Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:The surface areas of two spheres are in the ratio 16 : 9. The ratio of theirvolumes is(a) 64 : 27 (b) 16 : 9 (c) 4 : 3 (d) 163: 93 Solution: Let the radius of the two spheres be $r$ and $R$. As, $\frac{\text { Surface area of the first sphere }}{\text { Surface area of the second sphere }}=\frac{16}{9}$ $\Rightarrow \frac{4 \pi R^{2}}{4 \pi r^{2}}=\frac{16}{9}$ $\Rightarrow\left(\frac{R}{r}\right)^{2}=\frac{16}{9}$ $\Rightarrow \frac{...
Read More →The population of a town increases at the rate of 50 per thousand.
Question: The population of a town increases at the rate of 50 per thousand. Its population after 2 years will be 22050. Find its present population. Solution: Population after two years $=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $22,050=\mathrm{P}\left(1+\frac{50}{1000}\right)^{2}$ $22,050=\mathrm{P}(1.05)^{2}$ $\mathrm{P}=\frac{22,050}{1.1025}$ $=20,000$ Thus, the population two years ago was 20,000 ....
Read More →The annual rate of growth in population of a certain city is 8%.
Question: The annual rate of growth in population of a certain city is 8%. If its present population is 196830, what it was 3 years ago? Solution: Population after three years $=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $196,830=\mathrm{P}\left(1+\frac{8}{100}\right)^{3}$ $196,830=\mathrm{P}(1.08)^{3}$ $\mathrm{P}=\frac{196,830}{1.259712}$ $=156,250$ Thus, the population three years ago was 156,250 ....
Read More →If A is a square matrix,
Question: If $A$ is a square matrix, then write the matrix adj $\left(A^{T}\right)-(\operatorname{adj} A)^{T}$. Solution: In a non - singular matrix, adj $A^{T}=(\operatorname{adj} A)^{T}$. $\Rightarrow\left(\operatorname{adj} \mathrm{A}^{\mathrm{T}}\right)-(\operatorname{adj} \mathrm{A})^{\mathrm{T}}=$ : matrix...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:The number of solid spheres, each of diameter 6 cm, that can be madeby melting a solid metal cylinder of height 45 cm and diameter 4 cm, is(a) 2 (b) 4 (c) 5 (d) 6 Solution: We have, Radius of the spher, $r=\frac{6}{2}=3 \mathrm{~cm}$, Height of the cylinder, $H=45 \mathrm{~cm}$ and Radius of the cylinder, $R=\frac{4}{2}=2 \mathrm{~cm}$ Now, The number of sphere that can be made $=\frac{\text { Volume of the cylinder }}{\text { Volume ...
Read More →If A is a square matrix,
Question: If $A$ is a square matrix, then write the matrix adj $\left(A^{T}\right)-(\operatorname{adj} A)^{T}$. Solution: In a non - singular matrix, adj $A^{T}=(\operatorname{adj} A)^{T}$. $\Rightarrow\left(\operatorname{adj} \mathrm{A}^{\mathrm{T}}\right)-(\operatorname{adj} \mathrm{A})^{\mathrm{T}}=$ : matrix...
Read More →In a factory the production of scooters rose to 46305 from 40000 in 3 years.
Question: In a factory the production of scooters rose to 46305 from 40000 in 3 years. Find the annual rate of growth of the production of scooters. Solution: Let the annual rate of growth be $R$. $\therefore$ Production of scooters after three years $=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $46,305=4,000\left(1+\frac{\mathrm{R}}{100}\right)^{3}$ $(1+0.01 \mathrm{R})^{3}=\frac{46,305}{40,000}$ $(1+0.01 \mathrm{R})^{3}=1.157625$ $(1+0.01 \mathrm{R})^{3}=(1.05)^{3}$ $1+0.01 \...
Read More →Are the following statements True’ or ‘False’?
Question: Are the following statements True or False? Justify your answer. (i) If the zeroes of a quadratic polynomial ax2+ bx + c are both positive, then a, b and c all have the same sign. (ii) If the graph of a polynomial intersects the X-axis at only one point, it cannot be a quadratic polynomial. (iii) If the graph of a polynomial intersects the X-axis at exactly two points, it need not be a quadratic polynomial. (iv) If two of the zeroes of a cubic polynomial are zero, then it does not have...
