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square-root-and-cube-root.json
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square-root-and-cube-root.json
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{
"formula": "SQUARE ROOT:\n\nIf a² = b, then b is the square of a and a is the square root of b.\na = √b\n\nExample:\n5² = 25\n√25 = 5\n\nCUBE ROOT:\n\nIf a³ = b, b is the cube of a and a is the cube root of b.\na = b\n\nExample:\n4³ = 64\nCube root of 64 = 4\n\n(i)√xy = √x * √y\n(ii)√(x/y) = √x / √y",
"problems": [
{
"id": 1,
"answerIndex": 0,
"answers": ["0.032", "0.0032", "0.0256", "0.00256"],
"question": "√0.001024 = ?",
"solution": "√0.001024\n= √(1024/1000000)\n= 32/1000\n= 0.032"
},
{
"id": 3,
"answerIndex": 0,
"answers": ["213444", "214344", "214434", "231444"],
"question": "The least perfect square, which is divisible by each of 21, 36 and 66 is",
"solution": "LCM(21, 36, 66) = 2772.\n\n2772 = 2 x 2 x 3 x 3 x 7 x 11\n\nTo make it a perfect square,\nit must be multiplied by 7 x 11.\n\nSo, required number = 2772*77 = 213444"
},
{
"id": 2,
"answerIndex": 1,
"answers": ["0.1", "0.01", "0.001", "0.0001"],
"question": "∛0.000001 = ?",
"solution": "∛0.000001\n= ∛(1/10⁶)\n= 1/10²\n= 0.01"
},
{
"id": 4,
"answerIndex": 3,
"answers": ["38", "28", "48", "42"],
"question": "The square root of 1764 is",
"solution": "1764 = 2*2*3*3*7*7\n\n√1764 = 2*3*7 = 42"
},
{
"id": 5,
"answerIndex": 2,
"answers": ["3600", "2000", "400", "900"],
"question": "√100 + √10000 = √8100 + √x\n\nFind x.",
"solution": "√100 + √10000 = √8100 + √x\n\n10 + 100 = 90 + √x\n\n√x = 20\n\nx = 20² = 400"
},
{
"id": 6,
"answerIndex": 3,
"answers": ["53/7", "31/7", "15√2/7", "25/7"],
"question": "[√7 - (2/√7)]² equals",
"solution": "[√7 - (2/√7)]²\n= (√7)² + (2/√7)² - (2*√7*2/√7)\n= 7 + (4/7) - 4\n= 3 + 4/7\n= 25/7"
},
{
"id": 7,
"answerIndex": 0,
"answers": ["10", "12", "11", "13"],
"question": "If x = 5+2√6, find the value of x + 1/x.",
"solution": "x = 5+2√6\n\n1/x = 1/(5+2√6)\n = [1/(5+2√6)] * [(5-2√6)/(5-2√6)]\n = [(5-2√6)/(25-4*6)]\n = (5-2√6)\n\nx + 1/x\n= 5+2√6 + 5-2√6\n= 10"
},
{
"id": 8,
"answerIndex": 2,
"answers": ["5", "3√5", "√5", "5√3"],
"question": "The square root of (9+2√19)(9-2√19) is",
"solution": "(9+2√19)(9-2√19)\n= 9² - (2√19)²\n= 81 - (4*19)\n= 81 - 76\n= 5\n\n√[(9+2√19)(9-2√19)]\n= √5\n"
},
{
"id": 9,
"answerIndex": 2,
"answers": ["1(1/8)", "2(1/16)", "1(3/4)", "1(1/4)"],
"question": "√[3(1/16)] = ",
"solution": "√[3(1/16)]\n= √(49/16)\n= 7/4\n= 1(3/4)"
},
{
"id": 10,
"answerIndex": 1,
"answers": ["281", "301", "331", "321"],
"question": "Find the square root of 90601.",
"solution": "√90601 = 301"
},
{
"id": 11,
"answerIndex": 0,
