ACT · Practice Sheet 50

1. 1.

   Mathematics

   For the quadratic 2x² + 5x − 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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2. 2.

   Mathematics

   Find the distance between the points (4, -5) and (-1, -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

   Evaluate ∫ from 2 to 3 of 2x^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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4. 4.

   Quantitative Aptitude

   Two varieties costing $16 and $95 per kg are mixed to give a mixture worth $57 per kg. Find the mixing ratio.

   Formula:Alligation (Mixtures)

   $\dfrac{q_1}{q_2} = \dfrac{p_2-m}{m-p_1}$

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5. 5.

   Mathematics

   In an arithmetic progression with first term 12 and common difference 5, find term number 27.

   Formula:AP nth Term

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

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6. 6.

   Physics

   Find the voltage across a 9 Ω resistor carrying a current of 16 A.

   Formula:Ohm's Law

   $V = IR$

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7. 7.

   Quantitative Aptitude

   The sum of the ages of two people is 72 years, and one is 18 years older than the other. Find their ages.

   Formula:Ages (linear equations)

   $x + (x+d) = S$

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8. 8.

   Quantitative Aptitude

   In how many ways can 4 objects be chosen from 9 distinct objects? (Find ⁿᴄᵣ for n = 9, r = 4.)

   Formula:Combinations

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

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9. 9.

   Quantitative Aptitude

   What is 60% of 1,000?

   Formula:Percentage of a Number

   $\text{Value} = \dfrac{P}{100}\times N$

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10. 10.

    Mathematics

    Find the sum of the first 7 terms of a GP with first term 2 and common ratio 3.

    Formula:GP Sum of n Terms

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

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11. 11.

    Mathematics

    In an arithmetic progression with first term 14 and common difference 6, find term number 19.

    Formula:AP nth Term

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

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12. 12.

    Physics

    A projectile is launched at 19 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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13. 13.

    Physics

    Find the kinetic energy of a 3 kg object moving at 8 m/s.

    Formula:Kinetic Energy

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

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14. 14.

    Physics

    Find the density of an object with mass 407 g and volume 45 cm³.

    Formula:Density

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

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15. 15.

    Mathematics

    Find the distance between the points (5, -1) and (6, 7).

    Formula:Distance Between Two Points

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

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16. 16.

    Physics

    Find the force required to accelerate a 4 kg mass at 6 m/s².

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

    $F = ma$

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17. 17.

    Physics

    Find the momentum of a 31 kg body moving at 30 m/s.

    Formula:Linear Momentum

    $p = mv$

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18. 18.

    Mathematics

    Find the sum of the first 15 terms of an AP with first term 12 and common difference 2.

    Formula:AP Sum of n Terms

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

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19. 19.

    Mathematics

    Evaluate ∫ from 0 to 5 of 1x^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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20. 20.

    Quantitative Aptitude

    A vehicle covers 72 km in 2 hours. Find its average speed.

    Formula:Speed, Distance & Time

    $\text{Speed} = \dfrac{\text{Distance}}{\text{Time}}$

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