Two schools A and B want to award their selected students on the values of sincerity,
Question: Two schoolsAandBwant to award their selected students on the values of sincerity, truthfulness and helpfulness. The schoolAwants to award₹xeach,₹yeach and₹zeach for the three respective values to 3, 2 and 1 students respectively with a total award money of₹1,600.SchoolBwants to spend ₹2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹900, using matr...
Read More →Find the probability that a leap year selected at random will contain 53 Sundays.
Question: Find the probability that a leap year selected at random will contain 53 Sundays. Solution: A leap year has 366 days with 52 weeks and 2 days.Now, 52 weeks conatins 52 sundays. The remaining two days can be:(i) Sunday and Monday(ii) Monday and Tuesday(iii) Tuesday and Wednesday(iv) Wednesday and Thursday(v) Thursday and Friday(vi) Friday and Saturday(vii) Saturday and Sunday Out of these 7 cases, there are two cases favouring it to be Sunday. $\therefore \mathrm{P}$ (a leap year having...
Read More →Two schools A and B want to award their selected students on the values of sincerity,
Question: Two schoolsAandBwant to award their selected students on the values of sincerity, truthfulness and helpfulness. The schoolAwants to award₹xeach,₹yeach and₹zeach for the three respective values to 3, 2 and 1 students respectively with a total award money of₹1,600.SchoolBwants to spend ₹2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹900, using matr...
Read More →The area of a rhombus is 84 m
Question: The area of a rhombus is 84 m2. If its perimeter is 40 m, then find its altitude. Solution: Given: Area of the rhombus $=84 \mathrm{~m}^{2}$ Perimeter $=40 \mathrm{~m}$ Now, we know: Perimeter of the rhombus $=4 \times$ Side $\therefore 40=4 \times$ Side Side $=\frac{40}{4}=10 \mathrm{~m}$ Again, we know: Area of the rhombus $=$ Side $\times$ Altitude $\Rightarrow 84=10 \times$ Altitude Altitude $=\frac{84}{10}=8.4 \mathrm{~m}$ Hence, the altitude of the rhombus is $8.4 \mathrm{~m}$....
Read More →What is the probability that an ordinary year has 53 Mondays?
Question: What is the probability that an ordinary year has 53 Mondays? Solution: An ordinary year has 365 days consisting of 52 weeks and 1 day.This day can be any day of the week. $\therefore \mathrm{P}$ (of this day to be Monday) $=\frac{1}{7}$ Thus, the probability that an ordinary year has 53 Mondays is $\frac{1}{7}$....
Read More →The quadratic equation 2x2 – √5x + 1 = 0 has
Question: The quadratic equation 2x2 5x + 1 = 0 has (a) two distinct real roots (b) two equal real roots (c) no real roots (d) more than 2 real roots Solution: (c) Given equation is $2 x^{2}-\sqrt{5} x+1=0$. On comparing with $a x^{2}+b x+c=0$, we get $a=2, b=-\sqrt{5}$ and $c=1$ $\therefore$ Discriminant, $\quad D=b^{2}-4 a c=(-\sqrt{5})^{2}-4 \times(2) \times(1)=5-8$ $=-30$ Since, discriminant is negative, therefore quadratic equation $2 x^{2}-\sqrt{5} x+1=0$ has no real roots i.e., imaginary ...
Read More →In exchange of a square plot one of whose sides is 84 m,
Question: In exchange of a square plot one of whose sides is 84 m, a man wants to buy a rectangular plot 144 m long and of the same area as of the square plot. Find the width of the rectangular plot. Solution: Given: Side of the square plot $=84 \mathrm{~m}$ Now, the man wants to exchange it with a rectangular plot of the same area with length 144 . Area of the square plot $=84 \times 84=7056 \mathrm{~m}^{2}$ $\therefore$ Area of the rectangular plot $=$ Length $\times$ Width $7056=144 \times$ W...
Read More →The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18.
Question: The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap? Solution: Total number of apples = 900.P(a rotten apple) = 0.18 $\Rightarrow \frac{\text { number of rotten apples }}{\text { total number of apples }}=0.18$ $\Rightarrow \frac{\text { number of rotten apples }}{900}=0.18$ $\Rightarrow$ The number of rotten apples $=0.18 \times 900=162$ apples....
Read More →The cost of fencing a square field at 60 paise per metre is Rs 1200.
Question: The cost of fencing a square field at 60 paise per metre is Rs 1200. Find the cost of reaping the field at the rate of 50 paise per 100 sq. metres. Solution: Given: Cost of fencing 1 metre of a square field $=60$ paise And, the total cost of fencing the entire field $=$ Rs $1200=1,20,000$ paise $\therefore$ Perimeter of the square field $=\frac{120000}{60}=2000$ metres Now, perimeter of a square $=4 \times$ side For the given square field : $4 \times$ Side $=2000 \mathrm{~m}$ Side $=\f...
