If α and β are the zeroes of the quadratic polynomial
Question: If and are the zeroes of the quadratic polynomialf(x) =ax2+bx+c, then evaluate : (i) $\alpha-\beta$ (ii) $\frac{1}{\alpha}-\frac{1}{\beta}$ (iii) $\frac{1}{\alpha}+\frac{1}{\beta}-2 \alpha \beta$ (iv) $a^{2} \beta-a \beta^{2}$ (v) $\alpha^{4}+\beta^{4}$ (vi) $\frac{1}{a \alpha+b}+\frac{1}{a \beta+b}$ (vii) $\frac{\beta}{a \alpha+b}+\frac{\alpha}{a \beta+b}$ (viii) $a\left(\frac{\alpha^{2}}{\beta}+\frac{\beta^{2}}{\alpha}\right)+b\left(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\right)$ S...
Read More →Prove that the following functions do not have maxima or minima:
Question: Prove that the following functions do not have maxima or minima: (i) $f(x)=e^{x}$ (ii) $g(x)=\log x$ (iii) $h(x)=x^{3}+x^{2}+x+1$ Solution: i. We have, $f(x)=\mathrm{e}^{\mathrm{x}}$ $\therefore f^{\prime}(x)=e^{x}$ Now, if $f^{\prime}(x)=0$, then $e^{x}=0 .$ But, the exponential function can never assume 0 for any value of $x$. Therefore, there does not exist $c \in \mathbf{R}$ such that $f^{\prime}(c)=0$. Hence, functionfdoes not have maxima or minima. ii. We have, $g(x)=\log x$ $\th...
Read More →Assuming that x, y, z are positive real numbers, simplify each of the following
Question: Assuming that x, y, z are positive real numbers, simplify each of the following (i) $(\sqrt{(\mathrm{x}-3)})^{5}$ (ii) $\sqrt{x^{3} y^{-2}}$ (iii) $\left(x^{-\frac{2}{3}} y^{-\frac{1}{2}}\right)^{2}$ (iv) $(\sqrt{x})^{-\frac{2}{3}} \sqrt{y^{4}} \div \sqrt{x y^{\frac{1}{2}}}$ (v) $\sqrt[5]{243 \times 10^{5} z^{10}}$ (vi) $\left(\frac{x-4}{y-10}\right)^{\frac{5}{4}}$ (vii) $\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{5}\left(\frac{6}{7}\right)^{2}$ Solution: (i) $(\sqrt{(x-3)})^{5}$ $(\sqrt{...
Read More →Find the local maxima and local minima, if any, of the following functions.
Question: Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (i). $f(x)=x^{2}$ (ii). $g(x)=x^{3}-3 x$ (iii). $h(x)=\sin x+\cos x, 0x\frac{\pi}{2}$ (iv). $f(x)=\sin x-\cos x, 0x2 \pi$ (v). $f(x)=x^{3}-6 x^{2}+9 x+15$ (vi). $g(x)=\frac{x}{2}+\frac{2}{x}, x0$ (vii). $g(x)=\frac{1}{x^{2}+2}$ (viii). $f(x)=x \sqrt{1-x}, x0$ Solution: (i) $f(x)=x^{2}$ $\therefore f^{\prime}(x)=2 x$ Now, $f^{\prime}(x...
Read More →Given that is the mean and σ2 is the variance of n observations x1, x2 … xn.
Question: Given that $\bar{x}^{-}$is the mean and $\sigma^{2}$ is the variance of $n$ observations $x_{1}, x_{2} \ldots x_{n}$. Prove that the mean and variance of the observations $a x_{1}, a x_{2}, a x_{3} \ldots a x_{n}$ are $a \bar{x}$ and $a^{2} \sigma^{2}$, respectively $(a \neq 0)$. Solution: The given $n$ observations are $x_{1}, x_{2} \ldots x_{n}$. Mean $=\bar{x}$ Variance $=\sigma^{2}$ $\therefore \sigma^{2}=\frac{1}{n} \sum_{t=1}^{n} y_{i}\left(x_{i}-\bar{x}\right)^{2}$ $\ldots(1)$ I...
