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The average monthly income of P and Q is Rs. 5050. The average monthly income of Q and R is Rs. 6250. The average monthly income of P and R is Rs. 5200. Determine P's monthly income.
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Solution: Step 1: Convert the average incomes into sums for each pair: P + Q = 5050 × 2 = 10100 (Equation 1) Q + R = 6250 × 2 = 12500 (Equation 2) P + R = 5200 × 2 = 10400 (Equation 3) Step 2: Add all three equations together: (P + Q) + (Q + R) + (P + R) = 10100 + 12500 + 10400 2P + 2Q + 2R = 33000 2(P + Q + R) = 33000 Step 3: Find the total sum of incomes for P, Q, and R: P + Q + R = 33000 / 2 = 16500 (Equation 4) Step 4: To find P's income, subtract Equation 2 (Q + R) from Equation 4 (P + Q + R). P = (P + Q + R) - (Q + R) P = 16500 - 12500 P = 4000 Rs.
2
Given x = (sqrt(5) + sqrt(3)) / (sqrt(5) - sqrt(3)) and y = (sqrt(5) - sqrt(3)) / (sqrt(5) + sqrt(3)), what is the value of (x + y)?
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Solution: Step 1: Rationalize the expression for x. x = (sqrt(5) + sqrt(3)) / (sqrt(5) - sqrt(3)) * (sqrt(5) + sqrt(3)) / (sqrt(5) + sqrt(3)) x = (sqrt(5) + sqrt(3))^2 / ((sqrt(5))^2 - (sqrt(3))^2) x = (5 + 3 + 2*sqrt(15)) / (5 - 3) x = (8 + 2*sqrt(15)) / 2 x = 4 + sqrt(15) Step 2: Rationalize the expression for y. y = (sqrt(5) - sqrt(3)) / (sqrt(5) + sqrt(3)) * (sqrt(5) - sqrt(3)) / (sqrt(5) - sqrt(3)) y = (sqrt(5) - sqrt(3))^2 / ((sqrt(5))^2 - (sqrt(3))^2) y = (5 + 3 - 2*sqrt(15)) / (5 - 3) y = (8 - 2*sqrt(15)) / 2 y = 4 - sqrt(15) Step 3: Calculate (x + y). x + y = (4 + sqrt(15)) + (4 - sqrt(15)) x + y = 8
3
If `(5^(3/2)) * 5^3 / 5^(-3/2) = 5^(a+2)`, find the value of `a`.
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Solution: Step 1: Rewrite all terms with a common base (5) using exponent rules. The expression `5\u221a5` is interpreted as `5^1 * 5^(1/2) = 5^(3/2)`. Step 2: Apply the laws of exponents `a^m * a^n = a^(m+n)` and `a^m / a^n = a^(m-n)`. Combine the exponents on the left side: `Exponent = (3/2) + 3 - (-3/2)` `Exponent = 3/2 + 3 + 3/2` `Exponent = (3/2 + 3/2) + 3` `Exponent = 6/2 + 3` `Exponent = 3 + 3 = 6`. Step 3: Equate the powers of 5 on both sides of the equation. `5^6 = 5^(a+2)`. Step 4: Since the bases are the same, equate the exponents. `6 = a + 2`. Step 5: Solve for `a`. `a = 6 - 2`. `a = 4`. Step 6: The value of `a` is 4.
4
A man possesses a total of Rs. 480, consisting of one-rupee, five-rupee, and ten-rupee notes. If the count of notes for each denomination is identical, what is the total number of notes he has?
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Solution: Step 1: Let 'x' be the number of notes of each denomination (one-rupee, five-rupee, and ten-rupee). Since the number of notes of each denomination is equal. Step 2: Calculate the total value contributed by each denomination: - Value from one-rupee notes = x * Rs. 1 = Rs. x - Value from five-rupee notes = x * Rs. 5 = Rs. 5x - Value from ten-rupee notes = x * Rs. 10 = Rs. 10x Step 3: The total amount of money the man has is the sum of these values, which is given as Rs. 480. Formulate the equation: x + 5x + 10x = 480 Step 4: Combine like terms and solve for x: 16x = 480 x = 480 / 16 x = 30 Step 5: So, there are 30 notes of each denomination. The total number of notes is the sum of notes across all denominations: Total notes = (Number of one-rupee notes) + (Number of five-rupee notes) + (Number of ten-rupee notes) Total notes = x + x + x = 3x Total notes = 3 * 30 = 90.
