LFM Pure

isi-entrance 2024 Q5 Computation of a Limit, Value, or Explicit Formula View

Let $P ( x )$ be a polynomial with real coefficients. Let $\alpha _ { 1 } , \ldots , \alpha _ { k }$ be the distinct real roots of $P ( x ) = 0$. If $P ^ { \prime }$ is the derivative of $P$, show that for each $i = 1,2 , \ldots , k$,
$$\lim _ { x \rightarrow \alpha _ { i } } \frac { \left( x - \alpha _ { i } \right) P ^ { \prime } ( x ) } { P ( x ) } = r _ { i } ,$$
for some positive integer $r _ { i }$.

isi-entrance 2024 Q8 Direct Proof of an Inequality View

In a sports tournament involving $N$ teams, each team plays every other team exactly once. At the end of every match, the winning team gets 1 point and the losing team gets 0 points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows:
$$x _ { 1 } \geq x _ { 2 } \geq \cdots \geq x _ { N }$$
Prove that for any $1 \leq k \leq N$,
$$\frac { N - k } { 2 } \leq x _ { k } \leq N - \frac { k + 1 } { 2 } .$$

isi-entrance 2026 Q1 10 marks Existence Proof View

Suppose $f : \mathbb { R } \rightarrow \mathbb { R }$ is differentiable and $\left| f ^ { \prime } ( x ) \right| < \frac { 1 } { 2 }$ for all $x \in \mathbb { R }$. Show that for some $x _ { 0 } \in \mathbb { R } , f \left( x _ { 0 } \right) = x _ { 0 }$.

isi-entrance 2026 Q3 10 marks Direct Proof of a Stated Identity or Equality View

Suppose $f : [ 0,1 ] \rightarrow \mathbb { R }$ is differentiable with $f ( 0 ) = 0$. If $\left| f ^ { \prime } ( x ) \right| \leq f ( x )$ for all $x \in [ 0,1 ]$, then show that $f ( x ) = 0$ for all $x$.

isi-entrance 2026 Q5 10 marks Direct Proof of a Stated Identity or Equality View

Let $a , b , c$ be nonzero real numbers such that $a + b + c \neq 0$. Assume that $$\frac { 1 } { a } + \frac { 1 } { b } + \frac { 1 } { c } = \frac { 1 } { a + b + c }$$ Show that for any odd integer $k$, $$\frac { 1 } { a ^ { k } } + \frac { 1 } { b ^ { k } } + \frac { 1 } { c ^ { k } } = \frac { 1 } { a ^ { k } + b ^ { k } + c ^ { k } }$$

isi-entrance 2026 Q7 10 marks Existence Proof View

Consider a ball that moves inside an acute-angled triangle along a straight line, until it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence $=$ angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.

isi-entrance 2026 Q8 10 marks Direct Proof of an Inequality View

Let $n \geq 2$ and let $a _ { 1 } \leq a _ { 2 } \leq \cdots \leq a _ { n }$ be positive integers such that $\sum _ { i = 1 } ^ { n } a _ { i } = \Pi _ { i = 1 } ^ { n } a _ { i }$. Prove that $\sum _ { i = 1 } ^ { n } a _ { i } \leq 2n$ and determine when equality holds.

jee-advanced 2007 Q47 True/False Justification View

Let $f(x) = 2 + \cos x$ for all real $x$. STATEMENT-1: For each real $t$, there exists a point $c$ in $[t, t+\pi]$ such that $f'(c) = 0$. because STATEMENT-2: $f(t) = f(t+2\pi)$ for each real $t$.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True

jee-main 2013 Q72 Proof of Equivalence or Logical Relationship Between Conditions View

For integers $m$ and $n$, both greater than 1, consider the following three statements : $P : m$ divides $n$, $Q : m$ divides $n ^ { 2 }$, $R : m$ is prime, then
(1) $Q \wedge R \rightarrow P$
(2) $P \wedge Q \rightarrow R$
(3) $Q \rightarrow R$
(4) $Q \rightarrow P$

jee-main 2015 Q88 Direct Proof of a Stated Identity or Equality View

The negation of $\sim s \vee (\sim r \wedge s)$ is equivalent to:
(1) $s \wedge \sim r$
(2) $s \wedge (r \wedge \sim s)$
(3) $s \vee (r \vee \sim s)$
(4) $s \wedge r$

jee-main 2016 Q68 Direct Proof of a Stated Identity or Equality View

The Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ is equivalent to:
(1) $p \wedge q$
(2) $p \vee q$
(3) $p \vee \sim q$
(4) $\sim p \wedge q$

jee-main 2016 Q69 Direct Proof of a Stated Identity or Equality View

The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is:
(1) If the area of a square increases four times, then its side is not doubled.
(2) If the area of a square does not increase four times, then its side is not doubled.
(3) If the area of a square does not increase four times, then its side is doubled.
(4) If the side of a square is not doubled, then its area does not increase four times.

