If pth, qth, and rth terms of an A.P. and G.P.
Question: If pth, qth, and rthterms of an A.P. and G.P. are both a, b and c respectively, show that abc. bc a. ca b = 1 Solution: Let the first term of AP be $m$ and common difference as $d$ Let the GP first term as I and common ratio as s The $\mathrm{n}^{\text {th }}$ term of an AP is given as $\mathrm{t}_{\mathrm{n}}=\mathrm{a}+(\mathrm{n}-1) \mathrm{d}$ where $\mathrm{a}$ is the first term and $\mathrm{d}$ is the common difference The $n^{\text {th }}$ term of $a$ GP is given by $t_{n}=a r^{...
Read More →Find the maximum and minimum values
Question: Find the maximum and minimum values of $y=\tan x-2 x$. Solution: Given : $f(x)=y=\tan x-2 x$ $\Rightarrow f^{\prime}(x)=\sec ^{2} x-2$ For a local maxima or local minima, we must have $f^{\prime}(x)=0$ $\Rightarrow \sec ^{2} x-2=0$ $\Rightarrow \sec ^{2} x=2$ $\Rightarrow \sec x=\pm \sqrt{2}$ $\Rightarrow x=\frac{\pi}{4}$ and $\frac{3 \pi}{4}$ Thus, $x=\frac{\pi}{4}$ and $x=\frac{3 \pi}{4}$ are the possible points of local maxima or a local minima. Now, $f^{\prime \prime}(x)=2 \sec ^{2...
Read More →If the sum of p terms of an A.P. is q and
Question: If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is (p + q). Also, find the sum of first p q terms (p q). Solution: The sum of $n$ terms of an $A P$ is given by $\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}(2 \mathrm{a}+(\mathrm{n}-1) \mathrm{d})$ Where $a$ is the first term and $d$ is the common difference Given that $S_{p}=q$ and $S_{q}=p$ $\Rightarrow \mathrm{S}_{\mathrm{p}}=\frac{\mathrm{p}}{2}(2 \mathrm{a}+(\mathrm{p}-1) \mathrm{d...
Read More →Solve this
Question: If $(-1,3)$ and $(\alpha, \beta)$ are the extremities of the diameter of the circle $x^{2}+y^{2}-$ $6 x+5 y-7=0$, find the coordinates $(\alpha, \beta)$ Solution: Given $x^{2}+y^{2}-6 x+5 y-7=0$ Centre $\left(3,-\frac{5}{2}\right)$ As $(-1,3) \(\alpha, \beta)$ are the 2 extremities of the diameter, using mid - point formula we can write $\frac{\propto-1}{2}=3$ $\Rightarrow \propto=7$ and $\frac{\beta+3}{2}=-\frac{5}{2}$ $\Rightarrow \beta=-8$ $(\alpha, \beta)=(7,-8)$...
Read More →Show that the quadrilateral formed by the straight lines
Question: Show that the quadrilateral formed by the straight lines x y = 0, 3x + 2y = 5, x y = 10 and 2x + 3y = 0 is cyclic and hence find the equation of the circle. Solution: Solving the euations we get the coordinates of the quadrilateral. Slope of $\mathrm{AB}=\frac{1-0}{1-0}=1$ Slope of $C D=1$ $\mathrm{AB} \| \mathrm{CD}$ Slope of $B D=A C=-1$ $A C \| B D$ So they form a rectangle with all sides $=90^{\circ}$ The quadrilateral is cyclic as sum of opposite angles $=180^{\circ}$. Now, $A D=$...
Read More →If θ1, θ2, θ3, …, θn are in A.P.,
Question: If 1, 2, 3, , nare in A.P., whose common difference is d, show that Sec 1sec 2+ sec 2sec 3+ + sec n1sec n $=\frac{\tan \theta_{\mathrm{n}}-\tan \theta_{1}}{\sin \mathrm{d}}$ Solution: Given $\theta_{1}, \theta_{2}, \theta_{3}, \ldots, \theta_{n}$ are in A.P., and common difference is $d$, Now we have to prove that $\sec \theta_{1} \sec \theta_{2}+\sec \theta_{2} \sec \theta_{3}+\ldots+\sec \theta_{n-1} \sec \theta_{n}=\frac{\tan \theta_{n}-\tan \theta_{1}}{\sin d}$ On cross multiplicat...