Read More →Which constant must be added and subtracted
Question: Which constant must be added and subtracted to solve the quadratic equation $9 x^{2}+\frac{3}{4} x-\sqrt{2}=0$ by the method of completing the square? (a) $\frac{1}{8}$ (b) $\frac{1}{64}$ (C) $\frac{1}{4}$ (d) $\frac{9}{64}$ Solution: (b) Given equation is $9 x^{2}+\frac{3}{4} x-\sqrt{2}=0$. $(3 x)^{2}+\frac{1}{4}(3 x)-\sqrt{2}=0$ On putting $3 x=y$, we have $y^{2}+\frac{1}{4} y-\sqrt{2}=0$ $y^{2}+\frac{1}{4} y+\left(\frac{1}{8}\right)^{2}-\left(\frac{1}{8}\right)^{2}-\sqrt{2}=0$ $\lef...
Read More →Find the probability of getting the sum of two numbers, less than 3 or more than 11
Question: Find the probability of getting the sum of two numbers, less than 3 or more than 11, when a pair of distinct dice is thrown together. Solution: Favorable outcomes for sum of numbers on dice less than 3 or more than 11 are (1,1), (6,6). Required probability $=\mathrm{P}($ sum is less than 3 or more than 11$)=\frac{2}{36}=\frac{1}{18}$....
Read More →Find the area of the field in the form of a rhombus,
Question: Find the area of the field in the form of a rhombus, if the length of each side be 14 cm and the altitude be 16 cm. Solution: Given: Length of each side of a field in the shape of a rhombus $=14 \mathrm{~cm}$ Altitude $=16 \mathrm{~cm}$ Now, we know: Area of the rhombus $=$ Side $\times$ Altitude $\therefore$ Area of the field $=14 \times 16=224 \mathrm{~cm}^{2}$...
Read More →Peter throws two different dice together and finds the product of the two numbers obtained.
Question: Peter throws two different dice together and finds the product of the two numbers obtained. Rina throws a die and squares the number obtained. Who has the better chance to get the number 25? Solution: Total number of possible outcomes for Peter = 36.Possible outcomes for Peter to get product 25 is (5,5).Total number of possible outcomes for Rina = 6.Possible outcomes for Rina to get the number whose square is 25 is 5. Now, $\mathrm{P}$ (Peter will get 25) $=\frac{1}{36}$. And, $P($ Rin...
Read More →The length of a side of a square field is 4 m.
Question: The length of a side of a square field is 4 m. what will be the altitude of the rhombus, if the area of the rhombus is equal to the square field and one of its diagonal is 2 m? Solution: Given: Length of the square field $=4 \mathrm{~m}$ $\therefore$ Area of the square field $=4 \times 4=16 \mathrm{~m}^{2}$ Given : Area of the rhombus $=$ Area of the square field Length of one diagonal of the rhombus $=2 \mathrm{~m}$ $\therefore$ Side of the rhombus $=\frac{1}{2} \sqrt{d_{1}^{2}+d_{2}^...
Read More →Value(s) of k for which the quadratic
Question: Value(s) of k for which the quadratic equation 2x2-kx + k = 0 has equal roots is/are (a) 0 (b) 4 (c) 8 (d) 0, 8 Solution: (d) Given equation is 2x2 kx + k- 0 On comparing with ax2+ bx + c = 0, we get a = 2, b= k and c = k For equal roots, the discriminant must be zero. i.e., $D=b^{2}-4 a c=0$ $\Rightarrow \quad(-k)^{2}-4(2) k=0$ $\Rightarrow \quad k^{2}-8 k=0$ $\Rightarrow \quad k(k-8)=0$ $\therefore \quad k=0,8$ Hence, the required values of k are 0 and 8....
Read More →Peter throws two different dice together and finds the product of the two numbers obtained.
Question: Peter throws two different dice together and finds the product of the two numbers obtained. Rina throws a die and squares the number obtained. Who has the better chance to get the number 25? Solution: Total number of possible outcomes for Peter = 36.Possible outcomes for Peter to get product 25 is (5,5).Total number of possible outcomes for Rina = 6.Possible outcomes for Rina to get the number whose square is 25 is 5. Now, $\mathrm{P}$ (Peter will get 25) $=\frac{1}{36}$. And, $P($ Rin...
Read More →Two different dice are thrown together. Find the probability that the numbers obtained
Question: Two different dice are thrown together. Find the probability that the numbers obtained(i) have a sum less than 7(ii) have a product less than 16(iii) is a doublet of odd numbers. Solution: Total number of possible outcomes is 36.(i) The favorable outcomes for sum of numbers on dices less than 7 are (1,1), (1,2), (1,3), (1,4), (1,5), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (4,1), (4,2), (5,1). $P($ the sum of numbers appeared is less than 7$)=\frac{15}{36}=\frac{5}{12}$. (ii) T...