Read More →If a = xyp−1, b = xyq−1 and c = xyr−1,
Question: If $a=x y^{p-1}, b=x y^{q-1}$ and $c=x y^{r-1}$, prove that $a^{q-r} b^{r-p} c^{p-q}=1$ Solution: Given, $a=x y^{p-1}, b=x y^{q-1}$ and $c=x y^{r-1}$ To prove, $a^{q-r} b^{r-p} c^{p-q}=1$ Left hand side (LHS) = Right hand side (RHS) Considering LHS, $=a^{q-r} b^{r-p} c^{p-q} \ldots \ldots$ (i) By substituting the value of $a, b$ and $c$ in equation (i), we get $=\left(x y^{p-1}\right)^{q-r}\left(x y^{q-1}\right)^{r-p}\left(x y^{r-1}\right)^{p-q}$ $=x y^{p q-p r-q+r} x y^{q r-p q-r+p} x...
Read More →The mean and standard deviation of six observations are 8 and 4, respectively.
Question: The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations. Solution: Let the observations be $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$, and $x_{6}$. It is given that mean is 8 and standard deviation is 4. Mean, $\bar{x}=\frac{x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}}{6}=8$ If each observation is multiplied by 3 and the resulting observations areyi, then $y_{i}=3 ...
Read More →Given 4725 = 3a × 5b × 7c, find
Question: Given $4725=3 \times 5 \times 7$, find (i) The integral values of a, b and c (ii) The value of $2^{-a} \times 3^{b} \times 7^{c}$ Solution: (i) Taking out the LCM of 4725, we get $3^{3} \times 5^{2} \times 7^{1}=3^{a} \times 5^{b} \times 7^{c}$ By comparing, we get $a=3, b=2$ and $c=1$ (ii) The value of $2^{-a} \times 3^{b} \times 7^{c}$ Sol: $2^{-a} \times 3^{b} \times 7^{c}=2^{-3} \times 3^{2} \times 7^{1}$ $2^{-3} \times 3^{2} \times 7^{1}=1 / 8 \times 9 \times 7$ $63 / 8$...
Read More →The mean and variance of 7 observations are 8 and 16,
Question: The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12 and 14. Find the remaining two observations. Solution: Let the remaining two observations bexandy. The observations are 2, 4, 10, 12, 14,x,y. Mean, $\bar{x}=\frac{2+4+10+12+14+x+y}{7}=8$ $\Rightarrow 56=42+x+y$ $\Rightarrow x+y=14$ ...(1) Variance $=16=\frac{1}{n} \sum_{i=1}^{7}\left(x_{i}-\bar{x}\right)^{2}$ $16=\frac{1}{7}\left[(-6)^{2}+(-4)^{2}+(2)^{2}+(4)^{2}+(6)^{2}+x^{...
Read More →If 1176 = 2a × 3b × 7c, Find a, b, and c.
Question: If $1176=2^{a} \times 3^{b} \times 7$, Find $a$, $b$, and $c$. Solution: Given that 2,3 and 7 are factors of 1176 . Taking out the LCM of 1176 , we get $2^{3} \times 3^{1} \times 7^{2}=2^{a} \times 3^{b} \times 7^{c}$ By comparing, we get $a=3, b=1$ and $c=2$...
Read More →If 49392 = a4b2c3, find the values
Question: If $49392=a^{4} b^{2} c^{3}$, find the values of $a$, $b$ and $c$, where $a, b$ and $c$, where $a, b$, and $c$ are different positive primes. Solution: Taking out the LCM, the factors are $2^{4}, 3^{2}$ and $7^{3} a^{4} b^{2} c^{3}=2^{4}, 3^{2}$ and $7^{3}$ $a=2, b=3$ and $c=7$ [Since, $a, b$ and $c$ are primes $]$...
