If the solve the problem
Question: If $y=\left|\log _{e} x\right|$, find $\frac{d^{2} y}{d x^{2}}$ Solution: Given: $y=\left|\log _{e} x\right| \forall x0$ $y=\log _{e} x$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\mathrm{x}}=\mathrm{x}^{-1}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=(-1) \mathrm{x}^{-2}$...
Read More →The process of fixing a point
Question: The process of fixing a point with the help of the coordinates is known as________of the point. Solution: plotting To locate the exact position of a point we need two coordinates viz, the x-coordinate and the y-coordinate and this process of finding the position or representing the numbers on a graph sheet with the help of the coordinates is known as plotting of the point....
Read More →The horizontal and vertical lines
Question: The horizontal and vertical lines in a graph are usually called________and Solution: X-axis, Y-axis To draw a graph we need pairs of points known as coordinates and to plot a point we require two mutually perpendicular axes, also known as the X- axis (horizontal line) and the Y-axis (vertical line)....
Read More →If the solve the problem
Question: If $y=\left|x-x^{2}\right|$, then find $\frac{d^{2} y}{d x^{2}}$ Solution: Given: $y=\left|x-x^{2}\right|$ $y=\left\{\begin{array}{l}x-x^{2} ; x \geq 0 \\ x^{2}-x ; x \leq 0\end{array}\right.$ $\frac{d y}{d x}=\left\{\begin{array}{c}1-2 x ; x \geq 0 \\ 2 x-1 ; x \leq 0\end{array}\right.$ $\frac{d^{2} y}{d x^{2}}=\left\{\begin{array}{c}-2 ; x \geq 0 \\ 2 ; x \leq 0\end{array}\right.$...
Read More →If a, b, c are in GP, prove that
Question: If $a, b, c$ are in GP, prove that $a^{2}, b^{2}, c^{2}$ are in GP. Solution: To prove: $a^{2}, b^{2}, c^{2}$ are in GP Given: $a, b, c$ are in GP Proof: As a, b, c are in GP $\Rightarrow \mathrm{b}^{2}=\mathrm{ac} \ldots$ (i) Considering $b^{2}, c^{2}$ $\frac{c^{2}}{b^{2}}=$ common ratio $=r$ $\Rightarrow \frac{c^{2}}{a c}[$ From eqn. (i)] $\Rightarrow \frac{c}{a}=r$ Considering $a^{2}, b^{2}$ $\frac{b^{2}}{a^{2}}=$ common ratio $=r$ $\Rightarrow \frac{\mathrm{ac}}{\mathrm{a}^{2}}$ [F...
Read More →A point in which the x-coordinate
Question: A point in which the x-coordinate is zero and y-coordinate is non- zero will lie on the________. Solution: Y-axis Since, the x-coordinate is zero, i.e. the distance of the point from Y-axis is 0, the point lies on the Y-axis at a certain distance from the origin, which is given by y-coordinate....
Read More →We need________coordinates for representing
Question: We need________coordinates for representing a point on the graph sheet. Solution: pair of (or two) To represent the position of a point on the graph sheet we require two measures namely x-coordinate and y-coordinate....
Read More →If the solve the problem
Question: If $y=x+e^{x}$, find $\frac{d^{2} x}{d y^{2}}$ Solution: Given: $y=x+e^{x}$ $\frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{d}^{2} \mathrm{y}}=\frac{1}{\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}}$ $\frac{d y}{d x}=1+e^{x}$ $\frac{d^{2} y}{d x^{2}}=e^{x}$ $\frac{d^{2} x}{d^{2} y}=\frac{1}{e^{x}}$ $=e^{-x}$...
Read More →The relation between dependent
Question: The relation between dependent and independent variables is shown through a________. Solution: graph . Various types of graph depicts the relation between two variables, one of them is independent and the other is dependent. A graph shows how a change in the independent variable changes the dependent variable....
Read More →________.displays data that changes
Question: ________.displays data that changes continuously over periods of time. Solution: Line graph We have already discussed in the previous question that a line graph displays data that changes continuously over periods of time....
