If G(−2, 1) is the centroid of a ∆ABC and two of its vertices are A(1, −6) and B(−5, 2),
Question: If G(2, 1) is the centroid of a ∆ABCand two of its vertices areA(1, 6) andB(5, 2), find the third vertex of the triangle. Solution: Two vertices of ∆ABCareA(1, 6) andB(5, 2). Let the third vertex beC(a,b).Then the coordinates of its centroid are $C\left(\frac{1-5+a}{3}, \frac{-6+2+b}{3}\right)$ $C\left(\frac{-4+a}{3}, \frac{-4+b}{3}\right)$ But it is given thatG(2, 1) is the centroid. Therefore, $-2=\frac{-4+a}{3}, 1=\frac{-4+b}{3}$ $\Rightarrow-6=-4+a, 3=-4+b$ $\Rightarrow-6+4=a, 3+4=...
Read More →The following bodies,
Question: The following bodies, of same mass ' $m$ ' and radius ' $R$ ' are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is [Mark the body as per their respective numbering given in the question] a ringa disca solid cylindera solid sphere.Correct Option: , 4 Solution: (4) $\mathrm{Mg} \sin \theta \mathrm{R}=\left(\mathrm{mk}^{2}+\mathrm{mR}^{2}\right)^{\alpha}$ $\alpha=\frac{R g \sin \...
Read More →Write the sum of the series
Question: Write the sum of the series 12 22+ 32 42+ 52 62+ ... + (2n 1)2 (2n)2. Solution: The given series can be rewritten as: $3(-1)+7(-1)+11(-1)+\ldots+(4 n-1)(-1)$ $=-[3+7+11+\ldots+(4 n-1)]$ $=-\left[\frac{n}{2}\{3 \times 2+(n-1) 4\}\right]$ $=-\left[\frac{n}{2}(4 n+2)\right]$ $=-n(2 n+1)$...
Read More →If the centroid of a triangle is (1, 4)
Question: If the centroid of a triangle is (1, 4) and two of its vertices are (4, 3) and (9, 7), then the area of the triangle is (a) 183 sq. units (b) $\frac{183}{2}$ sq. units (c) 366 sq. units (d) $\frac{183}{4}$ sq. units Solution: We have to find the co-ordinates of the third vertex of the given triangle. Let the co-ordinates of the third vertex be $(x, y)$. The co-ordinates of other two vertices are (4,3) and (9, 7) The co-ordinate of the centroid is (1, 4) We know that the co-ordinates of...
Read More →Write the sum of the series 2 + 4 + 6 + 8 + ...
Question: Write the sum of the series 2 + 4 + 6 + 8 + ... + 2n. Solution: $S_{n}=2+4+6+8+\ldots+2 n$ $=\frac{n}{2}[4+(n-1) 2]$ $=\frac{n}{2}[2+2 n]$ $=n[n+1]$...
Read More →Find the centroid of ∆ABC whose vertices are A(−1, 0), B(5, −2) and C(8, 2).
Question: Find the centroid of ∆ABCwhose vertices areA(1, 0),B(5, 2) andC(8, 2). Solution: Here, (x1= 1,y1= 0), (x2= 5,y2= 2) and (x3= 8,y3= 2).LetG(x,y) be the centroid of the ∆ABC.Then, $x=\frac{1}{3}\left(x_{1}+x_{2}+x_{3}\right)=\frac{1}{3}(-1+5+8)=\frac{1}{3}(12)=4$ $y=\frac{1}{3}\left(y_{1}+y_{2}+y_{3}\right)=\frac{1}{3}(0-2+2)=\frac{1}{3}(0)=0$ Hence, the centroid of ∆ABCisG(4, 0)....
Read More →Find the lengths of the median of ∆ABC whose vertices are A(0, −1), B(2, 1) and C(0, 3).
Question: Find the lengths of the median of ∆ABCwhose vertices areA(0, 1),B(2, 1) andC(0, 3). Solution: The vertices of ∆ABCareA(0, 1),B(2, 1) andC(0, 3).LetAD,BEandCFbe the medians of ∆ABC. LetDbe the midpoint ofBC. So, the coordinates ofDare $D\left(\frac{2+0}{2}, \frac{1+3}{2}\right)$ i. e. $D\left(\frac{2}{2}, \frac{4}{2}\right)$ i. e. $D(1,2)$ LetEbe the midpoint ofAC.So, the coordinates ofEare $E\left(\frac{0+0}{2}, \frac{-1+3}{2}\right)$ i. e. $E\left(\frac{0}{2}, \frac{2}{2}\right)$ i. e...
