Three numbers are in the ratio of 4:5:6. If the sum of the largest and smallest equals the sum of the third and 55, find the numbers.
Let the numbers be 4x, 5x and 6x. 6x + 4x = 5x + 55 10x – 5x = 55 ... View Article
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A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained.
The volume of the cone is obtained as follows. \(\begin{array}{l}\text { Solid so obtained is a cone with } \mathbf{r}=\mathbf{5}... View Article
Solve the equations using the cross multiplication method: 3x + 2y = 10 and 4x − 2y = 4.
The given equations are \(\begin{array}{l}3 x+2 y=10—(1)\\ 4 x-2 y=4–(2)\\ \frac{\mathbf{x}}{\left(\mathbf{b}_{1} \mathbf{c}_{2}-\mathbf{b}_{2} \mathbf{c}_{1}\right)}=\frac{\mathbf{y}}{\left(\mathbf{c}_{1} \mathbf{a}_{2}-\mathbf{a}_{1} \mathbf{c}_{2}\right)}=\frac{-\mathbf{1}}{\left(\mathbf{a}_{1} \mathbf{b}_{2}-\mathbf{a}_{2} \mathbf{b}_{1}\right)}\\ \text { So,... View Article
If the radius of one circle is twice the radius of another circle whose centres are (2, 0) and (1, 2) respectively are cutting orthogonally, then the radius of the second circle is
If angle between circle is 90, then the circles are cutting orthogonally. By applying Pythagoras theorem \(\begin{array}{l}r_{1}^{2}+r_{2}^{2}=\left(C_{1} C_{2}\right)^{2}\\ r_{2}=2 r_{1}\\... View Article
Solve the following equations using matrix inversion method. 2x − 3y + 6 = 0 and 6x + y + 8 = 0.
The given equations are \(\begin{array}{l}2 \mathrm{x}-3 \mathrm{y}+6=0 \Longrightarrow 2 \mathrm{x}-3 \mathrm{y}=-6\\ 6 \mathrm{x}+\mathrm{y}+8=0 \Longrightarrow 6 \mathrm{x}+\mathrm{y}=-8\\ \text { The corresponding... View Article
Show that the local maximum value of x + (1 / x) is less than the local minimum value.
\(\begin{array}{l}\text { Let } y=x+\frac{1}{x} \Rightarrow \frac{d y}{d x}=1-\frac{1}{x^{2}} \\ \frac{d y}{d x}=0 \Rightarrow x^{2}=1 \Rightarrow x=\pm 1 \\ \frac{d^{2}... View Article
From a point in the interior of an equilateral triangle perpendiculars are drawn on the
From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides.
Which number would replace the question mark in the series? 7, 12, 19, ?, 39
7 + 5 = 12 12 + 7 = 19 19 + 9 = 28 28 + 11 = 39... View Article
6, 13, 25, 51, 101, ?
The pattern is * 2 + 1, * 2 – 1, * 2 + 1, * 2 – 1 ,…..... View Article
The instrument used to measure or construct angle is
The instrument used to measure or construct angle is a protractor.
If a number is divisible by 2 and 3, then it satisfies the divisibility rule of
If the number is divisible by 2 and 3, then it satisfies the divisibility rule of 6. Example: 306 is... View Article
Divide the sum of 65 / 12 and 12 / 7 by their difference.
The sum is = (65 / 12) + (12 / 7) The difference is = (65 / 12) – (12... View Article
Find the general term in the expansion of (1 – x)-3.
The r+1th term is given by \(\begin{array}{l}\mathbf{T}_{\mathrm{r}+1}=\frac{\mathbf{n}(\mathbf{n}-\mathbf{1})(\mathbf{n}-\mathbf{2}) \ldots(\mathbf{n}-\mathbf{r}+\mathbf{1})}{\mathbf{r} !} \mathbf{x}^{\mathbf{r}}\\ Now, \mathbf{T}_{\mathrm{r}+1}\\ =\frac{-3(-3-1)(-3-2)(-3-3) \ldots \ldots(-3-\mathbf{r}+1)}{\mathrm{r} !}(-\mathbf{x})^{\mathrm{r}}\\ =\frac{-3(-4)(-5)(-6) \ldots \ldots \cdot(-2-r)}{r... View Article
Find the eccentricity of the ellipse if the length of the latus rectum = half of the major axis.
Here \(\begin{array}{l}\begin{array}{l} \frac{2 b^{2}}{a}=a \Rightarrow 2 b^{2}=a^{2} \\ e=\sqrt{\left(1-\frac{b^{2}}{a^{2}}\right)}=\sqrt{\left(1-\frac{1}{2}\right)}=\frac{1}{\sqrt{2}} \end{array}\end{array} \)
Fill in the blank with >, < or = sign (- 25) - (- 42) ________ (- 42) - (- 25).
L.H.S = − 25 + 42 = 17 R.H.S = − 42 + 25 = − 17 L.H.S is greater... View Article
Find three positive numbers in the ratio 2 : 3 : 5, the sum of whose squares is 950.
The ratio of three positive numbers = 2 : 3 : 5 Sum of their squares = 950 Assume the... View Article
In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?
H = set of people speaking Hindi E = set of people speaking English n (H ∪ E) = 400;... View Article
Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
Let ABCD be the parallelogram with ∠A = ∠B. The sum of adjacent angles of a parallelogram is 180. ⇒ ... View Article
An exterior angle of a triangle measures 110° and its interior opposite angles are in the ratio 2: 3. Find the angles of the triangle.
Let the given interior opposite angles are (2x) and (3x). An exterior angle of a triangle is equal to the... View Article
Express as cm using decimals: 9 cm 8 mm.
1 mm = (1 / 10) cm 9 cm 8 mm = 9 + (1 / 10) x 8 =... View Article
