The value of f(0), so that the function
Question: The value of $f(0)$, so that the function $f(x)=\frac{\sqrt{a^{2}-a x+x^{2}}-\sqrt{a^{2}+a x+x^{2}}}{\sqrt{a+x}-\sqrt{a-x}}$ becomes continuous for all $x$, given by (a) $a^{3 / 2}$ (b) $a^{1 / 2}$ (c) $-a^{1 / 2}$ (d) $-a^{3 / 2}$ Solution: (C) $-a^{\frac{1}{2}}$ Given: $f(x)=\frac{\sqrt{a^{2}-a x+x^{2}}-\sqrt{a^{2}+a x+x^{2}}}{\sqrt{a+x}-\sqrt{a-x}}$ $\Rightarrow f(x)=\frac{\left(\sqrt{a^{2}-a x+x^{2}}-\sqrt{a^{2}+a x+x^{2}}\right)\left(\sqrt{a^{2}-a x+x^{2}}+\sqrt{a^{2}+a x+x^{2}}\r...
Read More →The coordinates of the point
Question: The coordinates of the point which is equidistant from the three vertices of the ΔAOB as shown in the figure is (a) $(x, y)$. (b) $(y, x)$ (c) $\left(\frac{x}{2}, \frac{y}{2}\right)$ (d) $\left(\frac{y}{2}, \frac{x}{2}\right)$ Solution: (a)Let the coordinate of the point which is equidistant from the three vertices 0(0, 0), A(0,2y) and B(2x, 0) is P(h,k). Then, PO = PA = PB ⇒ (PO) = (PA)= (PB)2 (i) By distance formula, $\left[\sqrt{(h-0)^{2}+(k-0)^{2}}\right]^{2}=\left[\sqrt{(h-0)^{2}+...
Read More →Decide which of the following statements is true and which is false. Give reasons for your answer.
Question: Decide which of the following statements is true and which is false. Give reasons for your answer. (i) A point whosex-coordinate is zero, will lie on they-axis. (ii) A point whosey-coordinate is zero, will lie onx-axis. (iii) The coordinates of the origin are (0, 0). (iv) Points whosexandycoordinates are equal, lie on a line passing through the origin. Solution: (i) The examples of points having x-coordinate as zero are (0,3), (0,6), (0,9). This can be represented in the following mann...
Read More →Write the coordinates of each of the vertices of each polygon in Fig. 27.9.
Question: Write the coordinates of each of the vertices of each polygon in Fig. 27.9. Solution: From the figure, we have: In polygon OXYZ: O lies on the origin and the coordinates of the origin are (0, 0). So, the coordinates of O are (0, 0). X lies on they-axis. So, thex-coordinate is 0. Hence, the coordinate of X is (0, 2). Also, YX is equal to 2 units and YZ is equal to 2 units. So, the coordinates of vertex Y are (2, 2). Z lies on thex-axis. So, they-coordinate is 0. Hence, the coordinates o...
Read More →Find the coordinates of points P, Q, R and S in Fig. 27.8.
Question: Find the coordinates of pointsP,Q,RandSin Fig. 27.8. Solution: Draw perpendiculars PA, QB, RC and SD from vertices P, Q, R and S on thex-axis. Also ,draw perpendiculars PE, QF, RG and SH on they-axis from these points. PE = 10 units and PA = 70 units Therefore, the coordinates of vertex P are (10, 70). QF = 12 units and QB = 80 units Therefore, the coordinates of vertex Q are (12, 80). RG = 16 units and RC = 100 units Therefore, the coordinates of vertex R are (16, 100). SH = 20 units ...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{ll}m x+1, x \leq \frac{\pi}{2} \\ \sin x+n, x\frac{\pi}{2}\end{array}\right.$ is continuous at $x=\frac{\pi}{2}$, then (a) $m=1, n=0$ (b) $m=\frac{n \pi}{2}+1$ (c) $n=\frac{m \pi}{2}$ (d) $m=n=\frac{\pi}{2}$ Solution: (c) $n=\frac{\mathrm{m} \pi}{2}$ Here, $f\left(\frac{\pi}{2}\right)=\frac{m \pi}{2}+1$ We have $\left(\mathrm{LHL}\right.$ at $\left.x=\frac{\pi}{2}\right)=\lim _{x \rightarrow \frac{\pi}{2}} f(x)=\lim _{h \rightarrow 0} f\left(\frac{\pi}{2}-...
