This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to questions 1 to 29: Question 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. Question 2: Find the resultant of velocities of 15 ms^-1 and 20 ms^-1 at right angle to each other. Step 1: When two velocities are at right angles, their resultant R is found using the Pythagorean theorem. R = sqrt(V_1^2 + V_2^2) Step 2: Substitute the given values V_1 = 15 ms^-1 and V_2 = 20 ms^-1. R = sqrt((15 ms)^-1)^2 + (20 ms^-1)^2 R = sqrt(225 m)^2s^-2 + 400 m^2s^-2 R = sqrt(625 m)^2s^-2 Step 3: Calculate the square root. R = 25 ms^-1 The correct answer is (a) 25 ms^-1. Question 3: Find the resultant of velocities of 6 ms^-1 and 8 ms^-1 at right angle to each other. Step 1: Use the Pythagorean theorem for resultant velocity at right angles. R = sqrt(V_1^2 + V_2^2) Step 2: Substitute V_1 = 6 ms^-1 and V_2 = 8 ms^-1. R = sqrt((6 ms)^-1)^2 + (8 ms^-1)^2 R = sqrt(36 m)^2s^-2 + 64 m^2s^-2 R = sqrt(100 m)^2s^-2 Step 3: Calculate the square root. R = 10 ms^-1 The correct answer is (a) 10 ms^-1. Question 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. Question 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. Question 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. Question 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. Question 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. Question 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. Question 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. Question 11: The 7^th term of a linear sequence is 33 and its 11^th term is 53. Find the common difference of the Sequence. Step 1: For a linear sequence (Arithmetic Progression), the n^th term is given by T_n = a + (n-1)d. Given T_7 = 33 a + 6d = 33 (Equation 1) Given T_11 = 53 a + 10d = 53 (Equation 2) Step 2: Subtract Equation 1 from Equation 2 to find d. (a + 10d) - (a + 6d) = 53 - 33 4d = 20 Step 3: Solve for d. d = (20)/(4) d = 5 The correct answer is (c) 5. Question 12: Find the median of 5, 2, 1, 3, 4, 6, 5, 7, 9, 8. Step 1: Arrange the numbers in ascending order. 1, 2, 3, 4, 5, 5, 6, 7, 8, 9 Step 2: Count the number of terms (N). There are 10 terms. Step 3: Since N is an even number, the median is the average of the two middle terms, which are the ((N)/(2))^th and ((N)/(2)+1)^th terms. The 5^th term is 5. The 6^th term is 5. Step 4: Calculate the median. Median = (5 + 5)/(2) = (10)/(2) = 5 The median is 5. (Note: This answer is not among the given options). Question 13: Find the mean of 16, 18, 20, 28, and 18. Step 1: Sum all the numbers. Sum = 16 + 18 + 20 + 28 + 18 = 100 Step 2: Count the number of terms. There are 5 terms. Step 3: Calculate the mean by dividing the sum by the number of terms. Mean = SumNumber of terms = (100)/(5) = 20 The correct answer is (a) 20. Question 14: Find the 21^st term of the A.P -4, -1.5, 1, 3.5 ... Step 1: Identify the first term (a) and the common difference (d). a = -4 d = -1.5 - (-4) = -1.5 + 4 = 2.5 Step 2: Use the formula for the n^th term of an A.P., T_n = a + (n-1)d. We need T_21. T_21 = -4 + (21-1)(2.5) T_21 = -4 + (20)(2.5) Step 3: Calculate the value. T_21 = -4 + 50 T_21 = 46 The correct answer is (b) 46. Question 15: Simplify sqrt(32) - 2sqrt(2) Step 1: Simplify sqrt(32). sqrt(32) = sqrt(16 × 2) = sqrt(16) × sqrt(2) = 4sqrt(2) Step 2: Substitute the simplified form back into the expression. 4sqrt(2) - 2sqrt(2) Step 3: Combine the terms. (4-2)sqrt(2) = 2sqrt(2) The correct answer is (a) 2sqrt(2). Question 16: The 3^rd and 7^th term of a G.P. are 81 and 16. Find the 5^th term. Step 1: For a Geometric Progression (GP), the n^th term is T_n = ar^n-1. T_3 = ar^2 = 81 (Equation 1) T_7 = ar^6 = 16 (Equation 2) Step 2: Divide Equation 2 by Equation 1 to find the common ratio r. (ar^6)/(ar^2) = (16)/(81) r^4 = (16)/(81) r = 4/(81) = (2)/(3) Step 3: Substitute r = (2)/(3) into Equation 1 to find the first term a. a((2)/(3))^2 = 81 a((4)/(9)) = 81 a = (81 × 9)/(4) = (729)/(4) Step 4: Calculate the 5^th term, T_5 = ar^4. T_5 = (729)/(4) × ((2)/(3))^4 T_5 = (729)/(4) × (16)/(81) T_5 = (729)/(81) × (16)/(4) T_5 = 9 × 4 = 36 The correct answer is (d) 36. Question 17: If = ^3/_5, 0^ < < 90^, Find . Step 1: Use the trigonometric identity ^2 + ^2 = 1. ((3)/(5))^2 + ^2 = 1 Step 2: Solve for ^2 . (9)/(25) + ^2 = 1 ^2 = 1 - (9)/(25) ^2 = (25 - 9)/(25) = (16)/(25) Step 3: Take the square root. Since 0^ < < 90^, is positive. = sqrt((16)/(25)) = (4)/(5) The correct answer is (a) ^4/_5. Question 18: Factorize x^2 - 4x - 21 Step 1: Find two numbers that multiply to -21 and add up to -4. The numbers are -7 and 3 (since -7 × 3 = -21 and -7 + 3 = -4). Step 2: Write the quadratic expression in factored form. (x-7)(x+3) The factorization is (x-7)(x+3). (Note: This answer is not among the given options). Question 19: Solve the quadratic equation 2x^2 - 3x - 5 = 0 Step 1: Factorize the quadratic equation. Find two numbers that multiply to 2 × (-5) = -10 and add to -3. These numbers are -5 and 2. 