GATE · Practice Sheet 27

1. 1.

   Mathematics

   For the quadratic 5x² − 6x + 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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2. 2.

   Mathematics

   Find the distance between the points (3, 7) and (2, -5).

   Formula:Distance Between Two Points

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

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3. 3.

   Quantitative Aptitude

   Find the simple interest on ₹21,000 at 6% per annum for 3 year(s).

   Formula:Simple Interest

   $SI = \dfrac{P\times R\times T}{100}$

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4. 4.

   Mathematics

   For the quadratic 3x² − 1x − 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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5. 5.

   Quantitative Aptitude

   In how many ways can 4 objects be chosen from 6 distinct objects? (Find ⁿᴄᵣ for n = 6, r = 4.)

   Formula:Combinations

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

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6. 6.

   Physics

   Find the kinetic energy of a 3 kg object moving at 9 m/s.

   Formula:Kinetic Energy

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

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7. 7.

   Mathematics

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

   Formula:Derivative of a Polynomial

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

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8. 8.

   Physics

   An object starts at 4 m/s and accelerates at 8 m/s² for 5 s. Find its final velocity.

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

   $v = u + at$

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9. 9.

   Physics

   If 488 J of work is done in 12 s, find the power.

   Formula:Power

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

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10. 10.

    Mathematics

    Find term number 3 of a geometric progression with first term 3 and common ratio 3.

    Formula:GP nth Term

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

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11. 11.

    Physics

    If 1,949 J of work is done in 45 s, find the power.

    Formula:Power

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

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12. 12.

    Physics

    A force of 613 N acts on an area of 43 m². Find the pressure.

    Formula:Pressure

    $P = \dfrac{F}{A}$

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13. 13.

    Physics

    Find the force required to accelerate a 3 kg mass at 19 m/s².

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

    $F = ma$

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14. 14.

    Physics

    If 1,414 J of work is done in 48 s, find the power.

    Formula:Power

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

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15. 15.

    Mathematics

    Find the sum of the first 9 terms of an AP with first term 8 and common difference 8.

    Formula:AP Sum of n Terms

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

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16. 16.

    Mathematics

    Find the slope of the line through (-4, -8) and (-3, 8).

    Formula:Slope of a Line

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

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17. 17.

    Mathematics

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

    Formula:Derivative of a Polynomial

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

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18. 18.

    Quantitative Aptitude

    A vehicle covers 225 km in 5 hours. Find its average speed.

    Formula:Speed, Distance & Time

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

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19. 19.

    Physics

    An object starts at 15 m/s and accelerates at 9 m/s² for 2 s. Find its final velocity.

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

    $v = u + at$

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20. 20.

    Mathematics

    If f(x) = 1x³ + 1x² + 3x, find f′(5).

    Formula:Derivative of a Polynomial

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

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