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Locus of all point P (x, y) satisfying x³ + y³ + 3xy = 1 consists of union of

(a) A line and an isolated point (b) A line pair and an isolated point (c) A line and a circle (d) A circle and an isolated point. Solution:... View Article

There is a boy who has six brothers. Each of them has 6 brothers and one sister. The boy’s father’s name is Mr. Patrick. How many children does Mr. Patrick has?

Solution: The boy has 6 brothers. Given that each of them has 6 brothers and one sister. So total number of children = 8. Number of... View Article

What is absolute value?

The absolute value of a number is the distance of a number from the origin of a number line. It is represented by |a|, where a is any integer.... View Article

What is solid angle formula?

The formula of solid angle(Ω) is, Ω = (Area of part of the spherical surface subtended)/(radius of the area of the part of the... View Article

In a 3-digit number, unit’s digit is one more than the hundred’s digit and ten’s digit is one less than that hundred’s digit. If the sum of the original 3-digit number and numbers obtained by changing the order of digits cyclically is 2664. What is the number?

Let hundred digit = b ten digit = b−1 unit digit = b+1 First form of Number =100×b+100(b−1)+b+1 = 111b − 9 Second... View Article

Write 14/9 as mixed fraction.

Given, 14/9 is an improper fraction Dividing 14 by 9 we get; quotient = 1 remainder = 5 Therefore, the required mixed fraction is 1 5/9... View Article

The following table gives the distribution of students of two sections according to the marks obtained by them. Represent the marks of the students of both the sections on the same graph by two frequency polygons. From the two polygons compare the performance of the two sections.

Given, Section A Section B Marks Frequency Marks Frequency 0-10 3 0-10 5 10-20 9 10-20 19 20-30 17 20-30 15 30-40 12 30-40 10... View Article

Find the domain and range of 5 sin(x – π/6 ).

The domain and range of 5 sin(x - π/6 ). Domain: ( − ∞ , ∞ ) , { x | x ∈ R } Range: [ − 5 , 5 ] , {... View Article

The number of different ways in which the letters of the word “YOGESH” can be placed in 8 boxes.

Given, the word YOGESH The total number of ways in which the rows can be filled = 8C6 =28 ways Now, as per the given figure, there will be... View Article

A random variable x follows binomial distribution with mean 3 and variance 3/2, and P(x ≤5) = (63/2^x). Find the value of x.

Let n and p be the parameters of the binomial distribution. Mean = np=3 Variance = npq = 3/2 (npq/np) = q = 1/2 p = 1 - 1/2 = 1/2 So, n = 6... View Article

A problem in statistics is given to five students A, B, C and D. Their chances of solving it are 1/3, 1/4, 1/5 and 1/6 respectively. What is the probability that the problem will be solved?

Probability of A solving = 1/3 P (A not solving) = 1 - 1/3 = 2/3 Similarly, P (B not solving) = 1 - 1/4 = 3/4 P (C not solving) = 1 -... View Article

If two tangents drawn from a point P to the parabola y^2=4x are at right angles, then the locus of P is (a) 2x+1=0 (b) x=-1 (c) 2x-1=0 (d) x=1

Answer: (b) x=-1 If tangents are perpendicular from point P then point P lies on directrix of parabola Locus of P = directrix of y2=4x... View Article

The graph of which of the following equations passes through the origin? (i) y = 2x + c (ii) y = 2x – c (iii) y = 2x

Answer: (iii) y = 2x If x = 0, then y = 2(0) = 0... View Article

Two varieties of oil are mixed in the ratio 4:3 to produce first quality and if they are mixed in the ratio 2:3, Second quality is obtained. How many kg of the first quality be mixed with 10 kg of the second quality so that a third quality having the 2 varieties in the ratio 5:4 may be produced?

Let the common ratio of the first quality oil be x. Given, the ratio is 4:3 Therefore, the total quantity of the first quality oil = 4x+3x =... View Article

Find the range of eccentricity of the ellipse (x^2/a^2) + (y^2/b^2) = 1 (a>b) such that the line segment joining the foci does not subtend a right angle at any point on the ellipse

Let the point on ellipse be P (a cosθ, b sinθ) Let A and B are the diameter of the ellipse If ∠APB = π/2, then P... View Article

A watch loses 3% time during the first week and then gains 2% time during the next one week. If the watch was set right at 12 noon on a Sunday, what will be the time that the watch will show exactly 14 days from the time it was set right ?

Number of hours in a week = 7 x 24 = 168 hours The watch loses 3% time during the first week. Therefore, 3% of 168 = 5.04 hours During the... View Article

If √(4 + √(8 – √(32 + √768))) = a√2 cos(11π/b), where a and b are natural numbers then (b – a) is divisible by:

Suppose, √768 = 32 cos θ 16√3 = 32 cos θ cos θ = √3/2 θ =... View Article

Let A be the set of all 3 × 3 skew-symmetric matrices whose entries are either −1, 0, or 1. If there are exactly three 0s, three 1s, and three (−1)s, then what is the number of such matrices?

Skew symmetric matrix can be written in the form of: Given, λ1, λ2, λ3 ∈{1,−1} Therefore,... View Article

If t_1,t_2 and t_3 are distinct, the points (t_1,2at_1 + at_1^3), (t_2, 2at_2 + at_2^3) and (t_3, 2at_3 + at_3^3) are collinear. Then what is the value of t_1 + t_2 + t_3?

Let the points are A(t1,2at1 + at13), B(t2, 2at2 + at23) and C(t3, 2at3 + at33) Given, A, B and C are collinear Area of triangle ABC = 0... View Article

For natural numbers m and n, if (1-y)^m (1+y)^n = 1 + a1y + a^2y^2+…, and a_1 = a_2 = 10, then (m,n) is:

Given, For natural numbers m and n, if (1-y)m (1+y)n = 1 + a1y + a2y2+..., and a1 = a2 = 10, Therefore, a1 = n - m = 10 and a2 = (m2-... View Article