Read More →If points (a, 0), (0, b) and (1, 1)
Question: If points $(a, 0),(0, b)$ and $(1,1)$ are collinear, then $\frac{1}{a}+\frac{1}{b}=$ (a) 1(b) 2(c) 0(d) 1 Solution: We have three collinear points $\mathrm{A}(a, 0) ; \mathrm{B}(0, b) ; \mathrm{C}(1,1)$. In general if $\mathrm{A}\left(x_{1}, y_{1}\right) ; \mathrm{B}\left(x_{2}, y_{2}\right) ; \mathrm{C}\left(x_{3}, y_{3}\right)$ are collinear then, $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$ So, $a(b-1)+0(1-0)+1(0-b)=0$ So, $a b=a+b$ D...
Read More →The angular speed of truck wheel is increased
Question: The angular speed of truck wheel is increased from $900 \mathrm{rpm}$ to $2460 \mathrm{rpm}$ in 26 seconds. The number of revolutions by the truck engine during this time is_______ (Assuming the acceleration to be uniform). Solution: (728) We know, $\theta=\left(\frac{\omega_{1}+\omega_{2}}{2}\right) t$ Let number of revolutions be $\mathrm{N}$ $\therefore 2 \pi \mathrm{N}=2 \pi\left(\frac{900+2460}{60 \times 2}\right) \times 26$ $\mathrm{N}=728$...
Read More →If the area of the triangle formed by the points (x, 2x),
Question: If the area of the triangle formed by the points (x, 2x), (2, 6) and (3, 1) is 5 square units , then x = (a) $\frac{2}{3}$ (b) $\frac{3}{5}$ (c) 3 (d) 5 Solution: We have the co-ordinates of the vertices of the triangle as $\mathrm{A}(x, 2 x) ; \mathrm{B}(-2,6) ; \mathrm{C}(3,1)$ which has an area of 5 sq.units. In general if $\mathrm{A}\left(x_{1}, y_{1}\right) ; \mathrm{B}\left(x_{2}, y_{2}\right) ; \mathrm{C}\left(x_{3}, y_{3}\right)$ are non-collinear points then area of the triang...
Read More →A mass M hangs on a massless rod of length l which rotates at a constant angular frequency.
Question: A mass $M$ hangs on a massless rod of length $l$ which rotates at a constant angular frequency. The mass $\mathrm{M}$ moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $\omega$. The angular momentum of $\mathrm{M}$ about point $\mathrm{A}$ is $\mathrm{L}_{\mathrm{A}}$ which lies in the positive $z$ direction and the angular momentum of $\mathrm{M}$ about $\mathrm{B}$ is $\mathrm{L}_{\mathrm{...
Read More →The sum of n terms of the series
Question: The sum of $n$ terms of the series $\frac{3}{1^{2}}+\frac{5}{1^{2}+2^{2}}+\frac{7}{1^{2}+2^{2}+3^{2}}+$______________ is_______________ . Solution: Sum of $n$ term of $\frac{3}{1^{2}}+\frac{5}{1^{2}+2^{2}}+\frac{7}{1^{2}+2^{2}+3^{2}}$ herenthtermtnfor above series is i. e $T_{n}=\frac{2 n+1}{1^{2}+2^{2}+\ldots .+n^{2}}$ $T_{n}=\frac{2 n+1}{\frac{\{n(n+1)(2 n+1)\}}{6}}=\frac{2 n+1}{\frac{\{(n+1)(2 n+1)\}}{6}}$ $T_{n}=\frac{6}{n(n+1)}$ i. e $T_{r}=\frac{6}{r(r+1)}$ Hence $\sum_{r=1}^{n} ...
Read More →In what ratio does the line x − y − 2 = 0 divide the line segment joining the points
Question: In what ratio does the linexy 2 = 0 divide the line segment joining the pointsA(3, 1) andB(8, 9)? Solution: Let the linexy 2 = 0 divide the line segment joining the pointsA(3, 1) andB(8, 9) in the ratiok: 1 atP.Then, the coordinates ofPare $P\left(\frac{8 k+3}{k+1}, \frac{9 k-1}{k+1}\right)$ Since,Plies on the linexy 2 = 0, we have: $\left(\frac{8 k+3}{k+1}\right)-\left(\frac{9 k-1}{k+1}\right)-2=0$ $\Rightarrow 8 k+3-9 k+1-2 k-2=0$ $\Rightarrow 8 k-9 k-2 k+3+1-2=0$ $\Rightarrow-3 k+2=...
