LFM Stats And Pure

jee-main 2023 Q66 Find a Specific Coefficient in a Single Binomial Expansion View

The absolute difference of the coefficients of $x ^ { 10 }$ and $x ^ { 7 }$ in the expansion of $\left( 2 x ^ { 2 } + \frac { 1 } { 2 x } \right) ^ { 11 }$ is equal to
(1) $13 ^ { 3 } - 13$
(2) $11 ^ { 3 } - 11$
(3) $10 ^ { 3 } - 10$
(4) $12 ^ { 3 } - 12$

jee-main 2023 Q66 Evaluate a Summation Involving Binomial Coefficients View

If $a _ { r }$ is the coefficient of $x ^ { 10 - r }$ in the Binomial expansion of $( 1 + x ) ^ { 10 }$, then $\sum _ { r = 1 } ^ { 10 } r ^ { 3 } \left( \frac { a _ { r } } { a _ { r - 1 } } \right) ^ { 2 }$ is equal to
(1) 4895
(2) 1210
(3) 5445
(4) 3025

jee-main 2023 Q66 Determine Parameters from Conditions on Coefficients or Terms View

Let the coefficients of three consecutive terms in the binomial expansion of $( 1 + 2 x ) ^ { \mathrm { n } }$ be in the ratio $2 : 5 : 8$. Then the coefficient of the term, which is in the middle of these three terms, is

jee-main 2023 Q66 Evaluate a Summation Involving Binomial Coefficients View

If $\frac { 1 } { n + 1 } { } ^ { n } C _ { n } + \frac { 1 } { n } { } ^ { n } C _ { n - 1 } + \ldots + \frac { 1 } { 2 } { } ^ { n } C _ { 1 } + { } ^ { n } C _ { 0 } = \frac { 1023 } { 10 }$ then $n$ is equal to
(1) 9
(2) 8
(3) 7
(4) 6

jee-main 2023 Q67 Determine Parameters from Conditions on Coefficients or Terms View

If the coefficients of $x^{7}$ in $\left(ax^{2} + \frac{1}{2bx}\right)^{11}$ and $x^{-7}$ in $\left(ax - \frac{1}{3bx^{2}}\right)^{11}$ are equal, then
(1) $729ab = 32$
(2) $32ab = 729$
(3) $64ab = 243$
(4) $243ab = 64$

jee-main 2023 Q67 Find a Specific Coefficient in a Single Binomial Expansion View

The constant term in the expansion of $\left( 2 x + \frac { 1 } { x ^ { 7 } } + 3 x ^ { 2 } \right) ^ { 5 }$ is $\_\_\_\_$.

jee-main 2023 Q67 Determine Parameters from Conditions on Coefficients or Terms View

If the co-efficient of $x ^ { 9 }$ in $\left( \alpha x ^ { 3 } + \frac { 1 } { \beta x } \right) ^ { 11 }$ and the co-efficient of $x ^ { - 9 }$ in $\left( \alpha x - \frac { 1 } { \beta x ^ { 3 } } \right) ^ { 11 }$ are equal, then $( \alpha \beta ) ^ { 2 }$ is equal to

jee-main 2023 Q67 Determine Parameters from Conditions on Coefficients or Terms View

If the coefficients of three consecutive terms in the expansion of $( 1 + x ) ^ { n }$ are in the ratio $1 : 5 : 20$ then the coefficient of the fourth term is
(1) 2436
(2) 5481
(3) 1827
(4) 3654

jee-main 2023 Q67 Evaluate a Summation Involving Binomial Coefficients View

The sum, of the coefficients of the first 50 terms in the binomial expansion of $( 1 - x ) ^ { 100 }$, is equal to
(1) ${ } ^ { 101 } C _ { 50 }$
(2) ${ } ^ { 99 } C _ { 49 }$
(3) $- { } ^ { 101 } C _ { 50 }$
(4) $- { } ^ { 99 } C _ { 49 }$

