In what ratio does y-axis divide the line segment joining the points
Question: In what ratio doesy-axis divide the line segment joining the points (4, 7) and (3, 7)? Solution: Lety-axis divides the line segment joining the points (4, 7) and (3, 7) in the ratiok: 1. Then $0=\frac{3 k-4}{k+1}$ $\Rightarrow 3 k=4$ $\Rightarrow k=\frac{4}{3}$ Hence, the required ratio is 4 : 3....
Read More →If the points(x, 4) lies on a circle whose centre is at the origin and radius is 5, then x =
Question: If the points(x, 4) lies on a circle whose centre is at the origin and radius is 5, thenx=(a) 5(b) 3(c) 0(d) 4 Solution: It is given that the point A(x, 4)is at a distance of 5 units from origin O. So, apply the distance formula to get, $5^{2}=(x)^{2}+4^{2}$ Therefore, $x^{2}=9$ So, $x=\pm 3$ So the answer is (b)...
Read More →If P is a point on x-axis such that its distance from the origin is 3 units,
Question: IfPis a point on x-axis such that its distance from the origin is 3 units, then the coordinates of a pointQonOYsuch thatOP=OQ, are(a) (0, 3)(b) (3, 0)(c) (0, 0)(d) (0, 3) Solution: GIVEN: If P is a point onxaxis such that its distance from the origin is 3 units. TO FIND: The coordinates of a point Q on OY such that OP= OQ. Onxaxis y coordinates is 0. Hence the coordinates of point P will be (3, 0) as it is given that the distance from origin is 3 units. Now then the coordinates of Q on...
Read More →The distance of the point (4, 7) from the y-axis is
Question: The distance of the point (4, 7) from the y-axis is(a) 4(b) 7(c) 11 (d) $\sqrt{65}$ Solution: The distance of a point fromy-axis is given by abscissa of that point. So, distance of (4, 7) fromy-axis is. So the answer is (a)...
Read More →A solid cylinder of mass m is wrapped with an inextensible light string and,
Question: A solid cylinder of mass $m$ is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is : [The coefficient of static friction, $\mu_{\mathrm{s}}$, is $\left.0.4\right]$$\frac{7}{2} \mathrm{mg}$$5 \mathrm{mg}$$\frac{m g}{5}$0Correct Option: , 3 Solution: (3) Let's take solid cylinder is in equilibrium $\mathrm{T}+\mathrm{f}=\mathrm{mg} \sin 60 \ldots$ (i) $\mathr...
Read More →The distance of the point (4, 7) from the x-axis is
Question: The distance of the point (4, 7) from the x-axis is(a) 4(b) 7(c) 11 (d) $\sqrt{65}$ Solution: The ordinate of a point gives its distance from thex-axis. So, the distance of (4, 7) fromx-axis is So the answer is (b)...
Read More →If the centroid of the triangle formed by (7, x)
Question: If the centroid of the triangle formed by (7, x) (y, 6) and (9, 10) is at (6, 3), then (x, y) =(a) (4, 5)(b) (5, 4)(c) (5, 2)(d) (5, 2) Solution: We have to find the unknown co-ordinates. The co-ordinates of vertices are $\mathrm{A}(7, x) ; \mathrm{B}(y,-6) ; \mathrm{C}(9,10)$ The co-ordinate of the centroid is (6, 3) We know that the co-ordinates of the centroid of a triangle whose vertices are $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)$ is $\left(\...
Read More →If three consecutive vertices of a parallelogram ABCD are A(1, −2), B(3, 6) and C(5, 10), find its fourth vertex D.
Question: If three consecutive vertices of a parallelogramABCDareA(1, 2),B(3, 6) andC(5, 10), find its fourth vertexD. Solution: LetA(1, 2),B(3, 6) andC(5, 10) be the three vertices of a parallelogramABCDand the fourth vertex beD(a,b).JoinACandBDintersecting atO. We know that the diagonals of a parallelogram bisect each other.Therefore,Ois the midpoint ofACas well asBD. Midpoint of $A C=\left(\frac{1+5}{2}, \frac{-2+10}{2}\right)=\left(\frac{6}{2}, \frac{8}{2}\right)=(3,4)$ Midpoint of $B D=\lef...
