A TV tower stands vertically on a bank of canal. From a point on other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°.
Question: A TV tower stands vertically on a bank of canal. From a point on other bank directly opposite the tower, the angle of elevation of the top of the tower is 60. From another point 20 m away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30. Find the height of the tower and the width of the canal. Solution: Let PQ=hm be the height of the TV tower and BQ=xm be the width of the canal.We have, $\mathrm{AB}=20 \mathrm...
Read More →Write the coefficient of x2 in each of the following
Question: Write the coefficient of x2 in each of the following (i) $\frac{\pi}{6} x+x^{2}-1$ (ii) $3 x-5$ (iii) $(x-1)(3 x-4)$ (iv) $(2 x-5)\left(2 x^{2}-3 x+1\right)$ Solution: (i) The coefficient of $x^{2}$ in $\frac{\pi}{6} x+x^{2}-1$ is 1 . (ii) The coefficient of $x^{2}$ in $3 x-5$ is $0 .$ (iii) Let $p(x)=(x-1)(3 x-4)$ $=3 x^{2}-7 x+4$ $=3 x^{2}-4 x-3 x+4$ Hence, the coefficient of $x^{2}$ in $p(x)$ is 3 . (iv) Let $p(x)=(2 x-5)\left(2 x^{2}-3 x+1\right)$ $=2 x\left(2 x^{2}-3 x+1\right)-5\...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root0.27 Solution: The number $0.27$ can be written as $\frac{27}{100}$. now $\sqrt[3]{0.27}=\sqrt[3]{\frac{27}{100}}=\frac{\sqrt[3]{27}}{\sqrt[3]{100}}=\frac{3}{\sqrt[3]{100}}$ By cube root table, we have: $\sqrt[3]{100}=4.642$ $\therefore \sqrt[3]{0.27}=\frac{3}{\sqrt[3]{100}}=\frac{3}{4.642}=0.646$ Thus, the required cube root is 0.646....
Read More →Solve this
Question: $\left|\begin{array}{ccc}-a\left(b^{2}+c^{2}-a^{2}\right) 2 b^{3} 2 c^{3} \\ 2 a^{3} -b\left(c^{2}+a^{2}-b^{2}\right) 2 c^{3} \\ 2 a^{3} 2 b^{3} -c\left(a^{2}+b^{2}-c^{2}\right)\end{array}\right|=a b c\left(a^{2}+b^{2}+c^{2}\right)^{3}$ Solution: $\Delta=\left|\begin{array}{ccc}-a\left(b^{2}+c^{2}-a^{2}\right) 2 b^{3} 2 c^{3} \\ 2 a^{3} -b\left(c^{2}+a^{2}-b^{2}\right) 2 c^{3} \\ 2 a^{3} 2 b^{3} -c\left(a^{2}+b^{2}-c^{2}\right)\end{array}\right|$ $=a b c\left|\begin{array}{ccc}-b^{2}-c...
Read More →For the polynomial
Question: For the polynomial $\frac{x^{3}+2 x+1}{5}-\frac{7}{2} x^{2}-x^{6}$, then write (i) the degree of the polynomial (ii) the coefficient of $x^{3}$ (iii) the coefficient of $x^{6}$ (iv) the constant term Solution: Given, polynomial is, $\frac{x^{3}+2 x+1}{5}-\frac{7}{2} x^{2}-x^{6}$ $=\frac{1}{5} x^{3}+\frac{2 x}{5}+\frac{1}{5}-\frac{7}{2} x^{2}-x^{6}$ (i) Degree of the polynomial is the highest power of the variable $i, e, 6 .$ (ii) The coefficient of $x^{3}$ in given polynomial is $\frac...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root37800 Solution: We have: $37800=2^{3} \times 3^{3} \times 175 \Rightarrow \sqrt[3]{37800}=\sqrt[3]{2^{3} \times 3^{3} \times 175}=6 \times \sqrt[3]{175}$ Also $170175180 \Rightarrow \sqrt[3]{170}\sqrt[3]{175}\sqrt[3]{180}$ From cube root table, we have: $\sqrt[3]{170}=5.540$ and $\sqrt[3]{180}=5.646$ For the difference $(180-170)$, i.e., 10 , the difference in values $=5.646-5.540=0.106$ $\therefore$ For the difference of $(175-170)$...
