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Question: Let $f(x)=|x|+|x-1|$, then (a) $f(x)$ is continuous at $x=0$, as well as at $x=1$ (b) $f(x)$ is continuous at $x=0$, but not at $x=1$ (c) $f(x)$ is continuous at $x=1$, but not at $x=0$ (d) none of these Solution: (a) $f(x)$ is continuous at $x=0$, as well as at $x=1$ Since modulus function is everywhere continuous, $|x|$ and $|x-1|$ are also everywhere continuous. Also, It is known that if $f$ and $g$ are continuous functions, then $f+g$ will also be continuous. Thus, $|x|+|x-1|$ is e...
Read More →The fourth vertex D of a parallelogram ABCD
Question: The fourth vertex D of a parallelogram ABCD whose three vertices are A(- 2, 3), B(6, 7) and C(8, 3) is (a) (0,1) (b) (0,-1) (c) (-1,0) (d) (1,0) Solution: (b)Let the fourth vertex of parallelogram, D(x4 ,y4) and L, M be the middle points of AC and BD, respectively, Then, $L=\left(\frac{-2+8}{2}, \frac{3+3}{2}\right)=(3,3)$ and$M=\left(\frac{6+x_{4}}{2}, \frac{7+y_{4}}{2}\right)$ Since, ABCD is a parallelogram, therefore diagonals AC and BD will bisect each other. Hence, L and M are the...
Read More →The following table gives the information regarding length of a side of a square and its area:
Question: The following table gives the information regarding length of a side of a square and its area: Draw a graph to illustrate this information. Solution: Here, length of a sideis an independent variable and area of square is a dependent variable. So, we take length of a side on the x-axis and area of square on the y-axis. Let us choose the following scale: On x-axis: 2 cm = 1 cm Ony-axis: 1 cm = 2 cm2 Now we plot (1,1), (2,4), (3,9), (4,16), (5,25). These points are joined to get the graph...
Read More →If the function
Question: If the function $f(x)=\left\{\begin{array}{cl}(\cos x)^{1 / x}, x \neq 0 \\ k , x=0\end{array}\right.$ is continuous at $x=0$, then the value of $k$ is (a) 0 (b) 1 (c) $-1$ (d) $e$. Solution: (b) Given: $f(x)=\left\{\begin{array}{l}(\cos x)^{\frac{1}{x}} \\ k, x=0\end{array}, x \neq 0\right.$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0}(\cos x)^{\frac{1}{x}}=k$ If $\lim _{x \rightarrow a} f(x)=1$ and $\lim _{x \rightarr...
Read More →The following table gives the information regarding the number of
Question: The following table gives the information regarding the number of persons employed to a piece of work and time taken to complete the work: Plot a graph of this information. Solution: Here, number of persons is an independent variable and time taken is a dependent variable. So, we take the number of persons on thex-axis and time taken on they-axis. Let us choose the following scale: On x-axis: 2 cm = 2 persons Ony-axis: 2 cm = 2 days Now, let us plot (2, 12), (4, 6), (6, 4), (8, 3). The...
Read More →The point which lies on the perpendicular
Question: The point which lies on the perpendicular bisector of the line segment joining the points A(-2, 5) and B(2, 5) is (a) (0,0) (b) (0, 2) (c) (2, 0) (d)(-2,0) Solution: (a)We know that, the perpendicular bisector of the any line segment divides the^jjpe segment into two equal parts i.e., the perpendicular bisector of the line segment always passes through the mid-point of the line segment. Mid-point of the line segment joining the points A (-2, -5) and S(2, 5) $=\left(\frac{-2+2}{2}, \fra...
Read More →The following table shows the amount of rice grown by a farmer in different years:
Question: The following table shows the amount of rice grown by a farmer in different years: Plot a graph to illustrate this information. Solution: Here, year is an independent variable and quantity of rice grown is a dependent variable. So, we take years on thex-axis and quantity of rice grown on they-axis. Let us choose the following scale: Onx-axis: 2 cm = 1 year Ony-axis: 1 cm = 20 quintals Let us assume that the origin O represents the coordinates (1999, 160). Now, let us plot (2000, 200), ...
