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Mathematics.
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Grade 9 · Lahore Board · 2024 · Group I
Complete objective paper · English questions

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15 questions. 15 marks.

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QUESTION 1 · 1 MARKSimplify a/(9a² − b²) + 1/(3a − b) = ----- :

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Correct answer: C. (4a + b)/(9a² − b²)

Add algebraic fractions by taking a common denominator and factorizing a difference of squares.

  1. Factorize 9a² − b² as (3a − b)(3a + b).
  2. Rewrite 1/(3a − b) with denominator 9a² − b²: 1/(3a − b) = (3a + b)/(9a² − b²).
  3. Now add: a/(9a² − b²) + (3a + b)/(9a² − b²) = (a + 3a + b)/(9a² − b²).
  4. Simplify the numerator: a + 3a + b = 4a + b.
  5. Therefore the result is (4a + b)/(9a² − b²).
  6. Restriction: 3a − b ≠ 0 and 3a + b ≠ 0.
QUESTION 2 · 1 MARKCongruent triangles are -----

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Correct answer: B. Same size and shape

Congruent figures have exactly the same shape and exactly the same size.

  1. Two triangles are called congruent when all corresponding sides and angles are equal.
  2. That means the triangles match exactly in both shape and size.
  3. So the correct description is 'Same size and shape'.
QUESTION 3 · 1 MARKSymbol used for congruency of two triangles is ----

Source note: Option B is unclear in the scan. Option A appears to be the congruence symbol.

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Correct answer: A. ≅

Mathematical symbols have fixed meanings; the congruence sign is used for equal shape and size.

  1. The standard symbol for congruence is ≅.
  2. For triangles, we write statements like △ABC ≅ △DEF.
  3. The symbol Δ means 'triangle' only, and = is the ordinary equality sign.
QUESTION 4 · 1 MARKA triangle having two sides congruent is called ----

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Correct answer: D. Isosceles

Triangles are classified by their sides.

  1. A triangle with exactly two equal sides is called an isosceles triangle.
  2. A scalene triangle has no equal sides, and an equilateral triangle has all three sides equal.
  3. So the required name is isosceles.
QUESTION 5 · 1 MARKIdentity matrix of order 2-by-2 is :

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Correct answer: C. [[1, 0], [0, 1]]

The identity matrix has 1s on the main diagonal and 0s elsewhere.

  1. For a 2 × 2 identity matrix, the main diagonal entries are 1 and 1.
  2. All off-diagonal entries must be 0.
  3. Therefore the identity matrix is [[1, 0], [0, 1]].
QUESTION 6 · 1 MARKIf x is no larger than 10, then ----:

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Correct answer: B. x ≤ 10

Translate words into an inequality.

  1. 'No larger than 10' means 'not greater than 10'.
  2. So x can be 10 or any number less than 10.
  3. Therefore the correct inequality is x ≤ 10.
QUESTION 7 · 1 MARKThe bisector of the angles of triangle are ----

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Correct answer: D. Concurrent

The three angle bisectors of a triangle meet at one point called the incenter.

  1. Each angle bisector divides one angle of the triangle into two equal parts.
  2. In any triangle, the three angle bisectors intersect at a single common point.
  3. Lines that meet at one point are called concurrent.
QUESTION 8 · 1 MARKConjugate of surd x + √y ------

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Correct answer: B. x − √y

The conjugate of a binomial surd is formed by changing the sign between the two terms.

  1. Start with x + √y.
  2. Keep the same terms x and √y.
  3. Change the middle sign from + to −.
  4. So the conjugate is x − √y.
QUESTION 9 · 1 MARKTriangles on equal bases and of equal altitudes are ---- in area:

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Correct answer: A. Equal

Area of a triangle depends on base and altitude: Area = (1/2) × base × height.

  1. The area of a triangle is (1/2) × base × altitude.
  2. If two triangles have equal bases and equal altitudes, then both have the same value of (1/2) × base × altitude.
  3. Therefore their areas are equal.
QUESTION 10 · 1 MARKPoint (2, -3) lies in the quadrant -----:

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Correct answer: D. IV

Quadrants are determined by the signs of x-coordinate and y-coordinate.

  1. For the point (2, -3), the x-coordinate is positive and the y-coordinate is negative.
  2. Points with (+, −) signs lie in the fourth quadrant.
  3. Therefore (2, -3) lies in quadrant IV.
QUESTION 11 · 1 MARKlogₐ mⁿ = -------:

Source note: Option B is slightly blurred under the watermark; it appears to read 'm logₐ m'.

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Correct answer: A. n logₐ m

Use the power rule of logarithms: logₐ(M^k) = k logₐ(M).

  1. In logₐ(mⁿ), the whole power mⁿ is the argument of the logarithm.
  2. By the logarithm law, the exponent n comes down in front: logₐ(mⁿ) = n logₐ(m).
  3. Therefore the correct option is n logₐ m.
  4. Restrictions: a > 0, a ≠ 1, and m > 0.
QUESTION 12 · 1 MARKFind 'm' so that x² + 4x + m is a complete square:

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Correct answer: C. 4

A complete square trinomial has the form (x + a)² = x² + 2ax + a².

  1. Compare x² + 4x + m with x² + 2ax + a².
  2. From 2a = 4, we get a = 2.
  3. Then m must equal a² = 2² = 4.
  4. Check: x² + 4x + 4 = (x + 2)², which is a complete square.
QUESTION 13 · 1 MARKDistance between the points (0, 0) and (1, 1) is:

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Correct answer: D. √2

Use the distance formula between two points.

  1. Distance = √((x₂ − x₁)² + (y₂ − y₁)²).
  2. Here, (x₁, y₁) = (0, 0) and (x₂, y₂) = (1, 1).
  3. So distance = √((1 − 0)² + (1 − 0)²) = √(1² + 1²).
  4. Therefore distance = √(1 + 1) = √2.
QUESTION 14 · 1 MARKi¹⁹ = ------- :

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Correct answer: D. −i

Powers of i repeat in a cycle of 4.

  1. The pattern is i¹ = i, i² = −1, i³ = −i, i⁴ = 1.
  2. Divide 19 by 4: 19 = 4 × 4 + 3, so i¹⁹ = i³.
  3. Since i³ = −i, we get i¹⁹ = −i.
QUESTION 15 · 1 MARKIf two opposite sides of a quadrilateral are congruent and parallel, it is a--------:

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Correct answer: A. Parallelogram

A quadrilateral is a parallelogram if one pair of opposite sides is both equal and parallel.

  1. There is a standard test for parallelograms: if one pair of opposite sides of a quadrilateral is equal in length and parallel, the quadrilateral is a parallelogram.
  2. The given condition matches this test exactly.
  3. Therefore the quadrilateral must be a parallelogram.

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Original exam paper sourced from Ilmkidunya ↗. Pastpaper’s solutions are study support, not an official board marking key. Where an option is unclear in the scan, the source note is retained.