Read More →A straight highway leads to the foot of a tower.
Question: A straight highway leads to the foot of a tower. A man standing on the top of the tower observes a car at an angle of depression of 30, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60. Find the time taken by the car to reach the foot of the tower form this point. Solution: Let PQ be the tower. We have, $\angle \mathrm{PBQ}=60^{\circ}$ and $\angle \mathrm{PAQ}=30^{\circ}$ Let $\mathrm{PQ}=h, \mathrm...
Read More →Determine the degree of each of the following polynomials.
Question: Determine the degree of each of the following polynomials. (i) 2x-1 (ii) -10 (iii) x3-9x+ 3x5 (iv) y3(1-y4) Solution: (i) Degree of polynomial 2x-1 is one, Decause the maximum exponent of x is one. (ii) Degree of polynomial -10 or -10x is zero, because the exponent of x is zero. (iii) Degree of polynomial x3 9x + 3xs is five, because the maximum exponent of x is five. (iv) Degree of polynomial y3(1-y4) or y3 y7is seven, because the maximum exponent of y is seven....
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root133100 Solution: We have: $133100=1331 \times 100 \Rightarrow \sqrt[3]{133100}=\sqrt[3]{1331 \times 100}=11 \times \sqrt[3]{100}$ By cube root table, we have: $\sqrt[3]{100}=4.642$ $\therefore \sqrt[3]{133100}=11 \times \sqrt[3]{100}=11 \times 4.642=51.062$...
Read More →Classify the following polynomials as polynomials in one variable,
Question: Classify the following polynomials as polynomials in one variable, two variables etc. (i) x2+ x +1 (ii) y3 5y (iii) xy+ yz +zx (iv) x2 Zxy + y2+1 Solution: (i) Polynomial x2+ x+ 1 is a one variable polynomial, because it contains only one variable i.e., x. (ii) Polynomial y3 5y is a one variable polynomial, because it contains only one variable i.e., y. (iii) Polynomial xy+ yz+ zx is a three variables polynomial, because it contains three variables x, y and z. (iv) Polynomial x2 Zxy + ...
Read More →Prove the following identities:
Question: Prove the following identities: $\left|\begin{array}{ccc}y+z z y \\ z z+x x \\ y x x+y\end{array}\right|=4 x y z$ Solution: LHS : $\left|\begin{array}{ccc}y+z z y \\ z z+x x \\ y x x+y\end{array}\right|$ $=\left|\begin{array}{ccc}y+z-z-y z-z-x-x y-x-x-y \\ z z+x x \\ y x x+y\end{array}\right|$ [Applying $R_{1} \rightarrow R_{1}-R_{2}-R_{3}$ ] $=\left|\begin{array}{ccc}0 -2 x -2 x \\ z z+x x \\ y x x+y\end{array}\right|$ $=-2 x\left|\begin{array}{ccc}0 1 1 \\ z z+x x \\ y x x+y\end{arra...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root7342 Solution: We have: $730073427400 \Rightarrow \sqrt[3]{7000}\sqrt[3]{7342}\sqrt[3]{7400}$ From the cube root table, we have: $\sqrt[3]{7300}=19.39$ and $\sqrt[3]{7400}=19.48$ For the difference $(7400-7300)$, i.e., 100 , the difference in values $=19.48-19.39=0.09$ $\therefore$ For the difference of $(7342-7300)$, i.e., 42 , the difference in the values $=\frac{0.09}{100} \times 42=0.0378=0.037$ $\therefore \sqrt[3]{7342}=19.39+0...