Read More →Find the equation of the circle circumscribing the triangle formed by the lines
Question: Find the equation of the circle circumscribing the triangle formed by the lines x + $y=6,2 x+y=4$ and $x+2 y=5$ Solution: Solving the equations we get the coordinates of the triangle: The required circle equation $\left|\begin{array}{cccc}x^{2}+y^{2} x y 1 \\ (-2)^{2}+8^{2} -2 8 1 \\ 1^{2}+2^{2} 1 2 1 \\ 7^{2}+(-1)^{2} 7 -1 1\end{array}\right|=0$ Using Laplace Expansion $\left(x^{2}+y^{2}\right)\left|\begin{array}{ccc}-2 8 1 \\ 1 2 1 \\ 7 -1 1\end{array}\right|-x\left|\begin{array}{ccc...
Read More →Find the maximum and minimum values
Question: Find the maximum and minimum values of the function $f(x)=\frac{4}{x+2}+x$. Solution: Given : $f(x)=\frac{4}{x+2}+x$ $\Rightarrow f^{\prime}(x)=-\frac{4}{(x+2)^{2}}+1$ For a local maxima or a local minima, we must have $f^{\prime}(x)=0$ $\Rightarrow-\frac{4}{(x+2)^{2}}+1=0$ $\Rightarrow-\frac{4}{(x+2)^{2}}=-1$ $\Rightarrow(x+2)^{2}=4$ $\Rightarrow x+2=\pm 2$ $\Rightarrow x=0$ and $-4$ Thus, $x=0$ and $x=-4$ are the possible points of local maxima or local minima. Now, $f^{\prime \prime...
Read More →Find the equation of a circle passing through the origin and intercepting
Question: Find the equation of a circle passing through the origin and intercepting lengths a and b on the axes. Solution: From the figure AD = b units and AE = a units. D(0, b), E(a, 0) and A(0, 0) lies on the circle. C is the centre. The general equation of a circle: $(x-h)^{2}+(y-k)^{2}=r^{2}$ $\ldots(\mathrm{i})$, where $(\mathrm{h}, \mathrm{k})$ is the centre and $\mathrm{r}$ is the radius. Putting $A(0,0)$ in (i) $(0-h)^{2}+(0-k)^{2}=r^{2}$ $\Rightarrow \mathrm{h}^{2}+\mathrm{k}^{2}=\mathr...
Read More →If A is the arithmetic mean and G1,
Question: If A is the arithmetic mean and G1, G2 be two geometric means between any two numbers, then prove that $2 \mathrm{~A}=\frac{\mathrm{G}_{1}^{2}}{\mathrm{G}_{2}}+\frac{\mathrm{G}_{2}^{2}}{\mathrm{G}_{1}}$ Solution: Given $\mathrm{A}$ is the arithmetic mean and $\mathrm{G}_{1}, \mathrm{G}_{2}$ be two geometric means between any two numbers Let the two numbers be ' $\mathrm{a}$ ' and ' $\mathrm{b}$ ' The arithmetic mean is given by $A=\frac{a+b}{2}$ and the geometric mean is given by $\mat...
Read More →Show that
Question: Show that $\frac{\log x}{x}$ has a maximum value at $x=e$. Solution: Here, $f(x)=\frac{\log x}{x}$ $\Rightarrow f^{\prime}(x)=\frac{1-\log x}{x^{2}}$ For the local maxima or minima, we must have $f^{\prime}(x)=0$ $\Rightarrow \frac{1-\log x}{x^{2}}=0$ $\Rightarrow 1=\log x$ $\Rightarrow \log e=\log x$ $\Rightarrow x=e$ Now, $f^{\prime \prime}(x)=\frac{x^{2}\left(\frac{-1}{x}\right)-2 x(1-\log x)}{x^{4}}=\frac{-3+2 \log x}{x^{3}}$ $\Rightarrow f^{\prime \prime}(e)=\frac{-3+2 \log e}{e^{...
Read More →The function
Question: The function $y=a \log x+b x^{2}+x$ has extreme values at $x=1$ and $x=2$. Find $a$ and $b$ Solution: Given : $f(x)=y=a \log x+b x^{2}+x$ $\Rightarrow f^{\prime}(x)=\frac{a}{x}+2 b x+1$ Since, $f^{\prime}(x)$ has extreme value $s$ at $x=1$ and $x=2, f^{\prime}(1)=0 .$ $\Rightarrow \frac{a}{1}+2 b(1)+1=0$ $\Rightarrow a=-1-2 b$ .....(1) $f^{\prime}(2)=0$ $\Rightarrow \frac{a}{2}+2 b(2)+1=0$ $\Rightarrow a+8 b=-2$ $\Rightarrow a=-2-8 b$ .....(2) From eqs. (1) and (2), we get $-2-8 b=-1-2...