jee-main 2023 Q68 Evaluate a Summation Involving Binomial Coefficients View

Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $( 1 + x ) ^ { 99 }$. Let a be the middle term in the expansion of $\left( 2 + \frac { 1 } { \sqrt { 2 } } \right) ^ { 200 }$. If $\frac { { } ^ { 200 } C _ { 99 } K } { a } = \frac { 2 ^ { l } m } { n }$, where $m$ and $n$ are odd numbers, then the ordered pair $( l , \mathrm { n } )$ is equal to: (1) $( 50,51 )$ (2) $( 51,99 )$ (3) $( 50,101 )$ (4) $( 51,101 )$

jee-main 2023 Q68 Find a Specific Coefficient in a Single Binomial Expansion View

Let $[ t ]$ denote the greatest integer $\leq t$. If the constant term in the expansion of $\left( 3 x ^ { 2 } - \frac { 1 } { 2 x ^ { 5 } } \right) ^ { 7 }$ is $\alpha$ then $[ \alpha ]$ is equal to $\_\_\_\_$

jee-main 2023 Q84 Find a Specific Coefficient in a Single Binomial Expansion View

Let $\alpha > 0$ be the smallest number such that the expansion of $\left(x^{\frac{2}{3}} + \frac{2}{x^3}\right)^{30}$ has a term $\beta x^{-\alpha}$, $\beta \in \mathbb{N}$. Then $\alpha$ is equal to $\underline{\hspace{1cm}}$.

jee-main 2023 Q85 Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

The coefficient of $x^7$ in $(1 - x + 2x^3)^{10}$ is $\_\_\_\_$.

jee-main 2024 Q62 Determine Parameters from Conditions on Coefficients or Terms View

Let $0 \leq \mathrm { r } \leq \mathrm { n }$. If ${ } ^ { \mathrm { n } + 1 } \mathrm { C } _ { \mathrm { r } + 1 } : { } ^ { n } \mathrm { C } _ { \mathrm { r } } : { } ^ { \mathrm { n } - 1 } \mathrm { C } _ { \mathrm { r } - 1 } = 55 : 35 : 21$, then $2 \mathrm { n } + 5 \mathrm { r }$ is equal to:
(1) 50
(2) 62
(3) 55
(4) 60

jee-main 2024 Q63 Evaluate a Summation Involving Binomial Coefficients View

If $A$ denotes the sum of all the coefficients in the expansion of $\left( 1 - 3 x + 10 x ^ { 2 } \right) ^ { n }$ and $B$ denotes the sum of all the coefficients in the expansion of $\left( 1 + x ^ { 2 } \right) ^ { n }$, then:
(1) $\mathrm { A } = \mathrm { B } ^ { 3 }$
(2) $3 \mathrm {~A} = \mathrm { B }$
(3) $\mathrm { B } = \mathrm { A } ^ { 3 }$
(4) $\mathrm { A } = 3 \mathrm {~B}$

jee-main 2024 Q63 Determine Parameters from Conditions on Coefficients or Terms View

Suppose $28 - p,\ p,\ 70 - \alpha,\ \alpha$ are the coefficient of four consecutive terms in the expansion of $(1 + x)^n$. Then the value of $2\alpha - 3p$ equals
(1) 7
(2) 10
(3) 4
(4) 6

jee-main 2024 Q64 Find a Specific Coefficient in a Single Binomial Expansion View

Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3}x^{\frac{1}{3}} + \frac{1}{2x^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{n}{m}\right)^{\frac{1}{3}}$ is:
(1) $\frac{4}{9}$
(2) $\frac{1}{9}$
(3) $\frac{1}{4}$
(4) $\frac{1}{4}$

jee-main 2024 Q64 Determine Parameters from Conditions on Coefficients or Terms View

