Theory of Indices

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JEE Advanced 1980

The expression $\frac{12}{3 + 5 + 2 2}$ is equal to

1 - 5 + 2 + 10

(B)

1 + 5 + 2 - 10

(C)

1 + 5 - 2 + 10

(D)

1 - 5 - 2 + 10

JEE Advanced 1979

LEVELBoard

If $f (x) = x^{9} - 6 x^{8} - 2 x^{7} + 12 x^{6} + x^{4} - 7 x^{3} + 6 x^{2} + x - 3$ , find $f (6)$ .

JEE Advanced 1978

LEVELBoard

Solve for $x : 4^{x} - 3^{x - 1/2} = 3^{x + 1/2} - 2^{2 x - 1}$

JEE Advanced 1985

LEVELJEE Main

Use method of mathematical induction $2. 7^{n} + 3. 5^{n} - 5$ is divisible by 24 for all $n > 0$

JEE Advanced 1980

LEVELJEE Main

Given $n^{4} < 1 0^{n}$ for a fixed positive integer $n \geq 2$ , prove that $(n + 1)^{4} < 1 0^{n + 1}$ .

JEE Advanced 1978

LEVELJEE Main

Show that the square of $\frac{26 - 15 3}{5 2 - 38 + 5 3}$ is a rational number.

JEE Main 2020 - 7 Jan (Morning)

LEVELBoard

The logical statement $(p \to q) \land (q \to\sim p)$ is equivalent to

(A)

\sim p

(B)

p

(C)

p \land q

(D)

p \lor q

JEE Main 2020 - 2 Sep (Morning)

LEVELBoard

The contrapositive of the statement "If I reach the station in time, then I will catch the train" is:

(A)

If I will catch the train, then I reach the station in time

(B)

If I do not reach the station in time, then I will catch the train

(C)

If I do not reach the station in time, then I will not catch the train

(D)

If I will not catch the train, then I do not reach the station in time

JEE Main 2020 - 4 Sep (Morning)

LEVELBoard

Given the following two statements: $(s_{1}) : (q \lor p) \to (p \leftrightarrow\sim q)$ is a tautology : $(s_{2}) :\sim q \land (\sim p \leftrightarrow q)$ is a fallacy. Then :

(A)

only

(S_{1})

is correct.

(B)

both

(s_{1})

and

(s_{2})

are correct.

(C)

only

(S_{2})

is correct.

(D)

both

(S_{1})

and

(S_{2})

are not correct.

JEE Main 2020 - 6 Sep (Evening)

LEVELBoard

Consider the statement: "For an integer n, if $n^{3} - 1$ is even, then n is odd." The contrapositive statement of this statement is :

(A)

For an integer n, if n is odd, then

n^{3} - 1

is even

(B)

For an integer n, if n is even, then

n^{3} - 1

is even.

(C)

For an integer n, if n is even, then

n^{3} - 1

is odd.

(D)

For an integer n, if

n^{3} - 1

is not even, then n is not odd.

JEE Main 2021 (February)

LEVELBoard

The contrapositive of the statement "If you will work, you will earn money" is:

(A)

If you will not earn money, you will not work

(B)

You will earn money, if you will not work

(C)

If you will earn money, you will work

(D)

To earn money, you need to work

JEE Main 2022 (24 June Shift 2)

LEVELBoard

Consider the following statements : \\ $A$ : Rishi is a judge. \\ $B$ : Rishi is honest. \\ $C$ : Rishi is not arrogant. \\ The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is

(A)

B \to (A \lor C)

(B)

(\sim B) \land (A \land C)

(C)

B \to ((\sim A) \lor (\sim C))

(D)

B \to (A \land C)

JEE Main 2024 (29 Jan Shift 2)

LEVELJEE Main

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The expression 3+5​+22​12​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=x9−6x8−2x7+12x6+x4−7x3+6x2+x−3, find f(6).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Solve for x:4x−3x−1/2=3x+1/2−22x−1

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Use method of mathematical induction 2.7n+3.5n−5 is divisible by 24 for all n>0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given n4<10n for a fixed positive integer n≥2, prove that (n+1)4<10n+1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Show that the square of 52​−38+53​​26−153​​​ is a rational number.

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

The contrapositive of the statement "If I reach the station in time, then I will catch the train" is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given the following two statements: (s1​):(q∨p)→(p↔∼q) is a tautology : (s2​):∼q∧(∼p↔q) is a fallacy. Then :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the statement: "For an integer n, if n3−1 is even, then n is odd." The contrapositive statement of this statement is :

The contrapositive of the statement "If you will work, you will earn money" is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the set C={(x,y)∣x2−2y=2023,x,y∈N}. Then ∑(x,y)∈C​(x+y) is equal to

The expression $\frac{12}{3 + 5 + 2 2}$ is equal to

If $f (x) = x^{9} - 6 x^{8} - 2 x^{7} + 12 x^{6} + x^{4} - 7 x^{3} + 6 x^{2} + x - 3$ , find $f (6)$ .

Solve for $x : 4^{x} - 3^{x - 1/2} = 3^{x + 1/2} - 2^{2 x - 1}$

Use method of mathematical induction $2. 7^{n} + 3. 5^{n} - 5$ is divisible by 24 for all $n > 0$

Given $n^{4} < 1 0^{n}$ for a fixed positive integer $n \geq 2$ , prove that $(n + 1)^{4} < 1 0^{n + 1}$ .

Show that the square of $\frac{26 - 15 3}{5 2 - 38 + 5 3}$ is a rational number.

The logical statement $(p \to q) \land (q \to\sim p)$ is equivalent to

Given the following two statements: $(s_{1}) : (q \lor p) \to (p \leftrightarrow\sim q)$ is a tautology : $(s_{2}) :\sim q \land (\sim p \leftrightarrow q)$ is a fallacy. Then :

Consider the statement: "For an integer n, if $n^{3} - 1$ is even, then n is odd." The contrapositive statement of this statement is :

Let the set $C = {(x, y) ∣ x^{2} - 2^{y} = 2023, x, y \in N}$ . Then $\sum_{(x, y) \in C} (x + y)$ is equal to