Core Syllabus Module

Basic Mathematics

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 2022

LEVELJEE Main

The product of all positive real values of $x$ satisfying the equation $x^{(16 (l o g_{5} x)^{3} - 68 l o g_{5} x)} = 5^{- 16}$ is ______.

JEE Main 2023 (10 Apr Shift 1)

LEVELJEE Main

Let $a, b, c$ be the three distinct positive real numbers such that $(2 a)^{l o g_{e} a} = (b c)^{l o g_{e} b}$ and $b^{l o g_{e} 2} = a^{l o g_{e} c}$ . Then $6 a + 5 b c$ is equal to ______.

JEE Main 2023 (11 Apr Shift 1)

LEVELJEE Main

The number of integral solution $x$ of $lo g_{(x + \frac{7}{2})} (\frac{x - 7}{2 x - 3})^{2} \geq 0$ is

The equation $x^{3/4 (l o g_{2} x)^{2} + l o g_{2} x - 5/4} = 2$ has

* Multiple Correct Options

(A)

at least one real solution

(B)

exactly three solutions

(C)

exactly one irrational solution

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

JEE Main 2026 (24 January Shift 1)

LEVELJEE Main

The number of the real solutions of the equation: $x ∣ x + 3∣ + ∣ x - 1∣ - 2 = 0$ is

JEE Main 2021 (March)

LEVELJEE Main

The number of elements in the set $x \in R : (∣ x ∣ - 3) ∣ x + 4∣ = 6$ is equal to

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 Advanced 1995

LEVELJEE Main

The function $f (x) = ∣ p x - q ∣ + r ∣ x ∣, x \in (- \infty, \infty)$ where $p > 0, q > 0, r > 0$ assumes its minimum value only on one point if

(A)

p \neq = q

(B)

r \neq = q

(C)

r \neq = p

(D)

p = q = r

JEE Advanced 1987

LEVELJEE Main

Solve for $x$ the following equation : $lo g_{(2 x + 3)} (6 x^{2} + 23 x + 21) = 4 - lo g_{(3 x + 7)} (4 x^{2} + 12 x + 9)$ .

JEE Main 2025 April

LEVELJEE Main

The number of real roots of the equation $x ∣ x - 2∣ + 3∣ x - 3∣ + 1 = 0$ is :

JEE Main 2024 (29 Jan Shift 2)

LEVELJEE Main

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

JEE Advanced 1987

LEVELJEE Main

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

JEE Main 2023 (10 Apr Shift 1)

LEVELJEE Main

The number of elements in the set ${n \in Z : ∣ n^{2} - 10 n + 19∣ < 6}$ is

JEE Advanced 1980

LEVELJEE Main

The least value of the expression $2 lo g_{10} x - lo g_{x} (0.01)$ , for $x > 1$ , is

(A)

(B)

(C)

- 0.01

The solution of equation $lo g_{7} lo g_{5} (x + 5 + x) = 0$ is .........

JEE Main 2026 (23 January Shift 2)

LEVELJEE Main

The sum of all the real solutions of the equation $lo g_{(x + 3)} (6 x^{2} + 28 x + 30) = 5 - 2 lo g_{(6 x + 10)} (x^{2} + 6 x + 9)$ is equal to

The value of $((lo g_{2} 9)^{2})^{\frac{1}{l o g _{2} ( l o g _{2} 9 )}} \times (7)^{\frac{1}{l o g _{4} 7}}$ is ________.

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 2024

LEVELJEE Main

Let $a = 32$ and $b = \frac{1}{5 ^{1/6} 6}$ . If $x, y \in R$ are such that $3 x + 2 y = lo g_{a} (18)^{5/4}$ and $2 x - y = lo g_{b} (1080)$ , then $4 x + 5 y$ is equal to ________.

