If a, b and c are all non-zero
Question: If $a, b$ and $c$ are all non-zero and $a+b+c=0$, then prove that $\frac{a^{2}}{b c}+\frac{b^{2}}{a c}+\frac{c^{2}}{a b}=3$ Solution: To prove, $\frac{a^{2}}{b c}+\frac{b^{2}}{a c}+\frac{c^{2}}{a b}=3$\ We know that, $a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)$ $=0\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right) \quad[\because a+b+c=0$, given $]$ $=0$ $\Rightarrow \quad a^{3}+b^{3}+c^{3}=3 a b c$ On dividing both sides by $a b c$, we get $\frac{a^{3}}{a b c}+...
Read More →Using determinants, find the area of the triangle whose vertices
Question: Using determinants, find the area of the triangle whose vertices are $(1,4),(2,3)$ and $(-5,-3)$. Are the given points collinear? Solution: $\Delta=\frac{1}{2}\left|\begin{array}{ccc}1 4 1 \\ 2 3 1 \\ -5 -3 1\end{array}\right|$ $=\frac{1}{2}\left|\begin{array}{ccc}1 4 1 \\ 1 -1 0 \\ -5 -3 1\end{array}\right| \quad$ [Applying $R_{2} \rightarrow R_{2}-R_{1}$ ] $=\frac{1}{2}\left|\begin{array}{ccc}1 4 1 \\ 1 -1 0 \\ -6 -7 0\end{array}\right| \quad$ [Applying $R_{3} \rightarrow R_{3}-R_{1}...
Read More →Solve the following
Question: If $\overline{3 x 2}$ is a multiple of 11, where $x$ is a digit, what is the value of $x ?$ Solution: Sum of the digits at odd places $=3+2=5$ Sum of the digit at even place $=x$ $\therefore$ Sum of the digit at even place $-$ Sum of the digits at odd places $=(x-5)$ $\because(\mathrm{x}-5)$ must be multiple by 11 . $\therefore$ Possible values of $(x-5)$ are $0,11,22,33 \ldots$ But $\mathrm{x}$ is a digit; therefore $x$ must be $0,1,2,3 \ldots 9$. $\therefore x-5=0$ $\Rightarrow x=5$...
Read More →Prove the following
Question: Multiply x2+ 4y2+ z2+ 2xy + xz 2yz by (-z + x-2y). Solution: Now, $\left(x^{2}+4 y^{2}+z^{2}+2 x y+x z-2 y z\right)(-z+x-2 y)$ $=x^{2}(-z+x-2 y)+4 y^{2}(-z+x-2 y)+z^{2}(-z+x-2 y)+2 x y(-z+x-2 y)$ $+x z(-z+x-2 y)-2 y z(-z+x-2 y)$ $=-x^{2} z+x^{3}-2 x^{2} y-4 y^{2} z+4 x y^{2}-8 y^{3}-z^{3}+x z^{2}-2 y z^{2}-2 x y z+2 x^{2} y-4 x y^{2}$ $-x z^{2}+x^{2} z-2 x y z+2 y z^{2}-2 x y z+4 y^{2} z$ $=\left(-x^{2} z+x^{2} z\right)+x^{3}+\left(-2 x^{2} y+2 x^{2} y\right)+\left(-4 y^{2} z+4 y^{2} z...
Read More →Find the value of x if the area
Question: Find the value of $x$ if the area of $\Delta$ is 35 square $\mathrm{cms}$ with vertices $(x, 4),(2,-6)$ and $(5,4)$. Solution: $\Delta=\frac{1}{2}\left|\begin{array}{ccc}x 4 1 \\ 2 -6 1 \\ 5 4 1\end{array}\right|=\pm 35$ $=\frac{1}{2}\left|\begin{array}{ccc}x 4 1 \\ 2-x -10 0 \\ 5 4 1\end{array}\right|=\pm 35 \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$ $=\frac{1}{2}\left|\begin{array}{ccc}x 4 1 \\ 2-x -10 0 \\ 5-x 0 0\end{array}\right|=\pm 35 \quad$ [Apply...
