Class 11 Maths questions with answers

210 NCERT Class 11 Maths questions with answers, chapter by chapter, from Sets to Probability.

How to teach Class 11 Maths · Class 11 Maths common misconceptions

• Chapter 1: Sets

  15 questions, 4 with pictures · Writing a set in roster form and in set-builder form · A set is a well-defined collection, and membership is written with the element symbol · Finite and infinite sets, and the number of elements n(S) · The empty set · Equal sets · Establishing that two set expressions are equal · Subsets, proper subsets and supersets · Intervals as subsets of the real numbers · The universal set and its picture as a Venn diagram · The difference of two sets · The complement of a set and De Morgan's laws · The intersection of two sets, and disjoint sets

• Chapter 2: Relations and Functions

  15 questions, 2 with pictures · An ordered pair records two objects in a fixed order, and two pairs are equal only when both components match · Given the pairs of A × B, A is the set of first entries and B the set of second entries · The cartesian product A × B collects every ordered pair whose first entry comes from A and second from B · If A has p elements and B has q, then A × B has exactly pq elements · Repeating the product gives ordered triples, and over R it gives the plane and space · A cartesian product distributes over union and over intersection, reaching both terms · A × B and B × A have the same size but are different sets · A function is evaluated by substituting the input, and a linear rule can be recovered from its pairs · A relation from A to B is a subset of A × B, cut out by a stated rule · The domain holds the first entries, the range the second entries, and the codomain is the whole target set · A relation is a function when every element of the first set has one and only one image · Every subset of A × B is a relation, so there are 2^(pq) of them · The domain is every input the rule admits, and the range is every value it actually takes · Two real functions may be added, subtracted, scaled, multiplied and divided pointwise · Identity, constant, polynomial, rational, modulus, signum and greatest-integer functions each have a fixed rule and shape

• Chapter 3: Trigonometric Functions

  15 questions, 2 with pictures · 2sin²(x/2) = 1 − cos x and 2cos²(x/2) = 1 + cos x give the half-angle magnitudes, and the quadrant of x/2 decides their signs · An angle records an amount of rotation together with its direction, so it may exceed a full turn and may be negative · In a circle of radius r an arc of length l subtends l = rθ, with θ measured in radians · One radian is the angle an arc of length one subtends in a unit circle, so a radian measure is an arc divided by a radius · Since π radian = 180°, a degree measure is multiplied by π/180 and a radian measure by 180/π · At the quadrantal angles the point sits on an axis, and going round a whole turn returns every value unchanged · For the point P(a, b) at angle x on the unit circle, cos x is the first coordinate and sin x the second · Shifting by π/2 exchanges sine and cosine, while shifting by π or 2π keeps the function and only decides a sign · cosec, sec, tan and cot are built as quotients of sine and cosine, so each is undefined exactly where its denominator vanishes · One ratio fixes the magnitude of the rest through the identities, and the quadrant fixes their signs · Because the values repeat after 2π, any angle can be shifted by whole revolutions into a range where the table applies · The signs of the coordinates in each quadrant fix the sign of every function there, and reflecting in the x-axis gives the odd-even rules · The table for 0, π/6, π/4, π/3 and π/2 gives every value the chapter's evaluations are built from · cos(x ± y) and sin(x ± y) mix both functions of both angles, and tan(x ± y) has the product in its denominator · sin and cos take every real input and land in [−1, 1]; the four quotients drop the inputs their denominators kill, and sec and cosec never land inside (−1, 1)

• Chapter 4: Complex Numbers and Quadratic Equations

  15 questions, 2 with pictures · A complex number is a number of the form a + ib with a and b real; a is Re z and b — not ib — is Im z · One equation between complex numbers is two real equations: a + ib = c + id exactly when a = c AND b = d · The product is (ac − bd) + i(ad + bc): the minus sign in the real part is i² = −1 arriving · The real numbers leave x² = −1 unsolved, so a symbol i is defined with i² = −1 and the system is enlarged to contain it · z·z̄ = |z|² = a² + b²: multiplying by the conjugate is how a complex expression is turned into a real one · |a + ib| = √(a² + b²) is a single NON-NEGATIVE REAL number, not a complex one · Addition is defined part by part, and subtraction is adding the additive inverse, so the minus sign reaches BOTH parts · A quotient is made real-denominatored by multiplying above and below by the conjugate of the divisor · The powers of i cycle with period four — i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = −1, i⁴ᵏ⁺³ = −i — and negative powers cycle too · Because complex numbers obey the same laws, (z₁ ± z₂)² and ³ and z₁² − z₂² expand exactly as for reals — the only extra step is i² = −1 · Every non-zero complex number has an inverse a/(a²+b²) + i(−b)/(a²+b²), which is the conjugate divided by the modulus squared · Plotting x + iy at the point (x, y) turns the complex numbers into a plane, in which the modulus is a distance from the origin and the axes acquire names · Both pass straight through a product and a quotient — |z₁z₂| = |z₁||z₂| and conj(z₁z₂) = z̄₁z̄₂ — which is why they simplify what expansion would not · √−a means √a·i for positive a, and the rule √a·√b = √(ab) FAILS when both a and b are negative · The conjugate z̄ = a − ib reverses the sign of the IMAGINARY part only, and in the Argand plane it is the mirror image in the real axis

