Which of the following are sets? Justify your answer.
Question: Which of the following are sets? Justify your answer. The collection of all the questions in this chapter. Solution: As the collection of all questions in this chapter is known and can be counted .i.e. well defined. , this is a set...
Read More →Which of the following are sets? Justify your answer
Question: Which of the following are sets? Justify your answer The collection of good hockey players in India. Solution: As a collection of good hockey players in India may vary from person to person. So, it is not well defined , this is not a set....
Read More →Find the length of the longest rod that can be placed in a room 12 m long,
Question: Find the length of the longest rod that can be placed in a room 12 m long, 9 m broad and 8 m high. Solution: Length of the room $=12 \mathrm{~m}$ Breadth $=9 \mathrm{~m}$ Height $=8 \mathrm{~m}$ Since the room is cuboidal in shape, the length of the longest rod that can be placed in the room will be equal to the length of the diagonal between opposite vertices. Length of the diagonal of the floor using the Pythagorus theorem $=\sqrt{1^{2}+\mathrm{b}^{2}}$ $=\sqrt{(12)^{2}+(9)^{2}}$ $=\...
Read More →Which of the following are sets? Justify your answer.
Question: Which of the following are sets? Justify your answer. The collection of all whole numbers less than 10. Solution: Whole numbers are 0, 1, 2, 3, Whole numbers less than 10 are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 As the collection of all whole numbers, less than 10 is known and can be counted, i.e. well defined. , this is a set....
Read More →If the system of equations
Question: If the system of equations $x-k y-z=0, k x-y-z=0, x+y-z=0$ has a non-zero solution then the values of $k$ are________ Solution: The system of homogeneous equations $x-k y-z=0, k x-y-z=0$ and $x+y-z=0$ has a non-zero solution or an infinite many solutions. $\therefore \Delta=\left|\begin{array}{ccc}1 -k -1 \\ k -1 -1 \\ 1 1 -1\end{array}\right|=0$ $\Rightarrow 1(1+1)+k(-k+1)-1(k+1)=0$ $\Rightarrow 2-k^{2}+k-k-1=0$ $\Rightarrow k^{2}=1$ $\Rightarrow k=\pm 1$ Thus, the values of $k$ are $...
Read More →The central hall of a school is 80 m long and 8 m high.
Question: The central hall of a school is 80 m long and 8 m high. It has 10 doors each of size 3 m 1.5 m and 10 windows each of size 1.5 m 1 m. If the cost of white-washing the walls of the hall at the rate of Rs 1.20 per m2is Rs 2385.60, fidn the breadth of the hall. Solution: Suppose that the breadth of the hall is $b \mathrm{~m}$. Lenght of the hall $=80 \mathrm{~m}$ Height of the hall $=8 \mathrm{~m}$ Total surface area of 4 walls including doors and windows $=2 \times($ length $\times$ heig...
Read More →Justify whether it is true to say
Question: Justify whether it is true to say that the following are the nth terms of an AP. (i) 2n 3 (ii) 3n2+ 5 (iii) 1 + n + n2 Solution: (i) Yes, here $a_{n}=2 n-3$ Put $n=1, \quad a_{1}=2(1)-3=-1$ Put $n=2, \quad a_{2}=2(2)-3=1$ Put $n=3, \quad a_{3}=2(3)-3=3$ Put $n=4 . \quad a_{4}=2(4)-3=5$ List of numbers becomes $-1,1,3, \ldots$ Here, $a_{2}-a_{1}=1-(-1)=1+1=2$ $a_{3}-a_{2}=3-1=2$ $a_{4}-a_{3}=5-3=2$ $\because a_{2}-a_{1}=a_{3}-a_{2}=a_{4}-a_{3}=\ldots$ Hence, $2 n-3$ is the $n$th term of...
