Linear Inequalities

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JEE Main 2002

LEVELBoard

If $a, b, c$ are distinct +ve real numbers and $a^{2} + b^{2} + c^{2} = 1$ then $ab + b c + c a$ is

If $S$ is the set of all real $x$ such that $\frac{2 x - 1}{2 x ^{3} + 3 x ^{2} + x}$ is positive, then $S$ contains

* Multiple Correct Options

(A)

(- \infty, - 3/2)

(B)

(- 3/2, - 1/4)

(C)

(- 1/4, 1/2)

(D)

(1/2, 3)

Find the set of all $x$ for which $\frac{2 x}{2 x ^{2} + 5 x + 2} > \frac{1}{x + 1}$

JEE Advanced 1980

LEVELJEE Main

Let $y = \frac{( x + 1 ) ( x - 3 )}{( x - 2 )}$ . Find all the real values of $x$ for which $y$ takes real values.

JEE Advanced 1987

LEVELJEE Main

Prove by mathematical induction that $\frac{( 2 n )!}{2 ^{2 n} ( n ! ) ^{2}} \leq \frac{1}{( 3 n + 1 ) ^{1/2}}$ for all positive integers $n$ .

JEE Main 11 Jan 2019 (Evening)

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Contrapositive of the statement "If two numbers are not equal, then their squares are not equal." Is:

(A)

If the squares of two numbers are equal, then the numbers are equal.

(B)

If the squares of two numbers are equal, then the numbers are not equal.

(C)

If the squares of two numbers are not equal, then the numbers are equal.

(D)

If the squares of two numbers are not equal, then the numbers are not equal.

JEE Main 2020 (9 January Shift 2)

LEVELBoard

If $p \to (p \land \sim q)$ is false. Then the truth values of $p$ and $q$ are respectively

JEE Main 2020 - 9 Jan (Evening)

LEVELBoard

If $p \to (p \land \sim q)$ false, then the truth values of $p$ and $q$ are respectively :

JEE Main 2021 (20 July Shift 1)

LEVELJEE Main

The Boolean expression $(p \land \sim q) \Rightarrow (q \lor \sim p)$ is equivalent to:

(A)

q \Rightarrow p

(B)

p \Rightarrow q

(C)

\sim q \Rightarrow p

(D)

p \Rightarrow\sim q

JEE Main 2021 (20 July Shift 2)

LEVELBoard

Consider the following three statements: (A) If $3 + 3 = 7$ then $4 + 3 = 8$ . (B) If $5 + 3 = 8$ then earth is flat. (C) If both (A) and (B) are true then $5 + 6 = 17$ . Then, which of the following statements is correct?

(A)

(A) is false, but (B) and (C) are true

(B)

(A) and (C) are true while (B) is false

(C)

(A) is true while (B) and (C) are false

(D)

(A) and (B) are false while (C) is true

JEE Main 2021 (27 July Shift 2)

LEVELJEE Main

Which of the following is the negation of the statement "for all $M > 0$ , there exists $x \in S$ such that $x \geq M$ "?

(A)

there exists

M > 0

, such that

x < M

for all

x \in S

(B)

there exists

M > 0

, there exists

x \in S

such that

x \geq M

(C)

there exists

M > 0

, there exists

x \in S

such that

x < M

(D)

there exists

M > 0

, such that

x \geq M

for all

x \in S

JEE Main 2022 (28 July Shift 2)

LEVELBoard

Let $p$ : Ramesh listens to music. $q$ : Ramesh is out of his village $r$ : It is Sunday $s$ : It is Saturday Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as

(A)

((\sim q) \land (r \lor s)) \Rightarrow p

(B)

(q \land (r \lor s)) \Rightarrow p

(C)

p \Rightarrow (q \land (r \lor s))

(D)

p \Rightarrow ((\sim q) \land (r \lor s))

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are distinct +ve real numbers and a2+b2+c2=1 then ab+bc+ca is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If S is the set of all real x such that 2x3+3x2+x2x−1​ is positive, then S contains

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the set of all x for which 2x2+5x+22x​>x+11​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let y=(x−2)(x+1)(x−3)​. Find all the real values of x for which y takes real values.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove by mathematical induction that 22n(n!)2(2n)!​≤(3n+1)1/21​ for all positive integers n.

Contrapositive of the statement "If two numbers are not equal, then their squares are not equal." Is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If p→(p∧∼q) is false. Then the truth values of p and q are respectively

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If p→(p∧∼q) false, then the truth values of p and q are respectively :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Boolean expression (p∧∼q)⇒(q∨∼p) is equivalent to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which of the following is the negation of the statement "for all M>0, there exists x∈S such that x≥M"?

If $a, b, c$ are distinct +ve real numbers and $a^{2} + b^{2} + c^{2} = 1$ then $ab + b c + c a$ is

If $S$ is the set of all real $x$ such that $\frac{2 x - 1}{2 x ^{3} + 3 x ^{2} + x}$ is positive, then $S$ contains

Find the set of all $x$ for which $\frac{2 x}{2 x ^{2} + 5 x + 2} > \frac{1}{x + 1}$

Let $y = \frac{( x + 1 ) ( x - 3 )}{( x - 2 )}$ . Find all the real values of $x$ for which $y$ takes real values.

Prove by mathematical induction that $\frac{( 2 n )!}{2 ^{2 n} ( n ! ) ^{2}} \leq \frac{1}{( 3 n + 1 ) ^{1/2}}$ for all positive integers $n$ .

If $p \to (p \land \sim q)$ is false. Then the truth values of $p$ and $q$ are respectively

If $p \to (p \land \sim q)$ false, then the truth values of $p$ and $q$ are respectively :

The Boolean expression $(p \land \sim q) \Rightarrow (q \lor \sim p)$ is equivalent to:

Which of the following is the negation of the statement "for all $M > 0$ , there exists $x \in S$ such that $x \geq M$ "?