5
Given tan15° = 2 - √3, determine the value of tan15° cot75° + tan75° cot15°.
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Solution: Step 1: Use complementary angle identities to simplify cot75° and tan75°. cot75° = cot(90° - 15°) = tan15°. tan75° = tan(90° - 15°) = cot15°. Step 2: Substitute these into the expression. The expression becomes tan15° (tan15°) + (cot15°) cot15° = tan²15° + cot²15°. Step 3: Calculate cot15° using the given tan15°. cot15° = 1 / tan15° = 1 / (2 - √3). Rationalize the denominator by multiplying by the conjugate: cot15° = (1 / (2 - √3)) × ((2 + √3) / (2 + √3)) = (2 + √3) / (2² - (√3)²) = (2 + √3) / (4 - 3) = 2 + √3. Step 4: Substitute the values of tan15° and cot15° into the simplified expression. tan²15° + cot²15° = (2 - √3)² + (2 + √3)². Step 5: Expand and simplify. (2 - √3)² = 4 - 4√3 + 3 = 7 - 4√3. (2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3. Summing these: (7 - 4√3) + (7 + 4√3) = 14.
6
Solve for x in the equation: 3^x - 3^(x-1) = 486.
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Solution: Step 1: Rewrite the second term using the laws of exponents (a^(m-n) = a^m / a^n). 3^(x-1) = 3^x / 3^1 = 3^x / 3. Step 2: Substitute this back into the equation. 3^x - (3^x / 3) = 486. Step 3: Factor out 3^x from the terms on the left side. 3^x * (1 - 1/3) = 486 3^x * (3/3 - 1/3) = 486 3^x * (2/3) = 486. Step 4: Isolate 3^x. 3^x = 486 × (3/2) 3^x = 243 × 3 3^x = 729. Step 5: Express 729 as a power of 3. 729 = 3^6. Step 6: Equate the exponents since the bases are the same. 3^x = 3^6 x = 6. Step 7: The value of x is 6.
7
A total sum of Rs. 8,200 was allocated among A, B, and C such that A received Rs. 500 more than B, and C received Rs. 300 more than A. What was C's share in rupees?
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Solution: Step 1: Define the shares of A, B, and C using a single variable. Let B's share be 'x' rupees. A's share = x + 500 rupees. C's share = A's share + 300 = (x + 500) + 300 = x + 800 rupees. Step 2: Set up an equation for the total sum distributed. The total sum is Rs. 8,200. (x + 500) + x + (x + 800) = 8,200. Step 3: Solve the equation for 'x'. 3x + 1,300 = 8,200. 3x = 8,200 - 1,300. 3x = 6,900. x = 6,900 / 3 = 2,300. Step 4: Calculate C's share using the value of 'x'. C's share = x + 800 = 2,300 + 800 = Rs. 3,100.
8
If `4b^2 + 1/b^2 = 2`, determine the value of `8b^3 + 1/b^3`.
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Solution: Step 1: Rewrite the given equation to identify a perfect square form. Given: `4b^2 + 1/b^2 = 2` This can be written as `(2b)^2 + (1/b)^2 = 2`. Consider the identity `(A+B)^2 = A^2 + B^2 + 2AB`. Here `A=2b` and `B=1/b`. So, `(2b + 1/b)^2 = (2b)^2 + (1/b)^2 + 2(2b)(1/b) = 4b^2 + 1/b^2 + 4`. Step 2: Substitute the given value into the identity. ` (2b + 1/b)^2 = 2 + 4` ` (2b + 1/b)^2 = 6` ` 2b + 1/b = √6` Step 3: Now, we need to find `8b^3 + 1/b^3`. This can be written as `(2b)^3 + (1/b)^3`. Apply the sum of cubes identity: `A^3 + B^3 = (A + B)(A^2 - AB + B^2)` or `(A+B)^3 - 3AB(A+B)`. Using `(A+B)^3 = A^3 + B^3 + 3AB(A+B)`, we have `A^3 + B^3 = (A+B)^3 - 3AB(A+B)`. Let `A = 2b` and `B = 1/b`. `8b^3 + 1/b^3 = (2b + 1/b)^3 - 3(2b)(1/b)(2b + 1/b)` Step 4: Substitute the value of `(2b + 1/b)` from Step 2. `8b^3 + 1/b^3 = (√6)^3 - 3(2)( √6)` `8b^3 + 1/b^3 = 6√6 - 6√6` Step 5: Simplify the expression. `8b^3 + 1/b^3 = 0`.
