Class: XI Mathematics ST. MARGARET SR. SEC. SCHOOL PRE-FINAL EXAMINATION 2025-26

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ST MARGARET SR. SEC. SCHOOL

PRE-FINAL EXAMINATION 2025-26

Subject:	Mathematics
Class:	XI
Time:	3 Hours
Maximum Marks:	80

General Instructions:

This Question paper contains five sections A, B, C, D and E. Each section is compulsory.
Section A has 20 MCQs of 1 mark each.
Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
Section C has 6 Short Answer (SA)-type questions of 3 marks each.
Section D has 4 Long Answer (LA)-type questions of 5 marks each.
Section E has 3 case based assessment of 4 marks each.

SECTION A (1 Mark Each)

Q1. Find the octant in which the points (-3, 1, 2) and (-3, 1, -2) lie respectively.

(a) second, fourth
(b) sixth, second
(c) fifth, sixth
(d) second, sixth

Q2. The range of function f(x) = |x + 3| is

(a) R
(b) (−∞, 3]
(c) [0, ∞)
(d) [1, 3]

Q3. Two finite sets have m and n elements respectively. The total number of subsets of first set is 56 more than the total number of subsets of the second set. The values of m and n respectively are.

(a) 7, 6
(b) 5, 1
(c) 6, 3
(d) 8, 7

Q4. If sin θ = 3 sin(θ + 2α), then the value of tan(θ + α) + 2 tan α is

(a) 3
(b) 2
(c) -1
(d) 0

Q5. If z = (1 – i√3) ⁄ (√3(1 – i)) then |z| = ?

(a) 1⁄√2
(b) 1⁄(2√2)
(c) 1⁄√3
(d) 1

Q6. It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?

(a) 720
(b) 1200
(c) 2400
(d) 2880

Q7. The expression cos(36° – A)cos(36° + A) + cos(54° – A)cos(54° + A) is equal to:

(a) sin 2A
(b) cos 3A
(c) cos 2A
(d) sin 3A

Q8. If A, B, C are three mutually exclusive and exhaustive events of an experiment such that 3P(A) = 2P(B) = P(C), then P(A) is equal to

(a) 1/11
(b) 2/11
(c) 5/11
(d) 6/11

Q9. Solving the system of inequalities -15 < 3(x-2)/5 ≤ 0, we get

(a) -13 < x ≤ 13
(b) -23 ≤ x < 2
(c) -23 < x ≤ 2
(d) -23 < x ≤ 23

Q10. The value of lim(x→0) (√(3+x) – √3) ⁄ x is

(a) 1⁄4
(b) 1⁄3
(c) 1⁄(2√3)
(d) 1⁄(3√2)

Q11. A pair of dice is rolled. If the outcome is a doublet, a coin is tossed. Then, the total number of outcomes for this experiment is

(a) 40
(b) 42
(c) 41
(d) 43

Q12. In a GP, the 3rd term is 24 and the 6th term is 192. Then, the 10th term is

(a) 1084
(b) 3290
(c) 3072
(d) 2340

Q13. The intercept cut off by a line from y-axis is twice than that from x-axis, and the line passes through the point (1, 2). The equation of the line is

(a) 2x + y = 4
(b) 2x + y = -4
(c) 2x – y = 4
(d) 2x – y = -4

Q14. Find the equation of the parabola with vertex at (0, 0) and focus at (0, 2).

(a) x² = 8y
(b) x² = 2y
(c) y² = 4x
(d) y² = 8x

Q15. A set of n values x₁, x₂,…,xₙ has standard deviation σ. The standard deviation of n values x₁+k, x₂+k,…,xₙ+k will be

(a) σ
(b) σ + k
(c) σ – k
(d) kσ

Q16. Let f(x) = { 3-2x, x ≤ 1; 3x+5, 1 < x ≤ 2; 5x-2, x > 2 }, then 2f(0) + f(3) is equal to

(a) 19
(b) 24
(c) 31
(d) 17

Q17. The largest coefficient in the expansion of (a+b)¹⁸ is

(a) C(18,9)
(b) C(18,6)
(c) C(18,18)
(d) C(18,12)

Q18. In an examination there are three multiple choice questions and each question has 4 choices. Number of ways in which a student can fail to get all answers correct is

(a) 11
(b) 12
(c) 27
(d) 63

ASSERTION-REASON BASED QUESTIONS

Directions: In the following questions, a statement of assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices.

(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.

Q19. Assertion(A): The foot of perpendicular drawn from the point A(1, 2, 8) on the xy-plane is (1, 2, 0).
Reason(R): Equation of xy plane is y = 0.