Read More →If Points (1, 2) (−5, 6) and (a, −2) are collinear, then a =
Question: If Points (1, 2) (5, 6) and (a, 2) are collinear, then a =(a) 3(b) 7(c) 2(d) 2 Solution: We have three collinear points $\mathrm{A}(1,2) ; \mathrm{B}(-5,6) ; \mathrm{C}(a,-2)$. 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, $1(6+2)-5(-2-2)+a(2-6)=0$ So, $-4 a+8+20=0$ Therefore, $a=7$ So the...
Read More →Consider a uniform wire of mass M and length L.
Question: Consider a uniform wire of mass $\mathrm{M}$ and length $\mathrm{L}$. It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is :$\frac{1}{4} \frac{\mathrm{ML}^{2}}{\pi^{2}}$$\frac{2}{5} \frac{\mathrm{ML}^{2}}{\pi^{2}}$$\frac{\mathrm{ML}^{2}}{\pi^{2}}$$\frac{1}{2} \frac{\mathrm{ML}^{2}}{\pi^{2}}$Correct Option: , 3 Solution: (3) $\pi \mathrm{r}=\mathrm{L} \Rightarrow \mathrm{r}=\frac{\mathrm{L}}{\pi}$ $\mathrm{...
Read More →Solve the following
Question: LetSndenote the sum of the cubes of firstnnatural numbers andsndenote the sum of firstn natural numbers. Then, write the value of $\sum_{r=1}^{n} \frac{S_{r}}{s_{r}}$ Solution: We know that, $S_{r}=1^{3}+2^{3}+3^{3}+\ldots+r^{3}=\left[\frac{r(r+1)}{2}\right]^{2}$ And, $s_{r}=1+2+3+\ldots+r=\frac{r(r+1)}{2}$ As, $\frac{S_{r}}{s_{r}}=\frac{\left[\frac{r(r+1)}{2}\right]^{2}}{\left[\frac{r(r+1)}{2}\right]}=\frac{r(r+1)}{2}=\frac{1}{2}\left(r^{2}+r\right)$ Now, $\sum_{r=1}^{n} \frac{S_{r}}{...
Read More →If the centroid of the triangle formed by the points (a, b),
Question: If the centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is at the origin, thena3+b3+c3= (a)abc(b) 0(c)a+b+c(d) 3abc Solution: The co-ordinates of the vertices are (a, b); (b, c) and (c, a) The co-ordinate of the centroid is (0, 0) We know that the co-ordinates of the centroid of a triangle whose vertices are $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)$ is $\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right)$ So...
Read More →Write the 50th term of the series 2 + 3 + 6 + 11 + 18 + ...
Question: Write the 50th term of the series 2 + 3 + 6 + 11 + 18 + ... Solution: We have, $a_{1}=2$ $a_{2}=3=2+1$ $a_{3}=6=2+1+3$ $a_{4}=11=2+1+3+5$ . . $a_{50}=2+1+3+5+\ldots(50$ terms $)$ $=2+\frac{49}{2}[2 \times 1+(49-1) \times 2]$ (As, the terms apart 2 are in A.P. with $a=1$ and $d=2$ ) $=2+\frac{49}{2}(2+48 \times 2)$ $=2+\frac{49}{2} \times 98$ $=2+49^{2}$ $=2+2401$ $=2403$ So, the 50th term of the given series is 2403....
Read More →A thin circular ring of mass M and radius r
Question: A thin circular ring of mass $\mathrm{M}$ and radius $\mathrm{r}$ is rotating about its axis with an angular speed $\omega$. Two particles having mass $m$ each are now attached at diametrically opposite points. The angular speed of the ring will become:$\omega \frac{\mathrm{M}}{\mathrm{M}+\mathrm{m}}$$\omega \frac{\mathrm{M}+2 \mathrm{~m}}{\mathrm{M}}$$\omega \frac{\mathrm{M}}{\mathrm{M}+2 \mathrm{~m}}$$\omega \frac{\mathrm{M}-2 \mathrm{~m}}{\mathrm{M}+2 \mathrm{~m}}$Correct Option: , ...