Read More →Simplify the following
Question: Simplify (2x- 5y)3 (2x+ 5y)3. Solution: (2x -5y)3 (2x + 5y)3= [(2x)3 (5y)3 3(2x)(5y)(2x 5y)] -[(2x)3+ (5y)3+ 3(2x)(5y)(2x+5y)] [using identity, (a b)3= a3-b3 3ab and (a + b)3=a3+b3+3ab ] = (2x)3 (5y)3 30xy(2x 5y) (2x)3 (5y)3 30xy (2x + 5y) = -2 (5y)3 30xy(2x 5y + 2x + 5y) = -2 x 125y3 30xy(4x) = -250y3-120x2y...
Read More →Given that the number
Question: Given that the number $\overline{67 y 19}$ is divisible by 9 , where $y$ is a digit, what are the possible values of $y$ ? Solution: It is given that $\overline{67 \mathrm{y} 19}$ is a multiple of 9 . $\therefore(6+7+\mathrm{y}+1+9)$ is a multiple of 9 . $\therefore(23+\mathrm{y})$ is a multiple of 9 . $23+\mathrm{y}=0,9,18,27,36 \ldots$ But $\mathrm{x}$ is a digit. So, $\mathrm{x}$ can take values $0,1,2,3,4 \ldots 9$. $23+\mathrm{y}=27$ $\Rightarrow y=4$...
Read More →A ladder of length 6 metres makes an angle of 45° with the floor while leaning against one wall of a room.
Question: A ladder of length 6 metres makes an angle of 45 with the floor while leaning against one wall of a room. If the foot of the ladder is kept fixed on the floor and it is made to lean against the opposite wall of the room, it makes an angle of 60 with the floor. Find the distance between two walls of the room. Solution: Let AB and CD be the two opposite walls of the roomand the foot of the ladder be fixed at thepoint O on the ground.We have, $\mathrm{AO}=\mathrm{CO}=6 \mathrm{~m}, \angle...
Read More →Without actual division,
Question: Without actual division, prove that 2x4 5x3+ 2x2 x+ 2 is divisible by x2-3x+2 Thinking Process (i) firstly, determine the factors of quadratic polynomial by splitting middle term. (ii) The two different values of zeroes put in biquadratic polynomial. (iii) In both the case if remainder is zero, then biquadratic polynomial is divisible by quadratic polynomial. Solution: Let p(x) = 2x4 5x3+ 2x2 x+ 2 firstly, factorise x2-3x+2. Now, x2-3x+2 = x2-2x-x+2 [by splitting middle term] = x(x-2)-...
Read More →If x is a digit of the number
Question: If $x$ is a digit of the number $\overline{66784 x}$ such that it is divisible by 9 , find possible values of $x$. Solution: It is given that $\overline{66784 \mathrm{x}}$ is a multiple of 9 . Therefore, $(6+6+7+8+4+\mathrm{x})$ is a multiple of 9 . And, $(31+x)$ is a multiple of 9 . Possible values of $(31+\mathrm{x})$ are $0,9,18,27,36,45, \ldots$ But $\mathrm{x}$ is a digit. So, $\mathrm{x}$ can only take value $0,1,2,3,4, \ldots 9$. $\therefore 31+\mathrm{x}=36$ $\Rightarrow \mathr...
Read More →If x is a digit such that the number
Question: If $x$ is a digit such that the number $\overline{18 x} 71$ is divisible by 3, find possible values of $x .$ Solution: It is given that $\overline{18 \times 71}$ is a multiple of 3 . $\therefore(1+8+\mathrm{x}+7+1)$ is a multiple of 3 . $\therefore(17+x)$ is a multiple of 3 . $\therefore 17+x=0,3,6,9,12,15,18,21 \ldots$ But $x$ is a digit. So, $\mathrm{x}$ can take values $0,1,2,3,4 \ldots 9$. $17+x=18 \Rightarrow \mathrm{x}=1$ $17+x=21 \Rightarrow \mathrm{x}=4$ $17+x=24 \Rightarrow \m...
Read More →Find the value
Question: Find the value of $\lambda$ so that the points $(1,-5),(-4,5)$ and $(\lambda, 7)$ are collinear. Solution: If the points $(1,-5),(-4,5)$ and $(\lambda, 7)$ are collinear, then $\left|\begin{array}{ccc}1 -5 1 \\ -4 5 1 \\ \lambda 7 1\end{array}\right|=0$ $\Rightarrow\left|\begin{array}{rcc}1 -5 1 \\ -5 10 0 \\ \lambda 7 1\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$ $\Rightarrow\left|\begin{array}{ccc}1 -5 1 \\ -5 10 0 \\ \lambda-1 12 0\e...