"answers": ["11", "12", "14", "16"],
"question": "√(102+(√(77+√(21-√25))) = ",
"solution": "√(102+(√(77+√(21-√25)))\n= √(102+(√(77+√(21-5)))\n= √(102+(√(77+√16)))\n= √(102+(√(77+4))\n= √(102+√81)\n= √(102+9)\n= √121\n= 11"
},
{
"id": 12,
"answerIndex": 1,
"answers": ["57", "67", "77", "87"],
"question": "A group of students decided to collect as many paise from each member of group as is the number of members. If the total collection amounts to Rs. 44.89, the number of the member is the group is",
"solution": "Amount Collected\n= 44.89*100\n= 4489 paise\n\nNumber of members\n= √4489\n= 67"
},
{
"id": 13,
"answerIndex": 3,
"answers": ["9", "12", "15/2", "7"],
"question": "(√484 / 6) * (√324 / 11) * (14 / √144)",
"solution": "(√484 / 6) * (√324 / 11) * (14 / √144)\n= (22/6)*(18/11)*(14/12)\n= 7"
},
{
"id": 14,
"answerIndex": 1,
"answers": ["3600", "360", "36", "3.6"],
"question": "If √0.00001296 = 0.0036, find the square root of 129600.",
"solution": "√0.00001296 = 0.0036\n\n√1296 = 36\n\n√129600 = 360"
},
{
"id": 15,
"answerIndex": 2,
"answers": ["21.66", "31.29", "22.35", "20.46"],
"question": "If 3√5 + √125 = 17.88, then find the value of √80 + 6√5.",
"solution": "3√5 + √125 = 17.88\n\n3√5 + 5√5 = 17.88\n\n8√5 = 17.88\n\n√5 = 2.235\n\n√80 + 6√5\n= √(16*5) + 6√5\n= 4√5 + 6√5\n= 10√5\n= 10*2.235\n= 22.35"
},
{
"id": 16,
"answerIndex": 0,
"answers": ["0", "1", "2", "3"],
"question": "How many two-digit numbers satisfy this property: The last digit (unit's digit) of the square of the two-digit number is 8?",
"solution": "The last digit of a square number can never be 8."
},
{
"id": 17,
"answerIndex": 3,
"answers": ["10²", "10⁶", "10⁵", "10⁴"],
"question": "√(0.0576*x) = 24. Find x.",
"solution": "√(0.0576*x) = 24\n\n(0.0576*x) = 24² = 576\n\nx = 576/0.0576 = 10⁴"
},
{
"id": 18,
"answerIndex": 1,
"answers": ["5.59", "7.826", "8.944", "10.062"],
"question": "If √5 = 2.236, then the value of (√5/2) - (10/√5) + √125 is equal to",
"solution": "(√5/2) - (10/√5) + √125\n= (√5/2) - (2*√5*√5/√5) + √(25*5)\n= (√5/2)-2√5+5√5\n= √5(1/2 - 2 + 5)\n= 7√5/2\n= (7*2.236)/2\n= 7.826"
},
{
"id": 19,
"answerIndex": 0,
"answers": ["1000", "1221", "1331", "1464"],
"question": "The smallest 4-digit number which is a perfect cube is",
"solution": "1000 is the smallest 4-digit number which is a perfect cube."
},
{
"id": 20,
"answerIndex": 2,
"answers": ["0.1039", "0.2078", "1.1039", "2.1039"],
"question": "If a = 0.1039, then the value of √(4a² - 4a + 1) + 3a is",
"solution": "√(4a² - 4a + 1) + 3a\n= √(2a-1)² + 3a\n\n√(2a-1)² is either (2a-1) or -(2a-1).\n\nCase I:\n2a-1+3a = 5a-1 = -0.4805\n\nCase II:\n-(2a-1)+3a = a+1 = 1.1039\n\nOnly case II satisfies the options."
}
]
}