Point P(5, −3) is one of the two points of trisection of the line segment j
Question: PointP(5, 3) is one of the two points of trisection of the line segment joining the pointsA(7, 2) andB(1, 5) near toA. Find the coordinates of the other point of trisection. Solution: We are given a line segment joining pointsA(7, 2) andB(1, 5) P(5, 3) is one of the two points of trisection of line segmentAB Pis near toA We are to find the coordinates of other points of trisection Let the other point of trisection isQ Therefore AP=PQ=QB That isQis the mid point of line segmentPB We kno...
Read More →If 3 cot θ = 2, show that
Question: If $3 \cot \theta=2$, show that $\left(\frac{4 \sin \theta-3 \cos \theta}{2 \sin \theta+6 \cos \theta}\right)=\frac{1}{3}$. Solution: It is given that $\cot \theta=\frac{2}{3}$. $\mathrm{LHS}=\frac{4 \sin \theta-3 \cos \theta}{2 \sin \theta+6 \cos \theta}$ Dividing the above expression by $\sin \theta$, we get: $\frac{4-3 \cot \theta}{2+6 \cot \theta} \quad\left[\because \cot \theta=\frac{\cos \theta}{\sin \theta}\right]$ Now, substituting the values of cotin the above expression, we g...
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Question: Express the matrix $A=\left[\begin{array}{ll}3 -4 \\ 1 -1\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix. Solution: Given : $A=\left[\begin{array}{ll}3 -4 \\ 1 -1\end{array}\right]$ $A^{T}=\left[\begin{array}{cc}3 1 \\ -4 -1\end{array}\right]$ Let $X=\frac{1}{2}\left(A+A^{T}\right)=\frac{1}{2}\left(\left[\begin{array}{ll}3 -4 \\ 1 -1\end{array}\right]+\left[\begin{array}{cc}3 1 \\ -4 -1\end{array}\right]\right)=\left[\begin{array}{cc}3 \frac{-3}{2} \\ \frac{-3...
Read More →How many spherical lead shots each having diameter 3 cm
Question: How many spherical lead shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9 cm 11 cm 12 cm? Solution: Given a cuboidal lead solid with dimensions 9 cm 11 cm 12 cm We have to find the number of spherical lead shots each having a diameter of 3cm which can be made from the cuboidal lead solid. Let the length of cuboidal lead solid L=9 cm Let the breadth of cuboidal lead solid B=11 cm Let the length of cuboidal lead solid H =9 cm Let the number of spheric...
Read More →In Fig. 2, a circle of radius 7 cm is inscribed in a square.
Question: In Fig. 2, a circle of radius 7 cm is inscribed in a square. Find the area of the shaded region $\left(U\right.$ se $\left.\pi=\frac{22}{7}\right)$ Solution: It is given that a circle of radius 7 cm is inscribed in a square We have to find the area of shaded region shown in figure We are given the following figure Let the side of the square =acm Since the circle in inscribed in the square Diameter of the circle =acm Radius of circle $=\frac{a}{2} \mathrm{~cm}$ Given that radius of circ...
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Question: If $\sin \alpha=\frac{1}{2}$, prove that $\left(3 \cos \alpha-4 \cos ^{3} \alpha\right)=0$ Solution: LHS $=\left(3 \cos \alpha-4 \cos ^{3} \alpha\right)$ $=\cos \alpha\left(3-4 \cos ^{2} \alpha\right)$ $=\sqrt{1-\sin ^{2} \alpha}\left[3-4\left(1-\sin ^{2} \alpha\right)\right]$ $=\sqrt{1-\left(\frac{1}{2}\right)^{2}}\left[3-4\left(1-\left(\frac{1}{2}\right)^{2}\right)\right]$ $=\sqrt{\frac{1}{1}-\frac{1}{4}}\left[3-4\left(\frac{1}{1}-\frac{1}{4}\right)\right]$ $=\sqrt{\frac{3}{4}}\left[...
