From the top of a 7-metre-high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°.
Question: From the top of a 7-metre-high building, the angle of elevation of the top of a cable tower is 60 and the angle of depression of its foot is 45. Determine the height of the tower[Use $\sqrt{3}=1.732$ ] Solution: Let AB be the 7-m high building and CD be the cable tower.We have, $\mathrm{AB}=7 \mathrm{~m}, \angle \mathrm{CAE}=60^{\circ}, \angle \mathrm{DAE}=\angle \mathrm{ADB}=45^{\circ}$ Also, $\mathrm{DE}=\mathrm{AB}=7 \mathrm{~m}$ In $\triangle \mathrm{ABD}$, $\tan 45^{\circ}=\frac{\...
Read More →Without expanding, prove that
Question: Without expanding, prove that Solution: $\left|\begin{array}{lll}a b c \\ x y z \\ p q r\end{array}\right|=\left|\begin{array}{lll}x y z \\ p q r \\ a b c\end{array}\right|=\left|\begin{array}{lll}y b q \\ x a p \\ z c r\end{array}\right|$ $\left|\begin{array}{lll}x y z \\ p q r \\ a b c\end{array}\right| R_{2} \leftrightarrow R_{3}=-\left|\begin{array}{lll}x y z \\ a b c \\ p q r\end{array}\right| R_{1} \leftrightarrow R_{2}=\left|\begin{array}{lll}a b c \\ x y z \\ p q r\end{array}\r...
Read More →Prove the following identities:
Question: Prove the following identities: $\left|\begin{array}{lll}a^{3} 2 a \\ b^{3} 2 b \\ c^{3} 2 c\end{array}\right|=2(a-b)(b-c)(c-a)(a+b+c)$ Solution: LHS $=\left|\begin{array}{ccc}a^{3} 2 a \\ b^{3} 2 b \\ c^{3} 2 c\end{array}\right|$ $=\left|\begin{array}{ccc}a^{3} 2 a \\ b^{3}-a^{3} 0 b-a \\ c^{3}-a^{3} 0 c-a\end{array}\right| \quad\left[\right.$ Applying $R_{2} \rightarrow R_{2}-R_{1}$ and $\left.R_{3} \rightarrow R_{3}-R_{1}\right]$ $=-(a-b)(c-a)\left|\begin{array}{ccc}a^{3} 2 a \\ b^{...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root34.2 Solution: The number $34.2$ could be written as $\frac{342}{10}$. Now $\sqrt[3]{34.2}=\sqrt[3]{\frac{342}{10}}=\frac{\sqrt[3]{342}}{\sqrt[3]{10}}$ Also $340342350 \Rightarrow \sqrt[3]{340}\sqrt[3]{342}\sqrt[3]{350}$ From the cube root table, we have: $\sqrt[3]{340}=6.980$ and $\sqrt[3]{350}=7.047$ For the difference $(350-340)$, i.e., 10 , the difference in values $=7.047-6.980=0.067 .$ $\therefore$ For the difference $(342-340)...
Read More →By actual division,
Question: By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial x4 + 1 and x-1. Solution: Using long division method, $x-1) x^{4}+1\left(x^{3}+x^{2}+x+1\right.$ $\frac{x^{4}-x^{3}}{x^{3}+1}$ $\frac{x^{3}-x^{2}}{x^{2}+1}$ $\frac{x^{2}-x}{x+1}$ Hence, Quotient $=x^{3}+x^{2}+x+1$ and Remainder $=2$...
Read More →The angle of elevation of the top of a chimney from the foot of a tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°.
Question: The angle of elevation of the top of a chimney from the foot of a tower is60 and the angle of depression of the foot of the chimney from the topof the tower is 30. If the height of the tower is 40 metres, then find the heightof the chimney.According to pollution control norms, the minimum height of a smoke emitting chimney should be 100 metres. State if the height of the above mentioned chimney meets the pollution norms. What value is discussed in this question? Solution: Let PQ be the...
Read More →Find the zeroes of the polynomial
Question: Find the zeroes of the polynomial p(x)= (x 2)2 (x+ 2)2. Solution: Given, polynomial is p(x) = (x 2)2 (x+ 2)2 For zeroes of polynomial, put p(x) = 0 (x 2)2 (x+ 2)2=0 (x-2 + x+2)(x-2-x-2) = 0 [using identity, a2-b2=(a-b)(a + b)] = (2x)(-4) = 0...
