Find two numbers whose sum is 24 and whose product is as large as possible.
Question: Find two numbers whose sum is 24 and whose product is as large as possible. Solution: Let one number bex. Then, the other number is (24 x). LetP(x) denote the product of the two numbers. Thus, we have: $P(x)=x(24-x)=24 x-x^{2}$ $\therefore P^{\prime}(x)=24-2 x$ $P^{\prime \prime}(x)=-2$ $P^{n}(x)=-2$ Now, $P^{\prime}(x)=0 \Rightarrow x=12$ Also, $P^{\prime \prime}(12)=-20$ $\therefore$ By second derivative test, $x=12$ is the point of local maxima of $P$. Hence, the product of the numb...
Read More →If the sum of the zeros of the quadratic polynomial f(t)
Question: If the sum of the zeros of the quadratic polynomial $l(t)=k t^{2}+2 t+3 k$ is equal to their product, find the value of $k$. Solution: Let $\alpha, \beta$ be the zeros of the polynomial $f(t)=k t^{2}+2 t+3 k$. Then, $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $\alpha+\beta=\frac{-2}{k}$ $\alpha \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$ $\alpha \beta=\frac{3 k}{k}$ $\alpha \beta=3$ It is given that the sum of the zero of ...
Read More →Two dice are thrown. The events A, B and C are as follows:
Question: Two dice are thrown. The events A, B and C are as follows: A: getting an even number on the first die. B: getting an odd number on the first die. C: getting the sum of the numbers on the dice 5 State true or false: (give reason for your answer) (i) A and B are mutually exclusive (ii) A and B are mutually exclusive and exhaustive (iii) $\mathrm{A}=\mathrm{B}^{\prime}$ (iv) $\mathrm{A}$ and $\mathrm{C}$ are mutually exclusive (v) $A$ and $B^{\prime}$ are mutually exclusive (vi) $\mathrm{...
Read More →If the squared difference of the zeros of the quadratic polynomial f(x)
Question: If the squared difference of the zeros of the quadratic polynomial $1(x)=x^{2}+p x+45$ is equal to 144, find the value of $p$. Solution: Given $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=x^{2}+p x+45$ $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $=\frac{-p}{1}$ = -p $\alpha \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$ $=\frac{45}{1}$ =45 We have, $(\alpha-\beta)^{2}=\alpha^{2}+\beta^{2}-2 \alpha \bet...
Read More →Find the maximum and minimum values of x + sin 2x on [0, 2π].
Question: Find the maximum and minimum values ofx+ sin 2xon [0, 2]. Solution: Letf(x) =x+ sin 2x. $\therefore f^{\prime}(x)=1+2 \cos 2 x$ Now, $f^{\prime}(x)=0 \Rightarrow \cos 2 x=-\frac{1}{2}=-\cos \frac{\pi}{3}=\cos \left(\pi-\frac{\pi}{3}\right)=\cos \frac{2 \pi}{3}$ $2 x=2 n \pi \pm \frac{2 \pi}{3}, n \in \mathrm{Z}$ $\Rightarrow x=n \pi \pm \frac{\pi}{3}, n \in \mathrm{Z}$ $\Rightarrow x=\frac{\pi}{3}, \frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{5 \pi}{3} \in[0,2 \pi]$ Then, we evaluate the va...
Read More →Prove that
Question: Prove that (i) $\left(\sqrt{3 \times 5^{-3}} \div \sqrt[3]{3^{-1}} \sqrt{5}\right) \times \sqrt[5]{3 \times 5^{6}}=\frac{3}{5}$ (ii) $9^{3 / 2}-3 \times 5^{0}-(1 / 81)^{-1 / 2}$ (iii) $\frac{1^{2}}{4}-3 \times 8^{\frac{2}{3}} \times 4^{0}+\left(\frac{9}{16}\right)^{-\frac{1}{2}}$ (iv) $\frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 4^{\frac{1}{4}}}{10^{-\frac{1}{5}} \times 5^{\frac{3}{5}}} \div \frac{4^{\frac{4}{3}} \times 5^{-\frac{7}{5}}}{4^{-\frac{3}{5}} \times 6}$ (v) $\sqrt{\...
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 $l(x)=x^{2}-p x+q$, prove that $\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}=\frac{p^{2}}{q^{2}}-\frac{4 p^{2}}{q}+2$. Solution: $\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}=\frac{\left[(p)^{2}-2 q\right]^{2}-2(q)^{2}}{q^{2}}$Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=x^{2}-p x+q$ $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $...