9
If 10^x = 1/2, what is the value of 10^(-8x)?
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Solution: Step 1: Rewrite the expression 10^(-8x) using exponent rules. 10^(-8x) = (10^x)^(-8) Step 2: Substitute the given value of 10^x = 1/2 into the expression. (10^x)^(-8) = (1/2)^(-8) Step 3: Apply the exponent rule (a/b)^(-n) = (b/a)^n. (1/2)^(-8) = (2/1)^8 = 2^8 Step 4: Calculate 2^8. 2^8 = 256.
10
Simplify the expression: [(sqrt(3) + sqrt(2)) / (sqrt(3) - sqrt(2))] - [(sqrt(3) - sqrt(2)) / (sqrt(3) + sqrt(2))]
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Solution: Step 1: Let the two fractions be A and B. A = (sqrt(3) + sqrt(2)) / (sqrt(3) - sqrt(2)) B = (sqrt(3) - sqrt(2)) / (sqrt(3) + sqrt(2)) The expression is A - B. Step 2: Rationalize A. A = [(sqrt(3) + sqrt(2)) / (sqrt(3) - sqrt(2))] * [(sqrt(3) + sqrt(2)) / (sqrt(3) + sqrt(2))] A = (sqrt(3) + sqrt(2))^2 / ((sqrt(3))^2 - (sqrt(2))^2) A = (3 + 2 + 2*sqrt(6)) / (3 - 2) A = 5 + 2*sqrt(6) Step 3: Rationalize B. B = [(sqrt(3) - sqrt(2)) / (sqrt(3) + sqrt(2))] * [(sqrt(3) - sqrt(2)) / (sqrt(3) - sqrt(2))] B = (sqrt(3) - sqrt(2))^2 / ((sqrt(3))^2 - (sqrt(2))^2) B = (3 + 2 - 2*sqrt(6)) / (3 - 2) B = 5 - 2*sqrt(6) Step 4: Calculate A - B. A - B = (5 + 2*sqrt(6)) - (5 - 2*sqrt(6)) = 5 + 2*sqrt(6) - 5 + 2*sqrt(6) = 4*sqrt(6).
11
Determine the value of the expression: (tan²20° / cosec²70°) + (cot²20° / sec²70°) + 2tan15° tan45° tan75°.
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Solution: Step 1: Simplify the first term using complementary angle identities. cosec 70° = cosec (90° - 20°) = sec 20°. So, (tan²20° / cosec²70°) = (tan²20° / sec²20°) = (sin²20°/cos²20°) / (1/cos²20°) = sin²20°. Step 2: Simplify the second term. sec 70° = sec (90° - 20°) = cosec 20°. So, (cot²20° / sec²70°) = (cot²20° / cosec²20°) = (cos²20°/sin²20°) / (1/sin²20°) = cos²20°. Step 3: Combine the first two simplified terms using the identity sin²θ + cos²θ = 1. The sum is sin²20° + cos²20° = 1. Step 4: Simplify the third term. tan 45° = 1. tan 75° = tan (90° - 15°) = cot 15°. So, 2tan15° tan45° tan75° = 2tan15° × 1 × cot15°. Since tanθ · cotθ = 1, this simplifies to 2 × 1 × 1 = 2. Step 5: Add the results from Step 3 and Step 4. Total value = 1 + 2 = 3.
12
Find the value of x that satisfies the equation √((1+x)/x) - √(x/(1+x)) = 1/√6.
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Solution: Step 1: Let y = √((1+x)/x). Then √(x/(1+x)) = 1/y. Substitute these into the given equation. y - 1/y = 1/√6 Step 2: Simplify the equation involving y. (y² - 1) / y = 1/√6 √6(y² - 1) = y √6y² - y - √6 = 0 Step 3: Solve the quadratic equation for y using the quadratic formula. y = [ -(-1) ± √((-1)² - 4(√6)(-√6)) ] / (2√6) y = [ 1 ± √(1 + 24) ] / (2√6) y = [ 1 ± √25 ] / (2√6) y = (1 ± 5) / (2√6) Since y = √((1+x)/x) must be positive, choose the positive root for y. y = (1 + 5) / (2√6) = 6 / (2√6) = 3/√6 = √9/√6 = √(3/2). Step 4: Substitute y = √(3/2) back into y = √((1+x)/x). √(3/2) = √((1+x)/x) Square both sides: 3/2 = (1+x)/x Step 5: Solve for x. 3x = 2(1+x) 3x = 2 + 2x x = 2.
