If α and β are the zeros of the quadratic polynomial f(t)
Question: If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $t(t)=t^{2}-4 t+3$, find the value of $\alpha^{4} \beta^{3}+\alpha^{3} \beta^{4} .$ Solution: Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $p(y)=t^{2}-4 t+3$ $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $=\frac{-(-4)}{1}$ = 4 $\alpha \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$ $=\frac{3}{1}$ = 3 We have $\alpha^{4} \beta^{3}+\alpha^{3}...
Read More →The numbers 1, 2, 3 and 4 are written separately on four slips of paper.
Question: The numbers 1, 2, 3 and 4 are written separately on four slips of paper. The slips are put in a box and mixed thoroughly. A person draws two slips from the box, one after the other, without replacement. Describe the sample space for the experiment. Solution: If 1 appears on the first drawn slip, then the possibilities that the number appears on the second drawn slip are 2, 3, or 4. Similarly, if 2 appears on the first drawn slip, then the possibilities that the number appears on the se...
Read More →A coin is tossed. If the out come is a head, a die is thrown.
Question: A coin is tossed. If the out come is a head, a die is thrown. If the die shows up an even number, the die is thrown again. What is the sample space for the experiment? Solution: When a coin is tossed, the possible outcomes are head (H) and tail (T). When a die is thrown, the possible outcomes are 1, 2, 3, 4, 5, or 6. Thus, the sample space of this experiment is given by: S = {T, H1, H3, H5, H21, H22, H23, H24, H25, H26, H41, H42, H43, H44, H45, H46, H61, H62, H63, H64, H65, H66}...
Read More →Find both the maximum value and the minimum value of
Question: Find both the maximum value and the minimum value of $3 x^{4}-8 x^{3}+12 x^{2}-48 x+25$ on the interval $[0,3]$ Solution: Let $f(x)=3 x^{4}-8 x^{3}+12 x^{2}-48 x+25$ $\begin{aligned} \therefore f^{\prime}(x) =12 x^{3}-24 x^{2}+24 x-48 \\ =12\left(x^{3}-2 x^{2}+2 x-4\right) \\ =12\left[x^{2}(x-2)+2(x-2)\right] \\ =12(x-2)\left(x^{2}+2\right) \end{aligned}$ Now, $f^{\prime}(x)=0$ gives $x=2$ or $x^{2}+2=0$ for which there are no real roots. Therefore, we consider only $x=2 \in[0,3]$. Now...
Read More →Suppose 3 bulbs are selected at random from a lot.
Question: Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and classified as defective (D) or non-defective (N). Write the sample space of this experiment? Solution: 3 bulbs are to be selected at random from the lot. Each bulb in the lot is tested and classified as defective (D) or non-defective (N). The sample space of this experiment is given by S = {DDD, DDN, DND, DNN, NDD, NDN, NND, NNN}...
Read More →If α and β are the zeros of the quadratic polynomial
Question: If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=x^{2}-5 x+4$, find the value of $\frac{1}{\alpha}+\frac{1}{\beta}-2 \alpha \beta$. Solution: Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=x^{2}-5 x+4$ Therefore $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $=\frac{-(5)}{1}$ = 5 $\alpha \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$ $=\frac{4}{1}$ = 4 We have, $\frac{1}{\alpha}...
Read More →An experiment consists of tossing a coin and then throwing it second time if a head occurs.
Question: An experiment consists of tossing a coin and then throwing it second time if a head occurs. If a tail occurs on the first toss, then a die is rolled once. Find the sample space. Solution: A coin has two faces: head (H) and tail (T). A die has six faces that are numbered from 1 to 6, with one number on each face. Thus, in the given experiment, the sample space is given by S = {HH, HT, T1, T2, T3, T4, T5, T6}...
Read More →A box contains 1 red and 3 identical white balls.
Question: A box contains 1 red and 3 identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment. Solution: It is given that the box contains 1 red ball and 3 identical white balls. Let us denote the red ball with R and a white ball with W. When two balls are drawn at random in succession without replacement, the sample space is given by S = {RW, WR, WW}...
Read More →An experiment consists of recording boy-girl composition of families with 2 children.