Read More →If a, b, c are in GP, prove that
Question: If a, b, c are in GP, prove that $\frac{1}{(a+b)}, \frac{1}{(2 b)}, \frac{1}{(b+c)}$ are in AP. Solution: To prove: $\frac{1}{(a+b)}, \frac{1}{(2 b)}, \frac{1}{(b+c)}$ are in AP Given: $a, b, c$ are in GP Formula used: When $a, b, c$ are in GP, $b^{2}=a c$ When $a, b, c$ are in $G P, b^{2}=a c$ Taking $\frac{1}{(a+b)}$ and $\frac{1}{(b+c)}$ $\frac{1}{(a+b)}+\frac{1}{(b+c)}$ $\Rightarrow \frac{b+c+a+b}{(a+b)(b+c)}$ $\Rightarrow \frac{a+c+2 b}{a b+a c+b^{2}+b c}$ $\Rightarrow \frac{a+c+2...
Read More →If the solve the problem
Question: If $y=1-x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !} \ldots .$ to $\infty$, then write $\frac{d^{2} y}{d x^{2}}$ in terms of $y$ Solution: Given: $y=1-\frac{x}{1 !}+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\cdots \infty$ $\frac{d y}{d x}=0-1+\frac{2 x}{2 !}-\frac{3 x^{2}}{3 !}-\frac{4 x^{3}}{4 !}+\cdots \infty$ $\frac{d^{2} y}{d x^{2}}=0-0+1-\frac{2 x}{2 !}+\frac{3 x^{2}}{3 !}-\frac{4 x^{3}}{4 !}+\cdots \infty$ $=1-\frac{\mathrm{x}}{1 !}+\frac{\mathrm{x}^{2}}{2 !}-\frac{\mathrm{x...
Read More →The coordinates of a point at a distance
Question: The coordinates of a point at a distance of 3 units from the X-axis and 6 units from the y-axis are (a) (0,3) (b) (6,0) (c) (3,6) (d) (6,3) Solution: (d) We know that, the x-coordinate is the distance of the point from /-axis and the y-coordinate is the distance of the point from X-axis. Hence, the coordinates of the required point are (6,3)....
Read More →If the solve the problem
Question: If $x=f(t)$ and $y=g(t)$, then write the value of $\frac{d^{2} y}{d x^{2}}$ Solution: Given: $x=f(t)$ and $y=g(t)$ $\frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{f}^{\prime}(\mathrm{t}) ; \frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{g}^{\prime}(\mathrm{t})$ $\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{g^{\prime}(t)}{f^{\prime}(t)}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}{\frac{\mathrm{dx...
Read More →A point which lies
Question: A point which lies on both the axes is (a) (0,0) (b) (0,1) (c)(1,0) (d) (1,1) Solution: (a) We know that, the axes are two mutually perpendicular lines intersecting each other at the point (0,0) also known as the origin. Hence, the point which lies on both the axes is (0,0)....
Read More →If the solve the problem
Question: If $x=2 a t, y=a t^{2}$, where $a$ is a constant, then find $\frac{d^{2} y}{d x^{2}}$ at $x=\frac{1}{2}$. Solution: Given: $x=2 a t, y=a t^{2}$ $\frac{d x}{d t}=2 a ; \frac{d y}{d t}=2 a t$ $\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}$ $=\mathrm{t}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}{\frac{\mathrm{dx}}{\mathrm{dt}}}$ $=\frac{1}{2 a}$...
Read More →If the solve the problem
Question: If $x=t^{2}$ and $y=t^{3}$, where $a$ is a constant, then find $\frac{d^{2} y}{d x^{2}}$ at $x=\frac{1}{2}$ Solution: Given: $x=t^{2} ; y=t^{3}$ $\frac{\mathrm{dy}}{\mathrm{dt}}=3 \mathrm{t}^{2} ; \frac{\mathrm{dx}}{\mathrm{dt}}=2 \mathrm{t}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}=\frac{3 \mathrm{t}}{2}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{dy}}{\mathr...
Read More →If the solve the problem
Question: If $x=a \cos n t-b \sin n t$ and $\frac{d^{2} y}{d t^{2}}=\lambda x$, then find the value of $\lambda$. Solution: Given: $y=a \cos n t-b \sin n t$ $\frac{d y}{d t}=-a n \sin n t-b n \cos n t$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dt}^{2}}=-\mathrm{an}^{2} \cos \mathrm{n} \mathrm{t}+\mathrm{bn}^{2} \sin \mathrm{n} \mathrm{t}=\lambda \mathrm{y}$ $\lambda y=-n^{2}(a \cos n t-b \sin n t)$ $\lambda y=-n^{2} y$ $\lambda=-n^{2}$...