Read More →If A and B are two sets such that
Question: If $A$ and $B$ are two sets such that $n(A)=37, n(B)=26$ and $n(A \cup B)=51$, find $\mathbf{n}(\mathbf{A} \cap \mathbf{B})$ Solution: Given: $n(A)=37$ $n(B)=26$ $n(A \cup B)=51$ To Find: $n(A \cap B)$ We know that, $|A \cup B|=|A|+|B|-|A \cap B|$ (where $A$ and $B$ are two finite sets) Therefore, $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $51=37+26-n(A \cap B)$ $n(A \cap B)=63-51=12$ Therefore, $n(A \cap B)=12$...
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Question: Let $f(x)= \begin{cases}\frac{x-4}{|x-4|}+a, x4 \\ a+b , x=4 \\ \frac{x-4}{|x-4|}+b, x4\end{cases}$ Then, $f(x)$ is continuous at $x=4$ when (a) $a=0, b=0$ (b) $a=1, b=1$ (c) $a=-1, b=1$ (d) $a=1, b=-1$. Solution: (d) $a=1, b=-1$ Given: $f(x)=\left\{\begin{array}{c}\frac{x-4}{|x-4|}+a, \text { if } \mathrm{x}4 \\ a+b, \text { if } \mathrm{x}=4 \\ \frac{x-4}{|x-4|}+b, \text { if } \mathrm{x}4\end{array}\right.$ We have $(\mathrm{LHL}$ at $x=4)=\lim _{x \rightarrow 4^{-}} f(x)=\lim _{h \...
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Question: Let $f(x)= \begin{cases}\frac{x-4}{|x-4|}+a, x4 \\ a+b , x=4 \\ \frac{x-4}{|x-4|}+b, x4\end{cases}$ Then, $f(x)$ is continuous at $x=4$ when (a) $a=0, b=0$ (b) $a=1, b=1$ (c) $a=-1, b=1$ (d) $a=1, b=-1$. Solution: (d) $a=1, b=-1$ Given: $f(x)=\left\{\begin{array}{c}\frac{x-4}{|x-4|}+a, \text { if } \mathrm{x}4 \\ a+b, \text { if } \mathrm{x}=4 \\ \frac{x-4}{|x-4|}+b, \text { if } \mathrm{x}4\end{array}\right.$ We have $(\mathrm{LHL}$ at $x=4)=\lim _{x \rightarrow 4^{-}} f(x)=\lim _{h \...
Read More →The following table shows the number of patients discharged from a hospital with HIV diagnosis in different years:
Question: The following table shows the number of patients discharged from a hospital with HIV diagnosis in different years: Represent this information by a graph. Solution: Here, years is an independent variable and the number of patients is a dependent variable. So, we take years on thex-axis and the number of patients on they-axis. Let us choose the following scale: Onx-axis: 2 cm = 1 year On y-axis: 1 cm = 10 patients Also, let us assume that on thex-axis, origin (O) represents 2001 and on t...
Read More →The point which divides the line segment
Question: The point which divides the line segment joining the points (7, 6) and (3, 4) in ratio 1: 2 internally lies in the (a) I quadrant (b) II quadrant (c) III quadrant (d) IV quadrant Solution: (d)If P(x, y) divides the line segment joining A(x1,y2) and B(x2, y2) internally in the ratio $m: n$, then $x=\frac{m x_{2}+n x_{1}}{m+n}$ and $y=\frac{m y_{2}+n y_{1}}{m+n}$ Given that, $x_{1}=7, y_{1}=-6, x_{2}=3, y_{2}=4, m=1$ and $n=2$ $\therefore$$x=\frac{1(3)+2(7)}{1+2}, y=\frac{1(4)+2(-6)}{1+2...