Read More →If A is a square matrix of order 3 such that adj (2A) = k adj (A),
Question: If $A$ is a square matrix of order 3 such that $\operatorname{adj}(2 A)=k \operatorname{adj}(A)$, then write the value of $k$. Solution: For any matrtix $A$ of order $n$, adj $(\lambda A)=\lambda^{n-1}(\operatorname{adj} A)$, where $\lambda$ is a constant. Thus, for matrix $A$ of order 3, we have $\operatorname{adj}(2 A)=2^{3-1}(\operatorname{adj} A)$ $\Rightarrow \operatorname{adj}(2 A)=2^{2}(\operatorname{adj} A)$ $\Rightarrow \operatorname{adj}(2 A)=4 \operatorname{adj}(A)$ $\Righta...
Read More →There is a continuous growth in population of a village at the rate of 5% per annum.
Question: There is a continuous growth in population of a village at the rate of 5% per annum. If its present population is 9261, what it was 3 years ago? Solution: Population after three years $=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $9,261=\mathrm{P}\left(1+\frac{5}{100}\right)^{3}$ $9,261=\mathrm{P}(1.05)^{3}$ $\mathrm{P}=\frac{9,261}{1.157625}$ $=8,000$ Thus, the population three years ago was 8,000 ....
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:A rectangular sheet of paper 40 cm22 cm, is rolled to form a hollowcylinder of height 40 cm. The radius of the cylinder (in cm) is(a) 3.5 (b) 7 (c) 80 (d) 5 Solution: We have, Length of the rectangular sheet, $l=40 \mathrm{~cm}$, Width of the rectangular sheet, $b=22 \mathrm{~cm}$ and Height of the hollow cylinder, $h=40 \mathrm{~cm}$ Let the radius of the cylinder be $r$. As, $l=h$ So, the circumference of base of the cylinder $=b$ $...
Read More →If A is a square matrix of order 3 such that |A| = 3,
Question: If $A$ is a square matrix of order 3 such that $|A|=3$, then write the value of $\operatorname{adj}(\operatorname{adj} A)$. Solution: For any square matrixA, we have $|a d j(a d j A)|=|A|^{(n-1)^{2}}$ $\Rightarrow|a d j(a d j A)|=(3)^{4}=81$...
Read More →Three years ago, the population of a town was 50000.
Question: Three years ago, the population of a town was 50000. If the annual increase during three successive years be at the rate of 4%, 5% and 3% respectively, find the present population. Solution: Here, $\mathrm{P}=$ Initial population $=50,000$ $\mathrm{R}_{1}=4 \%$ $\mathrm{R}_{2}=5 \%$ $\mathrm{R}_{3}=3 \%$ $\mathrm{n}=$ Number of years $=3$ $\therefore$ Population after three years $=\mathrm{P}\left(1+\frac{\mathrm{R}_{1}}{100}\right)\left(1+\frac{\mathrm{R}_{2}}{100}\right)\left(1+\frac...
Read More →If A is a square matrix of order 3 such that |A| = 3,
Question: If $A$ is a square matrix of order 3 such that $|A|=3$, then write the value of $\operatorname{adj}(\operatorname{adj} A)$. Solution: For any square matrixA, we have $|a d j(a d j A)|=|A|^{(n-1)^{2}}$ $\Rightarrow|a d j(a d j A)|=(3)^{4}=81$...
Read More →The present population of a town is 25000.
Question: The present population of a town is 25000. It grows at 4%, 5% and 8% during first year, second year and third year respectively. Find its population after 3 years. Solution: Here, $\mathrm{P}=$ Initial population $=25,000$ $\mathrm{R}_{1}=4 \%$ $\mathrm{R}_{2}=5 \%$ $\mathrm{R}_{3}=8 \%$ $\mathrm{n}=$ Number of years $=3$ $\therefore$ Population after three years $=\mathrm{P}\left(1+\frac{\mathrm{R}_{1}}{100}\right)\left(1+\frac{\mathrm{R}_{2}}{100}\right)\left(1+\frac{\mathrm{R}_{3}}{...
Read More →If A is a square matrix of order 3 such that |A| = 2,
Question: If $A$ is a square matrix of order 3 such that $|A|=2$, then write the value of $\operatorname{adj}(\operatorname{adj} A)$. Solution: For any square matrixA, we have $\operatorname{adj}(\operatorname{adj} A)=|A|^{n-2} A$ $\Rightarrow \operatorname{adj}(\operatorname{adj} A)=2 A \quad[\because|A|=2$ and $n=3]$...
Read More →The population of a city is 125000.
