GATE · Practice Sheet 12

1. 1.

   Mathematics

   In an arithmetic progression with first term 20 and common difference 2, find term number 11.

   Formula:AP nth Term

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

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2. 2.

   Physics

   A projectile is launched at 21 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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3. 3.

   Mathematics

   If f(x) = 5x³ + 3x² + 1x, find f′(4).

   Formula:Derivative of a Polynomial

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

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4. 4.

   Mathematics

   Find the slope of the line through (5, 1) and (7, 4).

   Formula:Slope of a Line

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

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5. 5.

   Physics

   A body moving at 2 m/s accelerates at 7 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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6. 6.

   Quantitative Aptitude

   Find the HCF and LCM of 7 and 27.

   Formula:LCM & HCF

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

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7. 7.

   Physics

   An object starts at 6 m/s and accelerates at 3 m/s² for 8 s. Find its final velocity.

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

   $v = u + at$

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8. 8.

   Physics

   If 1,436 J of work is done in 35 s, find the power.

   Formula:Power

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

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9. 9.

   Quantitative Aptitude

   In how many ways can 2 objects be arranged out of 8 distinct objects? (Find ⁿᴘᵣ for n = 8, r = 2.)

   Formula:Permutations

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

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10. 10.

    Mathematics

    Evaluate log base 10 of 10,000.

    Formula:Logarithm Evaluation

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

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11. 11.

    Mathematics

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

    Formula:Distance Between Two Points

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

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12. 12.

    Physics

    If 3,140 J of work is done in 37 s, find the power.

    Formula:Power

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

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13. 13.

    Mathematics

    Evaluate log base 10 of 1,000,000.

    Formula:Logarithm Evaluation

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

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14. 14.

    Physics

    Find the voltage across a 35 Ω resistor carrying a current of 6 A.

    Formula:Ohm's Law

    $V = IR$

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15. 15.

    Mathematics

    Find term number 5 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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16. 16.

    Physics

    Find the force required to accelerate a 37 kg mass at 14 m/s².

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

    $F = ma$

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17. 17.

    Mathematics

    For the quadratic 3x² + 8x + 3 = 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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18. 18.

    Quantitative Aptitude

    Find the HCF and LCM of 28 and 9.

    Formula:LCM & HCF

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

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19. 19.

    Mathematics

    Evaluate log base 3 of 27.

    Formula:Logarithm Evaluation

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

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20. 20.

    Mathematics

    If f(x) = 2x³ + 3x² + 4x, find f′(5).

    Formula:Derivative of a Polynomial

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

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