Read More →The points (- 4, 0), (4, 0) and (0, 3)
Question: The points (- 4, 0), (4, 0) and (0, 3) are the vertices of a (a) right angled triangle (b) isosceles triangle (c) equilateral triangle (d) scalene triangle Solution: (b) Let A(- 4, 0), B(4, 0), C(0, 3) are the given vertices. Now, distance between A (-4, 0) and B (4, 0), $A B=\sqrt{[4-(-4)]^{2}+(0-0)^{2}}$ $\left[\because\right.$ distance between two points $\left(x_{1}, y_{1}\right)$ and $\left.\left(x_{2}, y_{2}\right), d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2...
Read More →Using Venn diagrams, show that
Question: Using Venn diagrams, show that $(A-B), A \cap B)$ and $(B-A)$ are disjoint sets,taking $A=\{2,4,6,8,10,12\}$ and $B=\{3,6,9,12,15,$,$} .$ Solution: A - B is denoted by the yellow region only B - A is denoted by the blue region only AB is denoted by the common region (blue +yellow) There is no intersection between these three regions Hence the three sets are disjoint sets....
Read More →Solve this
Question: The function $f(x)=\left\{\begin{array}{rr}\frac{e^{1 / x}-1}{e^{1 / x}+1}, x \neq 0 \\ 0 , x=0\end{array}\right.$ (a) is continuous at $x=0$ (b) is not continuous at $x=0$ (c) is not continuous at $x=0$, but can be made continuous at $x=0$ (d) none of these Solution: (b) is not continuous atx= 0 Given: $f(x)=\left\{\begin{array}{l}\frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1}, x \neq 0 \\ 0, x=0\end{array}\right.$ We have $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}\left(\frac{...
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Question: The function $f(x)=\left\{\begin{array}{rr}\frac{e^{1 / x}-1}{e^{1 / x}+1}, x \neq 0 \\ 0 , x=0\end{array}\right.$ (a) is continuous at $x=0$ (b) is not continuous at $x=0$ (c) is not continuous at $x=0$, but can be made continuous at $x=0$ (d) none of these Solution: (b) is not continuous atx= 0 Given: $f(x)=\left\{\begin{array}{l}\frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1}, x \neq 0 \\ 0, x=0\end{array}\right.$ We have $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}\left(\frac{...
Read More →The area of a triangle with vertices A(3,0),
Question: The area of a triangle with vertices A(3,0), B(7, 0) and C(8, 4) is (a) 14 (b) 28 (c) 8 (d) 6 Solution: (c)Area of Δ ABC whose Vertices A(x1,y1),B(x2,y2) and C(x3, y3) are given by $\Delta=\left|\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]\right|$ Here, $x_{1}=3, y_{1}=0, x_{2}=7, y_{2}=0, x_{3}=8$ and $y_{3}=4$ $\therefore \quad \Delta=\left|\frac{1}{2}[3(0-4)+7(4-0)+8(0-0)]\right|=\left|\frac{1}{2}(-12+28+0)\right|=...
Read More →Using Venn diagrams, show that
Question: Using Venn diagrams, show that $(A-B), A \cap B)$ and $(B-A)$ are disjoint sets, taking $A=\{2,4,6,8,10,12\}$ and $B=\{3,6,9,12,15,$,$} .$ Solution: A - B is denoted by the yellow region only B - A is denoted by the blue region only AB is denoted by the common region (blue +yellow) There is no intersection between these three regions Hence the three sets are disjoint sets....
Read More →The perimeter of a triangle with vertices (0, 4),
Question: The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is (a) 5 (b) 12 (c)11 (d)7+5 Solution: (b)we Further, adding all the distance of a triangle to get the perimeter of a triangle.We plot the vertices of a triangle i.e., (0, 4), (0,0) and (3,0) on the paper shown as given below Now,perimeter of ΔAOB=Sum of the length of all its sides = d(AO) + d(OB) + d(AB) Distance between the points (x1,y1) and (x2, y2), $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2...
