GATE · Practice Sheet 1

1. 1.

   Mathematics

   Find the distance between the points (-2, 8) and (5, -5).

   Formula:Distance Between Two Points

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

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2. 2.

   Mathematics

   Find the sum of the first 21 terms of an AP with first term 12 and common difference 5.

   Formula:AP Sum of n Terms

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

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3. 3.

   Mathematics

   For the quadratic 3x² + 9x + 8 = 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.

   Mathematics

   In an arithmetic progression with first term 18 and common difference 1, find term number 30.

   Formula:AP nth Term

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

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5. 5.

   Mathematics

   Evaluate log base 2 of 64.

   Formula:Logarithm Evaluation

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

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6. 6.

   Physics

   A constant force of 59 N moves an object 45 m in the direction of the force. Calculate the work done.

   Formula:Work Done

   $W = F\,d$

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7. 7.

   Physics

   A projectile is launched at 20 m/s at 45° 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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8. 8.

   Quantitative Aptitude

   What is 25% of 800?

   Formula:Percentage of a Number

   $\text{Value} = \dfrac{P}{100}\times N$

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9. 9.

   Quantitative Aptitude

   Find the simple interest on ₹7,000 at 12% per annum for 2 year(s).

   Formula:Simple Interest

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

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10. 10.

    Physics

    If 2,798 J of work is done in 50 s, find the power.

    Formula:Power

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

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11. 11.

    Quantitative Aptitude

    What is 25% of 260?

    Formula:Percentage of a Number

    $\text{Value} = \dfrac{P}{100}\times N$

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12. 12.

    Quantitative Aptitude

    Find the simple interest on ₹33,000 at 6% per annum for 6 year(s).

    Formula:Simple Interest

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

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13. 13.

    Physics

    A constant force of 95 N moves an object 14 m in the direction of the force. Calculate the work done.

    Formula:Work Done

    $W = F\,d$

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14. 14.

    Physics

    Find the kinetic energy of a 17 kg object moving at 20 m/s.

    Formula:Kinetic Energy

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

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15. 15.

    Quantitative Aptitude

    In how many ways can 4 objects be arranged out of 7 distinct objects? (Find ⁿᴘᵣ for n = 7, r = 4.)

    Formula:Permutations

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

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16. 16.

    Quantitative Aptitude

    Find the simple interest on ₹9,000 at 4% per annum for 6 year(s).

    Formula:Simple Interest

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

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17. 17.

    Mathematics

    Find term number 3 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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18. 18.

    Mathematics

    Find the slope of the line through (2, -4) and (-7, -3).

    Formula:Slope of a Line

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

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19. 19.

    Mathematics

    If f(x) = 5x³ + 6x² + 2x, find f′(3).

    Formula:Derivative of a Polynomial

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

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20. 20.

    Quantitative Aptitude

    In how many ways can 3 objects be chosen from 6 distinct objects? (Find ⁿᴄᵣ for n = 6, r = 3.)

    Formula:Combinations

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

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