Read More →The perpendicular bisector of the line segment
Question: The perpendicular bisector of the line segment joining the points A(1,5) and 8(4,6) cuts the y-axis at (a) (0,13) (b) (0,-13) (c) (0,12) (d) (13,0) Solution: (a)Firstly, we plot the points of the line segment on the paper and join them. We know that, the perpendicular bisector of the line segment AB bisect the segment AB, i.e., perpendicular bisector of line segment AB passes through the mid-point of AB. $\therefore \quad$ Mid-point of $A B=\left(\frac{1+4}{2}, \frac{5+6}{2}\right)$ $\...
Read More →Find the coordinates of points A, B, C, D in Fig. 27.7.
Question: Find the coordinates of pointsA,B,C,Din Fig. 27.7. Solution: Draw perpendiculars AP, BP, CQ and DR from A, B, C and D on thex-axis. Also, draw perpendiculars AW, BX, CY and DZ on they-axis. From the figure, we have: AW = 1 unit and AP= 1 unit So, the coordinates of vertex A are (1, 1). Similarly, BX=1 unit and BP= 4 units So, the coordinates of vertex B are (1, 4). CY = 4 units and CQ= 6 units So, the coordinates of vertex B are (4, 6). DZ = 5 units and DR= 3 units So, the coordinates ...
Read More →Locate the points:
Question: Locate the points: (i) (1, 1), (1, 2), (1, 3), (1, 4) (ii) (2, 1), (2, 2), (2, 3), (2, 4) (iii) (1, 3), (2, 3), (3, 3), (4, 3) (iv) (1, 4), (2, 4), (3, 4), (4, 4). Solution: (i) In order to plot these points, the given steps are to be followed: Take a point O on a graph paper and draw horizontal and vertical lines OX andOY respectively. Then, let onx-axis and y axis 1 cm represents 1 unit. In order to plot point (1, 1), we start from the origin O and move 2 cm along OX and 1 cm along O...
Read More →Plot the points (2, 8), (7, 8) and (12, 8). Join these points in pairs.
Question: Plot the points (2, 8), (7, 8) and (12, 8). Join these points in pairs. Do they lie on a line? What do you observe? Solution: Take a point O on the graph paper and draw the horizontal and vertical lines OX andOY respectively. Then, let on thex-axis and y axis 1 cm represents 1 unit. In order to plot point (2, 8), we start from the origin O and move 8 cm along OX. The point we arrive at is (2, 8). To plot point (7, 8), we move 7 cm along OX and 8 cm alongOY. The point we arrive at is (7...
Read More →Solve this
Question: If $f(x)=\left\{\begin{aligned} \frac{\sin (a+1) x+\sin x}{x}, x0 \\ c , x=0 \text { is continuous at } x=0, \text { then } \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x \sqrt{x}}, x0 \end{aligned}\right.$ (a) $a=-\frac{3}{2}, b=0, c=\frac{1}{2}$ (b) $a=-\frac{3}{2}, b=1, c=-\frac{1}{2}$ (c) $a=-\frac{3}{2}, b \in R-\{0\}, c=\frac{1}{2}$ (d) none of these Solution: (c) $a=\frac{-3}{2} \quad, b \in R-\{0\}, c=\frac{1}{2}$ The given function can be rewritten as $f(x)= \begin{cases}\frac{\sin (...
Read More →Solve this
Question: If $f(x)=\left\{\begin{aligned} \frac{\sin (a+1) x+\sin x}{x}, x0 \\ c , x=0 \text { is continuous at } x=0, \text { then } \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x \sqrt{x}}, x0 \end{aligned}\right.$ (a) $a=-\frac{3}{2}, b=0, c=\frac{1}{2}$ (b) $a=-\frac{3}{2}, b=1, c=-\frac{1}{2}$ (c) $a=-\frac{3}{2}, b \in R-\{0\}, c=\frac{1}{2}$ (d) none of these Solution: (c) $a=\frac{-3}{2} \quad, b \in R-\{0\}, c=\frac{1}{2}$ The given function can be rewritten as $f(x)= \begin{cases}\frac{\sin (...
Read More →Plot the points (5, 0), (5, 1), (5, 8). Do they lie on a line?
