Past Paper 2025 11th Class Karachi Board Mathematics Subjective (Science Pre-Engineering Group)

Paper Image

SECTION 'B' (Short-Answer Questions) (40)

NOTE: Attempt any TEN part questions in all. All ques ns carry equal marks. (i.e. 4 marks of each part).

2.

i) Separate the real and imaginary parts of (√2 + i)-2.

ii) Using properties, Prove that:

|

αβγαβγ
βγααβγ
γαβαβγ
|= |
αα2α3
ββ2β3
γγ2γ3
|

iii) Show that

1-2-6
-329
20-3

is a periodic matrix of period 2.

iv) Find the angle between the vectors 3î + 5ĵ - 3k̂ and î - 4ĵ + k̂.

v) Insert three arithmetic means between 3 and 11.

vi) In a G.P. a3 = 12 and a8 = 384. Find the nth term.

vii) Find the sum of the series ∑k=2∞ (1/(k2-1)).

viii) If nP3 = 12, find n.

ix) Prove that by using principle of mathematical induction (7n - 4n) divisible by 3. ---OR---Prove that by using principle of mathematical induction. 13 + 23 + 33 + ... + n3 = [n(n+1)2]2, for all natural numbers n.

x) Find inverse of f and determine domain and range f-1 for the real valued function f(x) = x-1x-3, x ≠ 3.

xi) Prove that: sin 6θ + sin 4θcos 6θ + cos 4θ = tan 5θ

xii) Find the smallest angle in ΔABC when a = 6cm, b = 8cm and c = 9.4cm.

xiii) Find the value of R and r in ΔABC when a = 40cm, b = 12cm and c = 38cm.

SECTION “C” (Detailed Answer Questions)(40)

NOTE: Attempt any FIVE question from this section. All questions carry equal marks.

3.

Use Gauss-Jordan method to solve the system of linear equation:

x - y + 4z = 4

2x + 2y - z = 2

3x - 2y + 3z = -3

4.

In a parallelogram ABCD, X is the mid point of AB and Y divides BC in 1:2, Show that if Z divides DX in 6:1, then it also divides AY in 3:4.

5.

Two events A and B sure that P(A) = 14, P(A/B) = 12 and P(B/A) = 23. Find P(A ∩ B) and P(B).

6.

If x = 13 + 1.33.6 + 1.3.53.6.9 + ⋯ then prove that x2 + 2x - 2 = 0.

7.

Show that 2 tan-1 (23) = cos-1 (513)

8.

Prove that: cos 20° cos 40° cos 60° cos 80° = 116.

9.

Find the point of intersection of the functions y = 2x + 1 and y = x2 - 4x + 6, ∀x ∈ R graphically.

Matric / Inter Past Papers

Punjab Educational Boards Past Papers:

KPK Educational Boards Past Papers:

University Past Papers

Class Wise Past Papers

Past Papers

Class Wise Past Papers

Entry Test Past Papers

Competitive Exams Past Papers

Technical Education Exams Past Papers

Punjab Examination Commission Past Papers

Past Papers feedback:

If you want to share some information with other students regarding past papers, you can send your valued feedback to ilmkidunya.com

Share your comments & questions here

Guest

  • No comments yet. Be the first to comment!