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The rate of change of displacement with time is called (a) velocity (b) Acceleration (c) speed (d) momentum

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A

velocity

Step-by-step solution

66 steps

Here are the solutions to questions 1 to 29:

Part 1

The rate of change of displacement with time is called

  • The rate of change of displacement with respect to time is defined as velocity.
  • The correct answer is (a) velocity.

Part 2· 3 steps

Find the resultant of velocities of 15ms−115 \text{ms}^{-1} and 20ms−120 \text{ms}^{-1} at right angle to each other.

  1. When two velocities are at right angles, their resultant RR is found using the Pythagorean theorem.

    $$
R = \sqrt{V_1^2 + V_2^2}
$$

  2. Substitute the given values V1=15 ms−1V_1 = 15 \text{ ms}^{-1} and V2=20 ms−1V_2 = 20 \text{ ms}^{-1}.

    $$
R = \sqrt{(15 \text{ ms}^{-1})^2 + (20 \text{ ms}^{-1})^2}
$$
$$
R = \sqrt{225 \text{ m}^2\text{s}^{-2} + 400 \text{ m}^2\text{s}^{-2}}
$$
$$
R = \sqrt{625 \text{ m}^2\text{s}^{-2}}
$$

  3. Calculate the square root.

    $$
R = 25 \text{ ms}^{-1}
$$
    • The correct answer is (a) 25ms−125 \text{ms}^{-1}.

Part 3· 3 steps

Find the resultant of velocities of 6ms−16 \text{ms}^{-1} and 8ms−18 \text{ms}^{-1} at right angle to each other.

  1. Use the Pythagorean theorem for resultant velocity at right angles.

    $$
R = \sqrt{V_1^2 + V_2^2}
$$

  2. Substitute V1=6 ms−1V_1 = 6 \text{ ms}^{-1} and V2=8 ms−1V_2 = 8 \text{ ms}^{-1}.

    $$
R = \sqrt{(6 \text{ ms}^{-1})^2 + (8 \text{ ms}^{-1})^2}
$$
$$
R = \sqrt{36 \text{ m}^2\text{s}^{-2} + 64 \text{ m}^2\text{s}^{-2}}
$$
$$
R = \sqrt{100 \text{ m}^2\text{s}^{-2}}
$$

  3. Calculate the square root.

    $$
R = 10 \text{ ms}^{-1}
$$
    • The correct answer is (a) 10ms−110 \text{ms}^{-1}.

Part 4

The rate of change of velocity with time is called

  • The rate of change of velocity with respect to time is defined as acceleration.
  • The correct answer is (b) acceleration.

Part 5

The difference between the least entry and the greatest entry is

  • In statistics, the range is the difference between the highest and lowest values in a data set.
  • The correct answer is (a) range.

Part 6

A graphic display of a frequency distribution is called

  • A histogram is a graphical representation of the distribution of numerical data, often used for frequency distributions.
  • The correct answer is (c) histogram.

Part 7

The sum of all the frequencies from the first to the last one is called

  • The sum of all frequencies in a distribution is the total number of observations, which is also the final value of the cumulative frequency.
  • The correct answer is (a) cumulative.

Part 8

All the following are measures of central tendency except

  • Mean, median, and mode are measures of central tendency. Range is a measure of dispersion.
  • The correct answer is (d) range.

Part 9

The value which occurs most frequently in the distribution is called

  • The mode is the value that appears most often in a set of data.
  • The correct answer is (b) mode.

Part 10

All the following are measures of dispersion except

  • Range, variance, and standard deviation are measures of dispersion. Mean is a measure of central tendency.
  • The correct answer is (d) mean.

Part 11· 3 steps

The 7th7^{th} term of a linear sequence is 33 and its 11th11^{th} term is 53. Find the common difference of the Sequence.

  1. For a linear sequence (Arithmetic Progression), the nthn^{th} term is given by Tn=a+(n−1)dT_n = a + (n-1)d.

    Given $T_7 = 33 \implies a + 6d = 33$ (Equation 1)  
Given $T_{11} = 53 \implies a + 10d = 53$ (Equation 2)

  2. Subtract Equation 1 from Equation 2 to find dd.

    $$
(a + 10d) - (a + 6d) = 53 - 33
$$
$$
4d = 20
$$

  3. Solve for dd.

    $$
d = \frac{20}{4}
$$
$$
d = 5
$$
    • The correct answer is (c) 5.

