Engineering Entrance · Practice Sheet 40

1. 1.

   Mathematics

   For the quadratic 5x² + 9x − 6 = 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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2. 2.

   Mathematics

   Evaluate ∫ from 2 to 5 of 3x^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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3. 3.

   Mathematics

   Find the binomial coefficient C(7, 2) — the coefficient of x^2 in (1 + x)^7.

   Formula:Binomial Coefficient

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

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4. 4.

   Physics

   An object starts at 20 m/s and accelerates at 7 m/s² for 2 s. Find its final velocity.

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

   $v = u + at$

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5. 5.

   Mathematics

   Find the slope of the line through (8, 8) and (4, -2).

   Formula:Slope of a Line

   $m = \dfrac{y_2-y_1}{x_2-x_1}$

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6. 6.

   Mathematics

   In an arithmetic progression with first term 4 and common difference 10, find term number 26.

   Formula:AP nth Term

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

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7. 7.

   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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8. 8.

   Mathematics

   Find term number 7 of a geometric progression with first term 5 and common ratio 2.

   Formula:GP nth Term

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

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9. 9.

   Mathematics

   In an arithmetic progression with first term 11 and common difference 7, find term number 14.

   Formula:AP nth Term

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

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10. 10.

    Mathematics

    Find the slope of the line through (-2, 7) and (-3, -4).

    Formula:Slope of a Line

    $m = \dfrac{y_2-y_1}{x_2-x_1}$

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11. 11.

    Mathematics

    For the quadratic 4x² + 6x − 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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12. 12.

    Mathematics

    If f(x) = 1x³ + 4x² + 6x, find f′(5).

    Formula:Derivative of a Polynomial

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

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13. 13.

    Chemistry

    61 mL of a 10 M solution is diluted to 377 mL. Find the new concentration.

    Formula:Dilution

    $M_1V_1 = M_2V_2$

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14. 14.

    Mathematics

    Evaluate log base 2 of 32.

    Formula:Logarithm Evaluation

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

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15. 15.

    Physics

    If 558 J of work is done in 17 s, find the power.

    Formula:Power

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

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16. 16.

    Chemistry

    How many moles are present in 360 g of glucose (C₆H₁₂O₆) (molar mass = 180 g/mol)?

    Formula:Number of Moles

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

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17. 17.

    Physics

    Find the density of an object with mass 320 g and volume 32 cm³.

    Formula:Density

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

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18. 18.

    Mathematics

    In an arithmetic progression with first term 3 and common difference 4, find term number 18.

    Formula:AP nth Term

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

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19. 19.

    Mathematics

    Evaluate log base 3 of 729.

    Formula:Logarithm Evaluation

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

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20. 20.

    Chemistry

    How many moles are present in 90 g of glucose (C₆H₁₂O₆) (molar mass = 180 g/mol)?

    Formula:Number of Moles

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

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