Engineering Entrance · Practice Sheet 16

1. 1.

   Mathematics

   Find the binomial coefficient C(7, 3) — the coefficient of x^3 in (1 + x)^7.

   Formula:Binomial Coefficient

   $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$

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2. 2.

   Mathematics

   If f(x) = 3x³ + 2x² + 6x, find f′(1).

   Formula:Derivative of a Polynomial

   $\dfrac{d}{dx}x^{n} = n\,x^{\,n-1}$

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3. 3.

   Mathematics

   Find term number 4 of a geometric progression with first term 4 and common ratio 2.

   Formula:GP nth Term

   $a_n = a\,r^{\,n-1}$

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4. 4.

   Physics

   A projectile is launched at 47 m/s at 30° to the horizontal. Find its range (g = 9.8 m/s²).

   Formula:Projectile Range

   $R = \dfrac{u^{2}\sin 2\theta}{g}$

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5. 5.

   Physics

   Find the kinetic energy of a 2 kg object moving at 7 m/s.

   Formula:Kinetic Energy

   $KE = \tfrac{1}{2}mv^{2}$

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6. 6.

   Chemistry

   Find the pressure of 5 mol of an ideal gas at 221 K in a 6 L container (R = 0.0821 L·atm·mol⁻¹·K⁻¹).

   Formula:Ideal Gas Law

   $PV = nRT$

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7. 7.

   Mathematics

   Find the binomial coefficient C(6, 4) — the coefficient of x^4 in (1 + x)^6.

   Formula:Binomial Coefficient

   $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$

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8. 8.

   Mathematics

   If f(x) = 3x³ + 5x² + 4x, find f′(1).

   Formula:Derivative of a Polynomial

   $\dfrac{d}{dx}x^{n} = n\,x^{\,n-1}$

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9. 9.

   Physics

   A body moving at 8 m/s accelerates at 1 m/s² for 2 s. Find the distance travelled.

   Formula:Equation of Motion (s = ut + ½at²)

   $s = ut + \tfrac{1}{2}at^{2}$

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10. 10.

    Chemistry

    Find the pH of a solution with [H⁺] = 2×10⁻2 mol/L.

    Formula:pH of a Solution

    $pH = -\log_{10}[H^{+}]$

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11. 11.

    Physics

    An object starts at 5 m/s and accelerates at 10 m/s² for 6 s. Find its final velocity.

    Formula:Equation of Motion (v = u + at)

    $v = u + at$

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12. 12.

    Physics

    Find the kinetic energy of a 13 kg object moving at 29 m/s.

    Formula:Kinetic Energy

    $KE = \tfrac{1}{2}mv^{2}$

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13. 13.

    Chemistry

    Find the pH of a solution with [H⁺] = 1×10⁻3 mol/L.

    Formula:pH of a Solution

    $pH = -\log_{10}[H^{+}]$

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14. 14.

    Physics

    An object starts at 7 m/s and accelerates at 8 m/s² for 9 s. Find its final velocity.

    Formula:Equation of Motion (v = u + at)

    $v = u + at$

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15. 15.

    Chemistry

    Find the pressure of 3 mol of an ideal gas at 348 K in a 15 L container (R = 0.0821 L·atm·mol⁻¹·K⁻¹).

    Formula:Ideal Gas Law

    $PV = nRT$

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16. 16.

    Physics

    An object starts at 4 m/s and accelerates at 5 m/s² for 3 s. Find its final velocity.

    Formula:Equation of Motion (v = u + at)

    $v = u + at$

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17. 17.

    Mathematics

    Find the sum of the first 7 terms of a GP with first term 2 and common ratio 2.

    Formula:GP Sum of n Terms

    $S_n = a\,\dfrac{r^{n}-1}{r-1}$

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18. 18.

    Mathematics

    If f(x) = 5x³ + 6x² + 6x, find f′(5).

    Formula:Derivative of a Polynomial

    $\dfrac{d}{dx}x^{n} = n\,x^{\,n-1}$

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19. 19.

    Mathematics

    Find the sum of the first 13 terms of an AP with first term 13 and common difference 2.

    Formula:AP Sum of n Terms

    $S_n = \dfrac{n}{2}\,[2a+(n-1)d]$

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20. 20.

    Chemistry

    26 mL of a 1 M solution is diluted to 289 mL. Find the new concentration.

    Formula:Dilution

    $M_1V_1 = M_2V_2$

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