Micro-Topic Strategy Matrix

Theory of Indices

Linear Inequalities

Properties of Modulus Function

Properties of Logarithms

Logarithmic Equations and Inequalities

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The product of all positive real values of x satisfying the equation x(16(log5​x)3−68log5​x)=5−16 is ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be the three distinct positive real numbers such that (2a)loge​a=(bc)loge​b and bloge​2=aloge​c. Then 6a+5bc is equal to ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of integral solution x of log(x+27​)​(2x−3x−7​)2≥0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation x3/4(log2​x)2+log2​x−5/4=2​ has

.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 number of the real solutions of the equation: x∣x+3∣+∣x−1∣−2=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of elements in the set x∈R:(∣x∣−3)∣x+4∣=6 is equal to

.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.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f(x)=∣px−q∣+r∣x∣,x∈(−∞,∞) where p>0,q>0,r>0 assumes its minimum value only on one point if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Solve for x the following equation : log(2x+3)​(6x2+23x+21)=4−log(3x+7)​(4x2+12x+9).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of real roots of the equation x∣x−2∣+3∣x−3∣+1=0 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

.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; } The number of elements in the set {n∈Z:∣n2−10n+19∣<6} is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The least value of the expression 2log10​x−logx​(0.01), for x>1, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The solution of equation log7​log5​(x+5​+x​)=0 is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of all the real solutions of the equation log(x+3)​(6x2+28x+30)=5−2log(6x+10)​(x2+6x+9) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of ((log2​9)2)log2​(log2​9)1​×(7​)log4​71​ is ________.

.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; } Let a=32​ and b=51/66​1​. If x,y∈R are such that 3x+2y=loga​(18)5/4 and 2x−y=logb​(1080​), then 4x+5y is equal to ________.

The product of all positive real values of $x$ satisfying the equation $x^{(16 (l o g_{5} x)^{3} - 68 l o g_{5} x)} = 5^{- 16}$ is ______.

Let $a, b, c$ be the three distinct positive real numbers such that $(2 a)^{l o g_{e} a} = (b c)^{l o g_{e} b}$ and $b^{l o g_{e} 2} = a^{l o g_{e} c}$ . Then $6 a + 5 b c$ is equal to ______.

The number of integral solution $x$ of $lo g_{(x + \frac{7}{2})} (\frac{x - 7}{2 x - 3})^{2} \geq 0$ is

The equation $x^{3/4 (l o g_{2} x)^{2} + l o g_{2} x - 5/4} = 2$ has

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

The number of the real solutions of the equation: $x ∣ x + 3∣ + ∣ x - 1∣ - 2 = 0$ is

The number of elements in the set $x \in R : (∣ x ∣ - 3) ∣ x + 4∣ = 6$ is equal to

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 $n$ .

The function $f (x) = ∣ p x - q ∣ + r ∣ x ∣, x \in (- \infty, \infty)$ where $p > 0, q > 0, r > 0$ assumes its minimum value only on one point if

Solve for $x$ the following equation : $lo g_{(2 x + 3)} (6 x^{2} + 23 x + 21) = 4 - lo g_{(3 x + 7)} (4 x^{2} + 12 x + 9)$ .

The number of real roots of the equation $x ∣ x - 2∣ + 3∣ x - 3∣ + 1 = 0$ 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

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

The number of elements in the set ${n \in Z : ∣ n^{2} - 10 n + 19∣ < 6}$ is

The least value of the expression $2 lo g_{10} x - lo g_{x} (0.01)$ , for $x > 1$ , is

The solution of equation $lo g_{7} lo g_{5} (x + 5 + x) = 0$ is .........

The sum of all the real solutions of the equation $lo g_{(x + 3)} (6 x^{2} + 28 x + 30) = 5 - 2 lo g_{(6 x + 10)} (x^{2} + 6 x + 9)$ is equal to

The value of $((lo g_{2} 9)^{2})^{\frac{1}{l o g _{2} ( l o g _{2} 9 )}} \times (7)^{\frac{1}{l o g _{4} 7}}$ is ________.

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

Let $a = 32$ and $b = \frac{1}{5 ^{1/6} 6}$ . If $x, y \in R$ are such that $3 x + 2 y = lo g_{a} (18)^{5/4}$ and $2 x - y = lo g_{b} (1080)$ , then $4 x + 5 y$ is equal to ________.