Read More →If the solve the problem
Question: (i) $f(x)=(x-1)(x-2)^{2}$ (ii) $\mathrm{f}(\mathrm{x})=x \sqrt{1-x}, x \leq 1$ (iii) $f(x)=-(x-1)^{3}(x+1)^{2}$ Solution: (i) Given: $f(x)=(x-1)(x-2)^{2}$ $=(x-1)\left(x^{2}-4 x+4\right)$ $=x^{3}-4 x^{2}+4 x-x^{2}+4 x-4$ $=x^{3}-5 x^{2}+8 x-4$ $\Rightarrow f^{\prime}(x)=3 x^{2}-10 x+8$ For the local maxima or minima, we must have $f^{\prime}(x)=0$ $\Rightarrow 3 x^{2}-10 x+8=0$ $\Rightarrow 3 x^{2}-6 x-4 x+8=0$ $\Rightarrow(x-2)(3 x-4)=0$ $\Rightarrow x=2$ and $\frac{4}{3}$ Thus, $x=2$...
Read More →Find the rth term of an A.P.
Question: Find the rthterm of an A.P. sum of whose first n terms is 2n + 3n2.[Hint:an= Sn Sn1] Solution: Sum of first n terms be Sngiven as Sn= 2n + 3n2 We have to find the rthterm that is ar Using the given hint nthterm is given as an= Sn Sn-1 ⇒ar= Sr Sr-1 Using Sn= 2n + 3n2 ⇒ar= 2r + 3r2 (2(r 1) + 3(r 1)2) ⇒ar= 2r + 3r2 (2r 2 + 3(r2 2r + 1)) ⇒ar= 2r + 3r2 (2r 2 + 3r2 6r + 3) ⇒ar= 6r 1 Hence the rthterm is 6r 1 Long Answer Type...
Read More →If the solve the problem
Question: (i) $f(x)=(x-1)(x-2)^{2}$ (ii) $\mathrm{f}(\mathrm{x})=x \sqrt{1-x}, x \leq 1$ (iii) $f(x)=-(x-1)^{3}(x+1)^{2}$ Solution: (i) Given: $f(x)=(x-1)(x-2)^{2}$ $=(x-1)\left(x^{2}-4 x+4\right)$ $=x^{3}-4 x^{2}+4 x-x^{2}+4 x-4$ $=x^{3}-5 x^{2}+8 x-4$ $\Rightarrow f^{\prime}(x)=3 x^{2}-10 x+8$ For the local maxima or minima, we must have $f^{\prime}(x)=0$ $\Rightarrow 3 x^{2}-10 x+8=0$ $\Rightarrow 3 x^{2}-6 x-4 x+8=0$ $\Rightarrow(x-2)(3 x-4)=0$ $\Rightarrow x=2$ and $\frac{4}{3}$ Thus, $x=2$...
Read More →Find the sum of the series
Question: Find the sum of the series (33 23) + (53 43) + (73 63) + to (i) n terms (ii) 10 terms Solution: Given $\left(3^{3}-2^{3}\right)+\left(5^{3}-4^{3}\right)+\left(7^{3}-6^{3}\right)+\ldots$ Let the series be $S=\left(3^{3}-2^{3}\right)+\left(5^{3}-4^{3}\right)+\left(7^{3}-6^{3}\right)+\ldots$ i) Generalizing the series in terms of i $\mathrm{S}=\sum_{\mathrm{i}=1}^{\mathrm{n}}\left[(2 \mathrm{i}+1)^{3}-(2 \mathrm{i})^{3}\right]$ Using the formula $a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\rig...
Read More →Find the equation of the circle which passes through the points
Question: Find the equation of the circle which passes through the points A(1, 1) and B(2, 2) and whose radius is 1. Show that there are two such circles. Solution: The general equation of a circle: $(x-h)^{2}+(y-k)^{2}=r^{2}$ $\ldots(\mathrm{i})$, where $(\mathrm{h}, \mathrm{k})$ is the centre and $\mathrm{r}$ is the radius. Putting $A(1,1)$ in (i) $(1-h)^{2}+(1-k)^{2}=1^{2}$ $\Rightarrow h^{2}+k^{2}+2-2 h-2 k=1$ $\Rightarrow h^{2}+k^{2}-2 h-2 k=-1 . .$ (ii) Putting $B(2,2)$ in (i) $(2-h)^{2}+(...