Read More →Define a symmetric matrix. Prove that
Question: Define a symmetric matrix. Prove that for $A=\left[\begin{array}{ll}2 4 \\ 5 6\end{array}\right], A+A^{T}$ is a symmetric matrix where $A^{T}$ is the transpose of $A$. Solution: $A$ square matrix $A$ is called a symmetric mat rix, if $A^{T}=A$. Given : $A=\left[\begin{array}{ll}2 4 \\ 5 6\end{array}\right]$ $A^{T}=\left[\begin{array}{ll}2 5 \\ 4 6\end{array}\right]$ Now, $A+A^{T}=\left[\begin{array}{ll}2 4 \\ 5 6\end{array}\right]+\left[\begin{array}{ll}2 5 \\ 4 6\end{array}\right]$ $\...
Read More →Two tangents PA and PB are drawn from
Question: Two tangentsPAandPBare drawn from an external pointPto a circle with centreO. Prove thatAOBPis a cyclic quadrilateral. Solution: We are given two tangentsPAandPBdrawn to a circle with centreOfrom external pointP We are to prove that quadrilateralAOBPis cyclic We know that tangent at a point to a circle is perpendicular to the radius through that point. Therefore from figure $O A \perp A P$ $O B \perp B P$ That is $\angle O A P=90^{\circ}$ $\angle O B P=90^{\circ}$ $\angle O A P+\angle ...
Read More →If the numbers x − 2, 4x − 1 and
Question: If the numbersx 2, 4x 1 and 5x+ 2 are in A.P. find the value ofx. Solution: It is given that the numbers We have to find the value ofx We know that ifx, yandzare in A.P, then $y-x=z-y$ $2 y=x+z$ Therefore for the given numbers $2(4 x-1)=x-2+5 x+2$ $8 x-2=6 x$ $8 x-6 x=2$ $2 x=2$ Hence $x=1$...
Read More →Find the roots of the following quadratic equation:
Question: Find the roots of the following quadratic equation: $\frac{2}{5} x^{2}-x-\frac{3}{5}=0$ Solution: It is given that: $\frac{2}{5} x^{2}-x-\frac{3}{5}=0$ We have to find the roots of above equation. $\frac{2}{5} x^{2}-x-\frac{3}{5}=0$ Multiplying both sides by 5 $2 x^{2}-5 x-3=20$ $2 x^{2}-6 x+x-3=0$ $2 x(x-3)+1(x-3)=0$ $(x-3)(2 x+1)=0$ $x=3, x=-\frac{1}{2}$ Therefore the roots of the equation are : $3,-\frac{1}{2}$...
Read More →Find two consecutive odd positive integers,
Question: Find two consecutive odd positive integers, sum of whose squares is 290. Solution: We have to find two consecutive integers sum of whose squares is 290. Let the two consecutive integers bexandx+2 According to the question $x^{2}+(x+2)^{2}=290$ $x^{2}+x^{2}+4+4 x=290$ $2 x^{2}+4 x+4=290$ Dividing both sides by 2 $x^{2}+2 x+2=145$ $x^{2}+2 x-143=0$ $x^{2}+13 x-11 x-143=0$ $x(x+13)-11(x+13)=0$ $(x+13)(x-11)=0$ $x=-13,11$ Therefore two consecutive integers are 11,13...
Read More →21 glass spheres, each of radius 2 cm
Question: 21 glass spheres, each of radius 2 cm are packed in a cuboidal box of internal dimensions 16 cm 8 cm 8 cm and, then the box is filled with water. Find the volume of water filled in the box. Solution: It is given that 21 glass spheres each of radius 2 cm are packed in a cuboidal box of inner dimensionsand then filled with water. We have to find the volume of water. Radius of glass spherer Volume of a glass sphere $=\frac{4}{3} \pi r^{3}$ Volume of 21 glass spheres $=21 \times \frac{4}{3...
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Question: Express the matrix $A=\left[\begin{array}{rrr}4 2 -1 \\ 3 5 7 \\ 1 -2 1\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix. Solution: Given : $A=\left[\begin{array}{ccc}4 2 -1 \\ 3 5 7 \\ 1 -2 1\end{array}\right]$ $A^{T}=\left[\begin{array}{ccc}4 3 1 \\ 2 5 -2 \\ -1 7 1\end{array}\right]$ Let $X=\frac{1}{2}\left(A+A^{T}\right)=\frac{1}{2}\left(\left[\begin{array}{ccc}4 2 -1 \\ 3 5 7 \\ 1 -2 1\end{array}\right]+\left[\begin{array}{ccc}4 3 1 \\ 2 5 -2 \\ -1 7 1\end{...
