LFM Stats And Pure

jee-main 2022 Q83 Finite Geometric Sum and Term Relationships View

If $\frac { 6 } { 3 ^ { 12 } } + \frac { 10 } { 3 ^ { 11 } } + \frac { 20 } { 3 ^ { 10 } } + \frac { 40 } { 3 ^ { 9 } } + \ldots + \frac { 10240 } { 3 } = 2 ^ { n } \cdot m$, where $m$ is odd, then $m \cdot n$ is equal to $\_\_\_\_$.

jee-main 2022 Q84 Find the Largest Term or Coefficient in a Binomial Expansion View

If the maximum value of the term independent of $t$ in the expansion of $\left( t ^ { 2 } x ^ { \frac { 1 } { 5 } } + \frac { 1 - x ^ { \frac { 1 } { 10 } } } { t } \right)^{10}$, $x \geq 0$, is $K$, then $8K$ is equal to $\_\_\_\_$.

jee-main 2022 Q84 Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

If the coefficients of $x$ and $x ^ { 2 }$ in the expansion of $( 1 + x ) ^ { p } ( 1 - x ) ^ { q } , p , q \leq 15$, are $-3$ and $-5$ respectively, then the coefficient of $x ^ { 3 }$ is equal to $\_\_\_\_$.

jee-main 2022 Q84 Determine Parameters from Conditions on Coefficients or Terms View

Let the coefficients of the middle terms in the expansion of $\left( \frac { 1 } { \sqrt { 6 } } + \beta x \right) ^ { 4 } , ( 1 - 3 \beta x ) ^ { 2 }$ and $\left( 1 - \frac { \beta } { 2 } x \right) ^ { 6 } , \beta > 0$, respectively form the first three terms of an A.P. If $d$ is the common difference of this A.P., then $50 - \frac { 2 d } { \beta ^ { 2 } }$ is equal to $\_\_\_\_$.

jee-main 2022 Q85 Evaluate a Summation Involving Binomial Coefficients View

If $1 + \left( 2 + { } ^ { 49 } C _ { 1 } + { } ^ { 49 } C _ { 2 } + \ldots + { } ^ { 49 } C _ { 49 } \right) \left( { } ^ { 50 } C _ { 2 } + { } ^ { 50 } C _ { 4 } + \ldots + { } ^ { 50 } C _ { 50 } \right)$ is equal to $2 ^ { n } \cdot m$, where $m$ is odd, then $n + m$ is equal to $\_\_\_\_$.

jee-main 2023 Q63 Evaluate a Summation Involving Binomial Coefficients View

The value of $\sum _ { r } ^ { 22 } = 0 { } ^ { 22 } C _ { r } \cdot { } ^ { 23 } C _ { r }$ is
(1) ${ } ^ { 45 } C _ { 23 }$
(2) ${ } ^ { 44 } C _ { 23 }$
(3) ${ } ^ { 45 } C _ { 24 }$
(4) ${ } ^ { 44 } C _ { 22 }$

jee-main 2023 Q63 Determine Parameters from Conditions on Coefficients or Terms View

If the coefficient of $x^7$ in $\left(ax - \frac{1}{bx^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax + \frac{1}{bx^2}\right)^{13}$ are equal, then $a^4 b^4$ is equal to:
(1) 11
(2) 44
(3) 22
(4) 33

jee-main 2023 Q64 Evaluate a Summation Involving Binomial Coefficients View

The value of $\frac{1}{1! \cdot 50!} + \frac{1}{3! \cdot 48!} + \frac{1}{5! \cdot 46!} + \ldots + \frac{1}{49! \cdot 2!} + \frac{1}{51! \cdot 1!}$ is
(1) $\frac{2^{50}}{50!}$
(2) $\frac{2^{50}}{51!}$
(3) $\frac{2^{51}}{51!}$
(4) $\frac{2^{51}}{50!}$

jee-main 2023 Q65 Evaluate a Summation Involving Binomial Coefficients View

If $\left( { } ^ { 30 } C _ { 1 } \right) ^ { 2 } + 2 \left( { } ^ { 30 } C _ { 2 } \right) ^ { 2 } + 3 \left( { } ^ { 30 } C _ { 3 } \right) ^ { 2 } \ldots\ldots.. 30 \left( { } ^ { 30 } C _ { 30 } \right) ^ { 2 } = \frac { \alpha 60! } { ( 30! ) ^ { 2 } }$, then $\alpha$ is equal to
(1) 30
(2) 60
(3) 15
(4) 10

jee-main 2023 Q65 Find a Specific Coefficient in a Single Binomial Expansion View

The coefficient of $x^{-6}$, in the expansion of $\left(\frac{4x}{5} + \frac{5}{2x^2}\right)^9$, is $\_\_\_\_$.

jee-main 2023 Q65 Determine Parameters from Conditions on Coefficients or Terms View

If the coefficients of $x$ and $x ^ { 2 }$ in $( 1 + x ) ^ { p } ( 1 - x ) ^ { q }$ are 4 and $-5$ respectively, then $2p + 3q$ is equal to
(1) 60
(2) 69
(3) 66
(4) 63

jee-main 2023 Q65 Find a Specific Coefficient in a Single Binomial Expansion View

The coefficient of $x ^ { 5 }$ in the expansion of $\left( 2 x ^ { 3 } - \frac { 1 } { 3 x ^ { 2 } } \right) ^ { 5 }$ is
(1) $\frac { 80 } { 9 }$
(2) 9
(3) 8
(4) $\frac { 26 } { 3 }$

jee-main 2023 Q65 Determine Parameters from Conditions on Coefficients or Terms View

Let $\left( a + b x + c x ^ { 2 } \right) ^ { 10 } = \sum _ { i = 10 } ^ { 20 } p _ { i } x ^ { i } , a , b , c \in \mathbb { N }$. If $p _ { 1 } = 20$ and $p _ { 2 } = 210$, then $2 ( a + b + c )$ is equal to
(1) 6
(2) 15
(3) 12
(4) 8

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

Let the sum of the coefficient of first three terms in the expansion of $\left( x - \frac { 3 } { x ^ { 2 } } \right) ^ { n } ; x \neq 0 , n \in N$ be 376. Then, the coefficient of $x ^ { 4 }$ is equal to:

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

If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2} - \frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3l}$ is $2^\alpha \beta$ where $\beta < 0$ is an odd number, then $|\alpha l - \beta|$ is equal to $\_\_\_\_$.

jee-main 2023 Q66 Finite Geometric Sum and Term Relationships View

If $(20)^{19} + 2(21)(20)^{18} + 3(21)^{2}(20)^{17} + \ldots + 20(21)^{19} = k(20)^{19}$, then $k$ is equal to $\_\_\_\_$.

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 Combinatorial Identity or Bijection Proof View

$\sum _ { k = 0 } ^ { 6 } { } ^ { 51 - k } C _ { 3 }$ is equal to
(1) ${ } ^ { 51 } C _ { 4 } - { } ^ { 45 } C _ { 4 }$
(2) ${ } ^ { 51 } C _ { 3 } - { } ^ { 45 } C _ { 3 }$
(3) ${ } ^ { 52 } C _ { 4 } - { } ^ { 45 } C _ { 4 }$
(4) ${ } ^ { 52 } C _ { 3 } - { } ^ { 45 } C _ { 3 }$

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

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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