Find the ratio in which the point (−3, k) divides the join of A(−5, −4) and B(−2, 3).

Question: Find the ratio in which the point (3,k) divides the join ofA(5, 4) andB(2, 3). Also, find the value ofk. Solution: Let the pointP(3,k) divide the line AB in the ratios: 1.Then, by the section formula: $x=\frac{m x_{2}+n x_{1}}{m+n}, y=\frac{m y_{2}+n y_{1}}{m+n}$ The coordinates of $P$ are $(-3, k)$. $-3=\frac{-2 s-5}{s+1}, k=\frac{3 s-4}{s+1}$ $\Rightarrow-3 s-3=-2 s-5, k(s+1)=3 s-4$ $\Rightarrow-3 s+2 s=-5+3, k(s+1)=3 s-4$ $\Rightarrow-s=-2, k(s+1)=3 s-4$ $\Rightarrow s=2, k(s+1)=3 s...

Consider a 20 kg uniform circular disk of radius 0.2 m.

Question: Consider a $20 \mathrm{~kg}$ uniform circular disk of radius $0.2 \mathrm{~m}$. It is pin supported at its center and is at rest initially. The disk is acted upon by a constant force $\mathrm{F}=20 \mathrm{~N}$ through a massless string wrapped around its periphery as shown in the figure. Suppose the disk makes $\mathrm{n}$ number of revolutions to attain an angular speed of $50 \mathrm{rads}^{-1}$. The value of $\mathrm{n}$, to the nearest integer, is_____ [Given : In one complete rev...

The coordinates of the point on X-axis which are equidistant

Question: The coordinates of the point on X-axis which are equidistant from the points (3, 4) and (2, 5) are (a) $(20,0)$ (b) $(-23,0)$ (c) $\left(\frac{4}{5}, 0\right)$ (d) None of these Solution: Let the point be Abe equidistant from the two given points P (3, 4) and Q (2, 5). So applying distance formula, we get, $\mathrm{AP}^{2}=\mathrm{AQ}^{2}$ Therefore, $(a+3)^{2}+(-4)^{2}=(a-2)^{2}+5^{2}$ $10 a=4$ $a=\frac{2}{5}$ Hence the co-ordinates of $A$ are $\left(\frac{2}{5}, 0\right)$ So the answ...

Find the ratio in which the point P(m, 6) divides the join of A(−4, 3) and B(2, 8). Also, find the value of m.

Question: Find the ratio in which the pointP(m, 6) divides the join ofA(4, 3) andB(2, 8). Also, find the value ofm. Solution: Let the pointP(m, 6) divide the lineABin the ratiok: 1.Then, by the section formula: $x=\frac{m x_{2}+n x_{1}}{m+n}, y=\frac{m y_{2}+n y_{1}}{m+n}$ The coordinates of $P$ are $(m, 6)$. $m=\frac{2 k-4}{k+1}, 6=\frac{8 k+3}{k+1}$ $\Rightarrow m(k+1)=2 k-4,6 k+6=8 k+3$ $\Rightarrow m(k+1)=2 k-4,6-3=8 k-6 k$ $\Rightarrow m(k+1)=2 k-4,2 k=3$ $\Rightarrow m(k+1)=2 k-4, k=\frac{...

Solve the following

Question: If $S_{2}$ and $S_{4}$ denote respectively the sum of the squares and the sum of the fourth powers of first $n$ natural numbers, then $\frac{S_{4}}{S_{2}}=$ __________________ . Solution: S2: Sum of the squares of firstnnatural numbers. S4: Sum of the fourth powers of firstnnatural numbers. To find :- $\frac{S_{4}}{S_{2}}$ Since $S_{4}=\frac{n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)}{30}$ $S_{2}=\frac{n(n+1)(2 n+1)}{6}$ Hence,$\frac{S_{4}}{S_{2}}=\frac{n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\ri...

Four equal masses, m each are placed at the corners of a square of length

Question: Four equal masses, $\mathrm{m}$ each are placed at the corners of a square of length $(l)$ as shown in the figure. The moment of inertia of the system about an axis passing through $\mathrm{A}$ and parallel to $\mathrm{DB}$ would be : $\mathrm{m} / 2$$2 \mathrm{ml}^{2}$$3 \mathrm{ml}^{2}$$\sqrt{3} \mathrm{ml}^{2}$Correct Option: , 3 Solution: (3) Moment of inertia of point mass $=$ mass $\times(\text { Perpendicular distance from axis })^{2}$ Moment of Inertia $=\mathrm{m}(0)^{2}+\math...

