An edge of a variable cube is increasing at the rate
Question: An edge of a variable cube is increasing at the rate of $3 \mathrm{~cm}$ per second. How fast is the volume of the cube increasing when the edge is $10 \mathrm{~cm}$ long? Solution: Let $x$ be the side and $V$ be the volume of the cube at any time $t$. Then, $V=x^{3}$ $\Rightarrow \frac{d V}{d t}=3 x^{2} \frac{d x}{d t}$ $\Rightarrow \frac{d V}{d t}=3 \times(10)^{2} \times 3$ $\left[\because x=10 \mathrm{~cm}\right.$ and $\left.\frac{d x}{d t}=3 \mathrm{~cm} / \mathrm{sec}\right]$ $\Ri...
Read More →Prove that
Question: If $\left(p^{2}+q^{2}\right),(p q+q r),\left(q^{2}+r^{2}\right)$ are in GP then prove that $p, q, r$ are in GP Solution: To prove: p, q, r are in GP Given: $\left(p^{2}+q^{2}\right),(p q+q r),\left(q^{2}+r^{2}\right)$ are in GP Formula used: When $a, b, c$ are in GP, $b^{2}=a c$ Proof: When $\left(p^{2}+q^{2}\right),(p q+q r),\left(q^{2}+r^{2}\right)$ are in GP, $(p q+q r)^{2}=\left(p^{2}+q^{2}\right)\left(q^{2}+r^{2}\right)$ $p^{2} q^{2}+2 p q^{2} r+q^{2} r^{2}=p^{2} q^{2}+p^{2} r^{2}...
Read More →The side of a square sheet is increasing at the rate
Question: The side of a square sheet is increasing at the rate of $4 \mathrm{~cm}$ per minute. At what rate is the area increasing when the side is $8 \mathrm{~cm}$ long? Solution: Given : $A=x^{2}$ and $\frac{d x}{d t}=4 \mathrm{~cm} / \mathrm{min}$ Let $x$ be the side of the square and $A$ be its area at any time $t .$ Then, $A=x^{2}$ $\Rightarrow \frac{d A}{d t}=2 x \frac{d x}{d t}$ $\Rightarrow \frac{d A}{d t}=2 \times 8 \times 4$ $\left[\because x=8 \mathrm{~cm}\right.$ and $\left.\frac{d x...
Read More →The y-coordinate of any point lying
Question: The y-coordinate of any point lying on the X-axis will be zero. Solution: True The distance of the points which lie on X-axis, will be zero from the X-axis, i.e. y-coordinate is zero for the points lying on X-axis....
Read More →If a, b, c, d are in GP, then prove that
Question: If a, b, c, d are in GP, then prove that $\frac{1}{\left(a^{2}+b^{2}\right)}, \frac{1}{\left(b^{2}+c^{2}\right)}, \frac{1}{\left(c^{2}+d^{2}\right)}$ are in GP Solution: To prove: $\frac{1}{\left(a^{2}+b^{2}\right)}, \frac{1}{\left(b^{2}+c^{2}\right)}, \frac{1}{\left(c^{2}+d^{2}\right)}$ are in GP. Given: $a, b, c, d$ are in GP Proof: When $a, b, c, d$ are in GP then $\Rightarrow \frac{b}{a}=\frac{c}{b}=\frac{d}{c}$ From the above, we can have the following conclusion $\Rightarrow \mat...
Read More →The points (3,5) and (5,3)
Question: The points (3,5) and (5,3) represent the same point. Solution: False Two ordered pairs are equal, if they have same numbers at corresponding slot, i.e. x-coordinates are equal and Y-coordinates are equal. Hence, (3,5) and (5,3) are different points....
Read More →The coordinates of the origin
Question: The coordinates of the origin are (0,0). Solution: True Origin is the point, where two axes meet and its coordinates are (0,0)....
Read More →In the point (2,3),
Question: In the point (2,3), 3 denotes the y-coordinate. Solution: True In the ordered pair (2,3), the second number is called the y-coordinate or ordinate of the number. Hence, 3 denotes the y-coordinate of the point (2,3)....
Read More →The ordinate of a point is
Question: The ordinate of a point is its distance from the Y-axis. Solution: False The ordinate of a point is nothing but ^coordinate of the point and the y-coordinate denotes the distance of a point from X-axis....
Read More →The distance of the point (3,5)
Question: The distance of the point (3,5) from the Y-axis is 5. Solution: False We know that the x-coordinate of a point represents the distance of the point from Y-axis. Here x-coordinate is 3, so the distance of the point (3,5) from the Y-axis is 3....
Read More →The distance of any point from
Question: The distance of any point from the X-axis is called the x-coordinate. Solution: False The distance of any point from the X-axis is called the y-coordinate....
Read More →A line graph can also be
Question: A line graph can also be a whole unbroken line. Solution: True A fine graph, which represents the variation of a quantity with respect to the other, may be an unbroken line....
