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...
Read More →Solve this
Question: If $A=\left[a_{i j}\right]=\left[\begin{array}{rrr}2 3 -5 \\ 1 4 9 \\ 0 7 -2\end{array}\right]$ and $B=\left[b_{i j}\right]=$ $\left[\begin{array}{cc}2 -1 \\ -3 4 \\ 1 2\end{array}\right]$ then find (i) $a_{22}+b_{21}$ (ii) $a_{11} b_{11}+a_{22} b_{22}$ Solution: (i) $a_{22}+b_{21}$ Here, $a_{22}=4$ and $b_{21}=-3$ $\Rightarrow a_{22}+b_{21}=4-3=1$ (ii) $a_{11} b_{11}+a_{22} b_{22}$ Here, $a_{11}=2, b_{11}=2, a_{22}=4$ and $b_{22}=4$ $\Rightarrow a_{11} b_{11}+a_{22} b_{22}=2 \times 2+...
Read More →Solve this
Question: If $\tan \theta=\frac{a}{b}$, show that $\left(\frac{a \sin \theta-b \cos \theta}{a \sin \theta+b \cos \theta}\right)=\frac{\left(a^{2}-b^{2}\right)}{\left(a^{2}+b^{2}\right)}$ Solution: It is given that $\tan \theta=\frac{a}{b}$. $\mathrm{LHS}=\frac{a \sin \theta-b \cos \theta}{a \sin \theta+b \cos \theta}$ Dividing the numerator and denominator by $\cos \theta$, we get: $\frac{a \tan \theta-b}{a \tan \theta+b} \quad\left(\because \tan \theta=\frac{\sin \theta}{\cos \theta}\right)$ No...
Read More →Simplify:
Question: Simplify: (i) $\frac{8}{9}+\frac{-11}{6}$ (ii) $3+\frac{5}{-7}$ (iii) $\frac{1}{-12}+\frac{2}{-15}$ (iv) $\frac{-8}{19}+\frac{-4}{57}$ (v) $\frac{7}{9}+\frac{3}{-4}$ (vi) $\frac{5}{26}+\frac{11}{-39}$ (vii) $\frac{-16}{9}+\frac{-5}{12}$ (viii) $\frac{-13}{8}+\frac{5}{36}$ (ix) $0+\frac{-3}{5}$ (x) $1+\frac{-4}{5}$ Solution: (i) $\frac{8}{9}+\frac{-11}{6}$ L.C.M. of the denominators 9 and 6 is $18 .$ Now, we will express $\frac{8}{9}$ and $\frac{-11}{6}$ in the form in which they take t...
Read More →Simplify:
Question: Simplify: (i) $\frac{8}{9}+\frac{-11}{6}$ (ii) $3+\frac{5}{-7}$ (iii) $\frac{1}{-12}+\frac{2}{-15}$ (iv) $\frac{-8}{19}+\frac{-4}{57}$ (v) $\frac{7}{9}+\frac{3}{-4}$ (vi) $\frac{5}{26}+\frac{11}{-39}$ (vii) $\frac{-16}{9}+\frac{-5}{12}$ (viii) $\frac{-13}{8}+\frac{5}{36}$ (ix) $0+\frac{-3}{5}$ (x) $1+\frac{-4}{5}$ Solution: (i) $\frac{8}{9}+\frac{-11}{6}$ L.C.M. of the denominators 9 and 6 is $18 .$ Now, we will express $\frac{8}{9}$ and $\frac{-11}{6}$ in the form in which they take t...
Read More →Solve this
Question: If $\sin \theta=\frac{a}{b}$, show that $(\sec \theta+\tan \theta)=\sqrt{\frac{b+a}{b-a}}$. Solution: $\mathrm{LHS}=(\sec \theta+\tan \theta)$ $=\frac{1}{\cos \theta}+\frac{\sin \theta}{\cos \theta}$ $=\frac{1+\sin \theta}{\cos \theta}$ $=\frac{1+\sin \theta}{\sqrt{1-\sin ^{2} \theta}}$ $=\frac{\left(1+\frac{a}{b}\right)}{\sqrt{1-\left(\frac{a}{b}\right)^{2}}}$ $=\frac{\left(\frac{1}{1}+\frac{a}{b}\right)}{\sqrt{\frac{1}{1}-\frac{a^{2}}{b^{2}}}}$ $=\frac{\left(\frac{b+a}{b}\right)}{\sq...
