GATE · Practice Sheet 9

1. 1.

   Mathematics

   Evaluate ∫ from 1 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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2. 2.

   Quantitative Aptitude

   A vehicle covers 536 km in 8 hours. Find its average speed.

   Formula:Speed, Distance & Time

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

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3. 3.

   Physics

   A projectile is launched at 12 m/s at 60° 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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4. 4.

   Mathematics

   Evaluate ∫ from 0 to 3 of 3x^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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5. 5.

   Mathematics

   Find term number 6 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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6. 6.

   Quantitative Aptitude

   Find the HCF and LCM of 9 and 29.

   Formula:LCM & HCF

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

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7. 7.

   Physics

   Find the force required to accelerate a 11 kg mass at 8 m/s².

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

   $F = ma$

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8. 8.

   Quantitative Aptitude

   In how many ways can 2 objects be arranged out of 7 distinct objects? (Find ⁿᴘᵣ for n = 7, r = 2.)

   Formula:Permutations

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

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9. 9.

   Mathematics

   Evaluate log base 3 of 729.

   Formula:Logarithm Evaluation

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

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10. 10.

    Physics

    A body moving at 13 m/s accelerates at 3 m/s² for 5 s. Find the distance travelled.

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

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

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11. 11.

    Physics

    If 2,150 J of work is done in 45 s, find the power.

    Formula:Power

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

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12. 12.

    Mathematics

    For the quadratic 3x² − 2x + 5 = 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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13. 13.

    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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14. 14.

    Quantitative Aptitude

    A vehicle covers 252 km in 3 hours. Find its average speed.

    Formula:Speed, Distance & Time

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

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15. 15.

    Physics

    Find the kinetic energy of a 2 kg object moving at 22 m/s.

    Formula:Kinetic Energy

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

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16. 16.

    Mathematics

    Find the slope of the line through (5, -3) and (7, -4).

    Formula:Slope of a Line

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

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17. 17.

    Mathematics

    Evaluate ∫ from 0 to 5 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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18. 18.

    Quantitative Aptitude

    Find the simple interest on ₹13,000 at 6% per annum for 5 year(s).

    Formula:Simple Interest

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

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19. 19.

    Mathematics

    If f(x) = 3x³ + 6x² + 8x, find f′(3).

    Formula:Derivative of a Polynomial

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

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20. 20.

    Physics

    A body moving at 14 m/s accelerates at 5 m/s² for 8 s. Find the distance travelled.

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

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

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