Read More →If the system of equations
Question: If the system of equations $2 x-y-z=12, x-2 y+z=-4, x+y+\lambda z=4$ has no solution, then $\lambda=$___________ Solution: The given system of equations $2 x-y-z=12, x-2 y+z=-4$ and $x+y+\lambda z=4$ has no solution. $\therefore \Delta=\left|\begin{array}{ccc}2 -1 -1 \\ 1 -2 1 \\ 1 1 \lambda\end{array}\right|=0$ $\Rightarrow 2(-2 \lambda-1)-(-1)(\lambda-1)-1(1+2)=0$ $\Rightarrow-4 \lambda-2+\lambda-1-3=0$ $\Rightarrow-3 \lambda-6=0$ $\Rightarrow-3 \lambda=6$ $\Rightarrow \lambda=-2$ Th...
Read More →The number of solutions of the system of the system
Question: The number of solutions of the system of the system of equations $x+2 y+z=3,2 x+3 y+z=3,3 x+5 y+2 z=1$ is___________ Solution: The given system of equations is $x+2 y+z=3,2 x+3 y+z=3$ and $3 x+5 y+2 z=1$. $\Delta=\left|\begin{array}{lll}1 2 1 \\ 2 3 1 \\ 3 5 2\end{array}\right|=1(6-5)-2(4-3)+1(4-3)=1-2+1=0$ $\Delta_{x}=\left|\begin{array}{lll}3 2 1 \\ 3 3 1 \\ 1 5 2\end{array}\right|=3(6-5)-2(6-1)+1(15-3)=3-10+12=5 \neq 0$ Here, $\Delta=0$ and at least one of $\Delta_{x}, \Delta_{y}$ a...
Read More →A classroom is 7 m long, 6 m broad and 3.5 m high.
Question: A classroom is 7 m long, 6 m broad and 3.5 m high. Doors and windows occupy an area of 17 m2. What is the cost of white-washing the walls at the rate of Rs 1.50 per m2. Solution: Length of the classroom $=7 \mathrm{~m}$ Breadth of the classroom $=6 \mathrm{~m}$ Height of the classroom $=3.5 \mathrm{~m}$ Total surface area of the classroom to be whitewashed $=$ areas of the 4 walls $=2 \times($ breadth $\times$ height $+$ length $\times$ height $)$ $=2 \times(6 \times 3.5+7 \times 3.5)$...
Read More →A cuboid has total surface area of 50 m
Question: A cuboid has total surface area of 50 m2and lateral surface area is 30 m2. Find the area of its base. Solution: Total sufrace area of the cuboid $=50 \mathrm{~m}^{2}$ Its lateral surface area $=30 \mathrm{~m}^{2}$ Now, total surface area of the cuboid $=2 \times($ surface area of the base $)+($ surface area of the 4 walls $)$ $\Rightarrow 50=2 \times($ surface area of the base $)+(30)$ $\Rightarrow 2 \times($ surface area of the base $)=50-30=20$ $\therefore$ Surface area of the base $...
Read More →If the system of equations x + y + z = 6,
Question: If the system of equations $x+y+z=6, x+2 y+3 z=10, x+2 y+\lambda z=12$ is inconsistent then $\lambda=$_____________ Solution: The system of equations $x+y+z=6, x+2 y+3 z=10, x+2 y+\lambda z=12$ is inconsistent. $\therefore \Delta=\left|\begin{array}{lll}1 1 1 \\ 1 2 3 \\ 1 2 \lambda\end{array}\right|=0$ $\Rightarrow 1(2 \lambda-6)-1(\lambda-3)+1(2-2)=0$ $\Rightarrow 2 \lambda-6-\lambda+3=0$ $\Rightarrow \lambda-3=0$ $\Rightarrow \lambda=3$ Also, for $\lambda=3$, $\Delta_{y}=\left|\begi...
Read More →In which of the following situations,
Question: In which of the following situations, do the lists of numbers involved form an AP? Give reasons for your answers. (i) The fee charged from a student every month by a school for the whole session, when the monthly fee is ₹ 400. (ii) The fee charged every month by a school from classes I to XII, When the monthly fee for class I is ₹ 250 and it increase by ₹ 50 for the next higher class. (iii) The amount of money in the account of Varun at the end of every year when ₹ 1000 is deposited at...
Read More →The walls and ceiling of a room are to be plastered.