Read More →In Fig. 1, PQ and PR are tangents to the circle
Question: In Fig. 1,PQandPRare tangents to the circle with centreOsuch that QPR= 50,(a) 25(b) 30(c) 40(d) 50 Solution: We are given the below figure in which PQandPRare tangents to the circle with centreOand We have to find PQis the tangent to circle Therefore[Since Radius of a circle is perpendicular to tangent] Similarly We know that sum of angles of a quadrilateral Therefore in Quadrilateral PQOR $\angle Q P R+\angle P R O+\angle R O Q+\angle O Q P=360^{\circ}$ $50^{\circ}+90^{\circ}+\angle R...
Read More →If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?
Question: If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements? Solution: We know that if a matrix is of order $m \times n$, then it has $m n$ elements. The possible orders of a matrix with 8 elements are given below: $1 \times 8,2 \times 4,4 \times 2,8 \times 1$ Thus, there are 4 possible orders of the matrix. The possible orders of a matrix with 5 elements are given below: $1 \times 5,5 \times 1$ Thus, there are 2 possible orders of the matrix....
Read More →If radii of the two concentric circles are 15 cm and 17 cm,
Question: If radii of the two concentric circles are 15 cm and 17 cm, then the length of each chord of one circle which is tangent to other is:(a) 8 cm(b) 16 cm(c) 30 cm(d) 17 cm Solution: We are given that radii of two concentric circles are 15 cm and 17 cm We have to find the length of each chord of one circle which is tangent to other LetAbe the centre of the two concentric circles LetBCbe the chord of bigger circle tangent to smaller atF ThereforeAFis the radius of smaller circle ABis the ra...
Read More →Solve this
Question: If $A=\left[\begin{array}{ll}2 3 \\ 4 5\end{array}\right]$, prove that $A-A^{T}$ is a skew-symmetric matrix. Solution: Given : $A=\left[\begin{array}{ll}2 3 \\ 4 5\end{array}\right]$ $A^{T}=\left[\begin{array}{ll}2 4 \\ 3 5\end{array}\right]$ Now, $\left(A-A^{T}\right)=\left[\begin{array}{ll}2 3 \\ 4 5\end{array}\right]-\left[\begin{array}{ll}2 4 \\ 3 5\end{array}\right]$ $\Rightarrow\left(A-A^{T}\right)=\left|\begin{array}{ll}2-2 3-4 \\ 4-3 5-5\end{array}\right|$ $\Rightarrow\left(A-A...
Read More →If 4tan θ = 3 then prove that
Question: If $4 \tan \theta=3$ then prove that $\sin \theta \cos \theta=\frac{12}{25}$. Solution: Given: $4 \tan \theta=3$ $\Rightarrow \tan \theta=\frac{3}{4}$ Since, $\tan \theta=\frac{P}{B}$ $\Rightarrow P=3$ and $B=4$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow 3^{2}+4^{2}=H^{2}$ $\Rightarrow H^{2}=9+16=25$ $\Rightarrow H=5$ Therefore, $\sin \theta=\frac{P}{H}=\frac{3}{5}$ $\cos \theta=\frac{B}{H}=\frac{4}{5}$ $\sin \theta \times \cos \theta=\frac{3}{5} \times \frac{4}{5}$ $=\...
Read More →The sum of first five multiples of 3 is:
Question: The sum of first five multiples of 3 is: (a) 45(b) 65(c) 75(d) 90 Solution: We have to find the sum of first five multiples of 3 First five multiples of 3 are 3, 6, 9, 12 and 15 It forms an Arithmetic Progression (A.P) with First term Common difference, $d=a_{2}-a$ $=6-3$ $=3$ We know that the sum of the $n$ terms of an arithmetic progression $\mathrm{S}_{n}=\frac{n}{2}(2 a+(n-1) d)$ Here $n=5$ Therefore $\mathrm{S}_{5}=\frac{5}{2}(2 \times 3+(5-1) 3)$ $=\frac{5}{2} \times 18$ $=45$ He...
