LFM Pure and Mechanics

jee-advanced 2023 Q10 4 marks Telescoping or Non-Standard Summation Involving an AP View

Let $7 \overbrace { 5 \cdots 5 } ^ { r } 7$ denote the $( r + 2 )$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5. Consider the sum $S = 77 + 757 + 7557 + \cdots + 7 \overbrace { 5 \cdots 5 } ^ { 98 } 7$. If $S = \frac { 7 \overbrace { 5 \cdots 5 } ^ { 99 } 7 + m } { n }$, where $m$ and $n$ are natural numbers less than 3000, then the value of $m + n$ is

jee-main 2011 Q64 Counting or Combinatorial Problems on APs View

A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after
(1) 19 months
(2) 20 months
(3) 21 months
(4) 18 months

jee-main 2012 Q61 Properties of AP Terms under Transformation View

If $a, b, c, d$ and $p$ are distinct real numbers such that $\left(a^{2}+b^{2}+c^{2}\right)p^{2} - 2p(ab+bc+cd) + \left(b^{2}+c^{2}+d^{2}\right) \leq 0$, then
(1) $a, b, c, d$ are in A.P.
(2) $ab = cd$
(3) $ac = bd$
(4) $a, b, c, d$ are in G.P.

jee-main 2012 Q64 Properties of AP Terms under Transformation View

If the A.M. between $p ^ { \text {th} }$ and $q ^ { \text {th} }$ terms of an A.P. is equal to the A.M. between $r ^ { \text {th} }$ and $s ^ { \text {th} }$ terms of the same A.P., then $p + q$ is equal to
(1) $r + s - 1$
(2) $r + s - 2$
(3) $r + s + 1$
(4) $r + s$

jee-main 2012 Q74 Find Specific Term from Given Conditions View

If $100$ times the $100^{\text{th}}$ term of an AP with non-zero common difference equals the $50$ times its $50^{\text{th}}$ term, then the $150^{\text{th}}$ term of this AP is
(1) $-150$
(2) 150 times its $50^{\text{th}}$ term
(3) 150
(4) zero

jee-main 2012 Q90 Properties of AP Terms under Transformation View

If $x, y, z$ are in AP and $\tan^{-1}x$, $\tan^{-1}y$ and $\tan^{-1}z$ are also in AP, then
(1) $x = y = z$
(2) $2x = 3y = 6z$
(3) $6x = 3y = 2z$
(4) $6x = 4y = 3z$

jee-main 2013 Q64 Compute Partial Sum of an Arithmetic Sequence View

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be an A.P, such that $\frac { a _ { 1 } + a _ { 2 } + \ldots + a _ { p } } { a _ { 1 } + a _ { 2 } + a _ { 3 } + \ldots + a _ { q } } = \frac { p ^ { 3 } } { q ^ { 3 } } ; p \neq q$. Then $\frac { a _ { 6 } } { a _ { 21 } }$ is equal to:
(1) $\frac { 41 } { 11 }$
(2) $\frac { 31 } { 121 }$
(3) $\frac { 11 } { 41 }$
(4) $\frac { 121 } { 1861 }$

jee-main 2013 Q64 Compute Partial Sum of an Arithmetic Sequence View

If $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { n } , \ldots$ are in A.P. such that $a _ { 4 } - a _ { 7 } + a _ { 10 } = m$, then the sum of first 13 terms of this A.P., is :
(1) 10 m
(2) 12 m
(3) 13 m
(4) 15 m

jee-main 2013 Q65 Properties of AP Terms under Transformation View

If $x, y, z$ are positive numbers in A.P. and $\tan^{-1}x, \tan^{-1}y$ and $\tan^{-1}z$ are also in A.P., then which of the following is correct.
(1) $6x = 3y = 2z$
(2) $6x = 4y = 3z$
(3) $x = y = z$
(4) $2x = 3y = 6z$

jee-main 2013 Q66 Telescoping or Non-Standard Summation Involving an AP View

The sum of first 20 terms of the sequence $0.7, 0.77, 0.777, \ldots\ldots$, is:
(1) $\frac{7}{81}\left(179 + 10^{-20}\right)$
(2) $\frac{7}{9}\left(99 + 10^{-20}\right)$
(3) $\frac{7}{81}\left(179 - 10^{-20}\right)$
(4) $\frac{7}{9}\left(99 - 10^{-20}\right)$

jee-main 2014 Q62 Find Specific Term from Given Conditions View

Let $\alpha$ and $\beta$ be the roots of equation $p x ^ { 2 } + q x + r = 0 , p \neq 0$. If $p , q , r$ are in A.P. and $\frac { 1 } { \alpha } + \frac { 1 } { \beta } = 4$, then the value of $| \alpha - \beta |$ is
(1) $\frac { \sqrt { 34 } } { 9 }$
(2) $\frac { 2 \sqrt { 13 } } { 9 }$
(3) $\frac { \sqrt { 61 } } { 9 }$
(4) $\frac { 2 \sqrt { 17 } } { 9 }$