Read More →If the circumference of a circle is equal to the perimeter
Question: If the circumference of a circle is equal to the perimeter of a square then the ratio their areas is:(a) 22 : 7(b) 14 :11(c) 7 :22(d) 7 :11 Solution: It is given that circumference of a circle = perimeter of a square We have to find the ratio of their areas Let the radius of circle =r Let the circumference of circle = Let the area of circle = Let the side of the square =b Let the perimeter of square = Let the area of square = Therefore Circumference of circle $P_{c}=2 \pi r$ Perimeter ...
Read More →Prove that
Question: If $\tan \theta=\frac{1}{2}$ then evaluate $\left(\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}\right)$ Solution: Given: $\tan \theta=\frac{1}{2}$ Since, $\tan \theta=\frac{P}{B}$ $\Rightarrow P=1$ and $B=2$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow 1^{2}+2^{2}=H^{2}$ $\Rightarrow H^{2}=1+4$ $\Rightarrow H^{2}=5$ $\Rightarrow H=\sqrt{5}$ Therefore, $\sin \theta=\frac{P}{H}=\frac{1}{\sqrt{5}}$ $\cos \theta=\frac{B}{H}=\frac{2}{\sqrt{5}}$ Now, $\left(...
Read More →The height of a cone is 60 cm.
Question: The height of a cone is $60 \mathrm{~cm}$. A small cone is cut off at the top by a plane parallel to the base and its volume is $\frac{1}{64} t h$ the volume of original cone. The height from the base at which the section is made is: Solution: It is given that the height of a cone = 60 cm The volume of a small cone cut on the top by plane parallel to base of given cone=volume of given cone We have to find the height from the base at which section is made. LetRbe the radius of base of g...
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Question: Let $A=\left[\begin{array}{rrr}3 2 7 \\ 1 4 3 \\ -2 5 8\end{array}\right]$. Find matrices $X$ and $Y$ such that $X+Y=A$, where $X$ is a symmetric and $Y$ is a skew-symmetric matrix. Solution: Given : $A=\left[\begin{array}{ccc}3 2 7 \\ 1 4 3 \\ -2 5 8\end{array}\right]$ $\Rightarrow A^{T}=\left[\begin{array}{ccc}3 1 -2 \\ 2 4 5 \\ 7 3 8\end{array}\right]$ Let $X=\frac{1}{2}\left(A+A^{T}\right)=\frac{1}{2}\left(\left[\begin{array}{ccc}3 2 7 \\ 1 4 3 \\ -2 5 8\end{array}\right]+\left[\be...
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Question: Let $A=\left[\begin{array}{rrr}3 2 7 \\ 1 4 3 \\ -2 5 8\end{array}\right]$. Find matrices $X$ and $Y$ such that $X+Y=A$, where $X$ is a symmetric and $Y$ is a skew-symmetric matrix. Solution: Given : $A=\left[\begin{array}{ccc}3 2 7 \\ 1 4 3 \\ -2 5 8\end{array}\right]$ $\Rightarrow A^{T}=\left[\begin{array}{ccc}3 1 -2 \\ 2 4 5 \\ 7 3 8\end{array}\right]$ Let $X=\frac{1}{2}\left(A+A^{T}\right)=\frac{1}{2}\left(\left[\begin{array}{ccc}3 2 7 \\ 1 4 3 \\ -2 5 8\end{array}\right]+\left[\be...
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Question: If the matrix $A=\left[\begin{array}{rrr}5 2 x \\ y z -3 \\ 4 t -7\end{array}\right]$ is a symmetric matrix, find $x, y, z$ and $t$. Solution: Given : $A=\left[\begin{array}{ccc}5 2 x \\ y z -3 \\ 4 t -7\end{array}\right]$ $\Rightarrow A^{T}=\left[\begin{array}{ccc}5 y 4 \\ 2 z t \\ x -3 -7\end{array}\right]$ Since $A$ is a symmetric matrix, $A^{T}=A$. $\Rightarrow\left[\begin{array}{ccc}5 y 4 \\ 2 z t \\ x -3 -7\end{array}\right]=\left[\begin{array}{ccc}5 2 x \\ y z -3 \\ 4 t -7\end{a...