Read More →The mean and variance of eight observations are 9 and 9.25, respectively.
Question: The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations. Solution: Let the remaining two observations bexandy. Therefore, the observations are 6, 7, 10, 12, 12, 13,x,y. Mean, $\bar{x}=\frac{6+7+10+12+12+13+x+y}{8}=9$ $\Rightarrow 60+x+y=72$ $\Rightarrow x+y=12$ ...(1) Variance $=9.25=\frac{1}{n} \sum_{i=1}^{8}\left(x_{i}-\bar{x}\right)^{2}$ $9.25=\frac{1}{8}\left[(-3)^{2}+(-2)^...
Read More →Find the maximum and minimum values, if any, of the following functions given by
Question: Find the maximum and minimum values, if any, of the following functions given by (i) $f(x)=|x+2|-1$ (ii) $g(x)=-|x+1|+3$ (iii) $h(x)=\sin (2 x)+5$ (iv) $f(x)=|\sin 4 x+3|$ (v) $h(x)=x+1, x \in(-1,1)$ Solution: (i) $f(x)=|x+2|-1$ We know that $|x+2| \geq 0$ for every $x \in \mathbf{R}$. Therefore, $f(x)=|x+2|-1 \geq-1$ for every $x \in \mathbf{R}$. The minimum value of $f$ is attained when $|x+2|=0$. $|x+2|=0$ $\Rightarrow x=-2$ $\therefore$ Minimum value of $f=f(-2)==|-2+2|-1=-1$ Hence...
Read More →Find the zeros of each of the following quadratic polynomial and verify the relationship between the zeros and their coefficients:
Question: Find the zeros of each of the following quadratic polynomial and verify the relationship between the zeros and their coefficients: (i) $f(x)=x^{2}-2 x-8$ (ii) $g(s)=4 s^{2}-4 s+1$ (iii) $h(t)=t^{2}-15$ (iv) $6 x^{2}-3-7 x$ (v) $p(x)=x^{2}+2 \sqrt{2} x-6$ (vi) $q(x)=\sqrt{3} x^{2}+10 x+7 \sqrt{3}$ (vii) $f(x)=x^{2}-(\sqrt{3}+1) x+\sqrt{3}$ (viii) $g(x)=a\left(x^{2}+1\right)-x\left(a^{2}+1\right)$ (ix) $h(s)=2 s^{2}-(1+2 \sqrt{2}) s+\sqrt{2}$ (x) $f(v)=v^{2}+4 \sqrt{3} v-15$ (xi) $p(y)=y...
Read More →Solve the following equations for x:
Question: Solve the following equations for x: (i) $2^{2 x}-2^{x+3}+2^{4}=0$ (ii) $3^{2 x+4}+1=2 \times 3^{x+2}$ Solution: (i) We have, $\Rightarrow 2^{2 x}-2^{x+3}+2^{4}=0$ $\Rightarrow 2^{2 x}+2^{4}=2^{x} \cdot 2^{3}$ $\Rightarrow$ Let $2^{x}=y$ $\Rightarrow y^{2}+2^{4}=y \times 2^{3}$ $\Rightarrow y^{2}-8 y+16=0$ $\Rightarrow y^{2}-4 y-4 y+16=0$ $\Rightarrow y(y-4)-4(y-4)=0$ $\Rightarrow y=4$ $\Rightarrow x^{2}=2^{2}$ $\Rightarrow x=2$ (ii) We have, $3^{2 x+4}+1=2 \times 3^{x+2}$ $\left(3^{x+...
