Engineering Entrance · Practice Sheet 13

1. 1.

   Mathematics

   Evaluate ∫ from 1 to 3 of 5x^2 dx.

   Formula:Definite Integral of a Power

   $\int_{p}^{q} a x^{n}\,dx = \dfrac{a}{n+1}\big[x^{n+1}\big]_{p}^{q}$

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2. 2.

   Mathematics

   Find the distance between the points (0, 8) and (6, -7).

   Formula:Distance Between Two Points

   $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

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3. 3.

   Mathematics

   In an arithmetic progression with first term 10 and common difference 9, find term number 11.

   Formula:AP nth Term

   $a_n = a + (n-1)d$

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4. 4.

   Physics

   Calculate the potential energy of a 6 kg object at a height of 41 m (g = 9.8 m/s²).

   Formula:Gravitational Potential Energy

   $PE = mgh$

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5. 5.

   Physics

   A projectile is launched at 39 m/s at 45° 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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6. 6.

   Physics

   An object starts at 2 m/s and accelerates at 1 m/s² for 11 s. Find its final velocity.

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

   $v = u + at$

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7. 7.

   Physics

   Find the kinetic energy of a 1 kg object moving at 14 m/s.

   Formula:Kinetic Energy

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

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8. 8.

   Physics

   A projectile is launched at 18 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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9. 9.

   Mathematics

   If f(x) = 4x³ + 6x² + 6x, find f′(2).

   Formula:Derivative of a Polynomial

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

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10. 10.

    Chemistry

    Find the pressure of 1 mol of an ideal gas at 270 K in a 11 L container (R = 0.0821 L·atm·mol⁻¹·K⁻¹).

    Formula:Ideal Gas Law

    $PV = nRT$

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11. 11.

    Chemistry

    Find the pH of a solution with [H⁺] = 5×10⁻4 mol/L.

    Formula:pH of a Solution

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

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12. 12.

    Mathematics

    For the quadratic 3x² − 3x − 4 = 0, find the sum and product of its roots.

    Formula:Vieta's Formulas (roots)

    $\alpha+\beta=-\dfrac{b}{a},\;\alpha\beta=\dfrac{c}{a}$

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13. 13.

    Physics

    If 968 J of work is done in 38 s, find the power.

    Formula:Power

    $P = \dfrac{W}{t}$

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14. 14.

    Mathematics

    For the quadratic 4x² − 3x + 4 = 0, find the sum and product of its roots.

    Formula:Vieta's Formulas (roots)

    $\alpha+\beta=-\dfrac{b}{a},\;\alpha\beta=\dfrac{c}{a}$

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15. 15.

    Physics

    An object starts at 19 m/s and accelerates at 7 m/s² for 6 s. Find its final velocity.

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

    $v = u + at$

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16. 16.

    Physics

    Find the kinetic energy of a 16 kg object moving at 24 m/s.

    Formula:Kinetic Energy

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

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17. 17.

    Mathematics

    Find the sum of the first 24 terms of an AP with first term 13 and common difference 9.

    Formula:AP Sum of n Terms

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

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18. 18.

    Physics

    If 1,751 J of work is done in 49 s, find the power.

    Formula:Power

    $P = \dfrac{W}{t}$

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19. 19.

    Chemistry

    How many moles are present in 64 g of oxygen gas (O₂) (molar mass = 32 g/mol)?

    Formula:Number of Moles

    $n = \dfrac{m}{M}$

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20. 20.

    Physics

    An object starts at 9 m/s and accelerates at 2 m/s² for 3 s. Find its final velocity.

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

    $v = u + at$

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