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Given that x^{3} + y^{3} = 72 and xy = 8 with x > y. Then the value of (x  y) is
Given: x^{3} + y^{3} = 72 and xy = 8
Solution: (x + y)^{3 }= x^{3 }+ y^{3} + 3xy(x + y)
=> (x + y)^{3} = 72 + 3.8(x + y)
=> (x + y)^{3} − 24(x + y) − 72 = 0
This is a cubic equation in terms of (x + y) which has one real root = 6
=> x + y = 6
Now, (x − y)^{2 }= (x + y)^{2} − 4xy
=> (x  y) = 2(x − y) = 2
Expression:
Putting x = 6
=> 8 + 2p = 12
=> p = 4/2 = 2
Clearly, the lines x = a and y = b meets at a point (a, b)
If sum of the roots of a quadratic equation is 7 and product of the roots is 12. Find the quadratic equation.
A quadratic equation is of the form: x^{2 }− (sum of roots) x + (product of roots) = 0
Let the roots of equation are α and β
=> Sum of roots = α + β = 7
Product of roots = αβ = 12
Equation: x^{2} − (α + β)x + αβ = 0
=> x^{2} − 7x + 12 = 0
=> Ans  (A)
The area of the triangle formed by the graphs of the equations x = 0, 2x + 3y = 6 and x + y = 3 is:
AC represents x + y = 3
BC represents 2x + 3y = 6
AB represents x = 0
=> ABC is the required triangle.
Base AB = 1 unit and height OC = 3 units
Which value among 3√5, 4√6, 6√12, 12√276 is the largest?
Values: 3√5, 4√6, 6√12, 12√276
Taking L.C.M. of exponents, => L.C.M.(3, 4, 6, 12) = 12
Now, multiplying all the exponents by 12, we get :
Values: (5)^{4}, (6)^{3}, (12)^{2}, (276)^{1}
= 625, 216, 144, 276
Thus, 625 ≡ 3√5 is the largest.
=> Ans  (A)
Given:
Squaring both sides,
Again squaring both sides, we get:
=> 25x^{2} − 36 = 324 + 25x^{2} − 180x
=> 180x = 324 + 36 = 360
=> x = 360/180 = 2
=> Ans  (C)
Substituting values from equations (ii) and (iii), we get:
x − (−x) = 2x
= ∛5  7
=> Ans  (B)
I multiplied a natural number by 18 and another by 21 and added the products. Which one of the following could be the sum?
Let's say one number is n and another number is p
so acc. to question sum will be 18n + 21p
and this number will be divisible by 3 so answer will be (A)
Which one of the following will completely divide 5^{71} + 5^{72} + 5^{73}?
Among all options only option C has unit digit 5, and in given equation unit digit will also be 5.
So only 155 can divide the given equation completely.
A certain number when divided by 899 leaves the remainder 65. When the same number is divided by 31, the remainder is:
If the number N is divided by 899 and leaves a remainder 65 then N = 899K + 65
and hence when N will be divided by 31
remainder of = Remainder of 65/31 as 899 is completely divisible by 31
and hence remainder is 3
Rationalizing each term, we get, the denominator of each term will be 1, we get:
= 2 + 3 = 5
By what least number should 675 be multiplied to obtain a number which is a perfect cube?
675 = 5^{2} × 3^{3}
and hence in order to make it a perfect cube we need to multiply it with 5.
675 x 5 = 5^{3} × 3^{3}
A manufacturer marked an article at Rs. 50 and sold it allowing 20% discount. If his profit was 25%, then the cost price of the article was
Given: Marked Price = 50
Discount = 20%of 50 = 50 × 0.5 = 10
Hence sold price = 50  10 = 40
Let's say cost price is x
Profit = 25% of x (Always remember profit and loss applicable only on cost price) = x/4x
Hence sold price will be
or 5x/4 = 40
x = 32
If a trader sold an article at Rs.3,060 after allowing 15% and 10% successive discounts on marked price, then the marked price is
Let marked price = 100x
After allowing discount of 15%, price =
After further allowing discount of 10%, price =
Now, selling price = 76.5x = 3060
=> x = 3060/76.5 = 40
∴ Marked Price = 100 * 40 = Rs 4,000
By selling an article, a man makes a profit of 25% of its selling price. His profit percent is
Given Profit on selling price is 25%
Suppose selling price is y
hence profit will be y/4y and cost price will be 3y/4
Now profit percentage on cost price will be
i.e. 100/3 = 33.33
A starts business with Rs. 7000 and after 5 months. B joined as a partner. After a year the profit is divided in the ratio 2 : 3. The capital of B is:
B's capital be x.
x = 18000
A shopkeeper earns a profit of 12% on selling a book at 10% discount on printed price. The ratio of the cost price to printed price of the book is
Let the printed price be 100x and the cost price be 100y
After discount of 10%, selling price of the book = = 90x
After earning a profit of 12%, selling price =
= 112y
Now, equating above equations, we get:
=> 90x = 112y
If the 3rd and the 5th term of an arithmetic progression are 13 and 21, what is the 13th term?
T_{3} = a + 2d = 13(1)
T_{5} = a + 4d = 21(2)
on solving (1) AND (2)
d = 4 & a = 5
T_{13 }= a + 12d = 5 + 12(4) = 5 + 48 = 53
So the answer is option A.
What is the sum of the first 12 terms of an arithmetic progression if the first term is 19 and last term is 36?
First term of AP = a = 19 and last term = l = 36
Number of terms = n = 12
= 17 × 6 = 102
=> Ans  (C)
What is the sum of the first 17 terms of an arithmetic progression if the first term is 20 and last term is 28?
First term of AP = a = 20 and last term = l = 28
Number of terms = n = 17
= 17 × 4 = 68
=> Ans  (A)
The 3rd and 8th term of an arithmetic progression are 13 and 2 respectively. What is the 14th term?
Let the first term of an AP = a and the common difference = d
3th term of AP = A_{3} = a + 2d = −13 (i)
8th term = A_{8} = a + 7d = 2 (ii)
Subtracting equation (i) from (ii), we get :
=> 7d − 2d = 2 − (−13)
=> 5d = 15
=> d = 15/5 = 3
Substituting it in equation (ii), => a = 2 − 7(3) = 2 − 21 = −19
∴ 14th term = A_{14} = a + 13d
= −19 + 13(3) = −19 + 39 = 20
=> Ans  (C)
The 4th and 7th term of an arithmetic progression are 11 and 4 respectively. What is the 15th term?
Let the first term of an AP = a and the common difference = d
4th term of AP = A_{4} = a + 3d = 11 (i)
7th term = A7 = a + 6d = −4 (ii)
Subtracting equation (i) from (ii), we get:
=> 6d − 3d = −4 − 11
=> 3d = −15
=> d = −15/3 = −5
Substituting it in equation (i), => a = 11 − 3(−5) = 11 + 15 = 26
∴ 15th term = A_{15} = a + 14d
= 26 + 14(−5) = 26 − 70 = −44
=> Ans  (B)
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