The greatest common factor of given numbers is called
The greatest common factor of given numbers is called HCF.
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Ritu is now four times as old as his brother Raju. In 4 years time, her age will be twice Raju’s age. What are their present ages?
Let the age of Raju be x years. The age of Ritu will be = 4 × x years In... View Article
Find the value of the cube root of the number 45. (Round off your number to the nearest whole number)
Take, n = 45, choose any starting value of x. \(\begin{array}{l}3^{3} = 27<45\\ x=3\\ \mathrm{x}_{\mathrm{next}}=\frac{2}{3} \mathrm{x}+\frac{\mathrm{n}}{3 \mathrm{x}^{2}}\\ \mathrm{x}_{\mathrm{next}}=\frac{2}{3} 3+\frac{45}{3 \times... View Article
A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.
No. Consider the positive integer 3q + 1, where q is a natural number. => (3q + 1)2 = 9q2... View Article
Form the differential equation of the family of hyperbolas having foci on the x-axis and centre at the origin.
The equation of hyperbola required is \(\begin{array}{l}\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \ldots(1) \text { a and b are parameters}\\ \text { Differentiating w.r.t }\mathrm{x},... View Article
What is the degree of the given monomial 3mn?
The degree of the monomial 3mn is 2.
Suresh’s age is double Kiran’s age. The sum of their ages is 24 years. How old is Suresh?
Let Kiran’s age be x years. So, the age of Suresh = 2x years x + 2x = 24 ⇒... View Article
Two numbers are respectively 30 per cent and 40 per cent less than a third number. What per cent is the second of the first?
Let the third number be x. \(\begin{array}{l}\text \ First \ number =x-\frac{30 x}{100}\\ =\frac{100 x-30 x}{100}=\frac{70 x}{100}=\frac{7 x}{10}\\ \text \... View Article
Find the value of tan-1 [tan (5π / 6)] + cos-1 [cos (13π / 6)].
Consider \(\begin{array}{l}\tan \left(\frac{5 \pi}{6}\right)=\tan \left(\pi-\frac{\pi}{6}\right) \\ =-\tan \frac{\pi}{6} \\ =-\frac{1}{\sqrt{3}} \\ \cos \left(13 \frac{\pi}{6}\right)=\cos \left(2 \pi+\frac{\pi}{6}\right) \\ =\tan ^{-1}\left(\frac{-1}{\sqrt{3}}\right)+\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)... View Article
Express tan theta in terms of sin theta.
tan theta can be expressed as \(\begin{array}{l}\tan \theta=\frac{\sin \theta}{\cos \theta}=\frac{\sin \theta}{\sqrt{1-\sin ^{2} \theta}}\end{array} \)
If 7 *8 = 49, 4 *4 = 12, 6 * 4 = 18 then, 9 * 6 = ?
7 * 8 = 56 – 7 = 49. 4 * 4 = 16 – 4 = 12. 6 *... View Article
In which quadrant or on which axis do each of the points (- 2, 4), (3, – 1), (- 1, 0), (1, 2) and (- 3, – 5) lie?
Point (1, 2) lies in the first quadrant. Point (- 2, 4) lies in the second quadrant. Point (- 3,... View Article
A right circular cone is 5.4cm high and the radius of its base is 2cm. It is melted and recast into another right circular one with a radius of base as 1.5 cm. find the height of the new cone formed.
Volume of the old cone = volume of the new cone \(\begin{array}{l}\frac{1}{3} \pi \times r_{1}^{2} h_{1}=\frac{1}{3} \pi \times r_{2}^{2} h_{2}... View Article
The natural number which is a multiple of every number is
The natural number which is a multiple of every number is 1.
Which term of 37, 32, 27, … is – 103?
The nth term of the given sequence is \(\begin{array}{l}\mathrm{t}_{\mathrm{n}}=-103 \\ \mathrm{a}=37, \mathrm{~d}=32-37=-5 \\ -103=37+(\mathrm{n}-1)(-5) \\ =37-5 \mathrm{n}+5 \\ 5 \mathrm{n}=145... View Article
A man earns Rs. 25440 per month. He spends 1 / 4 of it on house rent, 3 / 8 on food and clothes, 1 / 10 on insurance and 1 / 5 on other items. The rest he saves. How much does he save each month?
He earns Rs. 25440 per month. \(\begin{array}{l}\text { Rent }=\frac{1}{4} \times 25440=6360 \Rightarrow \frac{6360}{25440} \times 360=90^{\circ}\\ \text { Food and... View Article
nPr and nCr are equal when
They are equal when \(\begin{array}{l}^{\mathrm{n}} \mathrm{P}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \\ \frac{\mathrm{n} !}{(\mathrm{n}-\mathrm{r}) !}=\frac{\mathrm{n} !}{(\mathrm{n}-\mathrm{r}) ! \cdot \mathrm{r} !} \\ \mathrm{r} !=1... View Article
The equation of plane passing through a point A (2, – 1, 3) and parallel to the vectors a = (3, 0, – 1) and b = (- 3, 2, 2) is:
Let the direction ratios of the normal of the required plane be a, b, c. The plane is parallel to... View Article
64000 is obviously not a perfect square. Give reason.
64000 is not a perfect square as the number of zeroes are odd.
An average daily wage of 10 workers in a factory ‘A’ is Rs. 30 and an average daily wage of 20 workers in a factory ‘B’ is Rs. 15. Find the average daily wages of all workers of both the factories.
From the question, \(\begin{array}{l}n_{1}=10, \bar{x}_{1}=30, n_{2}=20, \bar{x}_{2}=15 \\ \Rightarrow \bar{x}_{12}=\frac{n_{1} \bar{x}_{1}+n_{2} \bar{x}_{2}}{n_{1}+n_{2}}=\frac{10 \times 30+20 \times 15}{10+20} \\ =\frac{600}{30}\\=20\end{array} \)