Read More →Add the following rational numbers:
Question: Add the following rational numbers: (i) $\frac{3}{4}$ and $\frac{-5}{8}$ (ii) $\frac{5}{-9}$ and $\frac{7}{3}$ (iii) $-3$ and $\frac{3}{5}$ (iv) $\frac{-7}{27}$ and $\frac{11}{18}$ (v) $\frac{31}{-4}$ and $\frac{-5}{8}$ (vi) $\frac{5}{36}$ and $\frac{-7}{12}$ (vii) $\frac{-5}{16}$ and $\frac{7}{24}$ (viii) $\frac{7}{-18}$ and $\frac{8}{27}$ Solution: (i) Clearly, denominators of the given numbers are positive. The L.C.M. of denominators 4 and 8 is 8 . Now, we will express $\frac{3}{4}$...
Read More →Add the following rational numbers:
Question: Add the following rational numbers: (i) $\frac{3}{4}$ and $\frac{-5}{8}$ (ii) $\frac{5}{-9}$ and $\frac{7}{3}$ (iii) $-3$ and $\frac{3}{5}$ (iv) $\frac{-7}{27}$ and $\frac{11}{18}$ (v) $\frac{31}{-4}$ and $\frac{-5}{8}$ (vi) $\frac{5}{36}$ and $\frac{-7}{12}$ (vii) $\frac{-5}{16}$ and $\frac{7}{24}$ (viii) $\frac{7}{-18}$ and $\frac{8}{27}$ Solution: (i) Clearly, denominators of the given numbers are positive. The L.C.M. of denominators 4 and 8 is 8 . Now, we will express $\frac{3}{4}$...
Read More →Solve this
Question: If $I_{i}, m_{i}, n_{j}, i=1,2,3$ denote the direction cosines of three mutually perpendicular vectors in space, prove that $A A^{T}=I$,$=l$, where $A=$$\left[\begin{array}{lll}l_{1} m_{1} n_{1} \\ l_{2} m_{2} n_{2} \\ l_{3} m_{3} n_{3}\end{array}\right]$ Solution: Given. $\left(l_{1}, m_{1}, n_{1}\right),\left(l_{2}, m_{2}, n_{2}\right),\left(l_{3}, m_{3}, n_{3}\right)$ are the direction cosines of three mutually perpendicular vectors in space. $\left.\begin{array}{l}l_{1}^{2}+m_{1}^{...
Read More →Prove that
Question: If $\sqrt{3} \tan \theta=1$ then evaluate $\left(\cos ^{2} \theta-\sin ^{2} \theta\right)$. Solution: Given: $\sqrt{3} \tan \theta=1$ $\Rightarrow \tan \theta=\frac{1}{\sqrt{3}}$ Since, $\tan \theta=\frac{P}{B}$ $\Rightarrow P=1$ and $B=\sqrt{3}$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow H^{2}=1+3=4$ $\Rightarrow H=2$ Therefore, $\sin \theta=\frac{P}{H}=\frac{1}{2}$ $\cos \theta=\frac{B}{H}=\frac{\sqrt{3}}{2}$ $\cos ^{2} \theta-\sin ^{2} \theta=\left(\frac{\sqrt{3}}{2}...
Read More →Which of the following equations has the sum of its roots as 3?
Question: Which of the following equations has the sum of its roots as 3? (a)x2+ 3x 5 = 0(b) x2+ 3x+ 3 = 0 (c) $\sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x-1=0$ (d) 3x2 3x 3 = 0 Solution: Given the following quadratic equations (a)x2+ 3x 5 = 0 (b) x2+ 3x+ 3 = 0 (c) $\sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x-1=0$ (d) $3 x^{2}-3 x-3=0$ We are to find out which of the above equations has sum of roots = 3. The sum of the roots of the quadratic equation $a x^{2}+b x+c=0$ is given by $-\frac{b}{a}$. For given equat...
Read More →Solve this
Question: If $\sin \theta=\frac{c}{\sqrt{c^{2}+d^{2}}}$, where $d0$ then find the values of $\cos \theta$ and $\tan \theta$. Solution: Given : $\sin \theta=\frac{c}{\sqrt{c^{2}+d^{2}}}$ Since, $\sin \theta=\frac{P}{H}$ $\Rightarrow P=c$ and $H=\sqrt{c^{2}+d^{2}}$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow c^{2}+B^{2}=c^{2}+d^{2}$ $\Rightarrow B^{2}=d^{2}$ $\Rightarrow B=d$ Therefore, $\cos \theta=\frac{B}{H}=\frac{d}{\sqrt{c^{2}+d^{2}}}$ $\tan \theta=\frac{P}{B}=\frac{c}{d}$ Henc...