Read More →Find the zeroes of the polynomial in each of the following,
Question: Find the zeroes of the polynomial in each of the following, (i) p(x)=x 4 (ii) g(x)= 3 6x (iii) q(x) = 2x 7 (iv) h(y) = 2y Solution: (i) Given, polynomial is p(x) = x- 4 For zero of polynomial, put p(x) = x-4 = 0 = x= 4 Hence, zero of polynomial is 4. (ii) Given, polynomial is g(x) = 3-6x For zero of polynomial, put g(x) = 0 3-6x= 0 = 6x =3 = x=1/2. Hence, zero of polynomial is X (iii) Given, polynomial is q(x) = 2x -7 For zero of polynomial, put q(x) = 2x-7 = 0 2x= 7 = x =7/2 Hence, ze...
Read More →Prove the following identities:
Question: Prove the following identities: $\left|\begin{array}{ccc}a+x y z \\ x a+y z \\ x y a+z\end{array}\right|=a^{2}(a+x+y+z)$ Solution: LHS $=\left|\begin{array}{ccc}a+x y z \\ x a+y z \\ x y a+z\end{array}\right|$ $=\left|\begin{array}{ccc}a+x+y+z y z \\ a+x+y+z a+y z \\ a+x+y+z y a+z\end{array}\right|$ $\left[\right.$ Applying $\left.C_{1} \rightarrow C_{1}+C_{2}+C_{3}\right]$ $=(a+x+y+z)\left|\begin{array}{ccc}1 y z \\ 1 a+y z \\ 1 y a+z\end{array}\right|$ [Taking $(a+x+y+z)$ common from...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root833 Solution: We have: $830833840 \Rightarrow \sqrt[3]{830}\sqrt[3]{833}\sqrt[3]{840}$ From the cube root table, we have: $\sqrt[3]{830}=9.398$ and $\sqrt[3]{840}=9.435$ For the difference $(840-830)$, i.e., 10 , the difference in values $=9.435-9.398=0.037$ $\therefore$ For the difference $(833-830)$, i.e., 3 , the difference in values $=\frac{0.037}{10} \times 3=0.0111=0.011$ (upto three decimal places) $\therefore \sqrt[3]{833}=9....
Read More →The horizontal distance between two towers is 60 metres.
Question: The horizontal distance between two towers is 60 metres. The angle of depression of the top of the first tower when seen from the top of the second tower is 30. If the height of the second tower is 90 metres, find the height of the first tower$[$ Use $\sqrt{3}=1.732]$ Solution: Let $D E$ be the first tower and $A B$ be the second tower. Now, $A B=90 \mathrm{~m}$ and $A D=60 \mathrm{~m}$ such that $C E=60 \mathrm{~m}$ and $\angle B E C=30^{\circ}$. Let $D E=h \mathrm{~m}$ such that $A C...
Read More →The angle of elevation of the top of a building from the foot of a tower is 30°.
Question: The angle of elevation of the top of a building from the foot of a tower is 30. The angle of elevation of the top of the tower from the foot of the building is 60. If the tower is 60 m high, then find the height of the building. Solution: Let AB be the building and PQ be the tower.We have, $\mathrm{PQ}=60 \mathrm{~m}, \angle \mathrm{APB}=30^{\circ}, \angle \mathrm{PAQ}=60^{\circ}$ In $\Delta \mathrm{APQ}$, $\tan 60^{\circ}=\frac{\mathrm{PQ}}{\mathrm{AP}}$ $\Rightarrow \sqrt{3}=\frac{60...
Read More →Verify whether the following are true or false.
Question: Verify whether the following are true or false. (i) -3 is a zero of at 3 (ii) -1/3 is a zero of 3x+1 (iii) -4/5 is a zero of 4 5y (iv) 0 and 2 are the zeroes of t2 2t (v) -3 is a zero of y2+y-6 Solution: (i) False, because zero of $x-3$ is $3 .$ $[\because x-3=0 \Rightarrow x=3]$ (ii) True, because zero of $3 x+1$ is $-\frac{1}{3}$.$\left[\because 3 x+1=0 \Rightarrow x=\frac{-1}{3}\right]$ (iii) False, because zero of $4-5 y$ is $\frac{4}{5}$.$\left[\because 4-5 y=0 \Rightarrow y=\frac...