Read More →Find the zeros of each of the following quadratic polynomial and verify the relationship between the zeros and their coefficients:
Question: Find the zeros of each of the following quadratic polynomial and verify the relationship between the zeros and their coefficients: (i)f(x) =x2 2x 8 (ii)g(s) = 4s2 4s+ 1 (iii)h(t) =t2 15 (iv) 6x2 3 7x (v)p(x)=x2+22x6px=x2+22x-6 (vi)q(x)=3x2+10x+73qx=3x2+10x+73 (vii)f(x)=x2(3+1)x+3fx=x2-3+1x+3 (viii)g(x) =a(x2+ 1) x(a2+ 1) (ix)h(s)=2s2(1+22)s+2hs=2s2-1+22s+2 (x) $f(v)=v^{2}+4 \sqrt{3} v-15$ (xi) $p(y)=y^{2}+\frac{3 \sqrt{5}}{2} y-5$ (xii) $q(y)=7 y^{2}-\frac{11}{3} y-\frac{2}{3}$ ghtr(xi...
Read More →Solve the following equations for x:
Question: Solve the following equations for x: (i) $7^{2 x+3}=1$ (ii) $2^{x+1}=4^{x-3}$ (iii) $2^{5 x+3}=8^{x+3}$ (iv) $4^{2 x}=1 / 32$ (v) $4^{x-1} \times(0.5)^{3-2 x}=(1 / 8)^{x}$ (vi) $2^{3 x-7}=256$ Solution: (i) We have, $\Rightarrow 7^{2 x+3}=1$ $\Rightarrow 7^{2 x+3}=7^{0}$ $\Rightarrow 2 x+3=0$ $\Rightarrow 2 x=-3$ $\Rightarrow x=-3 / 2$ (ii) We have, $=2^{x+1}=4^{x-3}$ $=2^{x+1}=2^{2 x-6}$ $=x+1=2 x-6$ $=x=7$ (iii) We have, $=2^{5 x+3}=8^{x+3}$ $=2^{5 x+3}=2^{3 x+9}$ $=5 x+3=3 x+9$ $=2 ...
Read More →The sum and sum of squares corresponding to length x (in cm) and weight y
Question: The sum and sum of squares corresponding to lengthx(in cm) and weighty (in gm) of 50 plant products are given below: $\sum_{i=1}^{50} x_{i}=212, \sum_{i=1}^{50} x_{i}^{2}=902.8, \sum_{i=1}^{50} y_{i}=261, \sum_{i=1}^{50} y_{i}^{2}=1457.6$ Which is more varying, the length or weight? Solution: $\sum_{i=1}^{50} x_{i}=212, \sum_{i=1}^{50} x_{i}^{2}=902.8$ Here, $N=50$ $\therefore$ Mean, $\bar{x}=\frac{\sum_{i=1}^{50} y_{i}}{N}=\frac{212}{50}=4.24$ Variance $\left(\sigma_{1}^{2}\right)=\fr...
Read More →Find the maximum and minimum values, if any,
Question: Find the maximum and minimum values, if any, of the following functions given by (i) $f(x)=(2 x-1)^{2}+3$ (ii) $f(x)=9 x^{2}+12 x+2$ (iii) $f(x)=-(x-1)^{2}+10$ (iv) $g(x)=x^{3}+1$ Solution: (i) The given function is $f(x)=(2 x-1)^{2}+3$. It can be observed that $(2 x-1)^{2} \geq 0$ for every $x \in \mathbf{R}$. Therefore, $f(x)=(2 x-1)^{2}+3 \geq 3$ for every $x \in \mathbf{R}$. The minimum value of $f$ is attained when $2 x-1=0$. $2 x-1=0 \Rightarrow x=\frac{1}{2}$ $\therefore$ Minimu...