13
A boy's CAT score was consistently 75 for four consecutive years. For each correct answer, 1 mark was awarded, and for each incorrect answer, 1/3 mark was deducted. He attempted all questions every year. Which of the given options cannot be the total number of questions asked in CAT during any year?
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Solution: Step 1: Let 'C' be the number of correct answers and 'W' be the number of wrong answers. Step 2: The total number of questions (T) is `T = C + W`. Step 3: The net score is given by: `C - (1/3)W = 75`. Step 4: Multiply the score equation by 3 to eliminate the fraction: `3C - W = 225`. Step 5: From Step 2, `W = T - C`. Substitute this into the equation from Step 4: `3C - (T - C) = 225` `3C - T + C = 225` `4C - T = 225` `4C = 225 + T`. Step 6: Since C (number of correct answers) must be an integer, `(225 + T)` must be divisible by 4. Step 7: Check each option for T: * If T = 231: `225 + 231 = 456`. `456 / 4 = 114`. (Possible) * If T = 163: `225 + 163 = 388`. `388 / 4 = 97`. (Possible) * If T = 150: `225 + 150 = 375`. `375` is not divisible by 4 (since it's an odd number). * If T = 123: `225 + 123 = 348`. `348 / 4 = 87`. (Possible) Step 8: The total number of questions must be such that `(225 + T)` is a multiple of 4. Option 150 does not satisfy this condition. Step 9: Therefore, 150 is not a possible total number of questions.
14
If log10(2) = 0.3010, what is the value of log2(10)?
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Solution: Step 1: Apply the change of base formula for logarithms in the form: log_b(a) = 1 / log_a(b). So, log2(10) = 1 / log10(2). Step 2: Substitute the given value of log10(2). log2(10) = 1 / 0.3010. Step 3: Perform the division. To handle the decimal, multiply the numerator and denominator by 10000. log2(10) = 10000 / 3010 = 1000 / 301.
15
Given (a + 1/a)² = 3, determine the value of a³° + a²´ + a¹¸ + a¹² + a¶ + 1.
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Solution: Step 1: Start with the given equation: (a + 1/a)² = 3. Step 2: Take the square root of both sides: a + 1/a = ±√3. Step 3: Cube both sides of the equation (we can use either +√3 or -√3 as the result will be the same): (a + 1/a)³ = (±√3)³. Step 4: Apply the identity (X + Y)³ = X³ + Y³ + 3XY(X + Y): a³ + (1/a)³ + 3 * a * (1/a) * (a + 1/a) = ±3√3. a³ + 1/a³ + 3(a + 1/a) = ±3√3. Step 5: Substitute a + 1/a = ±√3 back into the equation: a³ + 1/a³ + 3(±√3) = ±3√3. a³ + 1/a³ = 0. Step 6: Multiply by a³ to clear the denominator: a⁶ + 1 = 0. Step 7: Now consider the expression to be evaluated: a³° + a²´ + a¹¸ + a¹² + a¶ + 1. Step 8: Factor the expression using the term (a¶ + 1): a²´(a¶ + 1) + a¹²(a¶ + 1) + (a¶ + 1). Step 9: Substitute a¶ + 1 = 0 into the factored expression: a²´(0) + a¹²(0) + 0. = 0 + 0 + 0 = 0.
16
Calculate the value of the expression: (1018)^2 - 1019 × 1017 + 1015 × 1012 - 1016 × 1011.
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Solution: Step 1: Analyze the first two terms: (1018)^2 - 1019 × 1017. Let a = 1018. Then the expression is a^2 - (a+1)(a-1). Using the identity (a+1)(a-1) = a^2 - 1^2 = a^2 - 1. So, a^2 - (a^2 - 1) = a^2 - a^2 + 1 = 1. Step 2: Analyze the last two terms: + 1015 × 1012 - 1016 × 1011. Let b = 1016 and c = 1011. The terms can be rewritten as + (b-1) × (c+1) - bc. Expand (b-1)(c+1) = bc + b - c - 1. Substitute back: (bc + b - c - 1) - bc = b - c - 1. Substitute the values for b and c: 1016 - 1011 - 1 = 5 - 1 = 4. Step 3: Add the results from Step 1 and Step 2. Total value = 1 + 4 = 5. The value of the expression is 5.
