SPSC LECTURER MATHEMATICS PAST PAPER 2000 TO TILL DATE
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چار آپشن میں سے کسی ایک پر کلک کرنے سے جواب سرخ ہو جائے گا۔
- A parabola
- A circle
- An ellipse
- None of these
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Explanation
Last verified on 04-06-2026
- Infinite value
- Finite value
- Can't be found
- None of these
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Explanation
If aₙ means the coefficient of xⁿ in the power series/Taylor series of f(x) = x - x³, then:
f(x) = x - x³ = 0 + 1·x + 0·x² + (-1)·x³ + 0·x⁴ + 0·x⁵ + ...
So:
a₀ = 0
a₁ = 1
a₂ = 0
a₃ = -1
a₄ = 0, a₅ = 0, ... all higher terms = 0
0 is also a finite value.
Last verified on 04-06-2026
- Ā × B̄ = 0̄
- Ā + B̄ = 0̄
- Ā - B̄ = 0̄
- None of these
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Explanation
Formula: Ā × B̄ = |Ā||B̄| sinθ
For non-zero vectors, |Ā||B̄| is never 0.
So cross product = 0 only when sinθ = 0 → θ = 0° or 180°
That’s exactly same direction or opposite direction = parallel.
This works for ALL parallel cases.
So correct answer is = Ā × B̄ = 0̄
Last verified on 04-06-2026
- x⁴ - x²
- x⁴ - 2x³ - x + 2
- x⁴ + 2x³ - x - 2
- None of these
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Explanation
x + 2 is a factor = Put x = -2 in polynomial. If answer comes 0, then yes it’s a factor.
x⁴ + 2x³ - x - 2 → 16 - 16 + 2 - 2 = 0 → 0
Last verified on 04-06-2026
- Verb
- Adjective
- Conjunction
- Adverb
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Explanation
- “Because” joins two clauses: She went to the store + she needed groceries.
- It shows a reason (cause and effect relationship).
- So, it is a subordinating conjunction.
Last verified on 04-06-2026
- On
- In
- At
- Before
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Explanation
- “Before” is used when something happens earlier than a point in time.
- It correctly shows the package came earlier than expected.
Last verified on 04-06-2026
- One
- Two
- Three
- Four
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Explanation
For 3 non-collinear points, you can make a parallelogram in 3 different ways depending on which 2 points you take as ends of a diagonal.
The 3 possible D's:
- AB is one side → D = A + C - B = (-1+1-6, 2-20-2) = (-6, -20)
- AC is one side → D = A + B - C = (-1+6-1, 2+2+20) = (4, 24)
- BC is one side → D = B + C - A = (6+1+1, 2-20-2) = (8, -20)
Rule: In parallelogram, A + C = B + D or any permutation.
Since A, B, C are not in a straight line, all 3 D’s are different.
Last verified on 04-06-2026
- F(ln a)
- f(a)
- f(e^x)
- f(1/a)
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Explanation
f(x) = e^x - ax makes a U-shape because e^x is always curving up.
U-shape has only 1 lowest point at x = ln a.
So the minimum value = f(ln a) = a - a ln a
Last verified on 04-06-2026
- 1
- 32i
- 32
- i
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Explanation
(1 + i)¹⁰ = (√2)¹⁰ × [cos(45°×10) + i sin(45°×10)]
= 2⁵ × [cos450° + i sin450°]
= 32 × [cos90° + i sin90°] 450° = 360° + 90°
= 32 × [0 + i×1]
= 32i
Last verified on 04-06-2026
- [1 2, 2 1, 1 3] [x1, x2, x3] = [9 7]
- [3 1, 2 1, 1 2] [x1, x2, x3] = [9 7]
- [1 2, 2 1, 1 3] [x1, x2, x3] = [7 9]
- None of these
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Explanation
System:
x1 + 2x2 + x3 = 9
2x1 + x2 + 3x3 = 7
Matrix form AX = B:
[1 2 1] [x1] [9]
[2 1 3] [x2] = [7]
[x3]
Augmented matrix [A|B]:
[1 2 1 | 9]
[2 1 3 | 7]
Last verified on 04-06-2026
