IIT-JEE · Practice Sheet 23

1. 1.

   Mathematics

   Evaluate log base 2 of 64.

   Formula:Logarithm Evaluation

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

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2. 2.

   Quantitative Aptitude

   Two fair dice are rolled. What is the probability that the sum of the numbers is 7?

   Formula:Probability (two dice)

   $P = \dfrac{\text{favourable}}{\text{total}}$

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3. 3.

   Quantitative Aptitude

   A vehicle covers 300 km in 6 hours. Find its average speed.

   Formula:Speed, Distance & Time

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

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4. 4.

   Mathematics

   For the quadratic 1x² + 6x − 1 = 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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5. 5.

   Physics

   Find the kinetic energy of a 8 kg object moving at 9 m/s.

   Formula:Kinetic Energy

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

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6. 6.

   Mathematics

   Find the sum of the first 18 terms of an AP with first term 8 and common difference 2.

   Formula:AP Sum of n Terms

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

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7. 7.

   Mathematics

   Find the sum of the first 5 terms of a GP with first term 5 and common ratio 2.

   Formula:GP Sum of n Terms

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

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8. 8.

   Mathematics

   Find the distance between the points (2, 8) and (6, 2).

   Formula:Distance Between Two Points

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

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9. 9.

   Quantitative Aptitude

   Two fair dice are rolled. What is the probability that the sum of the numbers is 7?

   Formula:Probability (two dice)

   $P = \dfrac{\text{favourable}}{\text{total}}$

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10. 10.

    Mathematics

    Find the binomial coefficient C(10, 2) — the coefficient of x^2 in (1 + x)^10.

    Formula:Binomial Coefficient

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

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11. 11.

    Physics

    Find the force required to accelerate a 40 kg mass at 8 m/s².

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

    $F = ma$

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12. 12.

    Physics

    If 3,428 J of work is done in 7 s, find the power.

    Formula:Power

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

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13. 13.

    Physics

    Find the voltage across a 28 Ω resistor carrying a current of 4 A.

    Formula:Ohm's Law

    $V = IR$

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14. 14.

    Physics

    Find the kinetic energy of a 8 kg object moving at 15 m/s.

    Formula:Kinetic Energy

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

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15. 15.

    Mathematics

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

    Formula:Distance Between Two Points

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

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16. 16.

    Physics

    A body moving at 8 m/s accelerates at 1 m/s² for 6 s. Find the distance travelled.

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

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

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17. 17.

    Mathematics

    Find the slope of the line through (-6, 8) and (-5, -4).

    Formula:Slope of a Line

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

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18. 18.

    Chemistry

    Find the pressure of 4 mol of an ideal gas at 277 K in a 5 L container (R = 0.0821 L·atm·mol⁻¹·K⁻¹).

    Formula:Ideal Gas Law

    $PV = nRT$

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19. 19.

    Mathematics

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

    Formula:Derivative of a Polynomial

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

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20. 20.

    Chemistry

    19 mL of a 5 M solution is diluted to 348 mL. Find the new concentration.

    Formula:Dilution

    $M_1V_1 = M_2V_2$

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