Find the area of a rhombus whose side is 6 cm and whose altitude is 4 cm.
Question: Find the area of a rhombus whose side is 6 cm and whose altitude is 4 cm. If one of its diagonals is 8 cm long, find the length of the other diagonal. Solution: Given: Side of the rhombus $=6 \mathrm{~cm}$ Altitude $=4 \mathrm{~cm}$ One of the diagonals $=8 \mathrm{~cm}$ Area of the rhombus $=$ Side $\times$ Altitude $=6 \times 4=24 \mathrm{~cm}^{2} \quad \ldots \ldots \ldots$ (i) We know : Area of rhombus $=\frac{1}{2} \times \mathrm{d}_{1} \times \mathrm{d}_{2}$ Using (i): $24=\frac{...
Read More →Tickets numbered 2, 3, 4, 5, ..., 100, 101 are placed in a box and mixed thoroughly.
Question: Tickets numbered 2, 3, 4, 5, ..., 100, 101 are placed in a box and mixed thoroughly. One ticket is drawn at random from the box. Find the probability that the number on the ticket is(i) an even number(ii) a number less than 16(iii) a number which is a perfect square(iv) a prime number less than 40 Solution: All possible outcomes are 2, 3, 4, 5................101.Number of all possible outcomes = 100(i) Out of these the numbers that are even = 2, 4, 6, 8...................100 Let E1be t...
Read More →A box contains 25 cards numbered from 1 to 25. A card is drawn at random from the bag.
Question: A box contains 25 cards numbered from 1 to 25. A card is drawn at random from the bag. Find the probability that the number on the drawn card is(i) divisible by 2 or 3,(ii) a prime number. Solution: Total number of outcomes = 25(i)Let E1be the event of gettinga card divisible by 2 or 3.Out of the given numbers, numbers divisible by 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 and 24.Out of the given numbers, numbers divisible by 3 are 3, 6, 9, 12, 15, 18, 21 and 24.Out of the given num...
Read More →Two institutions decided to award their employees for the three values of resourcefulness,
Question: Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs.x,yandzrespectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then u...
Read More →Cards numbered 1 to 30 are put in a bag. A card is drawn at random from the bag.
Question: Cards numbered 1 to 30 are put in a bag. A card is drawn at random from the bag. Find the probability that the number on the drawn card is(i) not divisible by 3,(ii) a prime number greater than 7,(iii) not a perfect square number. Solution: Total number of outcomes = 30.(i) Let E1be the event of getting a number not divisible by 3.Out of these numbers, numbers divisible by 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27 and 30.Number of favourable outcomes = 30 10 = 20 $\therefore \mathrm{P}($ ...
Read More →Which of the following equations has 2 as a root?
Question: Which of the following equations has 2 as a root? (a) x2-4x + 5=0 (b) x2+ 3x-12 =0 (c) 2x2 7x + 6 = 0 (d) 3x2 6x 2 = 0 Solution: (c) (a) Substitutina $x=2$ in $x^{2}-4 x+5$, we get $(2)^{2}-4(2)+5$ $=4-8+5=1 \neq 0$ So, $x=2$ is not a root of $x^{2}-4 x+5=0$ (b) Substituting $x=2$ in $x^{2}+3 x-12$, we get $(2)^{2}+3(2)-12$ $=4+6-12=-2 \neq 0$ So, $x=2$ is not a root of $x^{2}+3 x-12=0$. (c) Substituting $x=2$ in $2 x^{2}-7 x+6$, we get $2(2)^{2}-7(2)+6=2(4)-14+6$ $=8-14+6=14-14=0$ So,...
Read More →A box contains cards numbered 3, 5, 7, 9, .... , 35, 37. A card is drawn at random from the box.
Question: A box contains cards numbered 3, 5, 7, 9, .... , 35, 37. A card is drawn at random from the box. Find the probability that the number on the card is a prime number. Solution: Given numbers3, 5, 7, 9, .... , 35, 37 form an AP witha= 3 andd= 2.LetTn= 37. Then,3 + (n 1)2 = 37⇒ 3 + 2n 2 = 37⇒ 2n= 36⇒n= 18Thus, total number of outcomes = 18.Let Ebe the event of getting a prime number.Out of these numbers, the prime numbers are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 and 37.Number of favourable...
Read More →A school wants to award its students for the values of Honesty,
Question: A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard wor...
Read More →Cards bearing numbers 1, 3, 5, .... , 35 are kept in a bag. A card is drawn at random from the bag.
Question: Cards bearing numbers 1, 3, 5, .... , 35 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card bearing(i) a prime number less than 15,(ii) a number divisible by 3 and 5. Solution: Given number 1, 3, 5, .... , 35 form an AP witha= 1 andd= 2.LetTn= 35. Then,1 + (n 1)2 = 35⇒ 1 + 2n 2 = 35⇒ 2n= 36⇒n= 18Thus, total number of outcomes = 18.(i) Let E1be the event of gettinga prime number less than 15.Out of these numbers, prime numbers less than 15 ...