Read More →If a1, a2, a3, …, an are in A.P.,
Question: If a1, a2, a3, , anare in A.P., where ai 0 for all i, show that $\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\cdots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}=\frac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n}}}$ Solution: Given $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ are in A.P., where $a_{i}0$ for all $i$ To prove that: $\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\cdots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}=\frac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n}}}$ $\Ri...
Read More →Prove that the centres of the three circles
Question: Prove that the centres of the three circles $x^{2}+y^{2}-4 x-6 y-12=0, x^{2}+y^{2}+$ $2 x+4 y-5=0$ and $x^{2}+y^{2}-10 x-16 y+7=0$ are collinear. Solution: Given, $x^{2}+y^{2}-4 x-6 y-12=0$ centre $\left(-g_{1},-f_{1}\right)=(2,3)$ $x^{2}+y^{2}+2 x+4 y-5=0$ centre $\left(-g_{2},-f_{2}\right)=(-1,-2)$ $x^{2}+y^{2}-10 x-16 y+7=0$ centre $\left(-g_{3},-f_{3}\right)=(5,8)$ to prove that the centres are collinear, $\left|\begin{array}{lll}\mathrm{x}_{1} \mathrm{y}_{1} 1 \\ \mathrm{x}_{2} \m...
Read More →Find the equation of the circle concentric with the circle
Question: Find the equation of the circle concentric with the circle $x^{2}+y^{2}-6 x+12 y+$ 15 = 0 and of double its area. Solution: 2 or more circles are said to be concentric if they have the same centre and different radii. Given, $x^{2}+y^{2}-6 x+12 y+15=0$ Radius r = $\sqrt{g^{2}+f^{2}-c}=\sqrt{\left(-3^{2}\right)+6^{2}-15}=\sqrt{30}$ The concentric circle will have the equation $x^{2}+y^{2}-6 x+12 y+c^{\prime}=0$ Also given area of circle $=2 \times$ area of the given circle. $\Rightarrow...
Read More →In a cricket tournament 16 school teams participated.
Question: In a cricket tournament 16 school teams participated. A sum of Rs 8000 is to be awarded among themselves as prize money. If the last placed team is awarded Rs 275 in prize money and the award increases by the same amount for successive finishing places, how much amount will the first place team receive? Solution: Let the amount received by first place team be a Rs and d be difference in amount As the difference is same hence the second-place team will receive a d and the third place a ...
Read More →In a potato race 20 potatoes are placed in a line
Question: In a potato race 20 potatoes are placed in a line at intervals of 4 metres with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes? Solution: Given at start he has to run 24m to get the first potato then 28 m as the next potato is 4m away from first and so on Hence the sequence of his running will be 24, 28, 32 There are 20 terms in sequence a...
Read More →Find the equation of the circle concentric with the circle
Question: Find the equation of the circle concentric with the circle $x^{2}+y^{2}-4 x-6 y-3$ = 0 and which touches the y-axis. Solution: The given image of the circle is: We know that the general equation of the circle is given by: $x^{2}+y^{2}+2 g x+2 f y+c=0$ Also, Radius r = $\sqrt{g^{2}+f^{2}-c}$ Now, $r=\sqrt{(2)^{2}+(3)^{2}-(-3)}$ $r=\sqrt{4+9+3}$ $r=4$ units. We need to the find the equation of the circle which is concentric to the qiven circle and touches y-axis. The centre of the circle...
Read More →We know the sum of the interior angles of a triangle is 180°.
Question: We know the sum of the interior angles of a triangle is 180. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, sides form an arithmetic progression. Find the sum of the interior angles for a 21-sided polygon Solution: Given the sum of interior angles of a polygon having n sides is given by (n 2) 180 Sum of angles with three sides that is n = 3 is (3 2) 180 = 180 Sum of angles with four sides that is n = 4 is (4 2) 180 = 360 Sum of angles with five sides that is n =...
Read More →A carpenter was hired to build 192 window frames.
Question: A carpenter was hired to build 192 window frames. The first day he made five frames and each day, thereafter he made two more frames than he made the day before. How many days did it take him to finish the job? Solution: Given first day he made 5 frames then two frames more than the previous that is 7 then 9 and so on Hence the sequence of making frames each day is 5, 7, 9 The sequence is AP with first term as a = 5 and common difference d = 2 Total number of frames to be made is 192 L...
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