Question: Plot the points (5, 0), (5, 1), (5, 8). Do they lie on a line? What is your observation? Solution: Take a point O on the graph paper and draw horizontal and vertical lines OX and OY respectively. Then, let on thex-axis and y axis 1 cm represents 1 unit. In order to plot point (5, 0), we start from the origin O and move 5 cm along OX. The point we arrive at is point (5,0). To plot point (5, 1), we move 5 cm along OX and 1 cm along OY. The point we arrive at is point (5,1). To plot point...
Read More →Solve this
Question: If $f(x)=\left\{\begin{aligned} \frac{\sin (a+1) x+\sin x}{x}, x0 \\ c , x=0 \text { is continuous at } x=0, \text { then } \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x \sqrt{x}}, x0 \end{aligned}\right.$ (a) $a=-\frac{3}{2}, b=0, c=\frac{1}{2}$ (b) $a=-\frac{3}{2}, b=1, c=-\frac{1}{2}$ (c) $a=-\frac{3}{2}, b \in R-\{0\}, c=\frac{1}{2}$ (d) none of these Solution: (c) $a=\frac{-3}{2} \quad, b \in R-\{0\}, c=\frac{1}{2}$ The given function can be rewritten as $f(x)= \begin{cases}\frac{\sin (...
Read More →If A and B are two sets such that
Question: If $A$ and $B$ are two sets such that $n(A-B)=24, n(B-A)=19$ and $n(A \cap B)=$ 11, find: (i) $n(A)$ (ii) $n(B)$ (iii) $n(A \cup B)$ Solution: Given: $n(A-B)=24, n(B-A)=19$ and $n(A \cap B)=11$ To Find: (i) n(A) We know that, $n(A)=n(A-B)+n(A \cap B)$ $=24+11$ $=35$ Therefore, $n(A)=35 \ldots(1)$ (ii) $n(B)$ We know that, $n(B)=n(B-A)+n(A \cap B)$ $=19+11$ $=30$ Therefore, $n(B)=30 \ldots(2)$ (iii) $n(A \cup B)$ We know that, $n(A \cup B)=n(A)+n(B)-n(A \cap B)\{$ From $(1) \(2) n(A)=35...
Read More →The runs scored by two teams A and B in first 10 overs are given below:
Question: The runs scored by two teamsAandBin first 10 overs are given below: Overs: I II III IV V VI VII VIII IX X Team A: 2 1 8 9 4 5 6 10 6 2 Team B: 5 6 2 10 5 6 3 4 8 10 Draw a graph depicting the data, making the graphs on the same axes in each case in two different ways as a graph and as a bar chart. Solution: Here, over is an independent variable and run is a dependent variable. So, we take overs on x-axis and runs ony-axis. Let us choose the following scale: On x-axis: 1 cm = 1 over Ony...
Read More →The runs scored by two teams A and B in first 10 overs are given below:
Question: The runs scored by two teamsAandBin first 10 overs are given below: Overs: I II III IV V VI VII VIII IX X Team A: 2 1 8 9 4 5 6 10 6 2 Team B: 5 6 2 10 5 6 3 4 8 10 Draw a graph depicting the data, making the graphs on the same axes in each case in two different ways as a graph and as a bar chart. Solution: Here, over is an independent variable and run is a dependent variable. So, we take overs on x-axis and runs ony-axis. Let us choose the following scale: On x-axis: 1 cm = 1 over Ony...
Read More →Prove the following
Question: If $P\left(\frac{1}{2}, 4\right)$ is the mid-point of the line segment joining the points $Q(-6,5)$ and $f(-2,3)$, then the value of a is (a)-4 (b) -12 (c) 12 (d) -6 Solution: (b) Given that, $P\left(\frac{1}{2}, 4\right)$ is the mid-point of the line segment joining the points $Q(-6,5)$ and $R(-2,3)$, which shows in the figure given below $\therefore$ Mid-point of $Q R=P\left(\frac{-6-2}{2}, \frac{5+3}{2}\right)=P(-4,4)$ $\left[\right.$ since, mid-point of line segment having points $...
Read More →The runs scored by a cricket team in first 15 overs are given below:
Question: The runs scored by a cricket team in first 15 overs are given below: Overs: I II III IV V VI VII VIII IX X XI XII XIII XIV XV Runs: 2 1 4 2 6 8 10 21 5 8 3 2 6 8 12 Draw the graph representing the above data in two different ways as a graph and as a bar chart. Solution: Here, over is an independent variable and run is a dependent variable. So, we take overs on the x-axis and runs the ony-axis.Let us choose the following scale: On x-axis: 1 cm = 1 over Ony-axis: 1 cm = 2 runs Now, let u...