Read More →If points (t, 2t), (−2, 6) and (3, 1)
Question: If points (t, 2t), (2, 6) and (3, 1) are collinear, thent= (a) $\frac{3}{4}$ (b) $\frac{4}{3}$ (c) $\frac{5}{3}$ (d) $\frac{3}{5}$ Solution: We have three collinear points $\mathrm{A}(t, 2 t) ; \mathrm{B}(-2,6) ; \mathrm{C}(3,1)$. In general if $\mathrm{A}\left(x_{1}, y_{1}\right) ; \mathrm{B}\left(x_{2}, y_{2}\right) ; \mathrm{C}\left(x_{3}, y_{3}\right)$ are collinear then, $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$ So, $t(6-1)-2(1-2...
Read More →In what ratio is the line segment joining the points A(−2, −3) and B(3, 7) divided by the y-axis?
Question: In what ratio is the line segment joining the pointsA(2, 3) andB(3, 7) divided by they-axis? Also, find the coordinates of the points of division. Solution: LetABbe divided by thex-axis in the ratiok: 1 at the pointP.Then, by section formula the coordinates ofPare $P\left(\frac{3 k-2}{k+1}, \frac{7 k-3}{k+1}\right)$ ButPlies on they-axis; so, its abscissa is 0. Therefore, $\frac{3 k-2}{k+1}=0$ $\Rightarrow 3 k-2=0 \Rightarrow 3 k=2 \Rightarrow k=\frac{2}{3} \Rightarrow k=\frac{2}{3}$ T...
Read More →If (x , 2), (−3, −4) and (7, −5) are collinear, then x =
Question: If (x, 2), (3, 4) and (7, 5) are collinear, thenx=(a) 60(b) 63(c) 63(d) 60 Solution: We have three collinear points $\mathrm{A}(x, 2) ; \mathrm{B}(-3,-4) ; \mathrm{C}(7,-5)$. In general if $\mathrm{A}\left(x_{1}, y_{1}\right) ; \mathrm{B}\left(x_{2}, y_{2}\right) ; \mathrm{C}\left(x_{3}, y_{3}\right)$ are collinear then, $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)=0$ So, $x(-4+5)-3(-5-2)+7(2+4)=0$ So, $x+42+21=0$ Therefore, $x=-63$ So the ...
Read More →A triangular plate is shown.
Question: A triangular plate is shown. A force $\vec{F}=4 \hat{i}-3 \hat{j}$ is applied at point $P$. The torque at point $P$ with respect to point ' $\mathrm{O}^{\prime}$ and $^{\prime} \mathrm{Q}^{\prime}$ are : (1) $-15-20 \sqrt{3}, 15-20 \sqrt{3}$(2) $15+20 \sqrt{3}, 15-20 \sqrt{3}$(3) $15-20 \sqrt{3}, 15+20 \sqrt{3}$(4) $-15+20 \sqrt{3}, 15+20 \sqrt{3}$Correct Option: 1 Solution: (1) $\overrightarrow{\mathrm{F}}=4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}$ $\overrightarrow{\mathrm{r}}_{1}=5 \hat{...
Read More →If the sum of the squares of first n natural numbers exceeds their sum by 330,
Question: If the sum of the squares of firstnnatural numbers exceeds their sum by 330, thenn= _______________. Solution: Sum of squares of first $n$ natural numbers is $\frac{n(n+1)(2 n+1)}{6}$ Sum of first $n$ natural number is $\frac{n(n+1)}{2}$ According to given condition, $\Sigma n^{2}-\Sigma n=330$ i. e $\frac{n(n+1)(2 n+1)}{6}-\frac{n(n+1)}{2}=330$ i. e $\frac{n(n+1)}{2}\left[\frac{2 n+1}{3}-1\right]=330$ i. e $\frac{n(n+1)}{2}\left[\frac{2 n+1-3}{3}\right]=330$ i. e $\frac{n(n+1)}{2}\lef...