jee-main 2016 Q82 Direct Proof of a Stated Identity or Equality View

The Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ is equivalent to: (1) $\sim p \wedge q$ (2) $p \wedge q$ (3) $p \vee q$ (4) $p \vee \sim q$

jee-main 2016 Q83 Direct Proof of a Stated Identity or Equality View

The contrapositive of the following statement, ``If the side of a square doubles, then its area increases four times'', is: (1) If the area of a square increases four times, then its side is not doubled. (2) If the area of a square does not increase four times, then its side is not doubled. (3) If the area of a square does not increase four times, then its side is doubled. (4) If the side of a square is not doubled, then its area does not increase four times.

jee-main 2017 Q79 True/False Justification View

The following statement $(p \to q) \to [ ( \sim p \to q ) \to q ]$ is:
(1) a fallacy
(2) a tautology
(3) equivalent to $\sim p \to q$
(4) equivalent to $p \to \sim q$

jee-main 2018 Q73 True/False Justification View

If $p \rightarrow ( \sim p \vee \sim q )$ is false, then the truth values of $p$ and $q$ are, respectively
(1) $F , F$
(2) $T , T$
(3) $F , T$
(4) $T , F$

jee-main 2019 Q74 True/False Justification View

Consider the statement: ``$P ( n ) : n ^ { 2 } - n + 41$ is prime''. Then which one of the following is true?
(1) $P ( 3 )$ is false but $P ( 5 )$ is true
(2) Both $P ( 3 )$ and $P ( 5 )$ are false
(3) Both $P ( 3 )$ and $P ( 5 )$ are true
(4) $P ( 5 )$ is false but $P ( 3 )$ is true

jee-main 2020 Q57 True/False Justification View

Which of the following statement is a tautology?
(1) $p \vee (\sim q) \rightarrow p \wedge q$
(2) $\sim(p \wedge \sim q) \rightarrow p \vee q$
(3) $\sim(p \vee \sim q) \rightarrow p \wedge q$
(4) $\sim(p \vee \sim q) \rightarrow p \vee q$

jee-main 2020 Q57 Direct Proof of a Stated Identity or Equality View

The contrapositive of the statement ``If I reach the station in time, then I will catch the train'' is
(1) If I do not reach the station in time, then I will catch the train.
(2) If I do not reach the station in time, then I will not catch the train.
(3) If I will catch the train, then I reach the station in time.
(4) If I will not catch the train, then I do not reach the station in time.

jee-main 2020 Q58 Direct Proof of a Stated Identity or Equality View

For two statements $p$ and $q$, the logical statement $(p \rightarrow q) \wedge (q \rightarrow \sim p)$ is equivalent to
(1) $p$
(2) $q$
(3) $\sim p$
(4) $\sim q$

jee-main 2020 Q58 Direct Proof of a Stated Identity or Equality View

Negation of the statement: $\sqrt { 5 }$ is an integer or 5 is irrational is:
(1) $\sqrt { 5 }$ is not an integer 5 is not irrational
(2) $\sqrt { 5 }$ is not an integer and 5 is not irrational
(3) $\sqrt { 5 }$ is irrational or 5 is an integer
(4) $\sqrt { 5 }$ is an integer and 5 irrational

jee-main 2020 Q58 Direct Proof of a Stated Identity or Equality View

Contrapositive of the statement: 'If a function $f$ is differentiable at $a$, then it is also continuous at $a$', is
(1) If a function $f$ is continuous at $a$, then it is not differentiable at $a$.
(2) If a function $f$ is not continuous at $a$, then it is not differentiable at $a$.
(3) If a function $f$ is not continuous at $a$, then it is differentiable at $a$.
(4) If a function $f$ is continuous at $a$, then it is differentiable at $a$.

jee-main 2020 Q58 Direct Proof of a Stated Identity or Equality View

Consider the statement: ``For an integer n, if $\mathrm{n}^{3}-1$ is even, then n is odd''. The contrapositive statement of this statement is:
(1) For an integer n, if n is even, then $\mathrm{n}^{3}-1$ is odd.
(2) For an integer n, if $\mathrm{n}^{3}-1$ is not even, then n is not odd.
(3) For an integer n, if n is even, then $\mathrm{n}^{3}-1$ is even.
(4) For an integer n, if n is odd, then $\mathrm{n}^{3}-1$ is even.

jee-main 2020 Q59 True/False Justification View

If $p \rightarrow ( p \wedge \sim q )$ is false, then the truth values of $p$ and $q$ are respectively
(1) $F , F$
(2) $T , F$
(3) $T , T$
(4) $F , T$

jee-main 2020 Q59 Direct Proof of a Stated Identity or Equality View

The proposition $p \rightarrow \sim ( p \wedge \sim q )$ is equivalent to:
(1) $q$
(2) $( \sim p ) \vee q$
(3) $( \sim p ) \wedge q$
(4) $( \sim p ) \vee ( \sim q )$

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144