jee-main 2014 Q64 Summation of Derived Sequence from AP View

If $( 10 ) ^ { 9 } + 2 ( 11 ) ^ { 1 } ( 10 ) ^ { 8 } + 3 ( 11 ) ^ { 2 } ( 10 ) ^ { 7 } + \ldots\ldots + 10 ( 11 ) ^ { 9 } = k ( 10 ) ^ { 9 }$, then $k$ is equal to:
(1) 100
(2) 110
(3) $\frac { 121 } { 10 }$
(4) $\frac { 441 } { 100 }$

jee-main 2014 Q64 Compute Partial Sum of an Arithmetic Sequence View

Let $f ( n ) = \left[ \frac { 1 } { 3 } + \frac { 3 n } { 100 } \right] n$, where $[ n ]$ denotes the greatest integer less than or equal to $n$. Then $\sum _ { n = 1 } ^ { 56 } f ( n )$ is equal to
(1) 56
(2) 1287
(3) 1399
(4) 689

jee-main 2014 Q65 Find Specific Term from Given Conditions View

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its $4^{\text{th}}$ term is:
(1) 8
(2) 24
(3) 20
(4) 16

jee-main 2014 Q65 Counting or Combinatorial Problems on APs View

The number of terms in an $A.P$. is even, the sum of the odd terms in it is 24 and that the even terms is 30 . If the last term exceeds the first term by $10 \frac { 1 } { 2 }$, then the number of terms in the $A.P$. is
(1) 4
(2) 8
(3) 16
(4) 12

jee-main 2014 Q66 Telescoping or Non-Standard Summation Involving an AP View

If the sum $\frac { 3 } { 1 ^ { 2 } } + \frac { 5 } { 1 ^ { 2 } + 2 ^ { 2 } } + \frac { 7 } { 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } } + \ldots + $ up to 20 terms is equal to $\frac { k } { 21 }$, then $k$ is equal to
(1) 240
(2) 120
(3) 60
(4) 180

jee-main 2015 Q66 Telescoping or Non-Standard Summation Involving an AP View

The sum of first 9 terms of the series $\frac { 1 ^ { 3 } } { 1 } + \frac { 1 ^ { 3 } + 2 ^ { 3 } } { 1 + 3 } + \frac { 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } } { 1 + 3 + 5 } + \ldots$ is
(1) 192
(2) 71
(3) 96
(4) 142

jee-main 2015 Q87 Counting or Combinatorial Problems on APs View

The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30. If the last term exceeds the first term by $10\frac{1}{2}$, then the number of terms in the A.P. is:
(1) 4
(2) 8
(3) 12
(4) 16

jee-main 2016 Q61 Arithmetic-Geometric Hybrid Problem View

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:
(1) $\frac{8}{5}$
(2) $\frac{4}{3}$
(3) $1$
(4) $\frac{7}{4}$

jee-main 2016 Q61 Arithmetic-Geometric Hybrid Problem View

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is: (1) $\frac{8}{5}$ (2) $\frac{4}{3}$ (3) $1$ (4) $\frac{7}{4}$

jee-main 2016 Q64 Compute Partial Sum of an Arithmetic Sequence View

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots a _ { n } , \ldots$, be in A.P. If $a _ { 3 } + a _ { 7 } + a _ { 11 } + a _ { 15 } = 72$, then the sum of its first 17 terms is equal to :
(1) 306
(2) 204
(3) 153
(4) 612

jee-main 2017 Q64 Properties of AP Terms under Transformation View

For any three positive real numbers $a, b$ and $c$. If $9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c)$. 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 2017 Q64 Arithmetic-Geometric Hybrid Problem View

If the arithmetic mean of two numbers $a$ and $b , a > b > 0$, is five times their geometric mean, then $\frac { a + b } { a - b }$ is equal to:
(1) $\frac { 7 \sqrt { 3 } } { 12 }$
(2) $\frac { 3 \sqrt { 2 } } { 4 }$
(3) $\frac { \sqrt { 6 } } { 2 }$
(4) $\frac { 5 \sqrt { 6 } } { 12 }$

jee-main 2017 Q65 Compute Partial Sum of an Arithmetic Sequence View

If the sum of the first $n$ terms of the series $\sqrt { 3 } + \sqrt { 75 } + \sqrt { 243 } + \sqrt { 507 } + \ldots$ is $435 \sqrt { 3 }$, then $n$ equals:
(1) 13
(2) 15
(3) 29
(4) 18

jee-main 2017 Q78 Summation of Derived Sequence from AP View

Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy,\ \forall x, y \in \mathbb{R}$, then $\displaystyle\sum_{n=1}^{10} f(n)$ is equal to:
(1) 330
(2) 165
(3) 190
(4) 255

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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