Read More →If the solve the problem
Question: If $y=a x^{n+1}+b x^{-n}$ and $x^{2} \frac{d^{2} y}{d x^{2}}=\lambda y$, then write the value of $\lambda$ Solution: Given: $y=a x^{n+1}+b x^{-n}$ $\frac{d y}{d x}=(n+1) a x^{n}+(-n) b x^{-n-1}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{n}(\mathrm{n}+1) \mathrm{ax}^{\mathrm{n}-1}+(-\mathrm{n})(-\mathrm{n}-1) \mathrm{bx}^{-\mathrm{n}-2}$ $x^{2} \frac{d^{2} y}{d x^{2}}=x^{2}\left\{n(n+1) a x^{n-1}+(-n)(-n-1) b x^{-n-2}\right\}=\lambda y$ $\lambda y=n(n+1) a x^{n-1+2}+n(n...
Read More →If $y=a x^{n+1}+b x^{-n}$ and $x^{2} rac{d^{2} y}{d x^{2}}=lambda y$, then write the value of $lambda$
Question: If $y=a x^{n+1}+b x^{-n}$ and $x^{2} \frac{d^{2} y}{d x^{2}}=\lambda y$, then write the value of $\lambda$ Solution: Given: $y=a x^{n+1}+b x^{-n}$ $\frac{d y}{d x}=(n+1) a x^{n}+(-n) b x^{-n-1}$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{n}(\mathrm{n}+1) \mathrm{ax}^{\mathrm{n}-1}+(-\mathrm{n})(-\mathrm{n}-1) \mathrm{bx}^{-\mathrm{n}-2}$ $x^{2} \frac{d^{2} y}{d x^{2}}=x^{2}\left\{n(n+1) a x^{n-1}+(-n)(-n-1) b x^{-n-2}\right\}=\lambda y$ $\lambda y=n(n+1) a x^{n-1+2}+n(n...
Read More →If a, b, c, d are in GP, prove that
Question: If a, b, c, d are in GP, prove that (i) $(b+c)(b+d)=(c+a)(c+a)$ (ii) $\frac{a b-c d}{b^{2}-c^{2}}=\frac{a+c}{b}$ (iii) $(a+b+c+d)^{2}=(a+b)^{2}+2(b+c)^{2}+(c+d)^{2}$ Solution: (i) $(b+c)(b+d)=(c+a)(c+a)$ To prove: $(b+c)(b+d)=(c+a)(c+a)$ 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 $\Rightarrow \mathrm{bc}=\mathrm{ad} \ldots$ (i) $\Rightarrow \mathrm{b}^{2}=\m...
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $y^{2}=a x^{2}+b x+c$, then $y^{3} \frac{d^{2} y}{d x^{2}}$ is A. a constant B. a function of $x$ only C. a function of $y$ only D. a function of $x$ and $y$ Solution: Given: $y^{2}=a x^{2}+b x+c$ $\left.y=\sqrt{(} a x^{2}+b x+c\right)$ $\frac{d y}{d x}=\frac{1}{2 \sqrt{\left(a x^{2}+b x+c\right)}} \times(2 a x+b)$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ $=\frac{1}{2}\left\{\frac{\left(2 a \times \sqrt{a x^{2}+b x+c}\right)...
Read More →The point (3,4) is at a distance of
Question: The point (3,4) is at a distance of (a) 3 from both the axes (b) 4 from both the axes (c) 4 from the X-axis and 3 from Y-axis (d) 3 from X-axis and from Y-axis Solution: (c) We know that, the x-coordinate is the distance of the point from Y-axis and that of y-coordinate is the distance from X-axis. . Hence, the point (3,4) is at a distance of 4 from the X-axis and 3 from Y-axis....
Read More →Write the correct alternative in the following:
Question: Write the correct alternative in the following: If $x y-\log _{e} y=1$ satisfies the equation $x\left(y y_{2}+y_{1}^{2}\right)-y_{2}+\lambda y y_{1}=0$, then $\lambda=$ A. $-3$ B. 1 C. 3 D. none of these Solution: Given: $x y-\log _{e} y=1$ $x y=\log _{e} y+1$ Differentiate w.r.t. ' $x$ ' on both sides; $\mathrm{y}+\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}}$ $\frac{d y}{d x}\left(\frac{1}{y}-x\right)=y$ $\frac{\mathrm{dy}}{\mathrm{dx...
Read More →A graph that displays data
Question: A graph that displays data that changes continuously over periods of timeis 451 (a) bar graph (b) pie chart (c) histogram (d) line graph Solution: (d) Line graph is an important way to represent and compare the data which varies continuously. A line graph displays the relation between two varying quantities. In a line graph, we connect all the points by a line segment while in bar graph and histogram, we use rectangles of uniform width....
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