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 14 to 30: 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 correct answer is (d) (x-7)(x+3). 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 correct answer is (d) 3sqrt(2)2. 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 correct answer is (c) 36a^8. 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 correct answer is (d) 9. 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 correct answer is (c) 3. 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 correct answer is (b) 5. Question 30: Given that (1)/(8^3y) = 2^y+2, find y. Step 1: Express both sides of the equation with the same base. The smallest common base is 2. 8 = 2^3 (1)/((2^3)^3y) = 2^y+2 Step 2: Apply the exponent rules (a^m)^n = a^mn and (1)/(a^m) = a^-m. (1)/(2^9y) = 2^y+2 2^-9y = 2^y+2 Step 3: Equate the exponents since the bases are the same. -9y = y+2 Step 4: Solve for y. Subtract y from both sides. -9y - y = 2 -10y = 2 Step 5: Divide by -10. y = (2)/(-10) y = -(1)/(5) My calculation yields y = -(1)/(5). This answer is not among the provided options (a) 1, (b) 2, (c) 3, (d) 4. There might be a typo in the question or the options. However, if we assume the question intended to be 8^y = 2^y+2 (a common type of simplification error in such problems), then: 2^3y = 2^y+2 3y = y+2 2y = 2 y = 1 Assuming this interpretation, the answer would be (a) 1. That's 2 down. 3 left today — send the next one.