Question: The walls and ceiling of a room are to be plastered. The length, breadth and height of the room are 4.5 m, 3 m and 350 cm, respectively. Find the cost of plastering at the rate of Rs 8 per square metre. Solution: Length of a room $=4.5 \mathrm{~m}$ Breadth $=3 \mathrm{~m}$ Height $=350 \mathrm{~cm}$ $=\frac{350}{100} \mathrm{~m} \quad(\because 1 \mathrm{~m}=100 \mathrm{~cm})$ $=3.5 \mathrm{~m}$ Since only the walls and the ceiling of the room are to be plastered, we have: So, total are...
Read More →Show that the product of the areas of the floor
Question: Show that the product of the areas of the floor and two adjacent walls of a cuboid is the square of its volume. Solution: Suppose that the length, breadth and height of the cuboidal floor are $l \mathrm{~cm}, b \mathrm{~cm}$ and $h \mathrm{~cm}$, respectively. Then, area of the floor $=l \times b \mathrm{~cm}^{2}$ Area of the wall $=b \times h \mathrm{~cm}^{2}$ Area of its adjacent wall $=l \times h \mathrm{~cm}^{2}$ Now, product of the areas of the floor and the two adjacent walls $=(...
Read More →If the system of equations x + ay = 0, az + y = 0, ax + z = 0 has
Question: If the system of equations $x+a y=0, a z+y=0, a x+z=0$ has infinitely many solutions then $a=$____________ Solution: The given system of homogeneous equationsx + ay= 0,az + y= 0,ax + z = 0 has infinitely many solutions. $\therefore\left|\begin{array}{lll}1 a 0 \\ 0 1 a \\ a 0 1\end{array}\right|=0$ $\Rightarrow 1(1-0)-a\left(0-a^{2}\right)+0(0-a)=0$ $\Rightarrow 1+a^{3}=0$ $\Rightarrow(1+a)\left(1-a+a^{2}\right)=0$ $\Rightarrow a+1=0 \quad\left(a^{2}-a+10 \forall a \in \mathrm{R}\right...
Read More →The taxi fare after each km, when the fare
Question: The taxi fare after each km, when the fare is ₹ 15 for the first km ahd ₹ 8 for each additional km, does not form an AP as the total fare (in ₹) after each km is 15, 8, 8, 8, . Is the statement true? Give reasons. Solution: No, because the total fare (in ?) after each km is 15,(15+ 8), (15 + 2 x 8), (15+ 3 x 8),= 15,23, 31, 39, Let $\quad t_{1}=15, t_{2}=23, t_{3}=31$ and $t_{4}=39$ Now, $t_{2}-t_{1}=23-15=8$ $t_{3}-t_{2}=31-23=8$ $t_{4}-t_{3}=39-31=8$ Since, all the successive terms o...
Read More →The perimeter of a floor of a room is 30 m and its height is 3 m.
Question: The perimeter of a floor of a room is 30 m and its height is 3 m. Find the area of four walls of the room. Solution: Perimeter of the floor of the room $=30 \mathrm{~m}$ Height of the oom $=3 \mathrm{~m}$ Perimeter of a rectangle $=2 \times($ length $+$ breadth $)=30 \mathrm{~m}$ So, area of the four walls $=2 \times($ length $\times$ height $+$ breadth $\times$ height $)$ $=2 \times($ length $+$ breadth $) \times$ height $=30 \times 3=90 \mathrm{~m}^{2}$...
Read More →Is 0 a term of the AP 31, 28, 25,…?
Question: Is 0 a term of the AP 31, 28, 25,? Justify your answer. Solution: Let 0 be the nth term of given AP i.e., an= 0. Given that, first term a = 31, common difference, d = 28 31 = 3 The nth terms of an AP, is $a_{n}=a+(n-1) d$ $\Rightarrow \quad 0=31+(n-1)(-3)$ $\Rightarrow \quad 3(n-1)=31$ $\Rightarrow \quad n-1=\frac{31}{2}$ $\therefore$$n=\frac{31}{3}+1=\frac{34}{3}=11 \frac{1}{3}$ Since, n should be positive integer. So, 0 is not a term of the given AP....
Read More →A swimming pool is 20 m long 15 m wide and 3 m deep.