Read More →let the function :
Question: Let $f(x)=\left\{\begin{array}{cc}\frac{x^{4}-5 x^{2}+4}{|(x-1)(x-2)|}, x \neq 1,2 \\ 6 , \quad x=1 \\ 12 , \quad x=2\end{array}\right.$. Then, $f(x)$ is continuous on the set (a) $R$ (b) $R-\{1\}$ (c) $R-\{2\}$ (d) $R-\{1,2\}$ Solution: (d) $R-\{1,2\}$ Given: $f(x)=\left\{\begin{array}{c}\frac{x^{4}-5 x^{2}+4}{|(x-1)(x-2)|}, x \neq 1,2 \\ 6, \quad x=1 \\ 12, \quad x=2\end{array}\right.$ Now, $x^{4}-5 x^{2}+4=x^{4}-x^{2}-4 x^{2}+4=x^{2}\left(x^{2}-1\right)-4\left(x^{2}-1\right)=\left(x...
Read More →Draw the velocity-time graph from the following data:
Question: Draw the velocity-time graph from the following data: Time (in hours): 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 Speed (in km/hr): 30 45 60 50 70 50 40 45 Solution: Here, time is an independent variable and speed is a dependent variable. So, we take time on the x-axis and speed on the y-axis. Let us choose the following scale: On x-axis: 2 big division = 1 hour Ony-axis: 1 big division = 10 km/hr Let us assume that on the x-axis, the coordinate of origin (O) is 7:00. So, the coordin...
Read More →Draw the temperature-time graph in each of the following cases:
Question: Draw the temperature-time graph in each of the following cases: (i) Time (in hours): 7:00 9:00 11:00 13:00 15:00 17:00 19:00 21:00 Temperature (F) in: 100 101 104 102 100 99 100 98 (ii) Solution: Here, time is an independent variable and temperature is a dependent variable. So, we take time on the x-axis and temperature on the y-axis. Let us choose the following scale: For point (i): On x-axis: 1 cm = 1 hours Ony-axis: 1 cm = 2F For point (ii): On x-axis: 2 cm = 2 hours On y-axis: 1 c...
Read More →If the point P(2,1) lies on the line segment
Question: If the point P(2,1) lies on the line segment joining points A(4, 2) and 6(8, 4), then (a) $A P=\frac{1}{3} A B$ (b) $A P=P B$ (c) $P B=\frac{1}{3} A B$ (d) $A P=\frac{1}{2} A B$ Solution: (d)Given that, the point P(2,1) lies on the line segment joining the points A(4,2) and 8(8, 4), which shows in the figure below: Now, distance between $A(4,2)$ and $(2,1), A P=\sqrt{(2-4)^{2}+(1-2)^{2}}$ $=\sqrt{(-2)^{2}+(-1)^{2}}=\sqrt{4+1}=\sqrt{5}$ Distance between $A(4,2)$ and $B(8,4)$, $A B=\sqrt...
Read More →If A and B are two sets such that
Question: If $A$ and $B$ are two sets such that $n(A)=24, n(B)=22$ and $n(A \cap B)=8$, find: (i) $n(A \cup B)$ (ii) $n(A-B)$ (iii) $n(B-A)$ Solution: Given: $n(A)=24, n(B)=22$ and $n(A \cap B)=8$ To Find: (i) $n(A \cup B)$ $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $=24+22-8$ $=38$ Therefore, $n(A \cup B)=38$ (ii) $n(A-B)$ We know that, $n(A-B)=n(A)-n(A \cap B)$ $=24-8$ $=16$ Therefore, $n(A-B)=16$ (iii) $n(B-A)$ We know that, $n(B-A)=n(B)-n(A \cap B)$ $=22-8$ $=14$ Therefore, $n(B-A)=14$...
Read More →The following table shows the sales of a commodity during the years 2000 to 2006.
Question: The following table shows the sales of a commodity during the years 2000 to 2006. Draw a graph of this information. Solution: Here, year is an independent variable and sales is a dependent variable. So, we take year on thex-axis and sales on they-axis.Let us choose the following scale: Onx-axis: 2 cm = 1 year Ony-axis: 2 cm = 1 lakh rupees Assume that onx-axis, origin (O) represents 1991. So, the coordinates of O are (1991,0). Now, let us plot (2000, 1.5), (2001, 1.8), (2002, 2.4), (20...
Read More →