This exercise focuses on combinations, which is the selection of objects where the order does not matter. The formula for the number of combinations of $n$ distinct objects taken $r$ at a time is $C(n, r) = \frac{n!}{r!(n-r)!}$. 1. If $C(n, 8) = C(n, 2)$, find $C(n, 2)$. We use the property that if $C(n, … Read more
This exercise focuses on permutations, which is the arrangement of objects where the order matters. The formula for the number of permutations of $n$ distinct objects taken $r$ at a time is $P(n, r) = \frac{n!}{(n-r)!}$. 1. 3-Digit Numbers from Digits 1 to 9 (No Repetition) We have $n=9$ digits and we are choosing and … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 6 Exercise 6.2 . Learn to evaluate and simplify expressions involving factorials (e.g., $8!$, $4! – 3!$). Practice algebraic simplification techniques for fractions of factorials, such as $\frac{8!}{6! 2!}$, by expanding the larger factorials. Solve linear equations involving factorials to find the unknown variable $\mathbf{x}$ … Read more
This exercise uses the Fundamental Principle of Counting (Multiplication Principle), where the total number of ways to perform a sequence of independent tasks is the product of the number of ways to perform each task. Class 11 Maths Exercise 6.1 Solutions: Counting Principles, Permutations & Combinations 1. 3-Digit Numbers from Digits 1, 2, 3, 4, … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 5 Miscellaneous Exercise . Learn to solve compound inequalities involving three parts (Q.1-6) and find the intersection of solution sets for multiple inequalities (Q.7-10), representing them graphically on the number line. Master real-world applications including temperature conversion (Fahrenheit/Celsius) and solution mixture/dilution problems involving percentages (Q.12-13) … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 5 Exercise 5.1 . Learn to solve linear inequalities for $\mathbf{x}$ as a natural number, integer, or real number (Q.1-4). Practice complex algebraic inequalities (Q.5-16), including those involving fractions and graphing solutions on the number line (Q.17-20). Also includes solving application problems involving averages, consecutive … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 4 Miscellaneous Exercise This exercise reviews various properties and calculations involving complex numbers, including powers, identities, modulus, and algebra. 1. Evaluate: $\left[\left(\frac{1}{i}\right)^{25}\right]^3 + i^{18}$ First, simplify the powers of $i$: Now add the results: $$\text{Expression} = i + (-1) = -1 + i$$ $$\mathbf{\text{Result}: -1 … Read more
Find comprehensive, step-by-step solutions for NCERT Class 11 Maths Chapter 4 Exercise 4.1 . Master the process of expressing complex numbers in the standard $\mathbf{a + ib}$ form, including simplifying powers of $i$ (e.g., $i^{-39}$, $i^{19}$) and expanding complex numbers raised to a power (e.g., $(1-i)^4$, $(1/3 + 3i)^3$). Learn how to efficiently calculate the … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 3 Miscellaneous Exercise . Master advanced trigonometric proofs involving extensive use of the Sum-to-Product formulas (e.g., $\sin C \pm \sin D$, $\cos C \pm \cos D$) and Half-Angle Identities. Learn how to determine the exact values of $\mathbf{\sin(x/2)}$, $\mathbf{\cos(x/2)}$, and $\mathbf{\tan(x/2)}$ when given $\tan x$, … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 3 Exercise 3.2. Learn to find the values of all five remaining trigonometric functions given one value and the quadrant of $x$. Master the use of trigonometric identities ($\sin^2 x + \cos^2 x = 1$) and the CAST rule to determine signs. Practice evaluating trig … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 3 Exercise 3.3 . Master essential Trigonometric Identities including: Sum and Difference formulas (Q.5, Q.6, Q.7), Product-to-Sum formulas (Q.11, Q.12, Q.13), and Double/Triple Angle expansions (Q.24, Q.25). Practice proving identities using standard values, angle transformation, and algebraic manipulation. Essential for developing proof skills in trigonometry. … Read more
Find comprehensive, solved answers for the NCERT Class 11 Maths Chapter 2 Miscellaneous Exercise . Master advanced concepts: proving a relation is not a function by checking overlapping intervals, finding the Domain and Range of complex functions ($f(x) = \frac{x^2}{1+x^2}$, $\sqrt{x-1}$), solving for unknown constants in a function, and checking the properties of relations (Reflexivity, … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 2 Exercise 2.3 . Learn the critical rules to identify a function from a relation and practice finding the Domain and Range of various real-valued functions, including quadratic, square root, and absolute value functions. Master function notation and practical applications like the Celsius-Fahrenheit conversion function. … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 2 Exercise 2.2. Learn how to define a Relation ($R$) from $A$ to $B$ in Roster Form and Set-Builder Form. Identify the Domain, Codomain, and Range of various relations, including those on Natural Numbers and Integers. Also, calculate the number of possible relations between two … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 1 Miscellaneous Exercise (Sets). Learn to prove fundamental set properties (Absorption Law, Distributive Law), demonstrate set equivalences (A⊂B⟺A∪B=B), and disprove statements using effective counterexamples. Essential practice for mastering subset relations and set algebra proofs. This exercise covers fundamental concepts and properties of Sets, including subsets, … Read more