Part 12· 4 steps

Find the median of 5, 2, 1, 3, 4, 6, 5, 7, 9, 8.

  1. Arrange the numbers in ascending order.

    $$
1, 2, 3, 4, 5, 5, 6, 7, 8, 9
$$

  2. Count the number of terms (NN).

    There are 10 terms.

  3. Since NN is an even number, the median is the average of the two middle terms, which are the (N2)th(\frac{N}{2})^{th} and (N2+1)th(\frac{N}{2}+1)^{th} terms.

    The $5^{th}$ term is 5.  
The $6^{th}$ term is 5.

  4. Calculate the median.

    $$
\text{Median} = \frac{5 + 5}{2} = \frac{10}{2} = 5
$$
    • The median is 5. (Note: This answer is not among the given options).

Part 13· 3 steps

Find the mean of 16, 18, 20, 28, and 18.

  1. Sum all the numbers.

    $$
\text{Sum} = 16 + 18 + 20 + 28 + 18 = 100
$$

  2. Count the number of terms.

    There are 5 terms.

  3. Calculate the mean by dividing the sum by the number of terms.

    $$
\text{Mean} = \frac{\text{Sum}}{\text{Number of terms}} = \frac{100}{5} = 20
$$
    • The correct answer is (a) 20.

Part 14· 3 steps

Find the 21st21^{st} term of the A.P -4, -1.5, 1, 3.5 ...

  1. Identify the first term (aa) and the common difference (dd).

    $$
a = -4
$$
$$
d = -1.5 - (-4) = -1.5 + 4 = 2.5
$$

  2. Use the formula for the nthn^{th} term of an A.P., Tn=a+(n−1)dT_n = a + (n-1)d.

    We need T21T_{21}. T21=−4+(21−1)(2.5) T_{21} = -4 + (21-1)(2.5) T21=−4+(20)(2.5) T_{21} = -4 + (20)(2.5)

  3. Calculate the value.

    $$
T_{21} = -4 + 50
$$
$$
T_{21} = 46
$$
    • The correct answer is (b) 46.

Part 15· 3 steps

Simplify 32−22\sqrt{32} - 2\sqrt{2}

  1. Simplify 32\sqrt{32}.

    $$
\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}
$$

  2. Substitute the simplified form back into the expression.

    $$
4\sqrt{2} - 2\sqrt{2}
$$

  3. Combine the terms.

    $$
(4-2)\sqrt{2} = 2\sqrt{2}
$$
    • The correct answer is (a) 222\sqrt{2}.

Part 16· 4 steps

The 3rd3^{rd} and 7th7^{th} term of a G.P. are 81 and 16. Find the 5th5^{th} term.

  1. For a Geometric Progression (GP), the nthn^{th} term is Tn=arn−1T_n = ar^{n-1}.

    $$
T_3 = ar^2 = 81 \quad (\text{Equation 1})
$$
$$
T_7 = ar^6 = 16 \quad (\text{Equation 2})
$$

  2. Divide Equation 2 by Equation 1 to find the common ratio rr.

    $$
\frac{ar^6}{ar^2} = \frac{16}{81}
$$
$$
r^4 = \frac{16}{81}
$$
$$
r = \sqrt[4]{\frac{16}{81}} = \frac{2}{3}
$$

  3. Substitute r=23r = \frac{2}{3} into Equation 1 to find the first term aa.

    $$
a\left(\frac{2}{3}\right)^2 = 81
$$
$$
a\left(\frac{4}{9}\right) = 81
$$
$$
a = \frac{81 \times 9}{4} = \frac{729}{4}
$$

  4. Calculate the 5th5^{th} term, T5=ar4T_5 = ar^4.

    $$
T_5 = \frac{729}{4} \times \left(\frac{2}{3}\right)^4
$$
$$
T_5 = \frac{729}{4} \times \frac{16}{81}
$$
$$
T_5 = \frac{729}{81} \times \frac{16}{4}
$$
$$
T_5 = 9 \times 4 = 36
$$
    • The correct answer is (d) 36.

Part 17· 3 steps

If sin⁡θ=3/5\sin \theta = ^3/_5, 0∘<θ<90∘0^\circ < \theta < 90^\circ, Find cos⁡θ\cos \theta.