Read More →Using Venn diagrams, show that
Question: Using Venn diagrams, show that $(A-B), A \cap B)$ and $(B-A)$ are disjoint sets, taking $A=\{2,4,6,8,10,12\}$ and $B=\{3,6,9,12,15,$,$} .$ Solution: A - B is denoted by the yellow region only B - A is denoted by the blue region only AB is denoted by the common region (blue +yellow) There is no intersection between these three regions Hence the three sets are disjoint sets....
Read More →When two dice are rolled:
Question: When two dice are rolled: (i) List the outcomes for the event that the total is odd. (ii) Find probability of getting an odd total. (iii) List the outcomes for the event that total is less than 5. (iv) Find the probability of getting a total less than 5? Solution: Possible outcomes when two dice are rolled : $\mathrm{S}=\{(1,1),(1,2),(1,3),(1,4), \cdots,(6,5),(6,6)\}$ Therefore, the number of possible outcomes in the sample space is 36 . (i) The outcomes for the event that the total is...
Read More →If AOBC is a rectangle whose three vertices
Question: If AOBC is a rectangle whose three vertices are A(0, 3), O(0, 0) and B(5, 0), then the length of its diagonal is (a) 5 (b) 3 (c) 34 (d) 4 Solution: Now, length of the diagonal AB = Distance between the points A(0, 3) and B(5, 0). Distance between the points (x,, y,) and (x2, y2), $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$ Here, $x_{1}=0, y_{1}=3$ and $x_{2}=5, y_{2}=0$ $\therefore$ Distance between the points $A(0,3)$ and $B(5,0)$ $A B=\sqrt{(5-0)^{2}+(0-3)^{2...
Read More →Let A = {2, 3, 5, 7, 11, 13}, B = {5, 7, 9, 11, 15} be subsets of U = {2, 3, 5, 7, 9,
Question: Let A = {2, 3, 5, 7, 11, 13}, B = {5, 7, 9, 11, 15} be subsets of U = {2, 3, 5, 7, 9, 11, 13, 15}. Using Venn diagrams, verify that: (i) $\left(A \cup B^{\prime}\right)=\left(A^{\prime} \cap B^{\prime}\right)$ (ii) $(A \cap B)^{\prime}=\left(A^{\prime} \cup B^{\prime}\right)$ Solution: (i) Here blue region denotes set A - B The green region denotes set B A The overlapping region denotes $A \cap B$, and the orange region denotes the universal set U. From the Venn diagram we get $\left(A...
Read More →Solve this
Question: If $f(x)=(x+1)^{\cot x}$ be continuous at $x=0$, then $f(0)$ is equal to (a) 0 (b) $1 / e$ (c) $\mathrm{e}$ (d) none of these Solution: (c) $e$ Suppose $f(x)$ is continuous at $x=0$. Given: $f(x)=(x+1)^{\cot x}$ $\log f(x)=(\cot x)(\log (x+1)) \quad[$ Taking $\log$ on both sides $]$ $\Rightarrow \lim _{x \rightarrow 0} \log f(x)=\lim _{x \rightarrow 0}(\cot x)(\log (x+1))$ $\Rightarrow \lim _{x \rightarrow 0} \log f(x)=\lim _{x \rightarrow 0}\left(\frac{\log (x+1)}{\tan x}\right)$ $\Ri...
Read More →If you have a spinning wheel with 3-green sectors, 1-blue sector and 1-red sector.
Question: If you have a spinning wheel with 3-green sectors, 1-blue sector and 1-red sector. What is the probability of getting a green sector? Is it the maximum? Solution: Number of green sectors in the wheel $=3$ Number of blue sectors in the wheel $=1$ Number of red sectors in the wheel $=1$ Total number of sectors in the wheel $=3+1+1=5$ $\therefore$ Number of possible outcomes $=5$ Let $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ be the events of getting a green, blue and red sector, respectiv...
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