SPSC MATHEMATICS 25 YEARS PAST PAPERS
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چار آپشن میں سے کسی ایک پر کلک کرنے سے جواب سرخ ہو جائے گا۔
- 96
- 97
- 98
- 99
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Explanation
96: Even, divisible by 2, 3, 4, 6... → not prime97: Only divisible by 1 and 97. No factor between 2 to 9 → prime
98: Even, divisible by 2, 7, 14... → not prime
99: Divisible by 3, 9, 11... → not prime
So 97 is the greatest prime in the list.
Last verified on 04-06-2026
- d²x/dt² = - m²x
- d²x/dt² = + m²x
- d²x/dt² = ± m²x
- None of these
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Explanation
x = A cos(mt - a) is a wave/oscillation.
If you differentiate it twice, you get the same x back but with a minus sign and m² in front.
So the equation becomes: d²x/dt² = -m²x
Clear rule: Cos or Sin → 2nd derivative = -ω² × function
Last verified on 04-06-2026
- Only sine terms
- Cosine terms and a constant
- Sine terms and a constant
- None of these
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Explanation
The Fourier series expansion of an even, periodic function contains only cosine terms and a constant term.
In other words, the sine terms are absent.
Even functions:
An even function is symmetric about the y-axis, meaning f(x) = f(-x).
Fourier series:
A Fourier series represents a periodic function as a sum of sine and cosine waves.
Last verified on 04-06-2026
- 4
- 2
- 3
- 1
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Explanation
If α, β, γ is the angle, a line makes with the co-ordinate axis,
∴ cos2α + cos2β + cos2γ = 1
⇒(1− sin2α) + (1− sin2β) + (1 − sin2γ) = 1
⇒3− (sin2α+ sin2β + sin2γ) = 1
∴ (sin2α + sin2β + sin2γ) = 2
Last verified on 04-06-2026
- 1/2
- 1
- 2
- 4
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Explanation
y = (1/8)x^2 + (1/2)x + 1
At (0,1)
1) Slope of tangent at (0,1)
y = (1/8)x² + (1/2)x + 1
y' = (1/4)x + 1/2
At x=0, m = 1/2
2) Tangent equation
y - 1 = (1/2)x → y = 0.5x + 1
3) Intercepts
x-intercept: 0 = 0.5x + 1 → x = -2
y-intercept: y = 1
4) Area
Triangle with base 2, height 1
Area = 1/2 × 2 × 1 = 1
Last verified on 04-06-2026
- 6
- 5
- 4
- None of these
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Explanation
For matrix A = [, [x, -1]] with eigenvalues λ₁ = -y and λ₂ = x-1[2][3]
Use 2 properties of eigenvalues:
1) Sum = Trace
λ₁ + λ₂ = 2 + (-1) = 1
→ -y + (x-1) = 1
→ x - y = 2 → y = x - 2
2) Product = Determinant
λ₁ × λ₂ = 2(-1) - 3(x) = -2 - 3x
→ (-y)(x-1) = -2 - 3x
Put y = x - 2:
-(x-2)(x-1) = -2 - 3x
-(x² - 3x + 2) = -2 - 3x
-x² + 3x - 2 = -2 - 3x
x² - 6x = 0
x(x - 6) = 0
x = 0 or x = 6
Last verified on 04-06-2026
- 2
- -2
- -1
- None of these
اس سوال کو وضاحت کے ساتھ پڑھیں
Explanation
Given: 3x⁵ + 96 = ∫_c^x f(y) dy for all x
1) Differentiate both sides w.r.t x
Use Fundamental Theorem of Calculus: d/dx ∫_c^x f(y) dy = f(x)
So: d/dx [3x⁵ + 96] = f(x)
→ f(x) = 15x⁴
2) Put x = c
LHS: 3c⁵ + 96
RHS: ∫_c^c f(y) dy = 0
So 3c⁵ + 96 = 0
→ c⁵ = -32
→ c = -2
Last verified on 04-06-2026
- 11
- 12
- 7
- 7 and 11
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Explanation
7 appears 3 times 11 appears 3 times
9, 10, 12 each appear once
Both 7 and 11 occur most often.
So the mode is 7 and 11 - it’s bimodal.
Last verified on 04-06-2026
- y = vx, dy = vdx + xdv
- x + y = v, dy = dx + dy
- v = x - y, dy = dx - dy
- x + y = y, dy = dx + xdy
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Explanation
For a homogeneous DE of the form M(x, y) dx + N(x, y) dy = 0, where M and N are homogeneous functions of the same degree, you use:y = vx, so dy = v dx + x dv
That substitution makes the equation separable in variables v and x.
Last verified on 04-06-2026
- Semi group
- Subgroup
- Cyclic group
- Abelian group
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Explanation
- The set of complex numbers {1, i, -i, -1} under multiplication operation is a cyclic group.
- Two generators i and -i will covers all the elements of this group.
- Hence, it is a cyclic group.
Last verified on 04-06-2026