  1. Use the trigonometric identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.

    $$
\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1
$$

  2. Solve for cos⁡2θ\cos^2 \theta.

    $$
\frac{9}{25} + \cos^2 \theta = 1
$$
$$
\cos^2 \theta = 1 - \frac{9}{25}
$$
$$
\cos^2 \theta = \frac{25 - 9}{25} = \frac{16}{25}
$$

  3. Take the square root.

    Since 0∘<θ<90∘0^\circ < \theta < 90^\circ, cos⁡θ\cos \theta is positive. cos⁡θ=1625=45 \cos \theta = \sqrt{\frac{16}{25}} = \frac{4}{5}

    • The correct answer is (a) 4/5^4/_5.

Part 18· 2 steps

Factorize x2−4x−21x^2 - 4x - 21

  1. Find two numbers that multiply to -21 and add up to -4.

    The numbers are -7 and 3 (since $-7 \times 3 = -21$ and $-7 + 3 = -4$).

  2. Write the quadratic expression in factored form.

    $$
(x-7)(x+3)
$$
    • The factorization is (x−7)(x+3)(x-7)(x+3). (Note: This answer is not among the given options).

Part 19· 3 steps

Solve the quadratic equation 2x2−3x−5=02x^2 - 3x - 5 = 0

  1. Factorize the quadratic equation.

    Find two numbers that multiply to 2×(−5)=−102 \times (-5) = -10 and add to -3. These numbers are -5 and 2. 2x2−5x+2x−5=0 2x^2 - 5x + 2x - 5 = 0

  2. Group terms and factor out common factors.

    $$
x(2x - 5) + 1(2x - 5) = 0
$$
$$
(x+1)(2x-5) = 0
$$

  3. Set each factor to zero and solve for xx.

    $$
x+1 = 0 \implies x = -1
$$
$$
2x-5 = 0 \implies 2x = 5 \implies x = \frac{5}{2}
$$
    • The correct answer is (a) −1-1 or 5/2^5/_2.

Part 20· 2 steps

Express 0.000724 in standard form

  1. Move the decimal point to the right until there is one non-zero digit before it.

    $$
0.000724 \implies 7.24
$$

  2. Count the number of places the decimal point moved.

    It moved 4 places to the right. Since it moved to the right, the exponent will be negative. 7.24×10−4 7.24 \times 10^{-4}

    • The correct answer is (c) 7.24×10−47.24 \times 10^{-4}.

Part 21· 4 steps

Solve the inequality −2(13+x)≥9+5x-2(13+x) \ge 9+5x.

  1. Distribute the -2 on the left side.

    $$
-26 - 2x \ge 9 + 5x
$$

  2. Gather xx terms on one side and constant terms on the other.

    Add 2x2x to both sides. −26≥9+5x+2x -26 \ge 9 + 5x + 2x −26≥9+7x -26 \ge 9 + 7x

  3. Subtract 9 from both sides.

    $$
-26 - 9 \ge 7x
$$
$$
-35 \ge 7x
$$

  4. Divide by 7.

    The inequality sign does not change because we are dividing by a positive number. −357≥x \frac{-35}{7} \ge x −5≥x -5 \ge x

    • This can also be written as x≤−5x \le -5.
    • The correct answer is (d) x≤−5x \le -5.

Part 22· 3 steps

Express 98\sqrt{98} in basic form

  1. Find the largest perfect square factor of 98.

    $$
98 = 49 \times 2
$$

  2. Rewrite the square root using this factor.

    $$
\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2}
$$

  3. Simplify the perfect square.

    $$
7\sqrt{2}
$$
    • The correct answer is (b) 727\sqrt{2}.

Part 23· 3 steps

Express 288\sqrt{288} in basic form

  1. Find the largest perfect square factor of 288.

    $$
288 = 144 \times 2
$$

  2. Rewrite the square root using this factor.

    $$
\sqrt{288} = \sqrt{144 \times 2} = \sqrt{144} \times \sqrt{2}
$$

  3. Simplify the perfect square.

    $$
12\sqrt{2}
$$
    • The correct answer is (b) 12212\sqrt{2}.