Read More →Which of the following is not a quadratic equation?
Question: Which of the following is not a quadratic equation? (a) 2 (x -1)2= 4x2 2x +1 (b) 2x x2= x2+ 5 (c) (-2X +3)2= 3x2 5x (d) (x2+ 2x)2= x4+ 3 + 4x2 Solution: (d) (a) Given that, $2(x-1)^{2}=4 x^{2}-2 x+1$ $\Rightarrow \quad 2\left(x^{2}+1-2 x\right)=4 x^{2}-2 x+1$ $\Rightarrow \quad 2 x^{2}+2-4 x=4 x^{2}-2 x+1$ $\Rightarrow \quad 2 x^{2}+2 x-1=0$ which represents a quadratic equation because it has the quadratic form $a x^{2}+b x+c=0, a \neq 0$ (b) Given that, $2 x-x^{2}=x^{2}+5$ $\Rightarr...
Read More →The diagonal of a quadrilateral shaped field is 24 m
Question: The diagonal of a quadrilateral shaped field is 24 m and the perpendiculars dropped on it from the remaining opposite vertices are 8 m and 13 m. Find the area of the field. Solution: Given: Diagonal of a quadrilateral shaped field $=24 \mathrm{~m}$ Perpendiculars dropped on it from the remaining opposite vertices are $8 \mathrm{~m}$ and $13 \mathrm{~m}$. Now, we know : Area $=\frac{1}{2} \times \mathrm{d} \times\left(\mathrm{h}_{1}+\mathrm{h}_{2}\right)$ $\therefore$ Area of the field ...
Read More →Cards marked with numbers 1, 3, 5, ..., 101 are placed in a bag and mixed thoroughly.
Question: Cards marked with numbers 1, 3, 5, ..., 101 are placed in a bag and mixed thoroughly. A card is drawn at random from the bag. Find the probability that the number on the drawn card is(i) less than 19,(ii) a prime number less than 20. Solution: Given number 1, 3, 5, ..., 101 form an AP witha= 1 andd= 2.LetTn= 101. Then,1 + (n 1)2 = 101⇒ 1 + 2n 2 = 101⇒ 2n= 102⇒n= 51Thus, total number of outcomes = 51.(i)Let E1be the event of gettinga number less than 19.Out of these numbers,numbers les...
Read More →The diagonals of a rhombus are 7.5 cm and 12 cm.
Question: The diagonals of a rhombus are 7.5 cm and 12 cm. Find its area. Solution: Given: Lengths of the diagonals of a rhombus are $7.5 \mathrm{~cm}$ and $12 \mathrm{~cm}$. Now, we know : Area $=\frac{1}{2}\left(\mathrm{~d}_{1} \times \mathrm{d}_{2}\right)$ $\therefore$ Area of rhombus $=\frac{1}{2}(7.5 \times 12)=45 \mathrm{~cm}^{2}$...
Read More →The management committee of a residential colony decided to award some of its members (say x) for honesty,
Question: The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for hel...
Read More →The area of a rhombus is 240 cm
Question: The area of a rhombus is 240 cm2and one of the diagonal is 16 cm. Find another diagonal. Solution: Given: Area of the rhombus $=240 \mathrm{~cm}$ Length of one of its diagonals $=16 \mathrm{~cm}$ We know that if the diagonals of a rhombus are $\mathrm{d}_{1}$ and $\mathrm{d}_{2}$, then the area of the rhombus is given by : Area $=\frac{1}{2}\left(\mathrm{~d}_{1} \times \mathrm{d}_{2}\right)$ Putting the given values: $240=\frac{1}{2}\left(16 \times \mathrm{d}_{2}\right)$ $240 \times 2=...
Read More →A box contains cards bearing numbers 6 to 70.
Question: A box contains cards bearing numbers 6 to 70. If one card is drawn at random from the box, find the probability that it bears(i) a one-digit number,(ii) a number divisible by 5,(iii) an odd number less than 30,(iv) a composite number between 50 and 70. Solution: Given number 6, 7, 8, .... , 70 form an AP witha= 6 andd= 1.LetTn= 70. Then,6 + (n 1)1 = 70⇒ 6 +n 1 = 70⇒n= 65Thus, total number of outcomes = 65.(i)Let E1be the event of gettinga one-digit number.Out of these numbers, one-dig...
Read More →Which of the following is a quadratic equation?
Question: Which of the following is a quadratic equation? (a) x2+ 2x + 1 = (4 x)2+ 3 (b) $-2 x^{2}=(5-x)\left(2 x-\frac{2}{5}\right.$ (c) $(k+1) x^{2}+-\frac{3}{2} x=7$, where $k=-1$ (d) $x^{3}-x^{2}=(x-1)^{3}$ Solution: (a) Given that, $\quad x^{2}+2 x+1=(4-x)^{2}+3$ $\Rightarrow \quad x^{2}+2 x+1=16+x^{2}-8 x+3$ $\Rightarrow \quad 10 x-18=0$ which is not of the form $a x^{2}+b x+c, a \neq 0$. Thus, the equation is not a quadratic equation. (b) Given that, $-2 x^{2}=(5-x)\left(2 x-\frac{2}{5}\r...
