Archives

Let y = (A + Bx)e^(3x) is a solution of the differential equation(d^2y)/dx^2 + m(dy/dx) + ny = 0; m, n ∈ I, then find m and n.

Given,y = (A + Bx)e3xDifferentiating w.r.t x on both sides,dy/dx = 3(A + Bx)e3x + Be3xAdding my on both sides,(dy/dx) + my = 3(A + Bx)e3x + Be3x... View Article

Solve: 2(sin x – cos 2x) – sin 2x(1 + 2 sin x) + 2 cos x = 0

2(sin x − cos 2x) − sin 2x(1 + 2 sin x) + 2 cos x = 0 ⇒ 2 sin x − sin 2x − 2 cos 2x − 2 sin x sin 2x + 2 cos x = 0 We know that, sin 2x = 2 sin x... View Article

If am = bl, then find whether the pair of linear equation ax + by = c and lx + my = n has no solution, a unique solution or infinitely many solutions.

Given, am = bl a/l = b/m....(i) And the given pair of linear equations are: ax + by = c lx + my = n Comparing with the standard form of pair of... View Article

In the figure given below, AD = 4 cm, BD = 3 cm and CB = 12 cm, then cot θ equals

(a) 3/4 (b) 5/12 (c) 4/3 (d) 12/5 Correct answer: (d) 12/5 Solution: Given, AD = 4 cm BD = 3 cm CB = 12 cm In right ∆ABC, AB2 = AD2 + BD2 AB2 =... View Article

What is the square root of 106?

The value of the square root of 106 is 10.295630141.... The square root of 106 can be estimated using the long division methods as given below:... View Article

What is a quadratic equation formula?

We know that the standard form of a quadratic equation is ax2 + bx + c = 0, where a, b, c are real numbers and a ≠ 0. If α, β are the root of a... View Article

What are the types of linear equations?

There are three types of linear equations as given below: 1. Conditional equation: This is the most commonly used linear equation. Also, the... View Article

The minute hand of a clock overtakes the hour hand at intervals of 65 minutes. How much in a day does the clock gain or lose?

Solution: The minute hand gains 55 min. spaces over the hour hand in 60 minutes (in a correct clock). To be together again, the minute hand... View Article

At what time between 4 and 5, will the hands of a clock coincide?

(a) 15.81 min (b) 21.81 min (c) 23.81 min (d) 33.48 min Solution: Angle divisions of a clock A close observation of a clock reveals that an... View Article

If the product of perpendiculars from the foci upon the polar of P be constant and equal to c², prove that the locus of P is the ellipse b⁴x²(c² + a²e²) + c²a⁴y² = a⁴b⁴.

Solution: Let the equation of the ellipse is (x2/a2) + (y2/b2) = 1 The coordinates of foci are (ae, 0) and (-ae, 0). Let the coordinates of... View Article

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