Engineering Entrance · Practice Sheet 42

1. 1.

   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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2. 2.

   Chemistry

   26 mL of a 2 M solution is diluted to 212 mL. Find the new concentration.

   Formula:Dilution

   $M_1V_1 = M_2V_2$

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3. 3.

   Mathematics

   For the quadratic 4x² − 2x + 9 = 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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4. 4.

   Mathematics

   If f(x) = 5x³ + 6x² + 8x, find f′(4).

   Formula:Derivative of a Polynomial

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

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5. 5.

   Physics

   Find the force required to accelerate a 34 kg mass at 5 m/s².

   Formula:Newton's Second Law (F = ma)

   $F = ma$

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6. 6.

   Mathematics

   Evaluate ∫ from 0 to 3 of 5x^3 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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7. 7.

   Mathematics

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

   Formula:Derivative of a Polynomial

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

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8. 8.

   Physics

   Find the force required to accelerate a 20 kg mass at 13 m/s².

   Formula:Newton's Second Law (F = ma)

   $F = ma$

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9. 9.

   Chemistry

   Find the pressure of 1 mol of an ideal gas at 361 K in a 10 L container (R = 0.0821 L·atm·mol⁻¹·K⁻¹).

   Formula:Ideal Gas Law

   $PV = nRT$

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10. 10.

    Physics

    Find the momentum of a 34 kg body moving at 19 m/s.

    Formula:Linear Momentum

    $p = mv$

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11. 11.

    Mathematics

    Find the binomial coefficient C(10, 2) — the coefficient of x^2 in (1 + x)^10.

    Formula:Binomial Coefficient

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

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12. 12.

    Mathematics

    Find term number 5 of a geometric progression with first term 2 and common ratio 2.

    Formula:GP nth Term

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

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13. 13.

    Physics

    Find the density of an object with mass 43 g and volume 5 cm³.

    Formula:Density

    $\rho = \dfrac{m}{V}$

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14. 14.

    Mathematics

    Find the binomial coefficient C(10, 4) — the coefficient of x^4 in (1 + x)^10.

    Formula:Binomial Coefficient

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

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15. 15.

    Mathematics

    If f(x) = 4x³ + 4x² + 6x, find f′(3).

    Formula:Derivative of a Polynomial

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

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16. 16.

    Mathematics

    Find the distance between the points (-7, -4) and (2, -8).

    Formula:Distance Between Two Points

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

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17. 17.

    Mathematics

    Find the sum of the first 6 terms of a GP with first term 4 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

    For the quadratic 1x² + 2x − 8 = 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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19. 19.

    Physics

    A constant force of 194 N moves an object 12 m in the direction of the force. Calculate the work done.

    Formula:Work Done

    $W = F\,d$

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20. 20.

    Physics

    An object starts at 9 m/s and accelerates at 10 m/s² for 12 s. Find its final velocity.

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

    $v = u + at$

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