Read More →The diameter of a wheel of a bus is 90 cm which makes 315 revolutions per minute.
Question: The diameter of a wheel of a bus is 90 cm which makes 315 revolutions per minute. Determine its speed in kilometres per hour. [Use= 22/7] Solution: It is given that the diameter of the wheel is $90 \mathrm{~cm}$. $\therefore$ Radius of the circular wheel, $\mathrm{r}=\frac{90}{2}=45 \mathrm{~cm}$. $\therefore$ Perimeter of the wheel $=2 \times \pi \times \mathrm{r}=2 \times \frac{22}{7} \times 45=282.857 \mathrm{~cm}$ It means the wheel travels $282.857 \mathrm{~cm}$ in a revolution. N...
Read More →The prices of three commodities P, Q and R are Rs x, y and z per unit respectively.
Question: The prices of three commoditiesP,QandRare Rsx,yandzper unit respectively.Apurchases 4 units ofRand sells 3 units ofPand 5 units ofQ.Bpurchases 3 units ofQand sells 2 units ofPand 1 unit ofR.Cpurchases 1 unit ofPand sells 4 units ofQand 6 units ofR. In the processA,BandCearn Rs 6000, Rs 5000 and Rs 13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.c Solution: The pri...
Read More →A box contains cords numbered from 1 to 20. A card is drawn at random from the box.
Question: A box contains cords numbered from 1 to 20. A card is drawn at random from the box. Find the probability that the number on the drawn card is (i) a prime number, (ii) a composite number, (iii) a number divisible by 3. Solution: There are 20 cards in the box. One card can drawn at random from the box in 20 ways. Total number of outcomes = 20(i) The prime number cards in the box are 2, 3, 5, 7, 11, 13, 17 and 19. There are 8 prime number cards in the box. So, there are 8 ways to draw a c...
Read More →Find the area of Fig. 20.25, in square cm,
Question: Find the area of Fig. 20.25, in square cm, correct to one place of decimal. (Take= 22/7) Solution: The given figure is: Construction : Connect A to D. Then, we have: Area of the given figure $=$ (Area of rectangle $\mathrm{ABCD}+$ Area of the semicircle) $-$ (Area of $\triangle \mathrm{AED}$ ). $\therefore$ Total area of the figure $=($ Area of rectangle with sides $10 \mathrm{~cm}$ and $10 \mathrm{~cm})$ $+\left(\right.$ Area of semicircle with radius $\left.=\frac{10}{2}=5 \mathrm{~c...
Read More →Cards, marked with numbers 5 to 50, are placed in a box and mixed thoroughly.
Question: Cards, marked with numbers 5 to 50, are placed in a box and mixed thoroughly. A card is drawn from the box at random. Find the probability that the number on the card is (i) a prime number less than 10 (ii) a perfect square. Solution: All possible outcomes are 5, 6, 7, 8...................50. Number of all possible outcomes = 46 (i) Out of the given numbers, the prime numbers less than 10 are 5 and 7.Let E1be the event of getting a prime number less than 10.Then, number of favourable o...
Read More →The inside perimeter of a running track (shown in Fig. 20.24) is 400 m.
Question: The inside perimeter of a running track (shown in Fig. 20.24) is 400 m. The length of each of the straight portion is 90 m and the ends are semi-circles. If track is everywhere 14 m wide, find the area of the track. Also, find the length of the outer running track. Solution: It is given that the inside perimeter of the running track is $400 \mathrm{~m}$. It means the length of the inner track is $400 \mathrm{~m}$. Let $\mathrm{r}$ be the radius of the inner semicircles. Observe: Perime...
Read More →A bag contains some balls of which x are white, 2x are black are 3x are red. A ball is selected at random.
Question: A bag contains some balls of whichxare white, 2xare black are 3xare red. A ball is selected at random. What is the probability that it is (i) not red? (ii) white? Solution: Number of white balls in the bag =xNumber of black balls in the bag = 2xNumber of red balls in the bag = 3xTotal number of balls in the bag =x+ 2x+ 3x= 6x Total number of outcomes = 6x(i) There are 3xnon-red balls (xwhite balls and 2xblack balls) in the bag. So, there are 3xways to draw a ball from the bag which is ...
Read More →Vijay had some bananas and he divided
Question: Vijay had some bananas and he divided them into two lots A and B. He sold the first lot at the rate of ₹ 2 for 3 bananas and the second lot at the rate of ₹ 1 per banana and got a total of ₹ 400 If he had sold the first lot at the rate of ₹ 1 per banana and the second lot at the rate of ₹ 4 for 5 bananas, his total collection would have been ₹ 460. Find the total nmber of bananas he had. Solution: Let the number of bananas in lots A and B be x and y, respectively Case I Cost of the fir...
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