GATE · Practice Sheet 37

1. 1.

   Physics

   If 4,661 J of work is done in 24 s, find the power.

   Formula:Power

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

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2. 2.

   Physics

   A force of 453 N acts on an area of 32 m². Find the pressure.

   Formula:Pressure

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

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3. 3.

   Mathematics

   If f(x) = 4x³ + 3x² + 3x, find f′(2).

   Formula:Derivative of a Polynomial

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

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4. 4.

   Mathematics

   Find the distance between the points (4, 1) and (2, -5).

   Formula:Distance Between Two Points

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

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5. 5.

   Quantitative Aptitude

   Find the HCF and LCM of 29 and 33.

   Formula:LCM & HCF

   $\text{LCM}\times\text{HCF} = a\times b$

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6. 6.

   Physics

   A force of 749 N acts on an area of 37 m². Find the pressure.

   Formula:Pressure

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

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7. 7.

   Mathematics

   Evaluate ∫ from 0 to 5 of 1x^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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8. 8.

   Mathematics

   For the quadratic 4x² + 8x + 4 = 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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9. 9.

   Mathematics

   Find the distance between the points (7, -3) and (2, -6).

   Formula:Distance Between Two Points

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

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10. 10.

    Mathematics

    Find the slope of the line through (-1, 7) and (1, 3).

    Formula:Slope of a Line

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

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11. 11.

    Mathematics

    In an arithmetic progression with first term 6 and common difference 7, find term number 7.

    Formula:AP nth Term

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

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12. 12.

    Quantitative Aptitude

    In how many ways can 4 objects be arranged out of 6 distinct objects? (Find ⁿᴘᵣ for n = 6, r = 4.)

    Formula:Permutations

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

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13. 13.

    Mathematics

    Find term number 6 of a geometric progression with first term 5 and common ratio 3.

    Formula:GP nth Term

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

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14. 14.

    Mathematics

    If f(x) = 3x³ + 3x² + 7x, find f′(4).

    Formula:Derivative of a Polynomial

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

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15. 15.

    Physics

    A body moving at 1 m/s accelerates at 4 m/s² for 2 s. Find the distance travelled.

    Formula:Equation of Motion (s = ut + ½at²)

    $s = ut + \tfrac{1}{2}at^{2}$

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16. 16.

    Quantitative Aptitude

    A vehicle covers 639 km in 9 hours. Find its average speed.

    Formula:Speed, Distance & Time

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

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17. 17.

    Quantitative Aptitude

    In how many ways can 2 objects be chosen from 8 distinct objects? (Find ⁿᴄᵣ for n = 8, r = 2.)

    Formula:Combinations

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

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18. 18.

    Physics

    Find the electrical power when 65 V drives a current of 8 A.

    Formula:Electrical Power

    $P = VI$

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19. 19.

    Quantitative Aptitude

    A vehicle covers 234 km in 3 hours. Find its average speed.

    Formula:Speed, Distance & Time

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

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20. 20.

    Mathematics

    For the quadratic 4x² − 2x + 7 = 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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