Read More →To draw a pair f tangents to a circle which are inclined
Question: To draw a pair f tangents to a circle which are inclined to each other at an angle of 100, It is required to draw tangents at end points of those two radii of the circle, the angle between which should be:(a) 100(b) 50(c) 80(d) 200 Solution: Given a pair of tangents to a circle inclined to each other at angle of 100 We have to find the angle between two radii of circle joining the end points of tangents that is we have to findin below figure. LetObe the center of the given circle LetAB...
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Question: If $\sin \theta=\frac{12}{13}$ then evaluate $\left(\frac{2 \sin \theta-3 \cos \theta}{4 \sin \theta-9 \cos \theta}\right)$ Solution: Given: $\sin \theta=\frac{12}{13}$ Since, $\sin \theta=\frac{P}{H}$ $\Rightarrow P=12$ and $H=13$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow 12^{2}+B^{2}=13^{2}$ $\Rightarrow B^{2}=169-144$ $\Rightarrow B^{2}=25$ $\Rightarrow B=5$ Therefore, $\cos \theta=\frac{B}{H}=\frac{5}{13}$ $\cos \theta=\frac{B}{H}=\frac{5}{13}$ Now, $\left(\frac{2 ...
Read More →Two tangents making an angle of 120° with each other are drawn
Question: Two tangents making an angle of 120 with each other are drawn to a circle of radius 6 cm, then the length of each tangent is equal to (a) $\sqrt{3} \mathrm{~cm}$ (b) $6 \sqrt{3} \mathrm{~cm}$ (c) $\sqrt{2} \mathrm{~cm}$ (d) $2 \sqrt{3} \mathrm{~cm}$ Solution: We are given two tangents to a circle making an angle of 120 with each other. The radius of circle is 6 cm We have to find the length of each tangent. LetObe the center of the given circle LetABandACbe the two tangents to the give...
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Question: If $A=\left[\begin{array}{ll}3 -4 \\ 1 -1\end{array}\right]$, show that $A-A^{T}$ is a skewsymmetric matrix. Solution: Given : $A=\left[\begin{array}{ll}3 -4 \\ 1 -1\end{array}\right]$ $A^{T}=\left[\begin{array}{cc}3 1 \\ -4 -1\end{array}\right]$ Now, $A-A^{T}=\left[\begin{array}{ll}3 -4 \\ 1 -1\end{array}\right]-\left[\begin{array}{cc}3 1 \\ -4 -1\end{array}\right]$ $\Rightarrow A-A^{T}=\left[\begin{array}{ll}3-3 -4-1 \\ 1+4 -1+1\end{array}\right]$ $\Rightarrow A-A^{T}=\left[\begin{ar...
Read More →Let A be a matrix of order 3 × 4. If R1 denotes the first row of A and C2 denotes its second column,
Question: Let $A$ be a matrix of order $3 \times 4$. If $R_{1}$ denotes the first row of $A$ and $C_{2}$ denotes its second column, then determine the orders of matrices $R_{1}$ and $C_{2}$ Solution: The order of $R_{1}$ is $1 \times 4$ and the order of $C_{2}$ is $3 \times 1$....
Read More →Add and express the sum as a mixed fraction:
Question: Add and express the sum as a mixed fraction: (i) $\frac{-12}{5}$ and $\frac{43}{10}$ (ii) $\frac{24}{7}$ and $\frac{-11}{4}$ (iii) $\frac{-31}{6}$ and $\frac{-27}{8}$ (iv) $\frac{101}{6}$ and $\frac{7}{8}$ Solution: (i) We have $\frac{-12}{2}+\frac{43}{10}$. L.C.M. of the denominators 5 and 10 is $10 .$ Now, we will express $\frac{-12}{5}$ in the form in which it takes the denominator 10 . $\frac{-12 \times 2}{5 \times 2}=\frac{-24}{10}$ $\therefore \frac{-12}{5}+\frac{43}{10}=\frac{-2...
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