Read More →Find the area of a rhombus,
Question: Find the area of a rhombus, each side of which measures 20 cm and one of whose diagonals is 24 cm. Solution: Given: Side of the rhombus $=20 \mathrm{~cm}$ Length of a diagonal $=24 \mathrm{~cm}$ We know : If $\mathrm{d}_{1}$ and $\mathrm{d}_{2}$ are the lengths of the diagonals of the rhombus, then side of the rhombus $=\frac{1}{2} \sqrt{\mathrm{d}_{1}^{2}+\mathrm{d}_{2}^{2}}$ So, using the given data to find the length of the other diagonal of the rhombus: $20=\frac{1}{2} \sqrt{24^{2}...
Read More →Two different dice are thrown together. Find the probability that the numbers obtained have
Question: Two different dice are thrown together. Find the probability that the numbers obtained have(i) even sum(ii) even product. Solution: Total number of possible outcomes is 36.(i) The favorable outcomes are (1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3), (3,5), (4,2), (4,4), (4,6), (5,1), (5,3), (5,5), (6,2), (6,4), (6,6). $P($ the sum is even $)=\frac{18}{36}=\frac{1}{2}$ (ii) The favorable outcomes are (1,2), (1,4), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,2), (3,4), (...
Read More →Which of the following equations
Question: Which of the following equations has the sum of its roots as 3 ? (a) $2 x^{2}-3 x+6=0$ (b) $-x^{2}+3 x-3=0$ (c) $\sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x+1=0$ (d) $3 x^{2}-3 x+3=0$ Solution: (b) (a) Given that, $2 x^{2}-3 x+6=0$ On comparing with $a x^{2}+b x+c=0$, we get $a=2, b=-3$ and $c=6$ $\therefore \quad$ Sum of the roots $=\frac{-b}{a}=\frac{-(-3)}{2}=\frac{3}{2}$ So, sum of the roots of the quadratic equation $2 x^{2}-3 x+6=0$ is not $3 ;$ so it is not the answer. (b) Given that, $...
Read More →A rectangular grassy plot is 112 m long and 78 m broad.
Question: A rectangular grassy plot is 112 m long and 78 m broad. It has a gravel path 2.5 m wide all around it on the side. Find the area of the path and the cost of constructing it at Rs 4.50 per square metre. Solution: Given: The length of a rectangular grassy plot is $112 \mathrm{~m}$ and its width is $78 \mathrm{~m}$. Also, it has a gravel path of width $2.5 \mathrm{~m}$ around it on the sides. Its rough diagram is given below : Length of the inner rectangular field $=112-(2 \times 2.5)=107...
Read More →In a family of 3 children, find the probability of having at least one boy.
Question: In a family of 3 children, find the probability of having at least one boy. Solution: All possible outcomes are BBB, BBG, BGB, GBB, BGG, GBG, GGB and GGG.Number of all possible outcomes = 8Let Ebe the event ofhaving at least one boy.Then the outcomes areBBB, BBG, BGB, GBB, BGG, GBG and GGB.Number of possible outcomes = 7 $\therefore P$ (Having at least one boy) $=P(E)=\frac{\text { Number of outcomes favourable to } E}{\text { Number of all possible outcomes }}$ $=\frac{7}{8}$ Thus, th...
Read More →Two factories decided to award their employees for three values
Question: Two factories decided to award their employees for three values of (a) adaptable tonew techniques, (b) careful and alert in difficult situations and (c) keeping clam in tense situations, at the rate of ₹x, ₹yand ₹zper person respectively. The first factory decided to honour respectively 2, 4 and 3 employees with a total prize money of ₹ 29000. The second factory decided to honour respectively 5, 2 and 3 employees with the prize money of ₹ 30500. If the three prizes per person together ...
Read More →The floor of a building consists of 3000 tiles which are rhombus shaped and each of its diagonals are 45 cm and 30 cm in length.
Question: The floor of a building consists of 3000 tiles which are rhombus shaped and each of its diagonals are 45 cm and 30 cm in length. Find the total cost of polishing the floor, if the cost per m2is Rs 4. Solution: Given: The floor consist of 3000 rhombus shaped tiles. The lengths of the diagonals of each tile are $45 \mathrm{~cm}$ and $30 \mathrm{~cm}$. $\therefore$ Area of a rhombus shaped tile $=\frac{1}{2} \times(45 \times 30)=675 \mathrm{~cm}^{2}$ $\therefore$ Area of the complete floo...
Read More →Prove the following
Question: If $\frac{1}{2}$ is a root of the equation $x^{2}+k x-\frac{5}{4}=0$, then the value of $k$ is (a) 2 (b) $-2$ (c) $\frac{1}{4}$ (d) $\frac{1}{2}$ Solution: (a) Since, $\frac{1}{2}$ is a root of the quadratic equation $x^{2}+k x-\frac{5}{4}=0$ Then, $\quad\left(\frac{1}{2}\right)^{2}+k\left(\frac{1}{2}\right)-\frac{5}{4}=0$ $\Rightarrow \quad \frac{1}{4}+\frac{k}{2}-\frac{5}{4}=0 \Rightarrow \frac{1+2 k-5}{4}=0$ $\Rightarrow \quad 2 k-4=0$ $\Rightarrow \quad 2 k=4 \Rightarrow k=2$...
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