LFM Pure and Mechanics

jee-main 2017 Q83 Properties of AP Terms under Transformation View

For any three positive real numbers $a$, $b$ and $c$, $9 ( 25 a ^ { 2 } + b ^ { 2 } ) + 25 ( c ^ { 2 } - 3 a c ) = 15 b ( 3 a + c )$. Then:
(1) $b$, $c$ and $a$ are in G.P.
(2) $b$, $c$ and $a$ are in A.P.
(3) $a$, $b$ and $c$ are in A.P.
(4) $a$, $b$ and $c$ are in G.P.

jee-main 2018 Q64 Summation of Derived Sequence from AP View

Let $A$ be the sum of the first 20 terms and $B$ be the sum of the first 40 terms of the series $1 ^ { 2 } + 2 \cdot 2 ^ { 2 } + 3 ^ { 2 } + 2 \cdot 4 ^ { 2 } + 5 ^ { 2 } + 2 \cdot 6 ^ { 2 } + \ldots$ If $B - 2 A = 100 \lambda$, then $\lambda$ is equal to :
(1) 496
(2) 232
(3) 248
(4) 464

jee-main 2018 Q65 Telescoping or Non-Standard Summation Involving an AP View

Let $\frac { 1 } { x _ { 1 } } , \frac { 1 } { x _ { 2 } } , \ldots , \frac { 1 } { x _ { n } } \left( x _ { i } \neq 0 \right.$ for $\left. i = 1,2 , \ldots , n \right)$ be in A.P. such that $x _ { 1 } = 4$ and $x _ { 21 } = 20$. If $n$ is the least positive integer for which $x _ { n } > 50$, then $\sum _ { i = 1 } ^ { n } \left( \frac { 1 } { x _ { i } } \right)$ is equal to
(1) 3
(2) $\frac { 1 } { 8 }$
(3) $\frac { 13 } { 4 }$
(4) $\frac { 13 } { 8 }$

jee-main 2018 Q65 Summation of Derived Sequence from AP View

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots , a _ { 49 }$ be in $A.P$. such that $\sum _ { k = 0 } ^ { 12 } a _ { 4 k + 1 } = 416$ and $a _ { 9 } + a _ { 43 } = 66$. If $a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + \ldots + a _ { 17 } ^ { 2 } = 140 m$, then $m$ is equal to:
(1) 33
(2) 66
(3) 68
(4) 34

jee-main 2018 Q65 Find Specific Term from Given Conditions View

If $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ and $\frac { 1 } { h _ { 1 } } , \frac { 1 } { h ^ { 2 } } , \ldots \ldots \frac { 1 } { h _ { n } }$ are two A.P's such that $x _ { 3 } = h _ { 2 } = 8$ and $x _ { 8 } = h _ { 7 } = 20$, then $x _ { 5 }$. $h _ { 10 }$ equals.
(1) 2560
(2) 2650
(3) 3200
(4) 1600

jee-main 2018 Q66 Find Specific Term from Given Conditions View

If $x _ { 1 } , x _ { 2 } , \ldots\ldots , x _ { n }$ and $\frac { 1 } { h _ { 1 } } , \frac { 1 } { h _ { 2 } } , \ldots\ldots , \frac { 1 } { h _ { n } }$ are two A.P.s such that $x _ { 3 } = h _ { 2 } = 8 \& x _ { 8 } = h _ { 7 } = 20$, then $x _ { 5 } \cdot h _ { 10 }$ is equal to
(1) 3200
(2) 1600
(3) 2650
(4) 2560

jee-main 2019 Q61 Arithmetic-Geometric Hybrid Problem View

If three distinct numbers $a , b , c$ are in G.P. and the equations $a x ^ { 2 } + 2 b x + c = 0$ and $d x ^ { 2 } + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?
(1) $\frac { d } { a } , \frac { e } { b } , \frac { f } { c }$ are in A.P.
(2) $d , e , f$ are in A.P.
(3) $d , e , f$ are in G.P.
(4) $\frac { d } { a } , \frac { e } { b } , \frac { f } { c }$ are in G.P.

jee-main 2019 Q63 Summation of Derived Sequence from AP View

The sum of the series $1 + 2 \times 3 + 3 \times 5 + 4 \times 7 + \ldots$ upto $11 ^ { \text {th} }$ term is:
(1) 945
(2) 916
(3) 946
(4) 915

jee-main 2019 Q64 Compute Partial Sum of an Arithmetic Sequence View

If $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ are in A.P. such that $a _ { 1 } + a _ { 7 } + a _ { 16 } = 40$, then the sum of the first 15 terms of this A.P. is

jee-main 2019 Q64 Compute Partial Sum of an Arithmetic Sequence View

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is
(1) 1356
(2) 1365
(3) 1256
(4) 1465

jee-main 2019 Q64 Summation of Derived Sequence from AP View

If the sum of the first 15 terms of the series $\left( \frac { 3 } { 4 } \right) ^ { 3 } + \left( 1 \frac { 1 } { 2 } \right) ^ { 3 } + \left( 2 \frac { 1 } { 4 } \right) ^ { 3 } + 3 ^ { 3 } + \left( 3 \frac { 3 } { 4 } \right) ^ { 3 } + \ldots$ is equal to 225 K , then $K$ is equal to :
(1) 9
(2) 27
(3) 54
(4) 108

