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📐 9th Math Professional Paper (2026)

Syllabus: New Book 2025-26 (13 Units) | Pattern: BISE Sahiwal (75 Marks)

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Unit 1: Real Numbers

MCQs (Select for Section A)

Which is irrational? Value of i² is: Conjugate of 2+3i: Real part of 2(3+i): Imaginary part of -i(3i+2): Every real number is: Multiplicative inverse of i:

Short Questions (Select for Sec-I)

Define Irrational Number. Simplify ∛(-125). Find conjugate of -3+4i. Solve |x-3|=5. Express 0.33... as fraction. Simplify i⁹+i¹⁹. Find x,y if x+iy+1=4-3i. Write standard form of -2+3i.

Long Questions (Select for Sec-II)

Prove √2 is irrational. Simplify (243)^-2/3 (32)^-1/5 / √(196)^-1. Show z + z̅ is real.

Unit 2: Logarithms

MCQs (Select for Section A)

Scientific notation of 0.005: log m^n equals: Base of natural log: log(xy) equals: Characteristic of 0.0035: log_b(a) x log_c(b) = ? Common log base:

Short Questions (Select for Sec-I)

Write 0.0067 in scientific notation. Find x: log₂ x = 5. State 1st Law of Logarithm. Evaluate log 81 base 3. Write as single log: log 25 - 2 log 3. Define Mantissa. Find log 0.00032.

Long Questions (Select for Sec-II)

Evaluate (0.678 x 9.01) / 0.0234 using log tables. Prove log_a(mn) = log_a m + log_a n.

Unit 3: Sets & Functions

MCQs (Select for Section A)

Power set of {a,b} has: A ∪ A = ? Point (-2, -3) is in Quadrant: If f: A->B, Domain is: Empty set is subset of: Order pair (x,y) = (2,3) means x=?

Short Questions (Select for Sec-I)

Define Power Set. If A={1,2}, find AxA. Define Onto Function. Find domain of R={(1,2), (2,3)}. Verify associative law for union. Define Bijective Function. If U={1..5}, A={1,2}, find A'.

Long Questions (Select for Sec-II)

Verify De Morgan’s Laws for U={1..10}, A={2,4,6}, B={1,3,5}. Show A u (B n C) = (A u B) n (A u C).

Unit 4: Factorization

MCQs (Select for Section A)

(a+b)²-(a-b)²=? Factors of x²-5x+6: a³-b³ equals: Degree of x²y+3: HCF of x²-4, x-2:

Short Questions (Select for Sec-I)

Factorize 4x²-12x+9. Find a²+b² if a+b=5, ab=4. Define Remainder Theorem. Factorize x³-y³. Find HCF of 12p³q², 8p²q³. Evaluate polynomial P(x)=x³-2x+1 at x=1. Simplify (x²-4)/(x+2).

Long Questions (Select for Sec-II)

Factorize x³-2x²-x+2 using Factor Theorem. Find square root by division: 4x⁴+12x³+x²+12x+4.

Unit 5: Linear Equations

MCQs (Select for Section A)

Solution of |x|=3: x=0 is equation of: Linear equation has degree: If x-2 > 0 then:

Short Questions (Select for Sec-I)

Solve |3x+2|=5. Define Linear Inequality. Solve 3(x-1) < 6. Check if x=2 is solution of 2x+1=5. Solve for x: 2x-3=7.

Long Questions (Select for Sec-II)

Solve and graph 2(x-1) ≤ 3(x+2). Solve |x+2| - 3 = 5 - |x+2|.

Unit 6: Trigonometry

MCQs (Select for Section A)

Tan θ = ? sin²θ + cos²θ = ? 1 + tan²θ = ? π radians = ? Reciprocal of sin θ:

Short Questions (Select for Sec-I)

Define Radian. Convert 45° to radians. Convert π/6 to degrees. Verify sin²30° + cos²30° = 1. Find r when l=5, θ=0.5. Prove tan θ cot θ = 1.

Long Questions (Select for Sec-II)

Prove (1-sin θ)(1+sin θ) = cos²θ. Find values of all trig ratios if sin θ = 12/13.

Unit 7: Coord Geometry

MCQs (Select for Section A)

Distance form: Midpoint of (2,2) and (0,0): Point (2, -3) lies in: Collinear points lie on:

Short Questions (Select for Sec-I)

Find distance between (2,3) and (5,7). Define Collinear Points. Find midpoint of (4, -2) and (-4, 2). Define Coordinate Geometry. Verify A(1,1), B(2,2) are collinear.

Long Questions (Select for Sec-II)

Show points A(0,0), B(3,0), C(0,4) form right triangle. Find k if points (2,k), (3,5), (4,6) are collinear.

Unit 8: Logic

MCQs (Select for Section A)

Negation of p: p ^ q is true if: Symbol for implication: Truth value of T v F:

Short Questions (Select for Sec-I)

Define Proposition. Make Truth Table for p v q. Define Negation. What is Tautology? Define Conjunction.

Long Questions (Select for Sec-II)

Construct truth table for ~(p ^ q) -> ~p v ~q. Prove p -> q is equivalent to ~p v q.

Unit 9: Similar Figures

MCQs (Select for Section A)

Symbol for similarity: Congruent triangles are: SSS stands for:

Short Questions (Select for Sec-I)

Define Similar Triangles. State AAA Postulate. Diff between Congruency and Similarity. Define Ratio.

Long Questions (Select for Sec-II)

Prove: Line intersecting two sides proportionally is parallel to third.

Unit 10: Graphs

MCQs (Select for Section A)

y=mx+c is: Horizontal line eq:

Short Questions (Select for Sec-I)

Draw y=2x. Define Abscissa. Check if (1,2) lies on y=x+1. Define Dependent Variable.

Long Questions (Select for Sec-II)

Solve graphically: x+y=3, 2x-y=3.

Unit 11: Loci

MCQs (Select for Section A)

Bisection means: Right bisectors of triangle are:

Short Questions (Select for Sec-I)

Define Locus. Define Circumcenter. Define Incenter. Define Centroid.

Long Questions (Select for Sec-II)

Prove: Point on right bisector is equidistant from endpoints.

Unit 12: Info Handling

MCQs (Select for Section A)

Most frequent value: Middle value is: Sum/Total = ? Max - Min = ?

Short Questions (Select for Sec-I)

Define Range. Find Mean of 2,4,6. Define Histogram. Define Frequency. Find Mode of 1,2,2,3.

Long Questions (Select for Sec-II)

Find Mean, Median, Mode of: 2,4,4,6,8,10. Construct Frequency Distribution Table.

Unit 13: Probability

MCQs (Select for Section A)

Prob of impossible event: Total prob equals: P(E) = ? Prob of Head in coin toss:

Short Questions (Select for Sec-I)

Define Sample Space. Prob of 6 on die? Define Event. Define Random Experiment. Prob of even number on die?

Long Questions (Select for Sec-II)

Two coins tossed. Find prob of at least 1 Head. A bag has 3 Red, 4 Blue balls. Find P(Red).