Q20. Assertion (A): If A = {1, 2, 3, 4} and B = {5, 6, 7, 8}, then A and B are disjoint sets.
Reason (R): If A ∩ B = ∅, then sets A and B are called disjoint sets.

SECTION B (2 Marks Each)

Q21. Three balls are drawn from a bag containing 5 red, 4 white, and 3 black balls. Find the number of ways in which this can be done if atleast 2 balls are red.

Q22. Find the sum to n terms of the sequence 0.8, 0.88, 0.888, 0.8888….

Find the values of p, if sum to infinity for the G.P. p, 1, 1/p, …… is 25/4.

Q23. What is represented by the shaded regions in each of the following Venn-diagrams?
(i) [Venn Diagram with intersection of A and B shaded] (ii) [Venn Diagram with intersection of A, B and C shaded]

Q24. If y = (cos x) ⁄ (1 + sin x), then find dy/dx.

Find derivative of f(x) = x²cos x + cot x.

Q25. Find the equation of the set of the points P such that its distances from the points M(2, 0, -5) and X(0, 1, 4) are equal.

SECTION C (3 Marks Each)

Q26. Solve for x ∈ R: (|x-3| – 1) ⁄ (|x-3| – 2) ≤ 0

Q27. Prove that sin 5A = 5 sin A – 20 sin³ A + 16 sin⁵ A.

If tan x = b/a, then express √( (a+b)/(a-b) ) + √( (a-b)/(a+b) ) in terms of x.

Q28. Evaluate lim(x→0) (1 – cos x √(cos 2x)) ⁄ x²

Find the derivative of f(x) = √(cos 2x) by first principle.

Q29. Show that the equation of the straight lines passing through the origin and making an angle θ with the line y = mx + c is y/x = (m ± tan θ) ⁄ (1 ∓ m tan θ).

Q30. Find the number of words with or without meaning which can be made using all the letters of the word AGAIN. If these words are written as in a dictionary, what will be the 47th word?

Find the number of digits greater than 7000 that can be formed with the digits 3, 5, 7, 8, 9 where no digits are repeated.

Q31. If a + ib = (c + i) ⁄ (c – i), then prove that b/a = 2c ⁄ (c² – 1).

SECTION D (5 Marks Each)

Q32. The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.

Calculate the mean deviation about the median for the following data:

Class	0-10	10-20	20-30	30-40	40-50	50-60
Frequency	6	7	15	16	4	2

Q33. If tan(θ/2) = √((a-b)/(a+b)) tan(φ/2), then show that cos θ = (a cos φ + b) ⁄ (a + b cos φ).

If angle θ is divided into two parts such that the tangent of one part is k times the tangent of other, and φ is their difference, then show that sin θ = ((k+1)/(k-1)) sin φ.

Q34. (a) A relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows: (x, y) ∈ R ⇔ x divides y. Express R as a set of ordered pairs and determine the domain and range of R.
(b) Find the domain and the range of the function: f(x) = √(x² – 4).

Q35. (a) Find derivative of f(x) = tan³[sin(x⁻³ + 3)] (b) Let f(x) = { (k cos 2x)/(π – 2x), when x ≠ π/2; 3, when x = π/2 }. Find ‘k’ if lim(x→π/2) f(x) = f(π/2).

SECTION E (Case Based Questions – 4 Marks Each)

Q36. Read the following passage and answer the questions given below.

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m.

Find the equation of the path traced by the man? (2)
Find the length of latus rectum? (1)
Find its eccentricity and the distance between the directrices? (1)

Q37. Two students, Anil and Vijay, appeared in a highly competitive examination. Anil has been preparing part-time while managing a job, which has left him with limited preparation time. On the other hand, Vijay, though dedicated, has struggled with certain key concepts. Based on their preparation and past performance, the probability that Anil will qualify the examination is estimated to be 0.05, and the probability that Vijay will qualify is estimated at 0.10. Additionally, the probability that both students will qualify together, due to their independent preparation and individual strengths, is calculated as 0.02.

Find the probability that at least one of them will qualify the exam. (1)
Find the probability that at least one of them will not qualify the exam. (1)
Find the probability that both Anil and Vijay will not qualify the exam. (1)
Find the probability that only one of them will qualify the exam. (1)

Q38. A polygon is regular when all angles are equal and all sides are equal (otherwise it is “irregular”). Below given figure is an equilateral triangle with sides 18cm. The midpoints of its sides are joined to form another triangle whose midpoints, in turn, are joined to form another triangle. The process is continued Indefinitely.

Answer the questions given below:

What type of sequence do the lengths of the sides of the triangle form? (1)
What is the sum of perimeters of all the triangles? (1)
Find the sum of areas of all the triangles? (2)