Read More →Write whether the following statements are true or false. Justify your answer. ’
Question: Write whether the following statements are true or false. Justify your answer. (i) A Binomial can have atmost two terms. (ii) Every polynomial is a Binomial. (iii) A binomial ipay have degree 5. (iv) Zero of a polynomial is always 0. (v) A polynomial cannot have more than one zero. (vi) The degree of the sum of two polynomials each of degree 5 is always 5. Solution: (i) False, because a binomial has exactly two terms. (ii) False, because every polynomial is not a binomial . e.g., (a) 3...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root732 Solution: We have: $730732740 \Rightarrow \sqrt[3]{730}\sqrt[3]{732}\sqrt[3]{740}$ From cube root table, we have: $\sqrt[3]{730}=9.004$ and $\sqrt[3]{740}=9.045$ For the difference $(740-730)$, i.e., 10 , the difference in values $=9.045-9.004=0.041$ $\therefore$ For the difference of $(732-730)$, i.e., 2 , the difference in values $=\frac{0.041}{10} \times 2=0.0082$ $\therefore \sqrt[3]{732}=9.004+0.008=9.012$...
Read More →Using properties of determinants prove that
Question: Using properties of determinants prove that $\left|\begin{array}{ccc}x+4 2 x 2 x \\ 2 x x+4 2 x \\ 2 x 2 x x+4\end{array}\right|=(5 x+4)(4-x)^{2}$ Solution: $\Delta=\left|\begin{array}{ccc}x+4 2 x 2 x \\ 2 x x+4 2 x \\ 2 x 2 x x+4\end{array}\right|$ $=\left|\begin{array}{ccc}5 x+4 5 x+4 5 x+4 \\ 2 x x+4 2 x \\ 2 x 2 x x+4\end{array}\right|$ [Applying $R_{1} \rightarrow R_{1}+R_{2}+R_{3}$ ] $=5 x+4\left|\begin{array}{ccc}1 1 1 \\ 2 x x+4 2 x \\ 2 x 2 x x+4\end{array}\right|$ [Take out $...
Read More →Which of the following expressions are polynomials?
Question: Which of the following expressions are polynomials? Justify your answer, (i) 8 (ii) $\sqrt{3} x^{2}-2 x$ (iii) $1-\sqrt{5} x$ (iv) $\frac{1}{5 x^{-2}}+5 x+7$ (v) $\frac{(x-2)(x-4)}{x}$ (vi) $\frac{1}{x+1}$ (vii) $\frac{1}{7} a^{3}-\frac{2}{\sqrt{3}} a^{2}+4 a-7$ (viii) $\frac{1}{2 x}$ Solution: (i) Polynomial, because the exponent of the variable of 8 or $8 x^{0}$ is 0 which is a whole number. (ii) Polynomial, because the exponent of the variable of $\sqrt{3} x^{2}-2 x$ is a whole numb...
Read More →From the top of a tower 100 m high, a man observes two cars on the opposite sides of the tower with angles of depression 30° and 45°, respectively.
Question: From the top of a tower 100 m high, a man observes two cars on the opposite sides of the tower with angles of depression 30 and 45, respectively. Find the distance between the cars.$[$ Take $\sqrt{3}=1.73]$ Solution: Let PQ be the tower.We have, $\mathrm{PQ}=100 \mathrm{~m}, \angle \mathrm{PAQ}=30^{\circ}$ and $\angle \mathrm{PBQ}=45^{\circ}$ In $\Delta \mathrm{APQ}$, $\tan 30^{\circ}=\frac{\mathrm{PQ}}{\mathrm{AP}}$ $\Rightarrow \frac{1}{\sqrt{3}}=\frac{100}{\mathrm{AP}}$ $\Rightarrow...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root9800 Solution: We have: $9800=98 \times 100$ $\therefore \sqrt[3]{9800}=\sqrt[3]{98 \times 100}=\sqrt[3]{98} \times \sqrt[3]{100}$ By cube root table, we have: $\sqrt[3]{98}=4.610$ and $\sqrt[3]{100}=4.642$ $\therefore \sqrt[3]{9800}=\sqrt[3]{98} \times \sqrt[3]{100}=4.610 \times 4.642=21.40$ (upto three decimal places) Thus, the required cube root is 21.40....
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root5112 Solution: By prime factorisation, we have: $5112=2^{3} \times 3^{2} \times 71 \Rightarrow \sqrt[3]{5112}=2 \times \sqrt[3]{9} \times \sqrt[3]{71}$' By the cube root table, we have: $\sqrt[3]{9}=2.080$ and $\sqrt[3]{71}=4.141$ $\therefore \sqrt[3]{5112}=2 \times \sqrt[3]{9} \times \sqrt[3]{71}=2 \times 2.080 \times 4.141=17.227$ (upto three decimal places) Thus, the required cube root is 17.227....