${ } ^ { n - 1 } C _ { r } = \left( k ^ { 2 } - 8 \right) ^ { n } C _ { r + 1 }$ if and only if:
(1) $2 \sqrt { 2 } < k \leq 3$
(2) $2 \sqrt { 3 } < \mathrm { k } \leq 3 \sqrt { 2 }$
(3) $2 \sqrt { 3 } < \mathrm { k } < 3 \sqrt { 3 }$
(4) $2 \sqrt { 2 } < \mathrm { k } < 2 \sqrt { 3 }$

jee-main 2024 Q64 Determine Parameters from Conditions on Coefficients or Terms View

If the coefficients of $x ^ { 4 } , x ^ { 5 }$ and $x ^ { 6 }$ in the expansion of $( 1 + x ) ^ { n }$ are in the arithmetic progression, then the maximum value of $n$ is:
(1) 7
(2) 21
(3) 28
(4) 14

jee-main 2024 Q64 Find a Specific Coefficient in a Single Binomial Expansion View

If the constant term in the expansion of $\left( \frac { \sqrt [ 5 ] { 3 } } { x } + \frac { 2 x } { \sqrt [ 3 ] { 5 } } \right) ^ { 12 } , x \neq 0$, is $\alpha \times 2 ^ { 8 } \times \sqrt [ 5 ] { 3 }$, then $25 \alpha$ is equal to :
(1) 724
(2) 742
(3) 639
(4) 693

jee-main 2024 Q64 Determine Parameters from Conditions on Coefficients or Terms View

If the term independent of $x$ in the expansion of $\left( \sqrt { \mathrm { a } } x ^ { 2 } + \frac { 1 } { 2 x ^ { 3 } } \right) ^ { 10 }$ is 105 , then $\mathrm { a } ^ { 2 }$ is equal to : (1) 2 (2) 4 (3) 6 (4) 9

jee-main 2024 Q68 Determine Parameters from Conditions on Coefficients or Terms View

Let $a$ be the sum of all coefficients in the expansion of $\left( 1 - 2 x + 2 x ^ { 2 } \right) ^ { 2023 } \left( 3 - 4 x ^ { 2 } + 2 x ^ { 3 } \right) ^ { 2024 }$ and $b = \lim _ { x \rightarrow 0 } \frac { \int _ { 0 } ^ { x } \frac { \log ( 1 + t ) } { t ^ { 2024 } + 1 } dt } { x ^ { 2 } }$. If the equations $c x ^ { 2 } + d x + e = 0$ and $2 b x ^ { 2 } + a x + 4 = 0$ have a common root, where $c , d , e \in R$, then $d : c : e$ equals
(1) $2 : 1 : 4$
(2) $4 : 1 : 4$
(3) $1 : 2 : 4$
(4) $1 : 1 : 4$

jee-main 2024 Q82 Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

The coefficient of $x ^ { 2012 }$ in the expansion of $1 - x ^ { 2008 } 1 + x + x ^ { 2007 }$ is equal to $\_\_\_\_$ .

jee-main 2024 Q82 Evaluate a Summation Involving Binomial Coefficients View

If $\mathrm { S } ( x ) = ( 1 + x ) + 2 ( 1 + x ) ^ { 2 } + 3 ( 1 + x ) ^ { 3 } + \cdots + 60 ( 1 + x ) ^ { 60 } , x \neq 0$, and $( 60 ) ^ { 2 } \mathrm {~S} ( 60 ) = \mathrm { a } ( \mathrm { b } ) ^ { \mathrm { b } } + \mathrm { b }$, where $a , b \in N$, then $( a + b )$ equal to $\_\_\_\_$

jee-main 2024 Q83 Evaluate a Summation Involving Binomial Coefficients View

If $\frac { { } ^ { 11 } C _ { 1 } } { 2 } + \frac { { } ^ { 11 } C _ { 2 } } { 3 } + \ldots . . + \frac { { } ^ { 11 } C _ { 9 } } { 10 } = \frac { n } { m }$ with $\operatorname { gcd } ( n , m ) = 1$, then $n + m$ is equal to

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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