Solve this following
Question: If $P=\left[\begin{array}{rr}3 4 \\ 2 -1 \\ 0 5\end{array}\right]$ and $P=\left[\begin{array}{rr}7 -5 \\ -4 0 \\ 2 6\end{array}\right]$, verify that $(P+Q)^{\prime}=\left(P^{\prime}+Q^{\prime}\right)$ Solution:...
Read More →Explain why the experiment of tossing a coin three times
Question: Explain why the experiment of tossing a coin three times is said to have binomial distribution. Solution: As the random variable X takes 0, 1, 2, 3, , n is said to be binomial distribution having parameters n and p, if the probability is given by P(X = r) =nCrprqn-r, where q = 1 p and r = 0, 1, 2, 3, Similarly, in case of tossing a coin 3 times, we have n = 3 and X has the values 0, 1, 2, 3 with p = , q = . Therefore, it is said that the experiment of tossing a coin three times have bi...
Read More →Solve this following
Question: If $A=\left[\begin{array}{ccc}3 2 -1 \\ -5 0 -6\end{array}\right]$ and $B=\left[\begin{array}{ccc}-4 -5 -2 \\ 3 1 8\end{array}\right]$, verify that $(A+B)^{\prime}=\left(A^{\prime}+B^{\prime}\right)$ Solution:...
Read More →Two dice are thrown together and the total score is noted.
Question: Two dice are thrown together and the total score is noted. The events E, F and G are a total of 4, a total of 9 or more, and a total divisible by 5, respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent. Solution: If two dice are thrown together, we have n(S) = 36 Now, lets consider: E = A total of 4 = {(2, 2), (1, 3), (3, 1)} ⇒ n(E) = 3 F = A total of 9 or more = {(3, 6), (6, 3), (5, 4), (4, 5), (5, 5), (4, 6), (6, 4), (5, 6), (6, 5), (6...
Read More →A bag contains 5 red marbles and 3 black marbles.
Question: A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marble is red? Solution: Let red marbles be presented with R and black marble with B. Also, let E be the event that at least one of the three marbles drawn be black when the first marble is red. Now, the following three conditions are possible, if atleast one of the three marbles drawn be bl...
Read More →Solve this following
Question: If $\mathrm{A}=\left[\begin{array}{cc}3 5 \\ -2 0 \\ 4 -6\end{array}\right]$, verify that $(2 \mathrm{~A})^{\prime}=2 \mathrm{~A}^{\prime}$ Solution: Given $A=\left[\begin{array}{cc}3 5 \\ -2 0 \\ 4 -6\end{array}\right]$...
Read More →The probability that at least one of the two events
Question: The probability that at least one of the two events $A$ and $B$ occurs is $0.6$. If $A$ and $B$ occur simultaneously with probability $0.3$, evaluate $\mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})$. Solution: W.k.t, A ⋃ B denotes that atleast one of the events occurs and A ⋂ B denotes that two events occur simultaneously....
Read More →Refer to Exercise 1 above.
Question: Refer to Exercise 1 above. If the die were fair, determine whether or not the events A and B are independent. Solution: According to the solution of exercise 1, we have A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}and n(6) and n(S) = 6 x 6 = 36 So, P(A) = n(A)/n(S) = 6/36 = 1/6 And, B = {(4, 6), (6, 4), (5, 5), (5, 6), (6, 5), (6, 6)}; n(B) = 6 and n(S) = 36 So, P(B) = n(B)/n(S) = 6/36 = 1/6 Now, A ⋂B = {(5, 5), (6, 6)} So, P(A ⋂B) = 2/36 = 1/18 Hence, if A and B are not indepen...
Read More →Solve this following
Question: If $A=\left[\begin{array}{ccc}2 -3 5 \\ 0 7 -4\end{array}\right]$, verify that $\left(A^{\prime}\right)^{\prime}=A$. Solution:...
Read More →For a loaded die, the probabilities of outcomes are given as under:
Question: For a loaded die, the probabilities of outcomes are given as under: P(1) = P(2) = 0.2, P(3) = P(5) = P(6) = 0.1 and P(4) = 0.3. The die is thrown two times. Let A and B be the events, same number each time, and a total score is 10 or more, respectively. Determine whether or not A and B are independent. Solution: Given that a loaded die is thrown such that P(1) = P(2) = 0.2, P(3) = P(5) = P(6) = 0.1 and P(4) = 0.3 and die is thrown two times. Also given that: A = same number each time a...
Read More →Solve this following
Question: If $\left[\begin{array}{cc}2 3 \\ 5 7\end{array}\right]\left[\begin{array}{cc}1 -3 \\ -2 4\end{array}\right]=\left[\begin{array}{cc}-4 6 \\ -9 x\end{array}\right]$, find the value of $x$ Solution:...
Read More →Solve this following
Question: If $A=\left[\begin{array}{cc}1 0 \\ -1 7\end{array}\right]$ and $B=\left[\begin{array}{cc}0 4 \\ -1 7\end{array}\right]$, find $\left(3 A^{2}-2 B+1\right)$ Solution: Given : $A=\left[\begin{array}{cc}1 0 \\ -1 7\end{array}\right]$ and $B=\left[\begin{array}{cc}0 4 \\ -1 7\end{array}\right]$,...