Read More →If a, b, c, d are in GP, prove that
Question: If $a, b, c, d$ are in GP, prove that $\left(a^{2}-b^{2}\right),\left(b^{2}-c^{2}\right),\left(c^{2}-d^{2}\right)$ are in GP. Solution: To prove: $\left(a^{2}-b^{2}\right),\left(b^{2}-c^{2}\right),\left(c^{2}-d^{2}\right)$ are in GP. Given: $a, b, c$ are in GP Formula used: When $a, b, c$ are in $G P, b^{2}=a c$ Proof: When $a, b, c, d$ are in GP then $\Rightarrow \frac{b}{a}=\frac{c}{b}=\frac{d}{c}$ From the above, we can have the following conclusion $\Rightarrow \mathrm{bc}=\mathrm{...
Read More →For fixing a point on the graph
Question: For fixing a point on the graph sheet we need two coordinates. Solution: True To plot a point on the graph sheet we require two numbers, known as coordinates viz. the x-coordinate and the y-coordinate....
Read More →The point where the two axes
Question: The point where the two axes intersect is called the________. Solution: origin The X-axis and y-axis intersect each other at the point representing the origin. Coordinates of origin are (0,0)....
Read More →In the coordinates of a point,
Question: In the coordinates of a point, the second number denotes the________. Solution: y-coordinate y ordinate As we have discussed in the above question, that the second number denotes the . y-coordinate, also known as the ordinate....
Read More →A point has 5 as its x-coordinate
Question: A point has 5 as its x-coordinate and 4 as its y-coordinate. Then, the coordinates of the point are given by________. Solution: (5,4) To denote a point in 2-D, we use two numbers viz. the x-coordinate, the y-coordinate. In the ordered pair, the x-coordinate is written in the first slot and the y-coordinate in the second slot separated by comma. Hence, the required point is (5,4)....
Read More →If a, b, c are in GP, prove that
Question: If $a, b, c$ are in GP, prove that $\left(a^{2}+b^{2}\right),(a b+b c),\left(b^{2}+c^{2}\right)$ are in GP. Solution: To prove: $\left(a^{2}+b^{2}\right),(a b+b c),\left(b^{2}+c^{2}\right)$ are in GP Given: $a, b, c$ are in GP Formula used: When $a, b, c$ are in GP, $b^{2}=a c$ Proof: When a,b,c are in GP, $b^{2}=a c \ldots(i)$ Considering $\left(a^{2}+b^{2}\right),(a b+b c),\left(b^{2}+c^{2}\right)$ $(a b+b c)^{2}=\left(a^{2} b^{2}+2 a b^{2} c+b^{2} c^{2}\right)$ $=\left(a^{2} b^{2}+a...
Read More →In the point (4,7),
Question: In the point (4,7), 4 denotes the________. Solution: x-coordinate (abscissa) First number (coordinate) of the ordered pair is called as the x-coordinate or abscissa. Hence, in the point (4,7),4 denotes the x-coordinate....
Read More →The y-coordinate of the point
Question: The y-coordinate of the point (2,4) is________. Solution: 4 In the ordered pair (2,4), i.e. coordinates of a point, the second number is called as the y-coordinate of the point. Hence, the y-coordinate of (2,4) is 4...
Read More →The x-coordinate of any point
Question: The x-coordinate of any point lying on the y-axis will be________. Solution: zero Since, the x-coordinate represents the distance of the point from Y-axis is zero, therefore the points lying on the Y-axis have x-coordinate as zero....
Read More →For the point (5,2),
Question: For the point (5,2), the distance from the X-axis is________units. Solution: 2 We know that, the y-coordinate represents the distance of the point from the X-axis. ;Hence, the point (5,2) is at a distance of 2 units from the X-axis....
Read More →If a, b, c are in GP, prove that
Question: If $a, b, c$ are in GP, prove that $a^{3}, b^{3}, c^{3}$ are in GP Solution: To prove: $a^{3}, b^{3}, c^{3}$ are in GP Given: $a, b, c$ are in GP Proof: As a, b, c are in GP $\Rightarrow b^{2}=a c$ Cubing both sides $\Rightarrow\left(b^{2}\right)^{3}=(a c)^{3}$ $\Rightarrow b^{6}=a^{3} c^{3}$ $\Rightarrow \frac{b^{3}}{a^{3}}=\frac{c^{3}}{b^{3}}=$ common ratio $=r$ From the above equation, we can say that $a^{3}, b^{3}, c^{3}$ are in GP...
Read More →All points with y-coordinate
Question: All points with y-coordinate as zero lie on the________. Solution: X-axis Since, y-coordinate is zero, i.e. the distance of the point from X-axis is zero. Hence, the points lies on the X-axis....
Read More →The distance of any point from
Question: The distance of any point from the y-axis is the________coordinate. Solution: x x-coordinate of a point is the distance of any point from the Y-axis....
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