GATE · Practice Sheet 39

1. 1.

   Quantitative Aptitude

   In how many ways can 3 objects be arranged out of 9 distinct objects? (Find ⁿᴘᵣ for n = 9, r = 3.)

   Formula:Permutations

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

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2. 2.

   Physics

   Find the force required to accelerate a 21 kg mass at 19 m/s².

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

   $F = ma$

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3. 3.

   Mathematics

   For the quadratic 3x² + 9x + 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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4. 4.

   Quantitative Aptitude

   A vehicle covers 272 km in 4 hours. Find its average speed.

   Formula:Speed, Distance & Time

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

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5. 5.

   Physics

   Find the electrical power when 170 V drives a current of 5 A.

   Formula:Electrical Power

   $P = VI$

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6. 6.

   Mathematics

   Find the distance between the points (5, 4) and (-6, 7).

   Formula:Distance Between Two Points

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

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7. 7.

   Mathematics

   Evaluate ∫ from 0 to 4 of 2x^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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8. 8.

   Physics

   An object starts at 9 m/s and accelerates at 3 m/s² for 11 s. Find its final velocity.

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

   $v = u + at$

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9. 9.

   Physics

   Find the electrical power when 45 V drives a current of 6 A.

   Formula:Electrical Power

   $P = VI$

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10. 10.

    Physics

    A projectile is launched at 34 m/s at 30° 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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11. 11.

    Mathematics

    Find term number 7 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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12. 12.

    Quantitative Aptitude

    In how many ways can 3 objects be arranged out of 6 distinct objects? (Find ⁿᴘᵣ for n = 6, r = 3.)

    Formula:Permutations

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

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13. 13.

    Quantitative Aptitude

    In how many ways can 4 objects be chosen from 8 distinct objects? (Find ⁿᴄᵣ for n = 8, r = 4.)

    Formula:Combinations

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

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14. 14.

    Mathematics

    Find the slope of the line through (7, -4) and (4, -1).

    Formula:Slope of a Line

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

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15. 15.

    Quantitative Aptitude

    In how many ways can 3 objects be arranged out of 7 distinct objects? (Find ⁿᴘᵣ for n = 7, r = 3.)

    Formula:Permutations

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

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16. 16.

    Mathematics

    For the quadratic 3x² − 2x + 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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17. 17.

    Physics

    A force of 274 N acts on an area of 34 m². Find the pressure.

    Formula:Pressure

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

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18. 18.

    Mathematics

    Evaluate ∫ from 0 to 5 of 5x^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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19. 19.

    Quantitative Aptitude

    Find the simple interest on ₹38,000 at 12% per annum for 5 year(s).

    Formula:Simple Interest

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

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20. 20.

    Mathematics

    Evaluate log base 3 of 27.

    Formula:Logarithm Evaluation

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

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