Read More →Two dice are thrown. The events A, B and C are as follows:
Question: Two dice are thrown. The events A, B and C are as follows: A: getting an even number on the first die. B: getting an odd number on the first die. C: getting the sum of the numbers on the dice 5 Describe the events (i) $A^{\prime}$ (ii) not $B$ (iii) $A$ or $B$ (iv) $A$ and $B$ (v) $A$ but not $C$ (vi) $B$ or $C$ (vii) $\mathrm{B}$ and $\mathrm{C}$ (viii) $\mathrm{A} \cap \mathrm{B}^{\prime} \cap \mathrm{C}^{\prime}$ Solution: When two dice are thrown, the sample space is given by $\mat...
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 $l(x)=x^{2}-p x+q$, prove that $\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}=\frac{p^{2}}{q^{2}}-\frac{4 p^{2}}{q}+2$. Solution: $\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}=\frac{\left[(p)^{2}-2 q\right]^{2}-2(q)^{2}}{q^{2}}$Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=x^{2}-p x+q$ $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $...
Read More →It is given that at the function
Question: It is given that at $x=1$, the function $x^{4}-62 x^{2}+a x+9$ attains its maximum value, on the interval $[0,2]$. Find the value of $a$. Solution: Let $f(x)=x^{4}-62 x^{2}+a x+9$ $\therefore f^{\prime}(x)=4 x^{3}-124 x+a$ It is given that functionfattains its maximum value on the interval [0, 2] atx= 1. $\therefore f^{\prime}(1)=0$ $\Rightarrow 4-124+a=0$ $\Rightarrow a=120$ Hence, the value ofais 120....
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 $l(x)=x^{2}-p x+q$, prove that $\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}=\frac{p^{2}}{q^{2}}-\frac{4 p^{2}}{q}+2$. Solution: Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=x^{2}-p x+q$ $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $=\frac{-(-p)}{1}$ = p $\alpha \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$ $=\frac{q}{...
Read More →Three coins are tossed. Describe
Question: Three coins are tossed. Describe (i) Two events which are mutually exclusive. (ii) Three events which are mutually exclusive and exhaustive. (iii) Two events, which are not mutually exclusive. (iv) Two events which are mutually exclusive but not exhaustive. (v) Three events which are mutually exclusive but not exhaustive. Solution: When three coins are tossed, the sample space is given by S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} (i) Two events that are mutually exclusive can be A: ...
Read More →Find the maximum value of
Question: Find the maximum value of $2 x^{3}-24 x+107$ in the interval $[1,3]$. Find the maximum value of the same function in $[-3,-1]$. Solution: Let $f(x)=2 x^{3}-24 x+107$ $\therefore f^{\prime}(x)=6 x^{2}-24=6\left(x^{2}-4\right)$ Now, $f^{\prime}(x)=0 \Rightarrow 6\left(x^{2}-4\right)=0 \Rightarrow x^{2}=4 \Rightarrow x=\pm 2$ We first consider the interval [1, 3]. Then, we evaluate the value offat the critical pointx= 2 [1, 3] and at the end points of the interval [1, 3]. $f(2)=2(8)-24(2)...
Read More →Three coins are tossed once.
Question: Three coins are tossed once. Let A denote the event three heads show, B denote the event two heads and one tail show. C denote the event three tails show and D denote the event a head shows on the first coin. Which events are (i) mutually exclusive? (ii) simple? (iii) compound? Solution: When three coins are tossed, the sample space is given by S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Accordingly, A = {HHH} B = {HHT, HTH, THH} C = {TTT} D = {HHH, HHT, HTH, HTT} We now observe that ...
Read More →If α and β are the zeros of the quadratic polynomial p(s)
Question: If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $p(s)=3 s^{2}-6 s+4$, find the value of $\frac{\alpha}{\beta}+\frac{\beta}{\mathrm{a}}+2\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)+3 \alpha \beta .$ Solution: Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $p(s)=3 s^{2}-6 s+4$ $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $\alpha+\beta=\frac{-(-6)}{3}$ $\alpha+\beta=2$ $\alpha \beta=\frac{\text { Constant term ...
Read More →An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:
Question: An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: A: the sum is greater than 8, B: 2 occurs on either die C: The sum is at least 7 and a multiple of 3. Which pairs of these events are mutually exclusive? Solution: When a pair of dice is rolled, the sample space is given by $\mathrm{S}=\{(x, y): x, y=1,2,3,4,5,6\}$ $=\left\{\begin{array}{lllll}(1,1), (1,2), (1,3), (1,4), (1,5), (1,6) \\ (2,1), (2,2), (2,3), (2,4), (2,5),...
Read More →What is the maximum value of the function sin x + cos x?
Question: What is the maximum value of the function sinx+ cosx? Solution: Letf(x) = sinx+ cosx. $\therefore f^{\prime}(x)=\cos x-\sin x$ $f^{\prime}(x)=0 \Rightarrow \sin x=\cos x \Rightarrow \tan x=1 \Rightarrow x=\frac{\pi}{4}, \frac{5 \pi}{4} \ldots$, $f^{\prime \prime}(x)=-\sin x-\cos x=-(\sin x+\cos x)$ Now, $f^{\prime \prime}(x)$ will be negative when $(\sin x+\cos x)$ is positive i.e., when $\sin x$ and $\cos x$ are both positive. Also, we know that $\sin x$ and $\cos x$ both are positive...