Get detailed, step-by-step solutions for the NCERT Class 11 Maths Chapter 14 Miscellaneous Exercise Solve complex probability problems using combinations ($\binom{n}{k}$), permutations ($P(n, k)$), and the complement rule to find probabilities for drawing marbles, selecting cards, and lottery tickets (Q.1-4). Apply the Principle of Inclusion-Exclusion to solve derangement problems (Q.6). Use the Addition Rule and … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 14 Exercise 14.2. Master the classical definition of probability and the conditions for a valid probability distribution (Q.1). Calculate probabilities for various experiments like coin tosses and dice rolls (Q.2-8). Learn to apply the Addition Rule of Probability ($P(A \cup B) = P(A) + P(B) … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 14 Exercise 14.1 Learn to define the Sample Space and precisely describe events for experiments involving single dice, a pair of dice, and three tossed coins (Q.2-3). Master the concept of mutually exclusive events ($E \cap F = \emptyset$) (Q.1, Q.4). Solutions cover finding unions … Read more
Get detailed, step-by-step solutions for the NCERT Class 11 Maths Chapter 13 Miscellaneous Exercise Master advanced statistical problems involving Mean ($\bar{x}$), Variance ($\sigma^2$), and Standard Deviation ($\sigma$). Learn how to find missing observations given the mean and variance (Q.1-2). Analyze the effect of scaling (multiplication) on the mean and standard deviation (Q.3-4). Practice calculating the … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 13 Exercise 13.1 The Mean Deviation (MD) is a measure of dispersion calculated as the average of the absolute differences between the data values and the mean or median. Learn to calculate Mean Deviation ($\text{MD}$) for ungrouped data about both the Mean ($\bar{x}$) (Q.1-2) and … Read more
This exercise focuses on calculating the mean ($\bar{x}$), variance ($\sigma^2$), and standard deviation ($\sigma$) for different types of data, including ungrouped data, discrete frequency distributions, and continuous frequency distributions. Mean and Variance for Ungrouped Data (Q. 1–3) The formulas for ungrouped data are: $$\bar{x} = \frac{\sum x_i}{n} \quad \text{and} \quad \sigma^2 = \frac{1}{n} \sum (x_i … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 12 Exercise 12.2 Master the concept of the derivative. Find derivatives of algebraic functions at specific points (Q.1-3). Learn to find derivatives using the first principle for $x^3-27$, $(x-1)(x-2)$, $1/x^2$, and $\frac{x+1}{x-1}$ (Q.4), and for $\cos x$ (Q.10). Apply the Power Rule, Product Rule, and … Read more
Get detailed, step-by-step solutions for the NCERT Class 11 Maths Chapter 12 Miscellaneous Exercise Master advanced differentiation techniques. Find derivatives from the first principle for algebraic and trigonometric functions like $-x$, $-1/x$, $\sin(x+1)$, and $\cos(x-\pi/8)$ (Q.1). Apply the Product Rule and Quotient Rule to complex combinations of algebraic and trigonometric functions, including $\frac{\cos x}{1+\sin x}$ … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 12 Exercise 12.1 Master techniques for evaluating limits, including direct substitution (Q.1-5), algebraic manipulation of indeterminate forms ($\frac{0}{0}$) using factorization (Q.7-8) and identities (Q.17), and applying the fundamental limit $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ (Q.13-20). Learn to evaluate limits for piecewise functions by … Read more
Get detailed, step-by-step solutions for the NCERT Class 11 Maths Chapter 11 Miscellaneous Exercise Learn to apply 3D formulas to find the fourth vertex of a parallelogram using the midpoint formula (Q.1). Calculate the lengths of medians of a triangle (Q.2). Use the centroid formula to find unknown coordinates when the centroid is the origin … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 11 Exercise 11.2 Master the distance formula in 3D to find the distance between pairs of points (Q.1). Apply the distance formula to test for collinearity (Q.2), and verify the properties of isosceles and right-angled triangles (Q.3). Find the equation of the locus of points … Read more
Get detailed solutions for NCERT Class 11 Maths Chapter 11 Exercise 11.1 Master the fundamental concepts of three-dimensional geometry. Learn to identify the coordinates of points lying on the coordinate axes (Q.1) and coordinate planes (Q.2). Understand the concept of the eight octants and how to determine the octant of a given point based on … Read more
Get detailed, step-by-step solutions for the NCERT Class 11 Maths Chapter 10 Miscellaneous Exercise . Solve application problems for parabolas including finding the focus of a reflector (Q.1), and determining the width/height of parabolic arches (Q.2, 3). Find the height of a semi-elliptical arch (Q.4) and the locus of a moving rod (Q.5). Analyze the … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 10 Exercise 10.4 This exercise focuses on identifying the key properties of hyperbolas from their equations and finding the equation given specific conditions. All hyperbolas here are centered at the origin $(0, 0)$. The key relation for any hyperbola is $c^2 = a^2 + b^2$, … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 10 Exercise 10.3 This exercise covers finding the key properties of ellipses from their equations and determining the equation of an ellipse from given properties. All ellipses here are centered at the origin $(0, 0)$. The key relation for any ellipse is $a^2 = b^2 … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 10 Exercise 10.2 . Learn to find the focus, axis, directrix, and length of the latus rectum for parabolas centered at the origin (Q.1-6). Determine the equation of a parabola given its focus and directrix, or its vertex, focus, and a point it passes through … Read more
Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 10 Exercise 10.1 . Learn to find the equation of a circle given its center $(h, k)$ and radius $r$ (Q.1-5, 14). Master converting the general equation to find the center and radius (Q.6-9). Solve problems for circles passing through given points and whose center … Read more
This exercise reviews all key concepts of the Straight Lines chapter, including different forms of lines, distances, angles, concurrency, and geometric applications.Get detailed, step-by-step solutions for the NCERT Class 11 Maths Chapter 9 Miscellaneous Exercise 1. Line Parallel to Axes and Passing Through Origin The given line is $L: (k-3) x – (4 – k^2) … Read more