Read More →Prove the following
Question: If both x 2 and x -(1/2) are factors of px2+ 5x+r, then show that p = r. Solution: Let $f(x)=p x^{2}+5 x+r$ Since, $x-2$ is a factor of $f(x)$ then $f(2)=0$ $\therefore \quad p(2)^{2}+5(2)+r=0$ $\Rightarrow \quad 4 p+10+r=0$ $\ldots$ (i) Since, $\quad x-\frac{1}{2}$ is a factor of $f(x)$, then $f\left(\frac{1}{2}\right)=0$ $\therefore$ $p\left(\frac{1}{2}\right)^{2}+5\left(\frac{1}{2}\right)+r=0$ $\Rightarrow$ $p \times \frac{1}{4}+\frac{5}{2}+r=0$ $\Rightarrow$ $p+10+4 r=0$ .......(ii...
Read More →The angles of elevation of the top of a tower from two points at distances of 4 m
Question: The angles of elevation of the top of a tower from two points at distances of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Show that the height of the tower is 6 metres. Solution: Let $A B$ be the tower and $C$ and $D$ be two points such that $A C=4 \mathrm{~m}$ and $A D=9 \mathrm{~m}$. Let: $A B=h \mathrm{~m}, \angle B C A=\theta$ and $\angle B D A=90^{\circ}-\theta$ In the right $\triangle B C A$, we have: $\tan \theta=\frac{A B}{A C...
Read More →Given that the number
Question: Given that the number $\overline{35 \alpha 64}$ is divisible by 3 , where $\alpha$ is a digit, what are the possible values of $\alpha$ ? Solution: It is given that $\overline{35 a 64}$ is a multiple of 3 . $\therefore(3+5+a+6+4)$ is a multiple of 3 . $\therefore(a+18)$ is a multiple of 3 . $\therefore(a+18)=0,3,6,9,12,15,18,21 \ldots$ But $a$ is a digit of number $\overline{35 a 64}$. So, $a$ can take value $0,1,2,3,4 \ldots 9$. $a+18=18 \Rightarrow \mathrm{a}=0$ $a+18=21 \Rightarrow ...
Read More →The polynomial
Question: The polynomial p{x) = x4-2x3+ 3x2 -ax+3a-7 when divided by x+1 leaves the remainder 19. Find the values of a. Also, find the remainder when p(x) is divided by x+ 2. Solution: Given, $\quad p(x)=x^{4}-2 x^{3}+3 x^{2}-a x+3 a-7$ When we divide $p(x)$ by $x+1$, then we get the remainder $p(-1)$. Now, $\quad p(-1)=(-1)^{4}-2(-1)^{3}+3(-1)^{2}-a(-1)+3 a-7$ $=1+2+3+a+3 a-7=4 a-1$ According to the question, $p(-1)=19$ $\Rightarrow \quad 4 a-1=19$ $\Rightarrow \quad 4 a=20$ $\therefore \quad a...
Read More →Using determinants prove that the points (a, b), (a', b')
Question: Using determinants prove that the points (a,b), (a',b') and (aa',bb') are collinear ifab' =a'b. Solution: $\left|\begin{array}{ccc}a b 1 \\ a^{\prime} b^{\prime} 1 \\ a-a^{\prime} b-b^{\prime} 1\end{array}\right|$ $\Rightarrow \Delta=\left|\begin{array}{ccc}a b 1 \\ a^{\prime}-a b^{\prime}-b 0 \\ a-a^{\prime} b-b^{\prime} 1\end{array}\right| \quad$ [Applying $R_{2} \rightarrow R_{2}-R_{1}$ ] $\Rightarrow \Delta=\left|\begin{array}{ccc}a b 1 \\ a^{\prime}-a b^{\prime}-b 0 \\ -a^{\prime}...
Read More →Using determinants prove that the points (a, b), (a', b') and (a − a', b − b') are collinear if ab' = a'b.
Question: Using determinants prove that the points (a,b), (a',b') and (aa',bb') are collinear ifab' =a'b. Solution: $\left|\begin{array}{ccc}a b 1 \\ a^{\prime} b^{\prime} 1 \\ a-a^{\prime} b-b^{\prime} 1\end{array}\right|$ $\Rightarrow \Delta=\left|\begin{array}{ccc}a b 1 \\ a^{\prime}-a b^{\prime}-b 0 \\ a-a^{\prime} b-b^{\prime} 1\end{array}\right| \quad$ [Applying $R_{2} \rightarrow R_{2}-R_{1}$ ] $\Rightarrow \Delta=\left|\begin{array}{ccc}a b 1 \\ a^{\prime}-a b^{\prime}-b 0 \\ -a^{\prime}...
