Prove that 5√2 is irrational.
Let 5√2 be a rational number. \(\begin{array}{l}5 \sqrt{2}=\frac{p}{q}, \text { where p and q are some integers and HCF(p, q)... View Article
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If (150)x = 7, then x is equal to:
The given expression is \(\begin{array}{l}(150)^{x}=7 \\ \Rightarrow \log _{150} 7=x \\ \Rightarrow \frac{\log 7}{\log 150}=x \\ \Rightarrow \frac{\log 7}{\log (3... View Article
ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that ∠ABD = ∠BAC.
Answer: A quadrilateral can be defined as a plane figure having 4 sides. There are mainly 6 types of quadrilaterals... View Article
Evaluate using suitable identities: (52)2.
The algebraic equations which are valid for all values of variables in them are called algebraic identities. Using the identity (a+b)2 = a2 + b2 + 2ab... View Article
Limit from x to 0 of [1 – cos (1 – cos x)] / x4 =
The given limit is as follows. \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^{4}} \\ =\lim _{x \rightarrow 0} \frac{1-\cos... View Article
Find the common factors of 12x, 36.
12x = 2 × 2 × 3 × x 36 = 2 × 2 × 3 × 3 Common factor... View Article
Which constant must be added and subtracted to solve the quadratic equation 9x2 + (3 / 4) x + 2 = 0 by the method of completing the square?
The given expression is \(\begin{array}{l}9 x^{2}+\frac{3}{4} x+2=0\\ \text { Dividing by 9 on both sides}, \mathbf{x}^{2}+\frac{\mathbf{x}}{\mathbf{1 2}}+\frac{2}{\mathbf{9}}=\mathbf{0}\\ \mathrm{x}^{2}+\frac{\mathrm{x}}{12}=-\frac{2}{9}\\ \text {... View Article
Solve the equation 3 (2p -1 ) = 5 – (3p – 2).
The given equation is \(\begin{array}{l}\begin{array}{l} 3(2 p-1)=5-(3 p-2) \\ 6 p-3=5-3 p+2 \\ 6 p+3 p=5+2+3 \\ 9 p=10 \\... View Article
Find the perpendicular distance from the origin of the line joining the points (cos θ, sin θ) and (cosφ, sinφ).
Let A = (cos θ, sin θ) and B = (cosφ, sinφ) be the given points. \(\begin{array}{l}\text { Then equation... View Article
Define coaxial circles and deduce their equation in the simplest form.
A family of circles, out of which every pair have the same radical axis, are called co-axial circles. Equation of... View Article
f (x) = x cosx. Find f′ (π).
Consider the given function. \(\begin{array}{l}\mathbf{f}(\mathbf{x})=\mathbf{x} \times \cos \mathbf{x} \\ \mathbf{f}^{\prime}(\mathbf{x})=\mathbf{1} \times \cos \mathbf{x}+\mathbf{x} \times(-\sin \mathbf{x}) \\ =\cos \mathbf{x}-\mathbf{x} \sin x... View Article
Write the following statement in the form of the equation: One-fourth of m is 3 more than 7.
One-fourth of m is m / 4. \(\begin{array}{l}\text { It is greater than 7 by 3}. \\ \text { The... View Article
Give reason: A square can be thought of as a special rhombus.
All sides of a rhombus and a square are the same. A rhombus with each angle as a right angle... View Article
Write the following rational numbers in ascending order: (1 / 3), (- 2 / 9), (- 4 / 3).
The given rational numbers are \(\begin{array}{l}\begin{array}{l} \frac{1}{3}, \frac{-2}{9}, \frac{-4}{3}\\ \Rightarrow \frac{3}{9}, \frac{-2}{9}, \frac{-12}{9} \text { (Converting into same denominator) }\\... View Article
Integral of x2ex dx
On applying integrating by parts, \(\begin{array}{l}\mathbf{u}=\mathbf{x}^{2}, \mathbf{v}=\mathbf{e}^{\mathbf{x}}\\ \mathbf{u}^{\prime}=2 \mathbf{x}, \int \mathbf{v d x}=\mathbf{e}^{\mathbf{x}} \Longrightarrow \mathrm{x}^{2} \mathrm{e}^{\mathrm{x}}-\int 2 \mathrm{xe}^{\mathrm{x}} \mathrm{d} \mathrm{x}\\... View Article
From a point in the interior of an equilateral triangle, perpendiculars are drawn on
From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides.
Find all the solutions of 4 cos2 xsinx – 2 sin2 x = 3 sinx.
The given expression can be written as \(\begin{array}{l}\begin{array}{l} \begin{array}{l} \Rightarrow 4\left(1-\sin ^{2} x\right) \sin x-2 \sin ^{2} x-3 \sin x=0... View Article
A rectangular plot is given for constructing a house, having a measurement of 40m long and 15m in the front. According to the laws, a minimum of 3m wide space should be left in the front and back each and 2m wide space on each of the other sides. Find the largest area where the house can be constructed.
Length of rectangular plot = 40m Width = 15m By keeping 3m wide space both in front and back The... View Article
A sum of money lent out at simple interest amounts to Rs 720 after 2 years and to Rs 1020 after a further period of 5 years. What is the sum?
Simple interest for 2 years = Rs. 720. Simple interest for 7 years = Rs. 1020 Simple interest for 5... View Article
By actual division find the quotient and the remainder when (x4 + 1) is divided by (x – 1).
The division is as follows. \(\begin{array}{l}\text { Quotient }=x^{3}+x^{2}+x+1 \\ \text { Remainder }=2\end{array} \)