If the points (k, 2k), (3k, 3k) and (3, 1) are collinear, then k

Question: If the points (k, 2k), (3k, 3k) and (3, 1) are collinear, thenk (a) $\frac{1}{3}$ (b) $-\frac{1}{3}$ (c) $\frac{2}{3}$ (d) $-\frac{2}{3}$ Solution: We have three collinear points $\mathrm{A}(k, 2 k) ; \mathrm{B}(3 k, 3 k) ; \mathrm{C}(3,1)$. 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, area of the triangle is 0 . $x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(...

Find the ratio in which the point

Question: Find the ratio in which the point $P\left(\frac{3}{4}, \frac{5}{12}\right)$ divides the line segment joining the points $A\left(\frac{1}{2}, \frac{3}{2}\right)$ and $(2,-5)$ Solution: Let $k: 1$ be the ratio in which the point $P\left(\frac{3}{4}, \frac{5}{12}\right)$ divides the line segment joining the points $A\left(\frac{1}{2}, \frac{3}{2}\right)$ and $(2,-5)$. Then $\left(\frac{3}{4}, \frac{5}{12}\right)=\left(\frac{k(2)+\frac{1}{2}}{k+1}, \frac{k(-5)+\frac{3}{2}}{k+1}\right)$ $\R...

If three points (0, 0),

Question: If three points $(0,0),(3, \sqrt{3})$ and $(3, \lambda)$ form an equilateral triangle, then $\lambda=$ (a) 2(b) 3(c) 4(d) None of these Solution: We have an equilateral triangle $\triangle \mathrm{ABC}$ whose co-ordinates are $\mathrm{A}(0,0) ; \mathrm{B}(3, \sqrt{3})$ and $\mathrm{C}(3, \lambda)$. Since the triangle is equilateral. So, $\mathrm{AB}^{2}=\mathrm{AC}^{2}$ So, $(3-0)^{2}+(\sqrt{3}-0)^{2}=(3-0)^{2}+(\lambda-0)^{2}$ Cancel out the common terms from both the sides, Therefore...

The sum of n terms of the series

Question: The sum ofnterms of the series 22+ 42+ 62+......, is _______________. Solution: Sum ofnterms of series 22+ 42+ 62+ ......... Letnthterm of series be denoted by⊤n i.e⊤n= (2n)2 then $\sum_{r=1}^{n} T_{r}=\sum_{r=1}^{n}(2 r)^{2}$ $=4 \sum_{r=1}^{n} r^{2}$ $=4\left[\frac{n(n+1)(2 n+1)}{6}\right]$ i.e sum of $n$ terms of $2^{2}+4^{2}+6^{2}=\frac{2}{3}[n(n+1)(2 n+1)]$...

In what ratio does the point P(2, 5) divide the join of A(8, 2) and B(−6, 9)?

Question: In what ratio does the pointP(2, 5) divide the join ofA(8, 2) andB(6, 9)? Solution: Let the pointP(2, 5) divideABin the ratiok: 1.Then, by section formula, the coordinates ofPare $x=\frac{-6 k+8}{k+1}, y=\frac{9 k+2}{k+1}$ It is given that the coordinates of $P$ are $P(2,5)$. $\Rightarrow 2=\frac{-6 k+8}{k+1}, 5=\frac{9 k+2}{k+1}$ $\Rightarrow 2 k+2=-6 k+8,5 k+5=9 k+2$ $\Rightarrow 2 k+6 k=8-2,5-2=9 k-5 k$ $\Rightarrow 8 k=6,4 k=3$ $\Rightarrow k=\frac{6}{8}, k=\frac{3}{4}$ $\Rightarro...

If A (2, 2), B (−4, −4) and C (5, −8) are the vertices

Question: IfA(2, 2),B(4, 4) andC(5, 8) are the vertices of a triangle, than the length of the median through vertexCis (a) $\sqrt{65}$ (b) $\sqrt{117}$ (c) $\sqrt{85}$ (d) $\sqrt{113}$ Solution: We have a triangle $\triangle \mathrm{ABC}$ in which the co-ordinates of the vertices are $\mathrm{A}(2,2) \mathrm{B}(-4,-4)$ and $\mathrm{C}(5,-8)$. In general to find the mid-point $\mathrm{P}(x, y)$ of two points $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\left(x_{2}, y_{2}\right)$ we use se...