Question: A swimming pool is 20 m long 15 m wide and 3 m deep. Find the cost of repairing the floor and wall at the rate of Rs 25 per square metre. Solution: Length of the swimming pool $=20 \mathrm{~m}$ Breadth $=15 \mathrm{~m}$ Height $=3 \mathrm{~m}$ Now, surface area of the floor and all four walls of the pool $=($ length $\times$ breadth $)+2 \times($ breadth $\times$ height $+$ length $\times$ height $)$ $=(20 \times 15)+2 \times(15 \times 3+20 \times 3)$ $=300+2 \times(45+60)$ $=300+210$ ...
Read More →The system of equations:
Question: The system of equations: $x+y+z=5$ $x+2 y+3 z=9$ $x+3 y+\lambda z=\mu$ has a unique solution, if (a) $\lambda=5, \mu=13$ (b) $\lambda \neq 5$ (c) $\lambda=5, \mu \neq 13$ (d) $\mu \neq 13$ Solution: $(\mathrm{b}) \lambda \neq 5$ For a unique solution, $|A| \neq 0$. $\Rightarrow\left|\begin{array}{lll}1 1 1 \\ 1 2 3 \\ 1 3 \lambda\end{array}\right| \neq 0$ $\Rightarrow 1(2 \lambda-9)-1(\lambda-3)+1(3-2) \neq 0$ $\Rightarrow 2 \lambda-9-\lambda+3+1 \neq 0$ $\Rightarrow \lambda-5 \neq 0$ ...
Read More →The system of equations:
Question: The system of equations: $x+y+z=5$ $x+2 y+3 z=9$ $x+3 y+\lambda z=\mu$ has a unique solution, if (a) $\lambda=5, \mu=13$ (b) $\lambda \neq 5$ (c) $\lambda=5, \mu \neq 13$ (d) $\mu \neq 13$ Solution: $(\mathrm{b}) \lambda \neq 5$ For a unique solution, $|A| \neq 0$. $\Rightarrow\left|\begin{array}{lll}1 1 1 \\ 1 2 3 \\ 1 3 \lambda\end{array}\right| \neq 0$ $\Rightarrow 1(2 \lambda-9)-1(\lambda-3)+1(3-2) \neq 0$ $\Rightarrow 2 \lambda-9-\lambda+3+1 \neq 0$ $\Rightarrow \lambda-5 \neq 0$ ...
Read More →Two AP’s have the same common difference.
Question: Two APs have the same common difference. The first term of one AP is 2 and that of the other is 7. The difference between their 10th terms is the same as the difference between their 21st terms, which is the same as the difference between any two corresponding terms? Why? Solution: Let the same common difference of two APs isd, Given that, the first term of first AP and second AP are 2 and 7 respectively, then the APs are 2,2 + d,2 + 2d,2 + 3d,.,. and 7,7+ d, 7 +2d, 7+3d, Now, 10th ter...
Read More →A cloassroom is 11 m long, 8 m wide and 5 m high.
Question: A cloassroom is 11 m long, 8 m wide and 5 m high. Find the sum of the areas of its floor and the four walls (including doors, windows, etc.) Solution: Lenght of the classroom $=11 \mathrm{~m}$ Width $=8 \mathrm{~m}$ Height $=5 \mathrm{~m}$ We have to find the sum of the areas of its floor and the four walls (i.e., like an open box). $\therefore$ The sum of areas of the floor and the four walls $=($ length $\times$ width $)+2 \times($ width $\times$ height $+$ length $\times$ height $)$...
Read More →For the AP -3, – 7, – 11,…
Question: For the AP -3, 7, 11, can we find directly a30 a20without actually finding a30and a20? Give reason for your answer. Solution: True $\because n$th term of an AP, $a_{n}=a+(n-1) d$ $\therefore \quad a_{30}=a+(30-1) d=a+29 d$ and $\quad a_{20}=a+(20-1) d=a+19 d$ $\ldots(\mathrm{i})$ Now, $\quad a_{30}-a_{20}=(a+29 d)-(a+19 d)=10 d$ and from given AP common difference, $d=-7-(-3)=-7+3$ $=-4$ $\therefore \quad a_{30}-a_{20}=10(-4)=-40$ [from Eq. (i)]...
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