2x^2 - 5x + 2x - 5 = 0 Step 2: Group terms and factor out common factors. x(2x - 5) + 1(2x - 5) = 0 (x+1)(2x-5) = 0 Step 3: Set each factor to zero and solve for x. x+1 = 0 x = -1 2x-5 = 0 2x = 5 x = (5)/(2) The correct answer is (a) -1 or ^5/_2. Question 20: Express 0.000724 in standard form Step 1: Move the decimal point to the right until there is one non-zero digit before it. 0.000724 7.24 Step 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 The correct answer is (c) 7.24 × 10^-4. Question 21: Solve the inequality -2(13+x) 9+5x. Step 1: Distribute the -2 on the left side. -26 - 2x 9 + 5x Step 2: Gather x terms on one side and constant terms on the other. Add 2x to both sides. -26 9 + 5x + 2x -26 9 + 7x Step 3: Subtract 9 from both sides. -26 - 9 7x -35 7x Step 4: Divide by 7. The inequality sign does not change because we are dividing by a positive number. (-35)/(7) x -5 x This can also be written as x -5. The correct answer is (d) x -5. Question 22: Express sqrt(98) in basic form Step 1: Find the largest perfect square factor of 98. 98 = 49 × 2 Step 2: Rewrite the square root using this factor. sqrt(98) = sqrt(49 × 2) = sqrt(49) × sqrt(2) Step 3: Simplify the perfect square. 7sqrt(2) The correct answer is (b) 7sqrt(2). Question 23: Express sqrt(288) in basic form Step 1: Find the largest perfect square factor of 288. 288 = 144 × 2 Step 2: Rewrite the square root using this factor. sqrt(288) = sqrt(144 × 2) = sqrt(144) × sqrt(2) Step 3: Simplify the perfect square. 12sqrt(2) The correct answer is (b) 12sqrt(2). Question 24: Simplify sqrt(12) + sqrt(48)sqrt(24) Step 1: Simplify each surd in the expression. sqrt(12) = sqrt(4 × 3) = 2sqrt(3) sqrt(48) = sqrt(16 × 3) = 4sqrt(3) sqrt(24) = sqrt(4 × 6) = 2sqrt(6) Step 2: Substitute the simplified surds back into the expression. 2sqrt(3) + 4sqrt(3)2sqrt(6) = 6sqrt(3)2sqrt(6) Step 3: Simplify the fraction. 6sqrt(3)2sqrt(6) = 3sqrt(3)sqrt(6) Step 4: Rationalize the denominator by multiplying the numerator and denominator by sqrt(6). 3sqrt(3)sqrt(6) × sqrt(6)sqrt(6) = 3sqrt(18)6 Step 5: Simplify sqrt(18) and then the entire expression. sqrt(18) = sqrt(9 × 2) = 3sqrt(2) 3 × 3sqrt(2)6 = 9sqrt(2)6 = 3sqrt(2)2 The simplified expression is 3sqrt(2)2. (Note: This answer is not among the given options). Question 25: Simplify 9a^5 × 4a^3 Step 1: Multiply the coefficients and combine the variables using the exponent rule a^m × a^n = a^m+n. (9 × 4) × (a^5 × a^3) 36 × a^5+3 Step 2: Calculate the final expression. 36a^8 The simplified expression is 36a^8. (Note: This answer is not among the given options). Question 26: Simplify _4 12 - _4 36 + _4 192 Step 1: Use the logarithm properties _b x - _b y = _b ((x)/(y)) and _b x + _b y = _b (xy). _4 12 - _4 36 + _4 192 = _4 ((12)/(36)) + _4 192 Step 2: Simplify the fraction and then combine the remaining terms. = _4 ((1)/(3)) + _4 192 = _4 ((1)/(3) × 192) = _4 ((192)/(3)) = _4 64 Step 3: Determine the value of _4 64. We need to find x such that 4^x = 64. 4^x = 4^3 x = 3 The correct answer is (a) 3. Question 27: Simplify 3^n+1 × 3^1-n Step 1: Use the exponent rule a^m × a^n = a^m+n. 3^(n+1) + (1-n) Step 2: Simplify the exponent. 3^n+1+1-n = 3^2 Step 3: Calculate the value. 3^2 = 9 The simplified expression is 9. (Note: This answer is not among the given options). Question 28: Solve 4^x = 64 Step 1: Express both sides of the equation with the same base. 4^x = 4^3 Step 2: Equate the exponents. x = 3 The solution is x=3. (Note: This answer is not among the given options). Question 29: Solve 9^x+1 = 81^x-2 Step 1: Express both sides of the equation with the same base. Since 81 = 9^2, we can use base 9. 9^x+1 = (9^2)^x-2 Step 2: Apply the exponent rule (a^m)^n = a^mn. 9^x+1 = 9^2(x-2) 9^x+1 = 9^2x-4 Step 3: Equate the exponents. x+1 = 2x-4 Step 4: Solve for x. Subtract x from both sides. 1 = x-4 Step 5: Add 4 to both sides. 1+4 = x x = 5 The solution is x=5. (Note: This answer is not among the given options). Drop the next question.