Read More →If the points P(a, −11), Q(5, b), R(2, 15) and S(1, 1) are the vertices of a parallelogram PQRS,
Question: If the pointsP(a, 11),Q(5, b),R(2, 15) andS(1, 1) are the vertices of a parallelogramPQRS, find the values ofaandb. Solution: The points areP(a, 11),Q(5,b),R(2, 15) andS(1, 1). JoinPRandQS,intersecting atO.We know that the diagonals of a parallelogram bisect each other.Therefore,Ois the midpoint ofPRas well asQS. Midpoint of $P R=\left(\frac{a+2}{2}, \frac{-11+15}{2}\right)=\left(\frac{a+2}{2}, \frac{4}{2}\right)=\left(\frac{a+2}{2}, 2\right)$ Midpoint of $Q S=\left(\frac{5+1}{2}, \fra...
Read More →The ratio in which the x-axis divides the segment joining (3, 6) and (12, −3) is
Question: The ratio in which the x-axis divides the segment joining (3, 6) and (12, 3) is(a) 2: 1(b) 1 : 2(c) 2 : 1(d) 1 : 2 Solution: Let $P(x, 0)$ be the point of intersection of $x$-axis with the line segment joining $A(3,6)$ and $B(12,-3)$ which divides the line segment $A B$ in the ratio $\lambda: 1$. Now according to the section formula if point a point $P$ divides a line segment joining $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ in the ratio $m$ : $n$ internally than, $...
Read More →Write the sum of 20 terms of the series
Question: Write the sum of 20 terms of the series $1+\frac{1}{2}(1+2)+\frac{1}{3}(1+2+3)+\ldots$ Solution: Let the $n$th term be $a_{n}$. Here, $a_{n}=\frac{1}{n}(1+2+3+\ldots+n)=\left(\frac{n+1}{2}\right)$ We know: $S_{n}=\sum_{k=1}^{n} a_{k}$ Thus, we have: $S_{20}=\sum_{k=1}^{20} a_{k}$ $=\frac{1}{2}\left[\sum_{k=1}^{20}(k+1)\right]$ $=\frac{1}{2}\left[\sum_{k=1}^{20} k+20\right]$ $=\frac{1}{2}\left[\frac{20(21)}{2}+20\right]$ $=\frac{1}{2}[230]$ $=115$...
Read More →Show that the points A(3, 1), B(0, −2), C(1, 1) and D(4, 4)
Question: Show that the pointsA(3, 1),B(0, 2),C(1, 1) andD(4, 4) are the vertices of a parallelogramABCD. Solution: The points areA(3, 1),B(0, 2),C(1, 1) andD(4, 4).JoinACandBD, intersecting atO. We know that the diagonals of a parallelogram bisect each other. Midpoint of $A C=\left(\frac{3+1}{2}, \frac{1+1}{2}\right)=\left(\frac{4}{2}, \frac{2}{2}\right)=(2,1)$ Midpoint of $B D=\left(\frac{0+4}{2}, \frac{-2+4}{2}\right)=\left(\frac{4}{2}, \frac{2}{2}\right)=(2,1)$ Thus, the diagonalsACandBDhave...
Read More →If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers,
Question: If the sum of firstneven natural numbers is equal toktimes the sum of firstnodd natural numbers, then write the value ofk. Solution: According to the question, $2+4+\ldots+2 n=k(1+3+5+7+\ldots+(2 n-1))$ $\Rightarrow 2 \times \frac{n(n+1)}{2}=k\left[\frac{n}{2}\{2 \times 1+(n-1) \times 2\}\right]$ $\Rightarrow \frac{2 n(n+1)}{2}=k\left[\frac{n}{2}(2+2 n-2)\right]$ $\Rightarrow n(n+1)=k\left[\frac{n}{2}(2 n)\right]$ $\Rightarrow n^{2}+n=k n^{2}$ $\Rightarrow k=\frac{n+1}{n}$...