Question: An experiment consists of recording boy-girl composition of families with 2 children. (i) What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births? (ii) What is the sample space if we are interested in the number of girls in the family? Solution: (i) When the order of the birth of a girl or a boy is considered, the sample space is given by S = {GG, GB, BG, BB} (ii) Since the maximum number of children in each family is 2, a famil...
Read More →One die of red colour, one of white colour and one of blue colour are placed in a bag.
Question: One die of red colour, one of white colour and one of blue colour are placed in a bag. One die is selected at random and rolled, its colour and the number on its uppermost face is noted. Describe the sample space. Solution: A die has six faces that are numbered from 1 to 6, with one number on each face. Let us denote the red, white, and blue dices as R, W, and B respectively. Accordingly, when a die is selected and then rolled, the sample space is given by S = {R1, R2, R3, R4, R5, R6, ...
Read More →2 boys and 2 girls are in Room X, and 1 boy and 3 girls in Room Y.
Question: 2 boys and 2 girls are in Room X, and 1 boy and 3 girls in Room Y. Specify the sample space for the experiment in which a room is selected and then a person. Solution: Let us denote 2 boys and 2 girls in room $X$ as $B_{1}, B_{2}$ and $G_{1}, G_{2}$ respectively. Let us denote 1 boy and 3 girls in room $Y$ as $B_{3}$, and $G_{3}, G_{4}, G_{5}$ respectively. Accordingly, the required sample space is given by $S=\left\{X B_{1}, X B_{2}, X G_{1}, X G_{2}, Y B_{3}, Y G_{3}, Y G_{4}, Y G_{5...
Read More →If α and β are the zeros of the quadratic polynomial
Question: If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=x^{2}+x-2$, find the value of $\frac{1}{\alpha}-\frac{1}{\beta}$. Solution: Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomials $f(x)=x^{2}+x-2$ Sum of the zeros $=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $\alpha+\beta=-\left[\frac{1}{1}\right]$ $\alpha+\beta=-\frac{1}{1}$ $\alpha+\beta=-1$ Product if zeros $=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}...
Read More →Find the maximum profit that a company can make, if the profit function is given by
Question: Find the maximum profit that a company can make, if the profit function is given by $p(x)=41-72 x-18 x^{2}$ Solution: The profit function is given as $p(x)=41-72 x-18 x^{2}$. $\therefore p^{\prime}(x)=-72-36 x$ $\Rightarrow x=-\frac{72}{36}=-2$ Also, $p "(-2)=-360$ By second derivative test, $x=-2$ is the point of local maxima of $p$. $\therefore$ Maximum profit $=p(-2)$ $=41-72(-2)-18(-2)^{2}=41+144-72=113$ Hence, the maximum profit that the company can make is 113units.The solution g...
Read More →Describe the sample space for the indicated experiment:
Question: Describe the sample space for the indicated experiment: A coin is tossed and then a die is rolled only in case a head is shown on the coin. Solution: A coin has two faces: head (H) and tail (T). A die has six faces that are numbered from 1 to 6, with one number on each face. Thus, when a coin is tossed and then a die is rolled only in case a head is shown on the coin, the sample space is given by: S = {H1, H2, H3, H4, H5, H6, T}...
Read More →Describe the sample space for the indicated experiment: A coin is tossed and a die is thrown.
Question: Describe the sample space for the indicated experiment: A coin is tossed and a die is thrown. Solution: A coin has two faces: head (H) and tail (T). A die has six faces that are numbered from 1 to 6, with one number on each face. Thus, when a coin is tossed and a die is thrown, the sample space is given by: S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}...
Read More →Describe the sample space for the indicated experiment: A coin is tossed four times.
Question: Describe the sample space for the indicated experiment: A coin is tossed four times. Solution: When a coin is tossed once, there are two possible outcomes: head (H) and tail (T). When a coin is tossed four times, the total number of possible outcomes is $2^{4}=16$ Thus, when a coin is tossed four times, the sample space is given by: S = {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}...
Read More →Describe the sample space for the indicated experiment: A die is thrown two times.