• Chapter 5: Linear Inequalities

  15 questions, 2 with pictures · A solution is drawn by marking the boundary with an open circle for a strict inequality or a dark circle for a slack one and shading the side the solutions lie on · An inequality using less than or greater than is strict, and one using at most or at least is slack, which is what decides whether the boundary value is a solution · Many real statements fix a limit rather than a value, and a limit needs one of the symbols less than, greater than, at most or at least instead of an equals sign · A linear inequality is solved by clearing any denominators, collecting the variable on one side with Rule 1 and dividing by its coefficient with Rule 2 · A worded condition is turned into an inequality by naming the unknown, writing the quantity the words describe and choosing the symbol the words demand · An inequality between numbers alone is numerical, one containing a variable is literal, and one that bounds a quantity on both sides at once is double · An inequality is linear when the variables appear only to the first power, and the chapter names ax plus b compared with 0 as linear in one variable and ax plus by compared with c as linear in two · A solution is a value of the variable that makes the inequality a true statement, and the solution set is the collection of all of them · Two real numbers or two algebraic expressions related by less than, greater than, at most or at least form an inequality · The same inequality has different solution sets over the natural numbers, the integers and the real numbers, and over a restricted set it may have none at all · When both sides of an inequality are multiplied or divided by a negative number the inequality symbol is reversed · A solution over the real numbers is written as an interval, with a round bracket where the endpoint is excluded and a square bracket where it is included · A double inequality is solved by performing the same operation on all three parts at once, reversing both symbols if that operation is a multiplication or division by a negative number · Adding or subtracting the same number on both sides, and multiplying or dividing both sides by the same positive number, leave the inequality symbol unchanged · A system of two inequalities is solved by solving each one separately and keeping the values common to both

• Chapter 6: Permutations and Combinations

  15 questions, 1 with pictures · Choosing r objects is the same as rejecting the other n minus r, so n C r equals n C of n minus r; two equal C symbols therefore have lower indices that are either equal or add up to n, and n C r plus n C r minus one is n plus one C r · When one position is restricted -- the last digit must make the number even, the first digit may not be zero -- that position is filled first and the free positions are counted afterwards from what is left · A problem of arranging objects is counted by drawing one vacant place for each position and filling them in succession, each place taking one fewer object than the one before it when repetition is not allowed · When the outcomes fall into separate cases that cannot both happen, each case is counted on its own and the counts are added, which is how 'at least' and 'at most' questions are answered · Every combination of r objects yields r factorial permutations, so n P r equals n C r times r factorial, and n C r is n factorial divided by r factorial times n minus r factorial · If each of r places may take any of the n objects, with repetition allowed, the number of arrangements is n multiplied by itself r times, written n to the power r · The number of permutations of n different objects taken r at a time is the falling product n(n-1)(n-2)...(n-r+1), which equals n! divided by (n-r)! and is written n P r · The symbol n factorial denotes the product of the first n natural numbers, satisfies n! = n times (n-1)!, and is defined to be 1 when n is 0 · When one choice is followed by another, the number of ways the two happen in that order is the first count multiplied by the second, and the same holds for any number of successive choices · When so many objects must come from each of several groups, the separate selections are multiplied; if the chosen objects must then be arranged, that count multiplies again · When some objects are indistinguishable, the count of all arrangements is divided by the factorial of the number of objects of each repeated kind · A permutation is an arrangement of objects in a definite order, so two selections of the same objects in different orders are different permutations · An equation relating two P symbols is solved by writing each as a falling product, cancelling the factors common to both sides, and rejecting any root that cannot be a number of objects · Objects that must stay together are tied into a single block, arranged with the rest and then arranged inside the block; objects that must stay apart are placed into the gaps left by the others, or counted as the total minus the together count · A combination is a selection of objects in which the order is not important, so the same objects chosen in a different order are the same combination

• Chapter 7: Binomial Theorem

  15 questions, 1 with pictures · Adding or subtracting the expansions of a sum and a difference · Computing a numerical power by splitting the base · Expanding a trinomial by grouping it into a binomial · Expanding when the quantities themselves carry coefficients · How the two powers move through the expansion · Pascal's triangle and the addition rule that builds it · The entries of the triangle are the numbers nCr · Approximating a power with the first few terms · How many terms an expansion has · The statement of the binomial theorem and its proof by induction · Why expanding a binomial needs a theorem · The expansion of (1+x) to the n, and the two sums it gives · Why the signs alternate in the expansion of a difference · Expanding when one quantity is a fraction or a reciprocal · Comparing a large power with a number using only the first terms