Read More →The ratio in which (4, 5) divides the join of (2, 3) and (7, 8) is
Question: The ratio in which (4, 5) divides the join of (2, 3) and (7, 8) is (a) 2 : 3(b) 3 : 2(c) 3 : 2(d) 2 : 3 Solution: The co-ordinates of a point which divided two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ internally in the ratio $m: n$ is given by the formula, $(x, y)=\left(\left(\frac{m x_{2}+n x_{1}}{m+n}\right),\left(\frac{m y_{2}+n y_{1}}{m+n}\right)\right)$ Here it is said that the point (4, 5)divides the points A(2,3) and B(7,8). Substituting these values in...
Read More →Solve the following
Question: If $\sum_{r=1}^{n} r=55$, find $\sum_{r=1}^{n} r^{3} .$ Solution: $\sum_{r=1}^{n} r^{3}=1^{3}+2^{3}+3^{3}+\ldots+n^{3}$ $=\left[\frac{n(n+1)}{2}\right]^{2}$ $=\left[\sum_{r=1}^{n} r\right]^{2}$ $=[55]^{2}$ $=3025$...
Read More →A sphere of mass 2 kg and radius 0.5 m
Question: A sphere of mass $2 \mathrm{~kg}$ and radius $0.5 \mathrm{~m}$ is rolling with an initial speed of $1 \mathrm{~ms}^{-1}$ goes up an inclined plane which makes an angle of $30^{\circ}$ with the horizontal plane, without slipping. How low will the sphere take to return to the starting point $\mathrm{A}$ ? (1) $0.60 \mathrm{~s}$(2) $0.52 \mathrm{~s}$(3) $0.57 \mathrm{~s}$(4) $0.80 \mathrm{~s}$Correct Option: 3, Solution: (3) $a=\frac{g \sin \theta}{1+\frac{\mathrm{I}}{\mathrm{mR}^{2}}}=\f...
Read More →Find the third vertex of ∆ABC if two of its vertices are B(−3, 1) and C(0, −2)
Question: Find the third vertex of ∆ABCif two of its vertices areB(3, 1) andC(0, 2) and its centroid is at the origin. Solution: Two vertices of ∆ABCareB(3,1) andC(0, 2). Let the third vertex beA(a,b).Then, the coordinates of its centroid are $\left(\frac{-3+0+a}{3}, \frac{1-2+b}{3}\right)$ i. e. $\left(\frac{-3+a}{3}, \frac{-1+b}{3}\right)$ But it is given that the centroid is at the origin, that isG(0, 0). Therefore, $0=\frac{-3+a}{3}, 0=\frac{-1+b}{3}$ $\Rightarrow 0=-3+a, 0=-1+b$ $\Rightarro...
Read More →The line segment joining points (−3, −4),
Question: The line segment joining points (3, 4), and (1, 2) is divided by y-axis in the ratio(a) 1 : 3(b) 2 : 3(c) 3 : 1(d) 2 : 3 Solution: Let $\mathrm{P}(0, y)$ be the point of intersection of $y$-axis with the line segment joining $\mathrm{A}(-3,-4)$ and $\mathrm{B}(1,-2)$ which divides the line segment $\mathrm{AB}$ in the ratio $\lambda: 1$. Now according to the section formula if point a point $P$ divides a line segment joining $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$...
Read More →Write the sum to n terms of a series whose rth term is
Question: Write the sum tonterms of a series whoserthterm isr+ 2r. Solution: series whoserthterm isr+ 2r. $\left(1+2^{1}\right)+\left(2+2^{2}\right)+\left(3+2^{3}\right)+\left(4+2^{4}\right)+\ldots+\left(n+2^{n}\right)$ Thus, we have: $S_{n}=\left(1+2^{1}\right)+\left(2+2^{2}\right)+\left(3+2^{3}\right)+\left(4+2^{4}\right)+\ldots+\left(n+2^{n}\right)$ $=(1+2+3+4+\ldots+n)+\left(2+2^{2}+2^{3}+2^{4}+\ldots+2^{n}\right)$ $=\frac{n(n+1)}{2}+2\left(\frac{2^{n}-1}{2-1}\right)$ $=\frac{n(n+1)}{2}+2^{n...
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