Read More →A die is thrown. Describe the following events:
Question: A die is thrown. Describe the following events: (i) A: a number less than 7 (ii) B: a number greater than 7 (iii) C: a multiple of 3 (iv) D: a number less than 4 (v) E: an even number greater than 4 (vi) F: a number not less than 3 Also find $\mathrm{A} \cup \mathrm{B}, \mathrm{A} \cap \mathrm{B}, \mathrm{B} \cup \mathrm{C}, \mathrm{E} \cap \mathrm{F}, \mathrm{D} \cap \mathrm{E}, \mathrm{A}-\mathrm{C}, \mathrm{D}-\mathrm{E}, \mathrm{E} \cap \mathrm{F}^{\prime}, \mathrm{F}^{\prime}$ Sol...
Read More →A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even number”
Question: A die is rolled. Let E be the event die shows 4 and F be the event die shows even number. Are E and F mutually exclusive? Solution: When a die is rolled, the sample space is given by S = {1, 2, 3, 4, 5, 6} Accordingly, E = {4} and F = {2, 4, 6} It is observed that E F = {4} Φ Therefore, E and F are not mutually exclusive events....
Read More →Simplify
Question: Simplify (i) $\left(16^{-\frac{1}{5}}\right)^{\frac{5}{2}}$ (ii) $\sqrt[5]{(32)^{-3}}$ (iii) $\sqrt[3]{(343)^{-2}}$ (iv) $\frac{(25)^{\frac{3}{2}} \times(243)^{\frac{3}{5}}}{(16)^{\frac{5}{4}} \times(8)^{\frac{4}{3}}}$ (v) $\left(\frac{\sqrt{2}}{5}\right)^{8} \div\left(\frac{\sqrt{2}}{5}\right)^{13}$ (vi) $\left[\frac{5^{-1} \times 7^{2}}{5^{2} \times 7^{-4}}\right]^{\frac{7}{2}} \times\left[\frac{5^{-2} \times 7^{3}}{5^{3} \times 7^{-5}}\right]^{\frac{-5}{2}}$ Solution: (i) $\left(16^...
Read More →A die is thrown repeatedly until a six comes up.
Question: A die is thrown repeatedly until a six comes up. What is the sample space for this experiment? Solution: In this experiment, six may come up on the first throw, the second throw, the third throw and so on till six is obtained. Hence, the sample space of this experiment is given by S = {6, (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (1, 1, 6), (1, 2, 6), , (1, 5, 6), (2, 1, 6), (2, 2, 6), , (2, 5, 6), ,(5, 1, 6), (5, 2, 6), }...
Read More →A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls.
Question: A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls. If it shows head, we throw a die. Find the sample space for this experiment. Solution: The box contains 2 red balls and 3 black balls. Let us denote the 2 red balls as R1, R2and the 3 black balls as B1, B2, and B3. The sample space of this experiment is given by S = {TR1, TR2, TB1, TB2, TB3, H1, H2, H3, H4, H5, H6}...
Read More →If α and β are the zeros of the quadratic polynomial p(y)
Question: If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $p(y)=5 y^{2}-7 y+1$, find the value of $\frac{1}{\alpha}+\frac{1}{\beta}$. Solution: Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $p(y)=5 y^{2}-7 y+1$ $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $=-\frac{(-7)}{5}$ $=\frac{7}{5}$ $\alpha \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$ $=\frac{1}{5}$ We have, $\frac{1}{\alpha}+\frac{1}{\be...
Read More →An experiment consists of rolling a die and then tossing a coin once if the number on the die is even.
Question: An experiment consists of rolling a die and then tossing a coin once if the number on the die is even. If the number on the die is odd, the coin is tossed twice. Write the sample space for this experiment. Solution: A die has six faces that are numbered from 1 to 6, with one number on each face. Among these numbers, 2, 4, and 6 are even numbers, while 1, 3, and 5 are odd numbers. A coin has two faces: head (H) and tail (T). Hence, the sample space of this experiment is given by: S = {2...
Read More →At what points in the interval [0, 2π],
Question: At what points in the interval [0, 2], does the function sin 2xattain its maximum value? Solution: Let $f(x)=\sin 2 x$. $\therefore f^{\prime}(x)=2 \cos 2 x$ Now, $f^{\prime}(x)=0 \Rightarrow \cos 2 x=0$ $\Rightarrow 2 x=\frac{\pi}{2}, \frac{3 \pi}{2}, \frac{5 \pi}{2}, \frac{7 \pi}{2}$ $\Rightarrow x=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$ Then, we evaluate the values of $f$ at critical points $x=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}...
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