Read More →The area of the triangle formed by
Question: The area of the triangle formed by (a,b+c), (b,c+a) and (c,a+b)(a)a+b+c(b)abc(c) (a+b+c)2(d) 0 Solution: We have three non-collinear points $\mathrm{A}(a, b+c) ; \mathrm{B}(b, c+a) ; \mathrm{C}(c, a+b)$. In general if $\mathrm{A}\left(x_{1}, y_{1}\right) ; \mathrm{B}\left(x_{2}, y_{2}\right) ; \mathrm{C}\left(x_{3}, y_{3}\right)$ are non-collinear points then are of the triangle formed is given by- $\operatorname{ar}(\Delta \mathrm{ABC})=\frac{1}{2}\left|x_{1}\left(y_{2}-y_{3}\right)+x...
Read More →If A (5, 3), B (11, −5) and P (12, y)
Question: IfA(5, 3),B(11, 5) andP(12,y) are the vertices of a right triangle right angled atP, theny=(a) 2, 4(b) 2, 4(c) 2, 4(d) 2, 4 Solution: Disclaimer: option (b) and (c) are given to be same in the book. So, we are considering option (b) as 2, 4instead of 2, 4. We have a right angled triangle $\triangle \mathrm{APC}$ whose co-ordinates are $\mathrm{A}(5,3) ; \mathrm{B}(11,-5)$; $\mathrm{P}(12, y)$. So clearly the triangle is, right angled at A. So, $\mathrm{AP}^{2}=(12-5)^{2}+(y-3)^{2}$ $\m...
Read More →In what ratio is the line segment joining A(2, −3) and B(5, 6) divided by the x-axis?
Question: In what ratio is the line segment joiningA(2, 3) andB(5, 6) divided by thex-axis? Also, find the coordinates of the point of division. Solution: LetABbe divided by thex-axis in the ratiok: 1 at the pointP.Then, by section formula the coordinates ofPare $P=\left(\frac{5 k+2}{k+1}, \frac{6 k-3}{k+1}\right)$ ButPlies on thex-axis; so, its ordinate is 0. Therefore, $\frac{6 k-3}{k+1}=0$ $\Rightarrow 6 k-3=0 \Rightarrow 6 k=3 \Rightarrow k=\frac{3}{6} \Rightarrow k=\frac{1}{2}$ Therefore, t...
Read More →A solid disc of radius 'a' and mass 'm' rolls down
Question: A solid disc of radius 'a' and mass 'm' rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the disc will be $\frac{2}{\mathrm{~b}} \mathrm{~g} \sin \theta$ where $\mathrm{b}$ is (Round off to the Nearest Integer) $(\mathrm{g}=$ acceleration due to gravity $)$ $(\theta=$ angle as shown in figure $)$ Solution: (3) $\mathrm{a}=\frac{\mathrm{g} \sin \theta}{1+\frac{\mathrm{I}}{\mathrm{mR}^{2}}}=\frac{\mathrm{g} \sin \theta}{1+...
Read More →Solve this
Question: A force $\vec{F}=4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ is applied on an intersection point of $\mathrm{x}=2$ plane and $\mathrm{x}$-axis. The magnitude of torque of this force about a point $(2,3,4)$ is_____ (Round off to the Nearest Integer) Solution: $(20)$ $\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}$ $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}})-(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})=-3 \hat{\mathrm{j}}-4 \ha...
Read More →If (−1, 2), (2, −1) and (3, 1) are any three vertices of a parallelogram, then
Question: If (1, 2), (2, 1) and (3, 1) are any three vertices of a parallelogram, then (a)a= 2,b= 0(b)a= 2,b= 0(c)a= 2,b= 6(d)a= 6,b= 2 Solution: Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (1, 2); B (2,1) and C(3, 1). We have to find the co-ordinates of the forth vertex. Let the forth vertex be $\mathrm{D}(a, b)$ Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide. Now to find...
Read More →The value of
Question: Solution: $\frac{1^{3}+2^{3}+3^{3}+\ldots+10^{3}}{1+2+\ldots+10}$ Since $1^{3}+2^{3}+\ldots+n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}$ $\Rightarrow 1^{3}+2^{3}+\ldots+10^{3}=\left[\frac{10(10+1)}{2}\right]^{2}$ $=\left[\frac{10 \times 11}{2}\right]^{2}$ $=(55)^{2}$ also $1+2+\ldots+n=\frac{n(n+1)}{2}$ $\Rightarrow 1+2+\ldots+10=\frac{10(11)}{2}=\frac{5 \times 11}{2}=55$ $\Rightarrow \frac{1^{3}+2^{3}+\ldots+10^{3}}{1+2+. .+n}=\frac{(55)^{2}}{55}=55$...
Read More →