17
Given that sqrt(5) = 2.236 and sqrt(3) = 1.732, find the value of 1 / (sqrt(5) + sqrt(3)).
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Solution: Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of (sqrt(5) + sqrt(3)). Conjugate is (sqrt(5) - sqrt(3)). Expression = [1 / (sqrt(5) + sqrt(3))] * [(sqrt(5) - sqrt(3)) / (sqrt(5) - sqrt(3))] Step 2: Simplify the expression. = (sqrt(5) - sqrt(3)) / ((sqrt(5))^2 - (sqrt(3))^2) = (sqrt(5) - sqrt(3)) / (5 - 3) = (sqrt(5) - sqrt(3)) / 2 Step 3: Substitute the given values of sqrt(5) and sqrt(3). = (2.236 - 1.732) / 2 Step 4: Perform the arithmetic operations. = 0.504 / 2 = 0.252.
18
Pineapples are priced at Rs. 7 each, and watermelons at Rs. 5 each. If X spends a total of Rs. 38 on these two types of fruits, how many pineapples were purchased?
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Solution: Step 1: Let 'p' be the number of pineapples purchased and 'w' be the number of watermelons purchased. Step 2: Formulate a linear equation based on the total cost: 7p + 5w = 38 Step 3: Since 'p' and 'w' must be non-negative whole numbers (as they represent counts of fruits), we can test integer values for 'p' starting from 1. - If p = 1: 7(1) + 5w = 38 => 5w = 31. w = 31/5 (not an integer). - If p = 2: 7(2) + 5w = 38 => 14 + 5w = 38 => 5w = 24. w = 24/5 (not an integer). - If p = 3: 7(3) + 5w = 38 => 21 + 5w = 38 => 5w = 17. w = 17/5 (not an integer). - If p = 4: 7(4) + 5w = 38 => 28 + 5w = 38 => 5w = 10. w = 10/5 = 2 (This is an integer solution!). - If p = 5: 7(5) + 5w = 38 => 35 + 5w = 38 => 5w = 3. w = 3/5 (not an integer). - If p >= 6: 7p would be 42 or more, exceeding the total spent of Rs. 38, so no more solutions are possible. Step 4: The only valid integer solution for 'p' is 4. Step 5: Therefore, the number of pineapples purchased is 4.
19
If a^3 + 3a^2 + 9a = 1, then what is the value of a^3 + 3/a?
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Solution: Step 1: Given the equation: a^3 + 3a^2 + 9a = 1 (Equation 1). Step 2: We need to find the value of a^3 + 3/a. Step 3: From Equation 1, observe the coefficients 1, 3, 9. These are powers of 3: 3^0, 3^1, 3^2. This suggests a special algebraic identity. Step 4: Rewrite Equation 1 as a(a^2 + 3a + 9) = 1. Step 5: We know the identity for difference of cubes: (x - y)(x^2 + xy + y^2) = x^3 - y^3. If we consider (a - 3)(a^2 + 3a + 9), it equals a^3 - 3^3 = a^3 - 27. Step 6: From a(a^2 + 3a + 9) = 1, we have (a^2 + 3a + 9) = 1/a. Step 7: Multiply both sides of (a^2 + 3a + 9) = 1/a by (a - 3): (a - 3)(a^2 + 3a + 9) = (a - 3)(1/a) a^3 - 27 = 1 - 3/a Step 8: Rearrange the terms to get the desired expression: a^3 + 3/a = 1 + 27 a^3 + 3/a = 28. Step 9: The value of a^3 + 3/a is 28.
20
What value should replace the question mark in the equation: sqrt(0.0169 * ?) = 1.3?
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Solution: Step 1: Let the unknown value be represented by 'x'. The equation is: sqrt(0.0169 * x) = 1.3. Step 2: To eliminate the square root, square both sides of the equation: (sqrt(0.0169 * x))^2 = (1.3)^2. Step 3: This simplifies to: 0.0169 * x = 1.69. Step 4: To solve for 'x', divide both sides by 0.0169: x = 1.69 / 0.0169. Step 5: To simplify the division, multiply both the numerator and the denominator by 10000 (to remove the decimals): * Numerator: 1.69 * 10000 = 16900. * Denominator: 0.0169 * 10000 = 169. Step 6: Perform the division: x = 16900 / 169. Step 7: x = 100.
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