Read More →Simplify:
Question: Simplify: (i) $\frac{3^{n} \times 9^{n+1}}{3^{n-1} \times 9^{n-1}}$ (ii) $\frac{\left(5 \times 25^{\mathrm{n}+1}\right)\left(25 \times 5^{2 \mathrm{n}}\right)}{\left(5 \times 5^{2 \mathrm{n}+3}\right)-(25)^{\mathrm{n}+1}}$ (iii) $\frac{\left(5^{n+3}\right)-\left(6 \times 5^{n+1}\right)}{\left(9 \times 5^{n}\right)-\left(2^{2} \times 5^{n}\right)}$ Solution: (i) $\frac{3^{n} \times 9^{n+1}}{3^{n-1} \times 9^{n-1}}$ $=\frac{3^{\mathrm{n}} \times 9^{\mathrm{n}} \times 9}{\frac{3^{\mathrm{...
Read More →The following is the record of goals scored by team A in a football session:
Question: The following is the record of goals scored by team A in a football session: For the team B, mean number of goals scored per match was 2 with a standard deviation 1.25 goals. Find which team may be considered more consistent? Solution: The mean and the standard deviation of goals scored by team A are calculated as follows. Mean $=\frac{\sum_{i=1}^{5} f_{i} x_{i}}{\sum_{i=1}^{5} f_{i}}=\frac{50}{25}=2$ Thus, the mean of both the teams is same. $\sigma=\frac{1}{N} \sqrt{N \sum f_{i} x_{i...
Read More →The approximate change in the volume of a cube of side x metres
Question: The approximate change in the volume of a cube of sidexmetres caused by increasing the side by 3% is A. $0.06 x^{3} \mathrm{~m}^{3}$ B. $0.6 x^{3} \mathrm{~m}^{3}$ C. $0.09 x^{3} \mathrm{~m}^{3}$ D. $0.9 x^{3} \mathrm{~m}^{3}$ Solution: The volume of a cube $(V)$ of side $x$ is given by $V=x^{3}$. $\therefore d V=\left(\frac{d V}{d x}\right) \Delta x$ $=\left(3 x^{2}\right) \Delta x$ $=\left(3 x^{2}\right)(0.03 x) \quad[$ As $3 \%$ of $x$ is $0.03 x]$ $=0.09 x^{3} \mathrm{~m}^{3}$ Henc...
Read More →An analysis of monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results:
Question: An analysis of monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results: (i) Which firm A or B pays larger amount as monthly wages? (ii) Which firm, A or B, shows greater variability in individual wages? Solution: (i) Monthly wages of firm A = Rs 5253 Number of wage earners in firm A = 586 $\therefore$ Total amount paid $=$ Rs $5253 \times 586$ Monthly wages of firm B = Rs 5253 Number of wage earners in firm B = 648 $\therefore$ To...
Read More →if the
Question: If $f(x)=3 x^{2}+15 x+5$, then the approximate value of $f(3.02)$ is A.47.66B.57.66C.67.66D.77.66 Solution: Let $x=3$ and $\Delta x=0.02$. Then, we have: $f(3.02)=f(x+\Delta x)=3(x+\Delta x)^{2}+15(x+\Delta x)+5$ Now, $\Delta y=f(x+\Delta x)-f(x)$ $\begin{aligned} \Rightarrow f(x+\Delta x) =f(x)+\Delta y \\ \approx f(x)+f^{\prime}(x) \Delta x \end{aligned}$ $($ As $d x=\Delta x)$ $\Rightarrow f(3.02) \approx\left(3 x^{2}+15 x+5\right)+(6 x+15) \Delta x$ $=\left[3(3)^{2}+15(3)+5\right]+...
Read More →If abc = 1, show that
Question: If abc = 1, show that $\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}=1$ Solution: To prove, $\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}=1$ Left hand side (LHS) = Right hand side (RHS) Considering LHS, $=\frac{1}{1+a+\frac{1}{b}}+\frac{1}{1+b+\frac{1}{c}}+\frac{1}{1+c+\frac{1}{a}}$ $=\frac{b}{b+a b+1}+\frac{c}{c+b c+1}+\frac{a}{a+a c+1} \ldots .$(1) We know abc = 1 c = 1/ab By substituting the value c in equation (1), we get $=\frac{\mathrm{b}}{\math...
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