Class 11 MathematicsImportant Questions for Board Exams 2025-26

If A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}, find A − B, B − A, and A △ B (symmetric difference).

In a survey of 600 students in a school, 150 students were found to drink Tea and 225 students to drink Coffee, while 100 students drink both Tea and Coffee. Find the number of students who drink neither Tea nor Coffee.

Let A = {x : x ∈ N, x ≤ 6} and B = {x : x ∈ N, 3 < x < 8}. Find A ∪ B, A ∩ B, and verify De Morgan's law.

If A and B are two sets such that A ∪ B has 18 elements, A has 8 elements and B has 15 elements, how many elements does A ∩ B have?

Write the following sets in roster form: (i) A = {x : x is a two-digit natural number such that the sum of its digits is 8} (ii) B = {x : x ∈ N, x² < 25}.

If f(x) = x² − 3x + 2, find f(1), f(−1), and f(1/2). Also find the values of x such that f(x) = 0.

Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {(x, y) : y = x + 1}. Write R, domain, and range of R.

Find the domain and range of the function f(x) = (x² − 4)/(x − 2).

If f and g are two real functions defined as f(x) = √(x + 1) and g(x) = 1/(x − 2), find (f + g)(x), (fg)(x), and their domains.

The function f is defined by f(x) = 1 − x for x < 0; 1 for x = 0; x + 1 for x > 0. Draw the graph of this function.

Prove that: cos(π/4 + x) + cos(π/4 − x) = √2 cos x.

Find the general solution of: cos 4x = cos 2x.

Prove: (cos x − cos y)² + (sin x − sin y)² = 4 sin²((x − y)/2).

If tan x = 3/4 and x is in the third quadrant, find the values of sin x/2, cos x/2, and tan x/2.

Prove that: sin 20° sin 40° sin 80° = √3/8.

A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

Express (3 + 2i)(2 − 3i) in the form a + ib and find its modulus and argument.

Solve: x² + 3x + 5 = 0 over the complex number field.

If z₁ = 2 + 3i and z₂ = 1 − 2i, find z₁/z₂ in the form a + ib.

Find the modulus and argument of the complex number (1 + i)/(1 − i).

If (a + ib)² = x + iy, prove that (a² − b²)² + (2ab)² = (x² + y²).

Solve: 3x − 7 > 5x − 1 and represent the solution on a number line.

Solve the system of inequalities graphically: x + y ≤ 4, 2x − y > 0.

Solve: (x − 2)/(x + 5) > 2. Find the solution set and represent it on the number line.

A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid solution must be added so that the resulting mixture will contain more than 15% but less than 18% acid?

Solve: −5 ≤ (5 − 3x)/2 ≤ 8.

How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?

In how many ways can 6 people be seated at a round table? How many of these arrangements will have two specific people always together?

If ⁿC₈ = ⁿC₁₂, find ⁿC₁₇ and ²²Cₙ.

A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women?

How many 4-letter codes can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?

Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these, do all vowels always occur together?

Find the 4th term from the end in the expansion of (x³/2 − 2/x²)⁹.

If the coefficients of (r − 5)th and (2r − 1)th terms in the expansion of (1 + x)³⁴ are equal, find r.

Using the Binomial Theorem, evaluate (101)⁴.

Find the term independent of x in the expansion of (√x − √(3/x²))¹⁰.

If the 21st and 22nd terms in the expansion of (1 + x)⁴⁴ are equal, find x.

Prove that Σ(r=0 to n) 3ʳ · ⁿCᵣ = 4ⁿ.

The sum of first three terms of a GP is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio, and the sum to n terms of the GP.

Find the sum to n terms of the series: 3 × 8 + 6 × 11 + 9 × 14 + ...

If A and G be respectively the AM and GM between two positive numbers, prove that the numbers are A ± √(A² − G²).

The sum of three numbers in GP is 56. If we subtract 1, 7, 21 from these numbers in that order, we get an AP. Find the numbers.

Find the 20th term and sum to 20 terms of the series 2 + 6 + 18 + 54 + ...

Evaluate: Σ(k=1 to 10) (2 + 3k).

Find the equation of the line passing through (−1, 1) and (2, −4). Find its x-intercept and y-intercept.

Find the distance of the point (3, −5) from the line 3x − 4y − 26 = 0.

Find the equation of the line which is equidistant from parallel lines 9x + 6y − 7 = 0 and 3x + 2y + 6 = 0.

Prove that the three lines 3x + y + 2 = 0, 2x − y + 3 = 0, and x + 2y − 3 = 0 form a triangle. Also find the area of the triangle.

Find the angles between the lines y = (2 − √3)x + 5 and y = (2 + √3)x − 7.

Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.

Find the coordinates of the foci, vertices, eccentricity, and length of latus rectum of the ellipse x²/25 + y²/9 = 1.

Find the equation of the parabola with focus (2, 0) and directrix x = −2.

Find the equation of the hyperbola with foci at (±5, 0) and the length of the transverse axis = 8.

Find the equation of the ellipse whose foci are (±4, 0) and the sum of distances from a point on the ellipse to the two foci is 10.

Find the area of the triangle formed by the lines joining the vertex of the parabola x² = 12y to the endpoints of the latus rectum.

Find the distance between the points (2, −1, 3) and (−2, 1, 3).

Show that the points (−2, 3, 5), (1, 2, 3), and (7, 0, −1) are collinear.

Find the coordinates of the point which divides the join of (−2, 3, 5) and (1, −4, 6) in the ratio 2:3 internally.

Find the centroid of a triangle whose vertices are (1, 1, 1), (0, −1, 0), and (2, 1, −1).

The mid-point of a segment joining A(a, b, c) and B is (4, −1, 2). If B = (6, −7, 4), find A.

Evaluate: lim(x→3) (x⁴ − 81)/(2x² − 5x − 3).

Find the derivative of f(x) = sin x from first principles.

Evaluate: lim(x→0) (sin 2x + sin 3x) / (2x + sin 3x).

Find the derivative of (x² + 5x + 3)(x³ − 7x + 1) using the product rule.

Evaluate: lim(x→π/2) (tan 2x) / (x − π/2).

Find the derivative of (5x³ + 3x − 1)(x − 1) and hence find the slope of the tangent at x = 1.

Find the mean deviation about the mean for the data: 4, 7, 8, 9, 10, 12, 13, 17.

Find the mean, variance, and standard deviation for the following data: 6, 7, 10, 12, 13, 4, 8, 12.

The mean and standard deviation of 20 observations are found to be 10 and 2 respectively. On rechecking it was found that an observation 8 was incorrect, the correct value is 12. Find the correct mean and standard deviation.

Calculate the coefficient of variation for scores in Mathematics (mean = 57.5, SD = 1.5) and Statistics (mean = 31.2, SD = 0.6). Which subject shows greater variability?

Find the standard deviation for the following frequency distribution: class 0-10 (f = 5), 10-20 (f = 8), 20-30 (f = 15), 30-40 (f = 16), 40-50 (f = 6).

A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are similar in shape and size. A disc is drawn at random. Find the probability that it is (i) red (ii) yellow (iii) blue (iv) not blue.

Two students Anil and Ashima appeared in an examination. The probability that Anil will qualify is 0.05 and Ashima will qualify is 0.10. The probability that both will qualify is 0.02. Find the probability that (a) both will not qualify (b) at least one of them will qualify.

In a class of 60 students, 30 opted for NCC, 32 opted for NSS, and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that the student opted for NCC or NSS.

Three coins are tossed simultaneously. Write the sample space and find the probability of getting (i) exactly two heads (ii) at most two heads.

One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting a king or a heart.