Read More →Give an example of three matrices A, B, C such that
Question: Give an example of three matrices A, B, C such that $\mathrm{AB}=\mathrm{AC}$ but $\mathrm{B} \neq \mathrm{C}$ Solution:...
Read More →A man rides his motorcycle at the speed of 50 km/hour.
Question: A man rides his motorcycle at the speed of 50 km/hour. He has to spend Rs 2 per km on petrol. If he rides it at a faster speed of 80 km/hour, the petrol cost increases to Rs 3 per km. He has at most Rs 120 to spend on petrol and one hours time. He wishes to find the maximum distance that he can travel. Express this problem as a linear programming problem. Solution: Lets assume the man covers x km on his motorcycle at the speed of 50km/hr and covers y km at the speed of 50 km/hr and cov...
Read More →A company manufactures two types of sweaters: type A and type B.
Question: A company manufactures two types of sweaters: type A and type B. It costs Rs 360 to make a type A sweater and Rs 120 to make a type B sweater. The company can make at most 300 sweaters and spend at most Rs 72000 a day. The number of sweaters of type B cannot exceed the number of sweaters of type A by more than 100. The company makes a profit of Rs 200 for each sweater of type A and Rs 120 for every sweater of type B. Formulate this problem as a LPP to maximize the profit to the company...
Read More →A company manufactures two types of screws A and B.
Question: A company manufactures two types of screws A and B. All the screws have to pass through a threading machine and a slotting machine. A box of Type A screws requires 2 minutes on the threading machine and 3 minutes on the slotting machine. A box of type B screws requires 8 minutes of threading on the threading machine and 2 minutes on the slotting machine. In a week, each machine is available for 60 hours. On selling these screws, the company gets a profit of Rs 100 per box on type A scr...
Read More →Given an example of two matrices A and B such that
Question: Given an example of two matrices $A$ and $B$ such that $\mathrm{A} \neq \mathrm{O}, \mathrm{B} \neq \mathrm{O}, \mathrm{AB}=\mathrm{O}$ and $\mathrm{BA} \neq \mathrm{O}$ Solution:...
Read More →A firm has to transport 1200 packages using large vans
Question: A firm has to transport 1200 packages using large vans which can carry 200 packages each and small vans which can take 80 packages each. The cost for engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3000 is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost. Solution: Lets consider x and y to be the number of large and small vans respecti...
Read More →Solve this following
Question: If $\mathrm{A}=\left[\begin{array}{ll}1 1 \\ 0 1\end{array}\right]$, prove that $\mathrm{A}^{\mathrm{n}}=\left[\begin{array}{ll}1 \mathrm{n} \\ 0 1\end{array}\right]$ for all $\mathrm{n} \in \mathrm{N}$ Solution:...
Read More →A manufacturer of electronic circuits has a stock of 200 resistors,
Question: A manufacturer of electronic circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resistors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs 50 and that on type B circuit is Rs 60, formulate this problem as a LPP so that the manufacturer can maximize his profit. Solution: Let x units of type A and y uni...
Read More →Solve this following
Question: If $A=\left[\begin{array}{cc}3 4 \\ -4 -3\end{array}\right]$, find $f(A)$, where $f(x)=x 2-5 x+7$ Solution:...
Read More →Find the values of a and b for which
Question: Find the values of $a$ and $b$ for which $\left[\begin{array}{cc}\mathrm{a} \mathrm{b} \\ -\mathrm{a} 2 \mathrm{~b}\end{array}\right]\left[\begin{array}{c}2 \\ -1\end{array}\right]=\left[\begin{array}{l}5 \\ 4\end{array}\right]$ Solution:...
Read More →Find the equation of the plane through the intersection of the planes
Question: Find the equation of the plane through the intersection of the planes $\vec{r} \cdot(\hat{i}+3 \hat{j})-6=0$ and $\vec{r} \cdot(3 \hat{i}-\hat{j}-4 \hat{k})=0$, whose perpendicular distance from origin is unity. Solution:...
Read More →Solve this following
Question: If $\left[\begin{array}{lll}x 4 1\end{array}\right]\left[\begin{array}{ccc}2 1 2 \\ 1 0 2 \\ 0 2 -4\end{array}\right]\left[\begin{array}{r}x \\ 4 \\ -1\end{array}\right]=O$, find $x$ Solution:...
Read More →The plane ax + by = 0 is rotated about its line of intersection
Question: The planeax+by= 0 is rotated about its line of intersection with the planez= 0 through an angle . Prove that the equation of the plane in its new position is $a x+b y \pm\left(\sqrt{a^{2}+b^{2}} \tan \alpha\right) z=0$ Solution: Given planes are: ax + by = 0 . (i) and z = 0 . (ii) Now, the equation of any plan passing through the line of intersection of plane (i) and (ii) is (ax + by) + kz = 0 ⇒ ax + by + kz = 0 . (iii)...
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