Read More →From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45° respectively.
Question: From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30 and 45 respectively. If the bridge is at a height of 2.5 m from the banks, find width of the river. Solution: Let $A$ and $B$ be two points on the banks on the opposite side of the river and $P$ be the point on the bridge at a height of $2.5 \mathrm{~m}$. Thus, we have: $D P=2.5 \mathrm{~m}, \angle P A D=30^{\circ}$ and $\angle P B D=45^{\circ}$ In the right $\triangle A...
Read More →Write true (T) or false (F) for the following statements:
Question: Write true (T) or false (F) for the following statements: (i) 392 is a perfect cube. (ii) 8640 is not a perfect cube. (iii) No cube can end with exactly two zeros. (iv) There is no perfect cube which ends in 4. (v) For an integera,a3is always greater thana2. (vi) Ifaandbare integers such thata2b2, thena3b3. (vii) Ifadividesb, thena3dividesb3. (viii) Ifa2ends in 9, thena3ends in 7. (ix) Ifa2ends in 5, thena3ends in 25. (x) Ifa2ends in an even number of zeros, thena3ends in an odd number...
Read More →If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.
Question: If the points (a, 0), (0,b) and (1, 1) are collinear, prove thata+b=ab. Solution: Ifthe points (a, 0), (0,b) and (1, 1) are collinear, then $\left|\begin{array}{lll}a 0 1 \\ 0 b 1 \\ 1 1 1\end{array}\right|=0$ $\Rightarrow\left|\begin{array}{ccc}a 0 1 \\ -a b 0 \\ 1 1 1\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$ $\Rightarrow\left|\begin{array}{ccc}a 0 1 \\ -a b 0 \\ 1-a 1 0\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{3} ...
Read More →If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.
Question: If the points (a, 0), (0,b) and (1, 1) are collinear, prove thata+b=ab. Solution: Ifthe points (a, 0), (0,b) and (1, 1) are collinear, then $\left|\begin{array}{lll}a 0 1 \\ 0 b 1 \\ 1 1 1\end{array}\right|=0$ $\Rightarrow\left|\begin{array}{ccc}a 0 1 \\ -a b 0 \\ 1 1 1\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$ $\Rightarrow\left|\begin{array}{ccc}a 0 1 \\ -a b 0 \\ 1-a 1 0\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{3} ...
Read More →If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.
Question: If the points (a, 0), (0,b) and (1, 1) are collinear, prove thata+b=ab. Solution: Ifthe points (a, 0), (0,b) and (1, 1) are collinear, then $\left|\begin{array}{lll}a 0 1 \\ 0 b 1 \\ 1 1 1\end{array}\right|=0$ $\Rightarrow\left|\begin{array}{ccc}a 0 1 \\ -a b 0 \\ 1 1 1\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$ $\Rightarrow\left|\begin{array}{ccc}a 0 1 \\ -a b 0 \\ 1-a 1 0\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{3} ...
Read More →If the polynomials
Question: If the polynomials az3+4z2+ 3z-4 and z3-4z + o leave the same remainder when divided by z 3, find the value of a. Solution: Let p1(z) = az3+4z2+ 3z-4 and p2(z) = z3-4z + o When we divide p1(z) by z 3, then we get the remainder p,(3). Now, p1(3) = a(3)3 + 4(3)2 + 3(3) 4 = 27a+ 36+ 9-4= 27a+ 41 When we divide p2(z) by z-3 then we get the remainder p2(3). Now, p2(3) = (3)3-4(3)+a = 27-12 + a = 15+a According to the question, both the remainders are same. p1(3)= p2(3) 27a+41 = 15+a 27a-a =...
Read More →Give possible expression for the length
Question: Give possible expression for the length and breadth of the rectangle whose area is given by 4a2+4a 3. Solution: Given, area of rectangle = 4a2+ 6a-2a-3 = 4a2+ 4a 3 [by splitting middle term] = 2a(2a + 3) -1 (2a + 3) = (2a 1)(2a + 3) Hence, possible length = 2a -1 and breadth = 2a + 3...
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