct set is equal to
(1) 10
(2) 36
(3) 43
(4) 60

jee-main 2022 Q68 View

Let the mean and the variance of 5 observations $x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } , x _ { 5 }$ be $\frac { 24 } { 5 }$ and $\frac { 194 } { 25 }$ respectively. If the mean and variance of the first 4 observation are $\frac { 7 } { 2 }$ and $a$ respectively, then ( $4 a + x _ { 5 }$ ) is equal to
(1) 13
(2) 15
(3) 17
(4) 18

jee-main 2022 Q68 View

The mean of the numbers $a , b , 8 , 5 , 10$ is 6 and their variance is 6.8. If $M$ is the mean deviation of the numbers about the mean, then $25M$ is equal to
(1) 60
(2) 55
(3) 50
(4) 75

jee-main 2022 Q69 View

The mean and variance of the data $4,5,6,6,7,8 , x , y$ where $x < y$ are 6 and $\frac { 9 } { 4 }$ respectively. Then $x ^ { 4 } + y ^ { 2 }$ is equal to
(1) 320
(2) 420
(3) 162
(4) 674

jee-main 2022 Q71 View

If the mean deviation about median for the number $3, 5, 7, 2k, 12, 16, 21, 24$ arranged in the ascending order, is 6 then the median is
(1) 11.5
(2) 10.5
(3) 12
(4) 11

jee-main 2022 Q85 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 standard deviation is $\_\_\_\_$.

jee-main 2023 Q68 View

The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are $1, 3, 5$, then the sum of cubes of the remaining two observations is
(1) 1072
(2) 1792
(3) 1216
(4) 1456

jee-main 2023 Q69 View

The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is
(1) 11
(2) 13
(3) 12
(4) 14

jee-main 2023 Q70 View

Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution

$X _ { i }$	0	1	2	3	4	5
$f _ { i }$	$k + 2$	$2k$	$k ^ { 2 } - 1$	$k ^ { 2 } - 1$	$k ^ { 2 } + 1$	$k - 3$

where $\Sigma f _ { i } = 62$. If $\lfloor x \rfloor$ denotes the greatest integer $\leq x$, then $\lfloor \mu ^ { 2 } + \sigma ^ { 2 } \rfloor$ is equal to
(1) 9
(2) 8
(3) 7
(4) 6

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 Q73 View

Let the positive numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ and $a _ { 5 }$ be in a G.P. Let their mean and variance be $\frac { 31 } { 10 }$ and $\frac { m } { n }$ respectively, where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac { 31 } { 10 }$ and $a _ { 3 } + a _ { 4 } + a _ { 5 } = 14$, then $m + n$ is equal to $\_\_\_\_$ .

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 Q87 View

If the mean of the frequency distribution

Class :	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$
Frequency :	2	3	$x$	5

is 28, then its variance is $\_\_\_\_$.

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LFM Stats And Pure

Completing the square and sketching 61

Inequalities 181

Discriminant and conditions for roots 65

Solving quadratics and applications 131

Curve Sketching 228

Factor & Remainder Theorem 61

Polynomial Division & Manipulation 35

Binomial Theorem (positive integer n) 197

Function Transformations 28

Composite & Inverse Functions 211

Partial Fractions 12

Generalised Binomial Theorem 10

Complex Numbers Arithmetic 207

Complex Numbers Argand & Loci 159

Permutations & Arrangements 255

Combinations & Selection 237

Measures of Location and Spread 181

Probability Definitions 212

Tree Diagrams 18

Principle of Inclusion/Exclusion 33

Independent Events 35

Conditional Probability 167

Discrete Probability Distributions 176

Uniform Distribution 3

Binomial Distribution 101

Geometric Distribution 8

Hypergeometric Distribution 9

Negative Binomial Distribution 1

Modelling and Hypothesis Testing 22

Hypothesis test of binomial distributions 2

Data representation 21

Continuous Probability Distributions and Random Variables 322

Continuous Uniform Random Variables 3

Geometric Probability 16

Normal Distribution 87

Approximating Binomial to Normal Distribution 4

Sign Change & Interval Methods 26

Fixed Point Iteration 9

Newton-Raphson method 6