• Chapter 8: Sequences and Series

  15 questions · A sequence as a function whose domain is the natural numbers · Finding which term of a G.P. a given value is · Finite and infinite sequences · How two given terms fix the whole G.P. · Reading terms off a formula for the nth term · The series associated with a sequence · What makes a list a sequence, and how its terms are named · A sequence given by a recurrence relation · The general term of a G.P. · Choosing three unknown terms of a G.P. symmetrically · Sigma notation for a series · How many terms of a G.P. reach a given sum · What makes a progression geometric · The sum of the first n terms of a G.P. · The geometric mean of two numbers, and inserting several

• Chapter 9: Straight Lines

  15 questions, 11 with pictures · The coordinate formulae the chapter recalls before it begins · Parallel lines and the equality of their slopes · Perpendicular lines and the product of their slopes · The inclination of a line and the definition of its slope · The angle between two lines from their slopes · Equations of horizontal and vertical lines · Computing the slope from the coordinates of any two points on the line · The point of intersection of two lines, and when three lines are concurrent · The point-slope form of the equation of a line · The two-point form of the equation of a line · The intercept form of the equation of a line · The slope-intercept form, with a y-intercept or an x-intercept · Feet of perpendiculars, mirror images, and distance measured along a given line · The distance between two parallel lines · The perpendicular distance of a point from a line

• Chapter 10: Conic Sections

  15 questions, 5 with pictures · How the cutting plane decides which conic appears · Fixing a circle from the conditions it must meet · Recovering a circle's centre and radius by completing the square · Writing a circle from its centre and radius · The eccentricity and latus rectum of a hyperbola · The focus and directrix that define a parabola · The latus rectum of a parabola · The latus rectum of an ellipse · The two foci that define an ellipse · Building an ellipse's equation from given measurements · How an ellipse's semi-axes, focal distance and eccentricity are tied together · Reading an ellipse off its standard equation · The four standard parabolas and what their equations say · The two foci that define a hyperbola · Conics that model a real object

• Chapter 11: Introduction to Three Dimensional Geometry

  15 questions, 2 with pictures · The three axes, the three coordinate planes and the origin · Deciding whether three points are collinear or form a right angled triangle · The form a point takes when it lies on an axis or in a coordinate plane · The octant a point lies in, read from the signs of its three coordinates · A point's triplet as its three perpendicular distances from the coordinate planes · The distance between two points in space · The foot of the perpendicular from a point onto a coordinate plane or axis · Naming a quadrilateral from its sides and diagonals, and finding a missing vertex · The equation of a set of points described by a condition on distances · The centroid of a triangle as the average of its three vertices

• Chapter 12: Limits and Derivatives

  15 questions, 3 with pictures · Matching the angle so that the standard limit applies · The average rate of change, the instantaneous rate, and the slope of the tangent · The left hand limit, the right hand limit, and when the limit exists · The power rule and the derivative of a polynomial · The product rule for derivatives · The quotient rule for derivatives · The standard limit of the difference of nth powers · The limit at a point and the value of the function there may differ · The algebra of limits: sum, difference, product, quotient and constant multiple · The derivative function from the first principle · The limit of a polynomial function is its value at the point · The limit of a rational function when the denominator does not vanish · The order of limits, the sandwich theorem, and the bounds on sine over x · The zero over zero form and cancelling the factor that causes it · The two standard trigonometric limits

• Chapter 13: Statistics

  15 questions · The range of a series · What adding to, or multiplying, every observation does to the spread · Why a measure of central tendency is not enough · Mean deviation about the mean for ungrouped data · The median of a list, and mean deviation about it · Deviations about a central value, and why their signs are dropped · The limitations of mean deviation · The step-deviation shortcut for the mean, variance and standard deviation · From squared deviations to the variance · The standard deviation and its units · The sum-of-squares formula for the variance · Mean deviation for a continuous frequency distribution · Mean deviation for a discrete frequency distribution · Variance and standard deviation of a frequency distribution · Recovering or correcting observations from a given mean and variance

• Chapter 14: Probability

  15 questions, 2 with pictures · An event is a subset of the sample space · Simple events and compound events · The impossible event and the sure event · The probability of an event as the sum of its outcome probabilities · The algebra of events: not A, A or B, A and B, A but not B · Exhaustive events, and mutually exclusive and exhaustive events · Mutually exclusive events · Deciding whether an assignment of probabilities is valid · The axioms of probability · Equally likely outcomes and the ratio of favourable outcomes · Probability when the outcomes have to be counted · The addition rule for the event A or B · The probability of the event not A · The probability that neither event occurs, and that atleast one occurs