JEE Main Mathematics — practice by topic

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Statistics and Probability (179)Determinants (132)Vector Algebra and 3D Geometry (126)Continuity, Differentiability & Derivatives (112)Differential Equations (102)Integrals and Applications (98)Matrices and Determinants (89)Relations, Functions & Inverse Trigonometry (84)Straight Lines and Conic Sections (77)Complex Numbers and Quadratic Equations (70)Mensuration: Surface Areas & Volumes (69)Sets and Relations (66)Integrals (61)Linear Programming (59)Limits and Derivatives (55)Trigonometric Functions (51)Matrices (47)Quadratic Equations (43)Relations and Functions (43)Unit 6A: Indefinite Integrals & Fundamental Theorem (43)Unit 9A: Parametric Equations and Plane Curves (43)Probability & Statistics (39)Unit 2: Differentiation and Fundamental Derivative Rules (39)Unit 4-5: Applications of Derivatives & Curve Sketching (39)Unit 7: Differential Equations and Modeling (38)Sequences and Series (37)Unit 8A: Areas Between Curves (37)Coordinate Geometry (36)Vector Algebra (34)Probability (33)Sets, Relations and Functions (32)Permutations and Combinations (30)Sequences (30)Mensuration: Perimeter and Area (28)Three Dimensional Geometry (28)Arithmetic Progression (27)Unit 6B: Definite Integration & Riemann Sums (27)Application of Integrals (26)Binomial Theorem (26)Straight Lines (26)JEE Main 2023 Mixed (25)Unit 1B1: Limit Concepts and Graphical Limits (25)Continuity and Differentiability (16)Inverse Trigonometric Functions (16)Unit 1A1: Relations, Functions and Graphs (16)Number Systems (15)Complex Numbers (14)Algebra: Expansions, Factorisation & Indices (12)Algebra: Simultaneous Linear Equations & Quadratics (12)Unit 3A: Chain Rule and Composite Functions (12)Unit 3B: Implicit Differentiation and Inverse Functions (9)Mathematics (8)Section Formula (8)Real Numbers and Polynomials (5)Introduction to Trigonometry (4)Definite Integration (3)Even/Odd Functions (2)Geometry: Circles (2)Integral Calculus (2)Limits and Continuity (2)

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From a standard deck of 52 playing cards, what is the probability of drawing a heart?Read the answer and worked solution →Which property of conditional probability is correct?Read the answer and worked solution →Using standard limits, evaluate $\lim_{x\to 0}\dfrac{a^x-1}{x}$ for $a>0$.Read the answer and worked solution →The derivative of $a^x$ for $a>0$ and $a\neq 1$ isRead the answer and worked solution →

Worked practice questions

From a standard deck of 52 playing cards, what is the probability of drawing a heart?

$\dfrac{1}{2}$
$\dfrac{1}{4}$
$\dfrac{13}{52}$
Both B and C are correct

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Correct answer: D. Both B and C are correct

A standard deck has 13 hearts out of 52 cards. So the probability is $\dfrac{13}{52}=\dfrac14$. **Hence both options B and C represent the same correct value, so D is the intended correct choice.**

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Which property of conditional probability is correct?

$P(A'/B)=P(A/B)$
$P(A'/B)=1-P(A/B)$
$P(A'/B)=1-P(B/A)$
$P(A'/B)=P(A\cap B)$

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Correct answer: B. $P(A'/B)=1-P(A/B)$

Within the reduced sample space where $B$ has occurred, the events $A$ and $A'$ are complementary. **Therefore their conditional probabilities add to 1, so $P(A'/B)=1-P(A/B)$.**

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Using standard limits, evaluate $\lim_{x\to 0}\dfrac{a^x-1}{x}$ for $a>0$.

$0$
$1$
$\ln a$
$a$

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Correct answer: C. $\ln a$

A fundamental limit is $\lim_{x\to 0}\dfrac{a^x-1}{x}=\ln a$. It is closely related to the derivative of $a^x$ at $x=0$.

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