Find the cubic polynomial with the sum,
Question: Find the cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and product of its zeros as 3, 1 and 3 respectively. Solution: If $\alpha, \beta$ and $\gamma$ are the zeros of a cubic polynomial $f(x)$, then $f(x)=k\left\{x^{3}-(\alpha+\beta+\gamma) x^{2}+(\alpha \beta+\beta \gamma+\gamma \alpha) x-\alpha \beta \gamma\right\}$ where $k$ is any non-zero real number. Here, $\alpha+\beta+\gamma=3$ $\alpha \beta+\beta \gamma+\gamma \alpha=-1$ $\alpha \beta \gam...
Read More →In an entrance test that is graded on the basis of two examinations,
Question: In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of passing at least one of them is 0.95. What is the probability of passing both? Solution: Let A and B be the events of passing first and second examinations respectively. Accordingly, P(A) = 0.8, P(B) = 0.7 and P(A or B) = 0.95 We know that P(A or B) = P(A) +...
Read More →If 3x+1 = 9x-2, find the value of 21+x
Question: If $3^{x+1}=9^{x-2}$, find the value of $2^{1+x}$ Solution: $3^{x+1}=9^{x-2}$ $3^{x+1}=3^{2 x-4}$ x + 1 = 2x - 4 x = 5 Therefore the value of $2^{1+x}=2^{1+5}=2^{6}=64$...
Read More →Prove that the volume of the largest cone that can be inscribed in a sphere of radius
Question: Prove that the volume of the largest cone that can be inscribed in a sphere of radius $R$ is $\frac{8}{27}$ of the volume of the sphere. Solution: Letrandhbe the radius and height of the cone respectively inscribed in a sphere of radiusR. Let $V$ be the volume of the cone. Then, $V=\frac{1}{3} \pi r^{2} h$ Height of the cone is given by, $h=R+\mathrm{AB}=R+\sqrt{R^{2}-r^{2}} \quad[\mathrm{ABC}$ is a right triangle $]$ $\begin{aligned} \therefore V =\frac{1}{3} \pi r^{2}\left(R+\sqrt{R^...
Read More →In Class XI of a school 40% of the students study Mathematics and 30% study Biology.
Question: In Class XI of a school 40% of the students study Mathematics and 30% study Biology. 10% of the class study both Mathematics and Biology. If a student is selected at random from the class, find the probability that he will be studying Mathematics or Biology. Solution: Let A be the event in which the selected student studies Mathematics and B be the event in which the selected student studies Biology. Accordingly, $P(A)=40 \%=\frac{40}{100}=\frac{2}{5}$ $P(B)=30 \%=\frac{30}{100}=\frac{...
Read More →Verify that the numbers given along side of the cubic polynomials below are their zeros.
Question: Verify that the numbers given along side of the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and coefficients in each case: (i) $f(x)=2 x^{3}+x^{2}-5 x+2 ; \frac{1}{2}, 1,-2$ (ii) $g(x)=x^{3}-4 x^{2}+5 x-2 ; 2,1,1$ Solution: We have, $f(x)=2 x^{3}+2 x^{2}-5 x+2$ $f\left(\frac{1}{2}\right)=2\left(\frac{1}{2}\right)^{3}+\left(\frac{1}{2}\right)^{2}-5\left(\frac{1}{2}\right)+2$ $f\left(\frac{1}{2}\right)=\frac{1}{4}+\frac{1}{4}-\frac{5}{2}+2$ $f...
Read More →Determine (8x)x, if 9x+2 = 240 + 9x.
Question: Determine $(8 x)^{x}$, if $9^{x+2}=240+9^{x}$. Solution: $9^{x+2}=240+9^{x}$ $9^{x} \cdot 9^{2}=240+9^{x}$ Let $9^{x}$ be $y$ 81y = 240 + y 81y - y = 240 80y = 240 y = 3 Since, y = 3 Then, $9^{x}=3$ $3^{2 x}=3$ Therefore, $x=1 / 2$ $(8 x)^{x}=(8 \times 1 / 2)^{1 / 2}$ $=(4)^{1 / 2}$ $=2$ Therefore $(8 x)^{x}=2$...
Read More →A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16.
Question: A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16. Determine (i) P(not A), (ii) P (not B) (iii) P(A or B). Solution: It is given that P(A) = 0.42, P(B) = 0.48, P(A and B) = 0.16 (i) P(not A) = 1 P(A) = 1 0.42 = 0.58 (ii) P(not B) = 1 P(B) = 1 0.48 = 0.52 (iii) We know that P(A or B) = P(A) + P(B) P(A and B) $\therefore P(A$ or $B)=0.42+0.48-0.16=0.74$...
Read More →Verify that the numbers given along side of the cubic polynomials below are their zeros.