Part 24· 5 steps

Simplify 12+4824\frac{\sqrt{12} + \sqrt{48}}{\sqrt{24}}

  1. Simplify each surd in the expression.

    $$
\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
$$
$$
\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}
$$
$$
\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}
$$

  2. Substitute the simplified surds back into the expression.

    $$
\frac{2\sqrt{3} + 4\sqrt{3}}{2\sqrt{6}} = \frac{6\sqrt{3}}{2\sqrt{6}}
$$

  3. Simplify the fraction.

    $$
\frac{6\sqrt{3}}{2\sqrt{6}} = \frac{3\sqrt{3}}{\sqrt{6}}
$$

  4. Rationalize the denominator by multiplying the numerator and denominator by 6\sqrt{6}.

    $$
\frac{3\sqrt{3}}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{3\sqrt{18}}{6}
$$

  5. Simplify 18\sqrt{18} and then the entire expression.

    $$
\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
$$
$$
\frac{3 \times 3\sqrt{2}}{6} = \frac{9\sqrt{2}}{6} = \frac{3\sqrt{2}}{2}
$$
    • The simplified expression is 322\frac{3\sqrt{2}}{2}. (Note: This answer is not among the given options).

Part 25· 2 steps

Simplify 9a5×4a39a^5 \times 4a^3

  1. Multiply the coefficients and combine the variables using the exponent rule am×an=am+na^m \times a^n = a^{m+n}.

    $$
(9 \times 4) \times (a^5 \times a^3)
$$
$$
36 \times a^{5+3}
$$

  2. Calculate the final expression.

    $$
36a^8
$$
    • The simplified expression is 36a836a^8. (Note: This answer is not among the given options).

Part 26· 3 steps

Simplify log⁡412−log⁡436+log⁡4192\log_4 12 - \log_4 36 + \log_4 192

  1. Use the logarithm properties log⁡bx−log⁡by=log⁡b(xy)\log_b x - \log_b y = \log_b (\frac{x}{y}) and log⁡bx+log⁡by=log⁡b(xy)\log_b x + \log_b y = \log_b (xy).

    $$
\log_4 12 - \log_4 36 + \log_4 192 = \log_4 \left(\frac{12}{36}\right) + \log_4 192
$$

  2. Simplify the fraction and then combine the remaining terms.

    $$
= \log_4 \left(\frac{1}{3}\right) + \log_4 192
$$
$$
= \log_4 \left(\frac{1}{3} \times 192\right)
$$
$$
= \log_4 \left(\frac{192}{3}\right)
$$
$$
= \log_4 64
$$

  3. Determine the value of log⁡464\log_4 64.

    We need to find xx such that 4x=644^x = 64. 4x=43 4^x = 4^3 x=3 x = 3

    • The correct answer is (a) 3.

Part 27· 3 steps

Simplify 3n+1×31−n3^{n+1} \times 3^{1-n}

  1. Use the exponent rule am×an=am+na^m \times a^n = a^{m+n}.

    $$
3^{(n+1) + (1-n)}
$$

  2. Simplify the exponent.

    $$
3^{n+1+1-n} = 3^2
$$

  3. Calculate the value.

    $$
3^2 = 9
$$
    • The simplified expression is 9. (Note: This answer is not among the given options).

Part 28· 2 steps

Solve 4x=644^x = 64

  1. Express both sides of the equation with the same base.

    $$
4^x = 4^3
$$

  2. Equate the exponents.

    $$
x = 3
$$
    • The solution is x=3x=3. (Note: This answer is not among the given options).

Part 29· 5 steps

Solve 9x+1=81x−29^{x+1} = 81^{x-2}

  1. Express both sides of the equation with the same base.

    Since 81=9281 = 9^2, we can use base 9. 9x+1=(92)x−2 9^{x+1} = (9^2)^{x-2}

  2. Apply the exponent rule (am)n=amn(a^m)^n = a^{mn}.

    $$
9^{x+1} = 9^{2(x-2)}
$$
$$
9^{x+1} = 9^{2x-4}
$$

  3. Equate the exponents.

    $$
x+1 = 2x-4
$$

  4. Solve for xx.

    Subtract xx from both sides. 1=x−4 1 = x-4

  5. Add 4 to both sides.

    $$
1+4 = x
$$
$$
x = 5
$$
    • The solution is x=5x=5. (Note: This answer is not among the given options).
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