The value of

Question: The value of $\sum_{r=1}^{n}\left\{(2 r-1)+\frac{1}{2^{r}}\right\}$ is _______________ . Solution: $\sum_{r=1}^{n}\left\{(2 r-1)+\frac{1}{2^{r}}\right\}$ For $\sum_{r=1}^{n}(2 r-1)=1+3+5+7+\ldots \ldots+2 n+1$ Here first term is 1 last term is 2n 1 i.e sum is $\frac{n}{2}$ (first term + last term) i.e $\frac{n}{2}(1+2 n-1)$ $=\frac{n}{2}(2 n)$ i. e $\sum_{r=1}^{n}(2 r-1)=n^{2} \quad \ldots(1)$ also $\sum_{r=1}^{n} \frac{1}{2^{r}}=\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\ldots+\frac...

Find the coordinates of a point A, where AB is the diameter of a circle with centre C(2, −3)

Question: Find the coordinates of a pointA, whereABis the diameter of a circle with centreC(2, 3) and the other end of the diameter isB(1, 4). Solution: C(2, 3) is the centre of the given circle. LetA(a,b) andB(1, 4) be the two end-points of the given diameterAB.Then, the coordinates ofCare $x=\frac{a+1}{2}, y=\frac{b+4}{2}$ It is given that $x=2$ and $y=-3$. $\Rightarrow 2=\frac{a+1}{2},-3=\frac{b+4}{2}$ $\Rightarrow 4=a+1,-6=b+4$ $\Rightarrow a=4-1, b=-6-4$ $\Rightarrow a=3, b=-10$ Therefore, ...

There is a small source of light at some depth below the

Question: There is a small source of light at some depth below the surface of water (refractive index $=\frac{4}{3}$ ) in a tank of large\ cross sectional surface area. Neglecting any reflection from the bottom and absorption by water, percentage of light that emerges out of surface is (nearly): [Use the fact that surface area of a spherical cap of height $h$ and radius of curvature $r$ is $2 \pi r h$ and radius of curvature $r$ is $2 \pi r h$ ]$21 \%$$34 \%$$17 \%$$50 \%$Correct Option: 3 Solut...

The perimeter of the triangle formed by the points

Question: The perimeter of the triangle formed by the points (0, 0), (0, 1) and (0, 1) is (a) $1 \pm \sqrt{2}$ (b) $\sqrt{2}+1$ (c) 3 (d) $2+\sqrt{2}$ Solution: We have a trianglewhose co-ordinates are A (0, 0); B (1, 0); C (0, 1). So clearly the triangle is right angled triangle, right angled at A. So, $\mathrm{AB}=1$ unit $\mathrm{AC}=1$ unit Now apply Pythagoras theorem to get the hypotenuse, $\mathrm{BC}=\sqrt{\mathrm{AB}^{2}+\mathrm{AC}^{2}}$ $=\sqrt{2}$ So the perimeter of the triangle is,...

The sum of first 25 odd natural numbers is

Question: The sum of first 25 odd natural numbers is _________. Solution: Sum of for 25 odd numbers :- Let S denote the sum of first 25 odd numbers Here number are 1, 3, 5, 7, 9 --------- 49. Here first term is 1 last term is 49 Common differencedis 2 n= 25 i. e Sum $=\frac{n}{2}\left(a+T_{n}\right)$ $=\frac{25}{2}(1+49)$ $=\frac{25 \times 50}{2}$ $=25 \times 25$ i.e sum of first 25 odd numbers = 625....

The line segment joining A(−2, 9) and B(6, 3) is a diameter of a circle with centre C.

Question: The line segment joiningA(2, 9) andB(6, 3) is a diameter of a circle with centreC. Find the coordinates ofC. Solution: The given points areA(2, 9) andB(6, 3).Then,C(x,y) is the midpoint ofAB. $x=\frac{x_{1}+x_{2}}{2}, y=\frac{y_{1}+y_{2}}{2}$ $\Rightarrow x=\frac{-2+6}{2}, y=\frac{9+3}{2}$ $\Rightarrow x=\frac{4}{2}, y=\frac{12}{2}$ $\Rightarrow x=2, y=6$ Therefore, the coordinates of pointCare (2, 6)....

A line segment is of length 10 units.

Question: A line segment is of length 10 units. If the coordinates of its one end are (2, 3) and the abscissa of the other end is 10, then its ordinate is(a) 9, 6(b) 3, 9(c) 3, 9(d) 9, 6 Solution: It is given that distance between $\mathrm{P}(2,-3)$ and $\mathrm{Q}(10, y)$ is 10 . In general, the distance between $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\left(x_{2}, y_{2}\right)$ is given by, $\mathrm{AB}^{2}=\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}$ So, $10^{2}=(10-...