Question: Describe the sample space for the indicated experiment: A die is thrown two times. Solution: When a die is thrown, the possible outcomes are 1, 2, 3, 4, 5, or 6. When a die is thrown two times, the sample space is given by S = {(x,y):x,y= 1, 2, 3, 4, 5, 6} The number of elements in this sample space is 6 6 = 36, while the sample space is given by: S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5),...
Read More →Describe the sample space for the indicated experiment: A coin is tossed three times.
Question: Describe the sample space for the indicated experiment: A coin is tossed three times. Solution: A coin has two faces: head (H) and tail (T). When a coin is tossed three times, the total number of possible outcomes is $2^{3}=8$ Thus, when a coin is tossed three times, the sample space is given by: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}...
Read More →If α and β are the zeros of the quadratic polynomial p(x)
Question: If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $p(x)=4 x^{2}-5 x-1$, find the value of $\alpha^{2} \beta+\alpha \beta^{2}$. Solution: Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomials $p(x)=4 x^{2}-5 x-1$ Sum of the zeros $=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $\alpha+\beta=-\left(-\frac{5}{4}\right)$ $\alpha+\beta=\frac{5}{4}$ Product of zeros $=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$ $\alpha...
Read More →The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively.
Question: The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted. Solution: Number of observations (n) = 100 Incorrect mean $(\bar{x})=20$ Incorrect standard deviation $(\sigma)=3$ $\Rightarrow 20=\frac{1}{100} \sum_{i=1}^{100} x_{i}$ $\Rightarrow \sum_{i=1}^{100} x_{j}=2...
Read More →Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
Question: Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i) $f(x)=x^{3}, x \in[-2,2]$ (ii) $f(x)=\sin x+\cos x, x \in[0, \pi]$ (iii) $f(x)=4 x-\frac{1}{2} x^{2}, x \in\left[-2, \frac{9}{2}\right]$ (iv) $f(x)=(x-1)^{2}+3, x \in[-3,1]$ Solution: (i) The given function is $f(x)=x^{3}$. $\therefore f^{\prime}(x)=3 x^{2}$ Now, $f^{\prime}(x)=0 \Rightarrow x=0$ Then, we evaluate the value offat critical pointx= 0 and at end points of ...
Read More →If α and β are the zeros of the quadratic polynomial f(x)
Question: If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $t(x)=x^{2}-x-4$, find the value of $\frac{1}{\alpha}+\frac{1}{\beta}-\alpha \beta$. Solution: Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomials $f(x)=x^{2}-x-4$ sum of the zeros $=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $\alpha+\beta=-\left[-\frac{1}{1}\right]$ $\alpha+\beta=\frac{1}{1}$ $\alpha+\beta=1$ Product if zeros $=\frac{\text { Constant term }}{\text { Coefficient...
Read More →The mean and standard deviation of marks obtained by 50 students of a class in three subjects,
Question: The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below: Which of the three subjects shows the highest variability in marks and which shows the lowest? Solution: Standard deviation of Mathematics = 12 Standard deviation of Physics = 15 Standard deviation of Chemistry = 20 The coefficient of variation (C.V.) is given by $\frac{\text { Standard deviation }}{\text { Mean }} \times 100$. C.V. (in Math...
Read More →The mean and standard deviation of 20 observations are found to be 10 and 2, respectively.
Question: The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases: (i)If wrong item is omitted. (ii)If it is replaced by 12. Solution: (i)Number of observations (n) = 20 Incorrect mean = 10 Incorrect standard deviation = 2 $\bar{x}=\frac{1}{n} \sum_{j=1}^{20} x_{i}$ $10=\frac{1}{20} \sum_{i=1}^{20} x_{i}$ $\Rightarrow...
Read More →If α and β are the zeros of the quadratic polynomial
Question: If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=6 x^{2}+x-2$, find the value of $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}$. Solution: Since $\alpha$ and $\beta$ are the zeros of the quadratics polynomial $f(x)=6 x^{2}+x-2$ sum of zeros $=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $\alpha+\beta=-\frac{1}{6}$ Product of the zeros $=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$ $\alpha \beta=-\frac{1}{3}$ We have, $\frac{...
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