Question: The population of a city is 125000. If the annual birth rate and death rate are 5.5% and 3.5% respectively, calculate the population of city after 3 years. Solution: Here, $\mathrm{P}=$ Initial population $=125,000$ Annual birth rate $=\mathrm{R}_{1}=5.5 \%$ Annual death rate $=\mathrm{R}_{2}=3.5 \%$ Net growth rate, $\mathrm{R}=\left(\mathrm{R}_{1}-\mathrm{R}_{2}\right)=2 \%$ $\mathrm{n}=$ Number of years $=3$ $\therefore$ Population after three years $=\mathrm{P}\left(1+\frac{\mathrm...
Read More →If A is symmetric matrix,
Question: If $A$ is symmetric matrix, write whether $A^{T}$ is symmetric or skew-symmetric. Solution: For any symmetric matrix, $A^{T}=A$. Hence, $A^{T}$ is also symmetric....
Read More →Answer the following and justify.
Question: Answer the following and justify. (i) Can x2-1 be the quotient on division of x6+2x3+x-l by a polynomial in x of degree 5? (ii) What will the quotient and remainder be on division of ox2+ bx + c by px3+qx2+ rx+ s, p 0 ? (iii) If on division of a polynomial p(x) by a polynomial g(x),the quotient is zero, what is the relation between the degree of p(x)and g(x)l (vi) If on division of a non-zero polynomial p(x)by a polynomial g(x),the remainder is zero, what is the relation between the de...
Read More →The present population of a town is 28000.
Question: The present population of a town is 28000. If it increases at the rate of 5% per annum, what will be its population after 2 years? Solution: Here, $\mathrm{P}=$ Initial population $=28,000$ $\mathrm{R}=$ Rate of growth of population $=5 \%$ per annum $\mathrm{n}=$ Number of years $=2$ $\therefore$ Population after two years $=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $=28,000\left(1+\frac{5}{100}\right)^{2}$ $=28,000(1.05)^{2}$ $=30,870$ Hence, the population after ...
Read More →Solve this
Question: If adj $A=\left[\begin{array}{cc}2 3 \\ 4 -1\end{array}\right]$ and adj $B=\left[\begin{array}{cc}1 -2 \\ -3 1\end{array}\right]$, find adj $A B$ Solution: Given: $a d j A=\left[\begin{array}{cc}2 3 \\ 4 -1\end{array}\right]$ $a d j B=\left[\begin{array}{cc}1 -2 \\ -3 1\end{array}\right]$ For any two non-singular matrices, $a d j(A B)=(a d j B) \times(a d j A)$ $\Rightarrow \operatorname{adj}(A B)=\left[\begin{array}{cc}-6 5 \\ -2 -10\end{array}\right]$...
Read More →Which of the following is not the
Question: Which of the following is not the graph of a quadratic polynomial? Solution: (d) For any quadratic polynomial ax2+ bx + c, a0, the graph of the Corresponding equation y = ax2+ bx + c has one of the two shapes either open upwards like u or open downwards like depending on whether a 0 or a 0. These curves are called parabolas. So, option (d) cannot be possible. Also, the curve of a quadratic polynomial crosses the X-axis on at most two points but in option (d) the curve crosses the X-axi...
Read More →Sum of money amounts to Rs 453690 in 2 years at 6.5% per annum compounded annually.
Question: Sum of money amounts to Rs 453690 in 2 years at 6.5% per annum compounded annually. Find the sum. Solution: $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $453,690=\mathrm{P}\left(1+\frac{6.5}{100}\right)^{2}$ $\mathrm{P}(1.065)^{2}=453,690$ $\mathrm{P}=\frac{453,690}{1.134225}$ $\mathrm{P}=400,000$ Thus, the required sum is Rs 400,000 ....
Read More →If A is a non-singular square matrix such that
Question: If $A$ is a non-singular square matrix such that $A^{-1}=\left[\begin{array}{cc}5 3 \\ -2 -1\end{array}\right]$, then find $\left(A^{T}\right)^{-1} .$ Solution: For any invertible matrixA, $\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$ We have $A^{-1}=\left[\begin{array}{cc}5 3 \\ -2 -1\end{array}\right]$ $\Rightarrow\left(A^{T}\right)^{-1}=\left[\begin{array}{ll}5 -2 \\ 3 -1\end{array}\right]$...
Read More →If one of the zeroes of a quadratic polynomial of the form
Question: If one of the zeroes of a quadratic polynomial of the form x2+ ax + b is the negative of the other, then it (a) has no linear term and the constant term is negative (b) has no linear term and the constant term is positive (c) can have a linear term but the constant term is negative (d) can have a linear term but the constant term is positive Solution: (a) Let p(x) = x2+ ax + b. Put a = 0, then, p(x) = x2+ b = 0 ⇒ x2= -b ⇒ x = b [b 0] Hence, if one of the zeroes of quadratic polynomial ...
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