Read More →Show that
Question: Show that $\left|\begin{array}{lll}y+z x y \\ z+x z x \\ x+y y z\end{array}\right|=(x+y+z)(x-z)^{2}$ Solution: Let $\Delta=\mid y+z \quad x \quad y$ $z+x \quad z \quad x$ $x+y \quad y \quad z \mid$ $\Rightarrow \Delta=\mid 2(x+y+z) \quad x+y+z \quad x+y+z$ $z+x \quad z \quad x$ $\begin{array}{llll}x+y y z \mid \text { [Applying } \mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3} \text { ] }\end{array}$ $=(x+y+z) \mid \begin{array}{lll}2 1 1\end{array}$ $\begin{arr...
Read More →Find p(0), p( 1) and p(-2) for the following polynomials
Question: Find p(0), p( 1) and p(-2) for the following polynomials (i) p(x) = 10x 4x2 3 (ii) p(y) = (y + 2)(y 2) Solution: (i) Given, polynomial is p(x) = 10x 4x2 3 On putting x = 0,1 and 2, respectively in Eq. (i), we get p(0) = 10(0)-4(0)2-3 = 0-0-3= -3 p(1) = 10 (1) 4 (1 )2-3 = 10-4-3= 10-7= 3 and p(-2) =10 (-2)- 4 (-2)2 3 = -20-44-3 =-20-16-3=-39 Hence, the values of p(0), p(1) and p(-2) are respectively, -3,3 and 39. (ii) Given, polynomial isp(y) = (y+2)(y-2) On putting y =0,1 and -2, respe...
Read More →Prove the following
Question: If $p(x)=x^{2}-4 x+3$, then evaluate $p(2)-p(-1)+p(1 / 2)$. Solution: Given, $p(x)=x^{2}-4 x+3$ Now, $p(2)=(2)^{2}-4 \times 2+3=4-8+3=-1$ $p(-1)=(-1)^{2}-4(-1)+3=1+4+3=8$ and $p\left(\frac{1}{2}\right)=\left(\frac{1}{2}\right)^{2}-4 \times \frac{1}{2}+3$ $=\frac{1}{4}-2+3=\frac{1-8+12}{4}=\frac{5}{4}$ $\therefore$ $p(2)-p(-1)+p\left(\frac{1}{2}\right)=-1-8+\frac{5}{4}$ $=-9+\frac{5}{4}=\frac{-36+5}{4}=\frac{-31}{4}$...
Read More →Prove the following identities:
Question: Prove the following identities: $\left|\begin{array}{ccc}2 y y-z-x 2 y \\ 2 z 2 z z-x-y \\ x-y-z 2 x 2 x\end{array}\right|=(x+y+z)^{3}$ Solution: LHS $=\left|\begin{array}{ccc}2 y y-z-x 2 y \\ 2 z 2 z z-x-y \\ x-y-z 2 x 2 x\end{array}\right|$ $=\left|\begin{array}{ccc}2 y+2 z+x-y-z y-z-x+2 z+2 x 2 y+z-x-y+2 x \\ 2 z 2 z z-x-y \\ x-y-z 2 x 2 x\end{array}\right|$ [Applying $\left.R_{1} \rightarrow R_{1}+R_{2}+R_{3}\right]$ $=\left|\begin{array}{ccc}x+y+z x+y+z x+y+z \\ 2 z 2 z z-x-y \\ x...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root7532 Solution: We have: $750075327600 \Rightarrow \sqrt[3]{7500}\sqrt[3]{7532}\sqrt[3]{7600}$ From the cube root table, we have: $\sqrt[3]{7500}=19.57$ and $\sqrt[3]{7600}=19.66$ For the difference $(7600-7500)$, i.e., 100 , the difference in values $=19.66-19.57=0.09$ $\therefore$ For the difference of $(7532-7500)$, i.e., 32 , the difference in values $=\frac{0.09}{100} \times 32=0.0288=0.029$ (up to three decimal places) $\therefo...