jee-main 2019 Q64 Find General Term Formula View

Let the sum of the first $n$ terms of a non-constant A.P., $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { n }$ be $50 n + \frac { n ( n - 7 ) } { 2 } A$, where $A$ is a constant. If $d$ is the common difference of this A.P., then the ordered pair $\left( d , a _ { 50 } \right)$ is equal to
(1) $( 50,50 + 46 A )$
(2) $( A , 50 + 45 A )$
(3) $( 50,50 + 45 A )$
(4) $( A , 50 + 46 A )$

jee-main 2019 Q64 Counting or Combinatorial Problems on APs View

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is
(1) 262
(2) 190
(3) 225
(4) 157

jee-main 2019 Q65 Telescoping or Non-Standard Summation Involving an AP View

The sum of the following series $1 + 6 + \frac{9\left(1^2 + 2^2 + 3^2\right)}{7} + \frac{12\left(1^2 + 2^2 + 3^2 + 4^2\right)}{9} + \frac{15\left(1^2 + 2^2 + \ldots + 5^2\right)}{11} + \ldots$ up to 15 terms, is:
(1) 7520
(2) 7510
(3) 7830
(4) 7820

jee-main 2019 Q65 Summation of Derived Sequence from AP View

The sum $\sum _ { k = 1 } ^ { 20 } k \frac { 1 } { 2 ^ { k } }$ is equal to
(1) $1 - \frac { 11 } { 3 ^ { 20 } }$
(2) $2 - \frac { 21 } { 2 ^ { 20 } }$
(3) $2 - \frac { 3 } { 2 ^ { 17 } }$
(4) $2 - \frac { 11 } { 2 ^ { 19 } }$

jee-main 2019 Q65 Find Specific Term from Given Conditions View

If the sum and product of the first three terms in an A.P. are 33 and 1155, respectively, then a value of its $11^{\text{th}}$ term is:
(1) $- 25$
(2) $- 35$
(3) $25$
(4) $- 36$

jee-main 2019 Q66 Arithmetic-Geometric Hybrid Problem View

Let $a, b$ and $c$ be the $7^{\text{th}}, 11^{\text{th}}$ and $13^{\text{th}}$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $\frac{a}{c}$ is equal to:
(1) 2
(2) $\frac{7}{13}$
(3) $\frac{1}{2}$
(4) 4

jee-main 2019 Q66 Properties of AP Terms under Transformation View

If ${ } ^ { n } C _ { 4 } , { } ^ { n } C _ { 5 }$ and ${ } ^ { n } C _ { 6 }$ are in A.P., then $n$ can be
(1) 9
(2) 14
(3) 12
(4) 11

jee-main 2019 Q66 Summation of Derived Sequence from AP View

The sum of the series $2\cdot{}^{20}C_0 + 5\cdot{}^{20}C_1 + 8\cdot{}^{20}C_2 + 11\cdot{}^{20}C_3 + \ldots + 62\cdot{}^{20}C_{20}$ is equal to
(1) $2^{26}$
(2) $2^{25}$
(3) $2^{24}$
(4) $2^{23}$

jee-main 2019 Q76 Find Specific Term from Given Conditions View

If the lengths of the sides of a triangle are in A.P and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is:
(1) $3 : 4 : 5$
(2) $5 : 6 : 7$
(3) $5 : 9 : 13$
(4) $4 : 5 : 6$

jee-main 2020 Q52 Optimization Involving an Arithmetic Sequence View

Let $f : R \rightarrow R$ be such that for all $x \in R$, $\left( 2 ^ { 1 + x } + 2 ^ { 1 - x } \right)$, $f ( x )$ and $\left( 3 ^ { x } + 3 ^ { - x } \right)$ are in A.P., then the minimum value of $f ( x )$ is
(1) 2
(2) 3
(3) 0
(4) 4

jee-main 2020 Q53 Compute Partial Sum of an Arithmetic Sequence View

If the $10^{\text{th}}$ term of an A.P. is $\frac{1}{20}$, and its $20^{\text{th}}$ term is $\frac{1}{10}$, then the sum of its first 200 terms is.
(1) 50
(2) $50\frac{1}{4}$
(3) 100
(4) $100\frac{1}{2}$

jee-main 2020 Q53 Find Common Difference from Given Conditions View

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 7 }$

jee-main 2020 Q53 Optimization Involving an Arithmetic Sequence View

If the sum of the series $20 + 19 \frac { 3 } { 5 } + 19 \frac { 1 } { 5 } + 18 \frac { 4 } { 5 } + \ldots\ldots\ldots$ up to $n ^ { \text {th} }$ term is 488 and the $n ^ { \text {th} }$ term is negative, then :
(1) $n ^ { \text {th} }$ term is $- 4 \frac { 2 } { 5 }$
(2) $n = 41$
(3) $n ^ { \text {th} }$ term is - 4
(4) $n = 60$

jee-main 2020 Q53 Find Specific Term from Given Conditions View

Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ be a given A.P. whose common difference is an integer and $S _ { n } = a _ { 1 } + a _ { 2 } + \ldots + a _ { n }$. If $a _ { 1 } = 1 , a _ { n } = 300$ and $15 \leq n \leq 50$, then the ordered pair $\left( \mathrm { S } _ { n - 4 } , a _ { n - 4 } \right)$ is equal to:
(1) $( 2490,249 )$
(2) $( 2480,249 )$
(3) $( 2480,248 )$
(4) $( 2490,248 )$

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40