Solve the following

Question: LetSnandS'4denote respectively the sum and the sum of the squares of firstnnatural numbers.If $a_{n}=\frac{S_{n}{ }^{\prime}}{S_{n}}, n \in N .$ Then $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}$, __________$a_{n,}$_____________ forms an ______ with ______ . Solution: Sn: The sum of firstnnatural numbers . Sn: Sum of squares of firstnnatural numbers. If $a_{n}=\frac{S_{n}}{S_{n}} ; n \in N$. $a_{1}=\frac{1}{1}=1$ $a_{2}=\frac{1^{2}+2^{2}}{1+2}=\frac{5}{3}$ $a_{3}=\frac{1^{2}+2^{2}+3...

Question: LetSnandS'4denote respectively the sum and the sum of the squares of firstnnatural numbers.If $a_{n}=\frac{S_{n}{ }^{\prime}}{S_{n}}, n \in N .$ Then $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}$, __________$a_{n,}$_____________ forms an ______ with ______ . Solution: Sn: The sum of firstnnatural numbers . Sn: Sum of squares of firstnnatural numbers. If $a_{n}=\frac{S_{n}}{S_{n}} ; n \in N$. $a_{1}=\frac{1}{1}=1$ $a_{2}=\frac{1^{2}+2^{2}}{1+2}=\frac{5}{3}$ $a_{3}=\frac{1^{2}+2^{2}+3...

The midpoint of the line segment joining A(2a, 4) and B(−2, 3b) is C(1, 2a + 1).

Question: The midpoint of the line segment joiningA(2a, 4) andB(2, 3b) isC(1, 2a+ 1). Find the values ofaandb. Solution: The points areA(2a, 4) andB(2, 3b).LetC(1, 2a+ 1) be the mid point ofAB. Then: $x=\frac{x_{1}+x_{2}}{2}, y=\frac{y_{1}+y_{2}}{2}$ $\Rightarrow 1=\frac{2 a+(-2)}{2}, 2 a+1=\frac{4+3 b}{2}$ $\Rightarrow 2=2 a-2,4 a+2=4+3 b$ $\Rightarrow 2 a=2+2,4 a-3 b=4-2$ $\Rightarrow a=\frac{4}{2}, 4 a-3 b=2$ $\Rightarrow a=2,4 a-3 b=2$ Putting the value of $a$ in the equation $4 a+3 b=2$, we...

If the distance between the points (4, p) and (1, 0) is 5, then p =

Question: If the distance between the points (4,p) and (1, 0) is 5, thenp = (a) 4(b) 4(c) 4(d) 0 Solution: It is given that distance between $\mathrm{P}(4, p)$ and $\mathrm{Q}(1,0)$ is 5 . In general, the distance between $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\left(x_{2}, y_{2}\right)$ is given by, $\mathrm{AB}^{2}=\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}$ So, $5^{2}=(4-1)^{2}+(p-0)^{2}$ On further simplification, $p^{2}=16$ $p=\pm 4$ So, $p=\pm 4$ So the answer i...

A vessel of depth 2 h is half filled with a liquid

Question: A vessel of depth $2 \mathrm{~h}$ is half filled with a liquid of refractive index $2 \sqrt{2}$ and the upper half with another liquid of refractive index $\sqrt{2}$. The liquids are immiscible. The apparent depth of the inner surface of the bottom of vessel will be:$\frac{h}{\sqrt{2}}$$\frac{h}{2(\sqrt{2}+1)}$$\frac{h}{3 \sqrt{2}}$$\frac{3}{4} h \sqrt{2}$Correct Option: 4 Solution: (4) Apparent depth, $D=\frac{t_{1}}{\mu_{1}}+\frac{t_{2}}{\mu_{2}}=\frac{h}{\sqrt{2}}+\frac{h}{2 \sqrt{2...

If (2, p) is the midpoint of the line segment joining the points A(6, −5) and B(−2, 11),

Question: If (2,p) is the midpoint of the line segment joining the pointsA(6, 5) andB(2, 11), find the value ofp. Solution: The given points areA(6, 5) andB(2, 11).Let (x,y) be the mid point ofAB.Then: $x=\frac{x_{1}+x_{2}}{2}, y=\frac{y_{1}+y_{2}}{2}$ $\Rightarrow x=\frac{6+(-2)}{2}, y=\frac{-5+11}{2}$ $\Rightarrow x=\frac{6-2}{2}, y=\frac{-5+11}{2}$ $\Rightarrow x=\frac{4}{2}, y=\frac{6}{2}$ $\Rightarrow x=2, y=3$ So, the midpoint ofABis (2, 3).But it is given that the midpoint ofABis (2,p).Th...