Read More →Find the value of the polynomial
Question: Find the value of the polynomial $3 x^{3}-4 x^{2}+7 x-5$, when $x=3$ and also when $x=-3$. Solution: Let p(x) =3x3 4x2+ 7x 5 At x= 3, p(3) = 3(3)3 4(3)2+ 7(3) 5 = 327-49 + 21-5 = 81-36+21-5 P( 3) =61 At x = -3, p(-3)= 3(3)3 4(-3)2+ 7(-3)- 5 = 3(-27)-49-21-5 = -81-36-21-5 = -143 p(-3) = -143 Hence, the value of the given polynomial at x = 3 and x = -3 are 61 and -143, respectively....
Read More →Give an example of a polynomial,
Question: Give an example of a polynomial, which is (i) monomial of degree 1. (ii) -binomial of degree 20. (iii) trinomial of degree 2. Solution: (i) The example of monomial of degree 1 is 5y or 10x. (ii) The example of binomial of degree 20 is 6x20+ x11or x20+1 (iii) The example of trinomial of degree 2 is x2 5x+ 4 or 2x2-x-1...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root8.65 Solution: The number $8.65$ could be written as $\frac{865}{100}$. Now $\sqrt[3]{8.65}=\sqrt[3]{\frac{865}{100}}=\frac{\sqrt[3]{865}}{\sqrt[3]{100}}$ Also $860865870 \Rightarrow \sqrt[3]{860}\sqrt[3]{865}\sqrt[3]{870}$ From the cube root table, we have: $\sqrt[3]{860}=9.510$ and $\sqrt[3]{870}=9.546$ For the difference (870-860), i.e., 10, the difference in values $=9.546-9.510=0.036$ $\therefore$ For the difference of $(865-860...
Read More →Classify the following as a constant, linear,
Question: Classify the following as a constant, linear, quadratic and cubic polynomials (i) $2-x^{2}+x^{3}$ (ii) $3 x^{3}$ (iii) $5 t-\sqrt{7}$ (iv) $4-5 y^{2}$ (v) 3 (vi) $2+x$ (vii) $y^{3}-y$ (viii) $1+x+x^{2}$ (ix) $t^{2}$ (x) $\sqrt{2} x-1$ Thinking Process (i) Firstly check the maximum exponent of the variable.. (ii) If the maximum exponent of a variable is 0 , then it is a constant polynomial. (iii) If the maximum exponent of a variable is 1 , then it is a linear polynomial. (iv) If the ma...
Read More →Evaluate the following determinant:
Question: Evaluate the following determinant: (i) $\left|\begin{array}{ccc}1+a 1 1 \\ 1 1+a 1 \\ 1 1 1+a\end{array}\right|=a^{3}+3 a^{2}$ (ii) $\left|\begin{array}{ccc}a^{2}+2 a 2 a+1 1 \\ 2 a+1 a+2 1 \\ 3 3 1\end{array}\right|=(a-1)^{3}$ Solution: (ii) To Prove: $\left|\begin{array}{ccc}a^{2}+2 a 2 a+1 1 \\ 2 a+1 a+2 1 \\ 3 3 1\end{array}\right|=(a-1)^{3}$ LHS $=\left|\begin{array}{ccc}a^{2}+2 a 2 a+1 1 \\ 2 a+1 a+2 1 \\ 3 3 1\end{array}\right|$ Applying $R_{1} \rightarrow R_{1}-R_{2}$ $=\left|...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root0.86 Solution: The number $0.86$ could be written as $\frac{86}{100}$. Now $\sqrt[3]{0.86}=\sqrt[3]{\frac{86}{100}}=\frac{\sqrt[3]{86}}{\sqrt[3]{100}}$ By cube root table, we have: $\sqrt[3]{86}=4.414$ and $\sqrt[3]{100}=4.642$ $\therefore \sqrt[3]{0.86}=\frac{\sqrt[3]{86}}{\sqrt[3]{100}}=\frac{4.414}{4.642}=0.951$ (upto three decimal places) Thus, the required cube root is 0.951....
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root8.6 Solution: The number $8.6$ can be written as $\frac{86}{10}$. Now $\sqrt[3]{8.6}=\sqrt[3]{\frac{86}{10}}=\frac{\sqrt[3]{86}}{\sqrt[3]{10}}$ By cube root table, we have: $\sqrt[3]{86}=4.414$ and $\sqrt[3]{10}=2.154$ $\therefore \sqrt[3]{8.6}=\frac{\sqrt[3]{86}}{\sqrt[3]{10}}=\frac{4.414}{2.154}=2.049$ Thus, the required cube root is 2.049....
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