Engineering Entrance · Practice Sheet 14

1. 1.

   Mathematics

   Evaluate ∫ from 0 to 4 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

   Evaluate log base 10 of 100.

   Formula:Logarithm Evaluation

   $\log_{b}(b^{k}) = k$

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3. 3.

   Mathematics

   If f(x) = 1x³ + 4x² + 8x, find f′(2).

   Formula:Derivative of a Polynomial

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

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4. 4.

   Chemistry

   Find the molarity of a solution containing 0.5 mol of solute in 1 L of solution.

   Formula:Molarity

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

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5. 5.

   Mathematics

   Find the binomial coefficient C(9, 4) — the coefficient of x^4 in (1 + x)^9.

   Formula:Binomial Coefficient

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

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6. 6.

   Physics

   Find the kinetic energy of a 17 kg object moving at 27 m/s.

   Formula:Kinetic Energy

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

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7. 7.

   Chemistry

   A solution is made by dissolving 35 g of solute in 83 g of solvent. Find the mass percentage of the solute.

   Formula:Mass Percentage

   $\%\,\text{mass} = \dfrac{\text{solute}}{\text{solution}}\times 100$

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8. 8.

   Physics

   Find the voltage across a 50 Ω resistor carrying a current of 6 A.

   Formula:Ohm's Law

   $V = IR$

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9. 9.

   Mathematics

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

   Formula:Derivative of a Polynomial

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

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10. 10.

    Chemistry

    How many moles are present in 160 g of oxygen gas (O₂) (molar mass = 32 g/mol)?

    Formula:Number of Moles

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

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11. 11.

    Chemistry

    Find the molarity of a solution containing 3 mol of solute in 2 L of solution.

    Formula:Molarity

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

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12. 12.

    Physics

    An object starts at 3 m/s and accelerates at 7 m/s² for 5 s. Find its final velocity.

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

    $v = u + at$

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13. 13.

    Chemistry

    Find the molarity of a solution containing 2 mol of solute in 2.5 L of solution.

    Formula:Molarity

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

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14. 14.

    Mathematics

    Find the binomial coefficient C(6, 2) — the coefficient of x^2 in (1 + x)^6.

    Formula:Binomial Coefficient

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

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15. 15.

    Mathematics

    For the quadratic 5x² + 8x − 3 = 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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16. 16.

    Physics

    Find the voltage across a 50 Ω resistor carrying a current of 16 A.

    Formula:Ohm's Law

    $V = IR$

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17. 17.

    Physics

    Find the momentum of a 23 kg body moving at 9 m/s.

    Formula:Linear Momentum

    $p = mv$

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18. 18.

    Mathematics

    Find the distance between the points (8, -6) and (-2, -8).

    Formula:Distance Between Two Points

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

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19. 19.

    Physics

    Calculate the potential energy of a 1 kg object at a height of 2 m (g = 9.8 m/s²).

    Formula:Gravitational Potential Energy

    $PE = mgh$

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20. 20.

    Mathematics

    Evaluate ∫ from 0 to 4 of 3x^1 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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