Read More →Solve this
Question: If $A=\left[\begin{array}{rr}\sin \alpha \cos \alpha \\ -\cos \alpha \sin \alpha\end{array}\right]$, verify that $A^{\top} A=I_{2}$. Solution: Given : $A=\left[\begin{array}{cc}\sin \alpha \cos \alpha \\ -\cos \alpha \sin \alpha\end{array}\right]$ $A^{T}=\left[\begin{array}{cc}\sin \alpha -\cos \alpha \\ \cos \alpha \sin \alpha\end{array}\right]$ Now, $A^{T} A=\left[\begin{array}{cc}\sin \alpha -\cos \alpha \\ \cos \alpha \sin \alpha\end{array}\right]\left[\begin{array}{cc}\sin \alpha ...
Read More →The following table gives the production yield
Question: The following table gives the production yield per hectare of wheat of 100 farms of a village. Change the above distribution to more than type distribution and draw its ogive. Solution: We have the following distribution We have to change the above distribution in to the more than type distribution and we have to draw its ogive. We have the following procedure to find the more than type distribution To draw its ogive, take the number of frames on y-axis and production yield onx-axis. M...
Read More →The following table gives the production yield
Question: The following table gives the production yield per hectare of wheat of 100 farms of a village. Change the above distribution to more than type distribution and draw its ogive. Solution: We have the following distribution We have to change the above distribution in to the more than type distribution and we have to draw its ogive. We have the following procedure to find the more than type distribution To draw its ogive, take the number of frames on y-axis and production yield onx-axis. M...
Read More →Add the following rational numbers.
Question: Add the following rational numbers. (i) $\frac{-5}{7}$ and $\frac{3}{7}$ (ii) $\frac{-15}{4}$ and $\frac{7}{4}$ (iii) $\frac{-8}{11}$ and $\frac{-4}{11}$ (iv) $\frac{6}{13}$ and $\frac{-9}{13}$ Solution: (i) $\frac{-5}{7}+\frac{3}{7}=\frac{-5+3}{7}=\frac{-2}{7}$ (ii) $\frac{-15}{4}+\frac{7}{4}=\frac{-15+7}{4}=\frac{-8}{4}=-2$ (iii) $\frac{-8}{11}+\frac{-4}{11}=\frac{-8-4}{11}=\frac{-12}{11}$ (iv) $\frac{6}{13}+\frac{-9}{13}=\frac{6-9}{13}=\frac{-3}{13}$...
Read More →Find ten rational numbers between
Question: Find ten rational numbers between $\frac{3}{5}$ and $\frac{3}{4}$. Solution: The L. C.M of the denominators 5 and 4 of both the fractions is 20 . We can write: $\frac{3}{5}=\frac{3 \times 4}{5 \times 4}=\frac{12}{20}$ $\frac{3}{4}=\frac{3 \times 5}{4 \times 5}=\frac{15}{20}$ Since the integers between the numerators 12 and 15 are not sufficient, we will multiply both the fractions by 5 . $\frac{12}{20}=\frac{12 \times 5}{20 \times 5}=\frac{60}{100}$ $\frac{15}{20}=\frac{15 \times 5}{20...
Read More →Draw the graphs of following equations:
Question: Draw the graphs of following equations: 2xy= 1 andx+ 2y= 13(i) find the solution of the equations from the graph.(ii) shade the triangular region formed by the lines and the y-axis. Solution: Here we have to draw the graph between two equations given by (1) (2) Also we have to find the solution of the given equations. The first equation can written as follow (3) Now we are going to find the value ofyat different value ofx Now mark the points (0,-1), (1,1) and (2,3) onxy-plane and we wi...
Read More →If sec θ + tan θ = p, prove that sin θ =
Question: If $\sec \theta+\tan \theta=p$, prove that $\sin \theta=\frac{p^{2}-1}{p^{2}+1}$ Solution: Given that: $\sec \theta+\tan \theta=p$, then we have to prove that $\sin \theta=\frac{p^{2}-1}{p^{2}+1}$ We can rewrite the given data as $p=\sec \theta+\tan \theta$ $=\frac{1}{\cos \theta}+\frac{\sin \theta}{\cos \theta}$ $=\frac{1+\sin \theta}{\cos \theta}$ Now we take the right hand side $R H S=\frac{p^{2}-1}{p^{2}+1}$ Now we are putting the value ofpin the above expression, we get $R H S=\fr...
Read More →