Question: Verify that the numbers given along side of the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and coefficients in each case: (i) $f(x)=2 x^{3}+x^{2}-5 x+2 ; \frac{1}{2}, 1,-2$ (ii) $g(x)=x^{3}-4 x^{2}+5 x-2 ; 2,1,1$ Solution: We have, $f(x)=2 x^{3}+2 x^{2}-5 x+2$ $f\left(\frac{1}{2}\right)=2\left(\frac{1}{2}\right)^{3}+\left(\frac{1}{2}\right)^{2}-5\left(\frac{1}{2}\right)+2$ $f\left(\frac{1}{2}\right)=\frac{1}{4}+\frac{1}{4}-\frac{5}{2}+2$ $f...
Read More →Events E and F are such that P(not E or not F) = 0.25,
Question: Events E and F are such that P(not E or not F) = 0.25, State whether E and F are mutually exclusive. Solution: It is given thatP (not E or not F) = 0.25 i.e., $P\left(E^{\prime} \cup F^{\prime}\right)=0.25$ $\Rightarrow \mathrm{P}(\mathrm{E} \cap \mathrm{F})^{\prime}=0.25 \quad\left[\mathrm{E}^{\prime} \cup \mathrm{F}^{\prime}=(\mathrm{E} \cap \mathrm{F})^{\prime}\right]$ Now, $P(E \cap F)=1-P(E \cap F)^{\prime}$ $\Rightarrow P(E \cap F)=1-0.25$ $\Rightarrow P(E \cap F)=0.75 \neq 0$ $\...
Read More →If x = 21/3 + 22/3, show that x3 − 6x = 6
Question: If $x=2^{1 / 3}+2^{2 / 3}$, show that $x^{3}-6 x=6$ Solution: $x^{3}-6 x=6$ $x=2^{1 / 3}+2^{2 / 3}$ Putting cube on both the sides, we get...
Read More →If E and F are events such that P(E) =, P(F) = and P(E and F) =, find:(i) P(E or F), (ii) P(not E and not F).
Question: If $E$ and $F$ are events such that $P(E)=\frac{1}{4}, P(F)=\frac{1}{2}$ and $P(E$ and $F)=\frac{1}{8}$, find:(i) $P(E$ or $F)$, (ii) $P($ not $E$ and not $F)$. Solution: Here, $P(E)=\frac{1}{4}, P(F)=\frac{1}{2}$, and $P(E$ and $F)=\frac{1}{8}$ (i) We know that $P(E$ or $F)=P(E)+P(F)-P(E$ and $F)$ $\therefore P(E$ or $F)=\frac{1}{4}+\frac{1}{2}-\frac{1}{8}=\frac{2+4-1}{8}=\frac{5}{8}$ (ii) From (i), $P(E$ or $F)=P(E \cup F)=\frac{5}{8}$ We have $(E \cup F)^{\prime}=\left(E^{\prime} \c...
Read More →A wire of length 28 m is to be cut into two pieces.
Question: A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum? Solution: Let a piece of lengthlbe cut from the given wire to make a square. Then, the other piece of wire to be made into a circle is of length (28 l) m. Now, side of square $=\frac{l}{4}$. Let $r$ be the radius of the circle. Then, $2 \pi r=28-l \Rig...
Read More →Given P(A) =and P(B) =. Find P(A or B), if A and B are mutually exclusive events.
Question: Given $P(A)=\frac{3}{5}$ and $P(B)=\frac{1}{5} .$ Find $P(A$ or $B)$, if $A$ and $B$ are mutually exclusive events. Solution: Here, $P(A)=\frac{3}{5}, P(B)=\frac{1}{5}$ For mutually exclusive events A and B, P(A or B) = P(A) + P(B) $\therefore P(A$ or $B)=\frac{3}{5}+\frac{1}{5}=\frac{4}{5}$...
Read More →Of all the closed cylindrical cans (right circular),
Question: Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area? Solution: Letrandhbe the radius and height of the cylinder respectively. Then, volume (V)of the cylinder is given by, $V=\pi r^{2} h=100$(given) $\therefore h=\frac{100}{\pi r^{2}}$ Surface area (S)of the cylinder is given by, $S=2 \pi r^{2}+2 \pi r h=2 \pi r^{2}+\frac{200}{r}$ $\therefore \frac{d S}{d r}=4 \pi r-\frac{200}{...
Read More →Find the values of x in each of the following
Question: Find the values of x in each of the following Solution: We have $2^{5 x} \div 2^{x}=\sqrt[5]{\left(2^{20}\right)}$ $=\frac{2^{5 x}}{2^{x}}=\left(2^{20}\right)^{\frac{1}{5}}$ $=2^{5 x-x}=2^{20 \times \frac{1}{5}}$ $=2^{4 x}=2^{4}$ = 4x = 4 [On equating exponent] x = 1 Hence the value of x is 1 (ii) $\left(2^{3}\right)^{4}=\left(2^{2}\right)^{x}$ We have $\left(2^{3}\right)^{4}=\left(2^{2}\right)^{x}$ $=2^{3 \times 4}=2^{2 \times x}$ 12 = 2x 2x = 12 [On equating exponents] x = 6 Hence th...
