GATE · Practice Sheet 10

1. 1.

   Quantitative Aptitude

   In how many ways can 2 objects be chosen from 7 distinct objects? (Find ⁿᴄᵣ for n = 7, r = 2.)

   Formula:Combinations

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

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2. 2.

   Physics

   A projectile is launched at 43 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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3. 3.

   Quantitative Aptitude

   Find the HCF and LCM of 17 and 15.

   Formula:LCM & HCF

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

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4. 4.

   Physics

   An object starts at 10 m/s and accelerates at 2 m/s² for 8 s. Find its final velocity.

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

   $v = u + at$

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5. 5.

   Mathematics

   Find the distance between the points (-4, 7) and (-8, -2).

   Formula:Distance Between Two Points

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

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6. 6.

   Physics

   Find the force required to accelerate a 46 kg mass at 1 m/s².

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

   $F = ma$

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7. 7.

   Quantitative Aptitude

   In how many ways can 3 objects be chosen from 9 distinct objects? (Find ⁿᴄᵣ for n = 9, r = 3.)

   Formula:Combinations

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

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8. 8.

   Mathematics

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

   Formula:GP nth Term

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

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9. 9.

   Quantitative Aptitude

   A vehicle covers 210 km in 3 hours. Find its average speed.

   Formula:Speed, Distance & Time

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

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10. 10.

    Quantitative Aptitude

    In how many ways can 4 objects be arranged out of 9 distinct objects? (Find ⁿᴘᵣ for n = 9, r = 4.)

    Formula:Permutations

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

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11. 11.

    Mathematics

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

    Formula:Derivative of a Polynomial

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

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12. 12.

    Physics

    Find the force required to accelerate a 10 kg mass at 20 m/s².

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

    $F = ma$

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13. 13.

    Physics

    A constant force of 170 N moves an object 24 m in the direction of the force. Calculate the work done.

    Formula:Work Done

    $W = F\,d$

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14. 14.

    Quantitative Aptitude

    Find the HCF and LCM of 21 and 14.

    Formula:LCM & HCF

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

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15. 15.

    Mathematics

    If f(x) = 4x³ + 6x² + 8x, find f′(2).

    Formula:Derivative of a Polynomial

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

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16. 16.

    Physics

    A projectile is launched at 42 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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17. 17.

    Physics

    An object starts at 5 m/s and accelerates at 8 m/s² for 6 s. Find its final velocity.

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

    $v = u + at$

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18. 18.

    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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19. 19.

    Mathematics

    For the quadratic 4x² − 3x + 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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20. 20.

    Mathematics

    Find the slope of the line through (4, -1) and (5, -6).

    Formula:Slope of a Line

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

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