Read More →Prove the following identities
Question: Prove the following identities $\left|\begin{array}{ccc}x+\lambda 2 x 2 x \\ 2 x x+\lambda 2 x \\ 2 x 2 x x+\lambda\end{array}\right|=(5 x+\lambda)(\lambda-x)^{2}$ Solution: LHS : $\left|\begin{array}{ccc}x+\lambda 2 x 2 x \\ 2 x x+\lambda 2 x \\ 2 x 2 x x+\lambda\end{array}\right|$ $=\left|\begin{array}{ccc}x+\lambda 2 x 2 x \\ 2 x-x-\lambda x+\lambda-2 x 0 \\ 2 x-x-\lambda 0 x+\lambda-2 x\end{array}\right| \quad$ [Applying $R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-...
Read More →Prove the following
Question: If $a+b+c=0$, then $a^{3}+b^{3}+c^{3}$ is equal to (a) 0 (b) $a b c$ (c) $3 a b c$ (d) $2 \mathrm{abc}$ Solution: (d) Now, a3+b3+ c3= (a+ b + c) (a2+ b2+ c2 ab be ca) + 3abc [using identity, a3+b3+ c3 3 abc = (a + b + c)(a2+ b2+ c2 ab be -ca)] = 0 + 3abc [ a + b + c = 0, given] a3+b3+ c3= 3abc...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root250 Solution: We have: $250=25 \times 100$ $\therefore$ Cube root of 250 would be in the column of $\sqrt[3]{10 x}$ against 25 . By the cube root table, we have: $\sqrt[3]{250}=6.3$ Thus, the required cube root is 6.3....
Read More →Prove the following
Question: If $49 x^{2}-b=\left(7 x+\frac{1}{2}\right)\left(7 x-\frac{1}{2}\right)$, then the value of $b$ is (a) 0 (b) $\frac{1}{\sqrt{2}}$ (c) $\frac{1}{4}$ (d) $\frac{1}{2}$ Solution: (c) Given, $\left(49 x^{2}-b\right)=\left(7 x+\frac{1}{2}\right)\left(7 x-\frac{1}{2}\right)$ $\Rightarrow \quad\left[49 x^{2}-(\sqrt{b})^{2}\right]=\left[(7 x)^{2}-\left(\frac{1}{2}\right)^{2}\right]\left[\right.$ [using identity, $\left.(a+b)(a-b)=a^{2}-b^{2}\right]$ $\Rightarrow \quad 49 x^{2}-(\sqrt{b})^{2}=4...
Read More →Two men are on opposite sides of a tower. they measure the angles of elevation of the top of the tower as 30° and 45° respectively.
Question: Two men are on opposite sides of a tower. they measure the angles of elevation of the top of the tower as 30 and 45 respectively. If the height of the tower is 50 metres, find the distance between the two men$[$ Take $\sqrt{3}=1.732]$ Solution: Let $C D$ be the tower and $A$ and $B$ be the positions of the two men standing on the opposite sides. Thus, we have: $\angle D A C=30^{\circ}, \angle D B C=45^{\circ}$ and $C D=50 \mathrm{~m}$ Let $A B=x \mathrm{~m}$ and $B C=y \mathrm{~m}$ suc...
Read More →Making use of the cube root table,
Question: Making use of the cube root table, find the cube root1346 Solution: By prime factorisation, we have: $1346=2 \times 673 \Rightarrow \sqrt[3]{1346}=\sqrt[3]{2} \times \sqrt[3]{673}$ Also $670673680 \Rightarrow \sqrt[3]{670}\sqrt[3]{673}\sqrt[3]{680}$ From the cube root table, we have: $\sqrt[3]{670}=8.750$ and $\sqrt[3]{680}=8.794$ For the difference (680-670), i.e., 10, the difference in the values $=8.794-8.750=0.044$ $\therefore$ For the difference of $(673-670)$, i.e., 3 , the diffe...
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