Read More →Show that the right circular cylinder of given surface and maximum volume
Question: Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base. Solution: Letrandhbe the radius and height of the cylinder respectively. Then, the surface area (S)of the cylinder is given by, $\begin{aligned} S= 2 \pi r^{2}+2 \pi r h \\ \Rightarrow h =\frac{S-2 \pi r^{2}}{2 \pi r} \\ =\frac{S}{2 \pi}\left(\frac{1}{r}\right)-r \end{aligned}$ LetVbe the volume of the cylinder. Then, $V=\pi r^{2} h=\pi r^{2}\left[\fra...
Read More →Show that of all the rectangles inscribed in a given fixed circle,
Question: Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area. Solution: Let a rectangle of lengthland breadthbbe inscribed in the given circle of radiusa. Then, the diagonal passes through the centre and is of length 2acm. Now, by applying the Pythagoras theorem, we have: $(2 a)^{2}=l^{2}+b^{2}$ $\Rightarrow b^{2}=4 a^{2}-l^{2}$ $\Rightarrow b=\sqrt{4 a^{2}-l^{2}}$ $\therefore$ Area of the rectangle, $A=I \sqrt{4 a^{2}-l^{2}}$ $\therefore \frac{d A...
Read More →If α and β are the zeros of the polynomial f(x)
Question: If $\alpha$ and $\beta$ are the zeros of the polynomial $1(x)=x^{2}+p x+q$, from a polynomial whose zeros are $(\alpha+\beta)^{2}$ and $(\alpha-\beta)^{2}$. Solution: If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=x^{2}+p x+q$ $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $=\frac{-p}{1}$ $\alpha \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$ $=\frac{q}{1}$ = q Let $S$ and $P$ denote respectively the sum...
Read More →Fill in the blanks in following table:
Question: Fill in the blanks in following table: Solution: (i) Here, $P(A)=\frac{1}{3}, P(B)=\frac{1}{5}, P(A \cap B)=\frac{1}{15}$ We know that $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$ $\therefore \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{1}{3}+\frac{1}{5}-\frac{1}{15}=\frac{5+3-1}{15}=\frac{7}{15}$ (ii) Here, $P(A)=0.35, P(A \cap B)=0.25, P(A \cup B)=0.6$ We know that $P(A \cup B)=P(A)+P(B)-P(A \cap B)$ $\t...
Read More →A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top,
Question: A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible? Solution: Let the side of the square to be cut off bexcm. Then, the height of the box isx, the length is 45 2x,and the breadth is 24 2x. Therefore, the volumeV(x) of the box is given by, $\begin{aligned} V(x) =x(45-2 x)(24-2 x) \\ =x\left...
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 $f(x)=x^{2}-2 x+3$, find a polynomial whose roots are $(i) \alpha+2, \beta+2$ (ii) $\frac{\alpha-1}{\alpha+1}, \frac{\beta-1}{\beta+1}$. Solution: (i) Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=x^{2}-2 x+3$ $\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$ $=\frac{-(-2)}{1}$ = 2 Product of the zeros $=\frac{\text { Constant term }}{\text { Coefficient of } x...
Read More →Check whether the following probabilities P(A) and P(B) are consistently defined
Question: Check whether the following probabilities P(A) and P(B) are consistently defined (i) P(A) = 0.5, P(B) = 0.7, P(A B) = 0.6 (ii) P(A) = 0.5, P(B) = 0.4, P(A B) = 0.8 Solution: (i) P(A) = 0.5, P(B) = 0.7, P(A B) = 0.6 It is known that if E and F are two events such that E F, then P(E) P(F). However, here, P(A B) P(A). Hence, P(A) and P(B) are not consistently defined. (ii)P(A) = 0.5, P(B) = 0.4, P(A B) = 0.8 It is known that if E and F are two events such that E F, then P(E) P(F). Here, i...
Read More →In a lottery, person choses six different natural numbers at random from 1 to 20,
Question: In a lottery, person choses six different natural numbers at random from 1 to 20, and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? [Hint:order of the numbers is not important.] Solution: Total number of ways in which one can choose six different numbers from 1 to $20={ }^{20} \mathrm{C}_{6}=\frac{\lfloor 20}{|6| 20-6}=\frac{\mid 20}{|6| 14}=\frac{20 \times 19 \times 1...
Read More →A letter is chosen at random from the word ‘ASSASSINATION’.
Question: A letter is chosen at random from the word ASSASSINATION. Find the probability that letter is (i) a vowel (ii) an consonant Solution: There are 13 letters in the word ASSASSINATION. $\therefore$ Hence, $n(S)=13$ (i) There are 6 vowels in the given word. $\therefore$ Probability (vowel) $=\frac{6}{13}$ (ii) There are 7 consonants in the given word. $\therefore$ Probability (consonant) $=\frac{7}{13}$...
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