If the zeroes of the cubic polynomial
Question: If the zeroes of the cubic polynomial x3 6x2+ 3x + 10 are of the form a,a + b and a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial. Solution: Let $\quad f(x)=x^{3}-6 x^{2}+3 x+10$ Given that, $a_{1}(a+b)$ and $(a+2 b)$ are the zeroes of $f(x)$. Then, Sum of the zeroes $=-\frac{\left(\text { Coefficient of } x^{2}\right)}{\left(\text { Coefficient of } x^{3}\right)}$ $\Rightarrow \quad a+(a+b)+(a+2 b)=-\frac{(-6)}{1}$ $\Right...
Read More →What will be the compound interest on Rs 4000
Question: What will be the compound interest on Rs 4000 in two years when rate of interest is 5% per annum? Solution: We know that amount $A$ at the end of $n$ years at the rate of $R \%$ per annum is given by $A=P\left(1+\frac{R}{100}\right)^{n}$. Given: $\mathrm{P}=\mathrm{Rs} 4,000$ $\mathrm{R}=5 \%$ p. a $\mathrm{n}=2$ years Now, $\mathrm{A}=4,000\left(1+\frac{5}{100}\right)^{2}$ $=4,000(1.05)^{2}$ $=\mathrm{Rs} 4,410$ And, $\mathrm{CI}=\mathrm{A}-\mathrm{P}$ $=\mathrm{Rs} 4,410-\mathrm{Rs} ...
Read More →The diameters of two circular ends of a bucket are 44 cm and 24 cm,
Question: The diameters of two circular ends of a bucket are 44 cm and 24 cm, and the height of the bucket is 35 cm. The capacity of the bucket is(a) 31.7 litres(b) 32.7 litres(c) 33.7 litres(d) 34.7 litres Solution: (b) 32.7 litresLetRandrbe the radii of the top and base of the bucket, respectively, and lethbe its height Then, $R=\frac{44}{2} \mathrm{~cm}=22 \mathrm{~cm}, r=\frac{24}{2} \mathrm{~cm}=12 \mathrm{~cm}, h=35 \mathrm{~cm}$ Capacity of the bucket = Volume of the frustum of the cone $...
Read More →Solve the following equations
Question: If $A=\left[\begin{array}{cc}2 3 \\ 5 -2\end{array}\right]$, write $A^{-1}$ in terms of $A$ Solution: $|A|=\left|\begin{array}{cc}2 3 \\ 5 -2\end{array}\right|=-19 \neq 0$ $A$ is a non-singular matri $x$. Therefore, it is invertible. Let $C_{i j}$ be a cofactor of $a_{i j}$ in $A$. The cofactors of element $A$ are given by $C_{11}=-2$ $C_{12}=-5$ $C_{21}=-3$ $C_{22}=2$ $\operatorname{adj} A=\left[\begin{array}{cc}-2 -5 \\ -3 2\end{array}\right]^{T}=\left[\begin{array}{cc}-2 -3 \\ -5 2\...
Read More →Find the compound interest when principal = Rs 3000,
Question: Find the compound interest when principal = Rs 3000, rate = 5% per annum and time = 2 years. Solution: Principal for the first year $=$ Rs 3,000 Interest for the first year $=\operatorname{Rs}\left(\frac{3,000 \times 5 \times 1}{100}\right)$ = Rs 150 Amount at the end of the first year $=\mathrm{Rs} 3,000+\mathrm{Rs} 150$ = Rs 3,150 Principal for the second year $=$ Rs 3,150 Interest for the second year $=\operatorname{Rs}\left(\frac{3,150 \times 5 \times 1}{100}\right)$ = Rs 157.50 Am...
Read More →Find the compound interest when principal = Rs 3000,
Question: Find the compound interest when principal = Rs 3000, rate = 5% per annum and time = 2 years. Solution: Principal for the first year $=$ Rs 3,000 Interest for the first year $=\operatorname{Rs}\left(\frac{3,000 \times 5 \times 1}{100}\right)$ = Rs 150 Amount at the end of the first year $=\mathrm{Rs} 3,000+\mathrm{Rs} 150$ = Rs 3,150 Principal for the second year $=$ Rs 3,150 Interest for the second year $=\operatorname{Rs}\left(\frac{3,150 \times 5 \times 1}{100}\right)$ = Rs 157.50 Am...
Read More →For each of the following, find a quadratic
Question: For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also, find the zeroes of these polynomials by factorisation. (i) $\frac{-8}{3}, \frac{4}{3}$ (ii) $\frac{21}{8}, \frac{5}{16}$ (iii) $-2 \sqrt{3},-9$ (iv) $\frac{-3}{2 \sqrt{5}},-\frac{1}{2}$ Solution: (i) Given that, sum of zeroes $(S)=-\frac{8}{3}$ and product of zeroes $(P)=\frac{4}{3}$ $\therefore$ Required quadratic expression, $f(x)=x^{2}-S x+P$ $=x^{2}+\frac{8}{3...
Read More →Solve the following equations
Question: If $A=\left[\begin{array}{cc}3 1 \\ 2 -3\end{array}\right]$, then find |adj $A \mid$. Solution: $|A|=\left|\begin{array}{cc}3 1 \\ 2 -3\end{array}\right|=-11$ $\therefore|\operatorname{adj} A|=|A|^{n-1}=(-11)^{2-1}=-11$...
Read More →Twelve solid spheres of the same size are made by melting a solid metallic
Question: Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter 2 cm and height 16 cm. The diameter of each sphere is(a) 2 cm(b) 3 cm(c) 4 cm(d) 6 cm Solution: (a) 2 cmLet the diameter of each sphere bedcm.LetrandRbe the radii of the sphere and the cylinder, respectively,andhbe the height of the cylinder. As $R=\frac{\text { Diameter }}{2}$, $R=\frac{2}{2} \mathrm{~cm}=1 \mathrm{~cm}$ $h=16 \mathrm{~cm}$ Therefore, $12 \times \frac{4}{3} \pi r^{3}=\...
Read More →Solve this
Question: If $A=\left[\begin{array}{ll}a b \\ c d\end{array}\right], B=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$, find adj $(A B)$ Solution: $A \times B=\left[\begin{array}{ll}a b \\ c d\end{array}\right]$ $A \times B$ is a non $-$ singular matrix. Therefore, it is invertible. Let $C_{i j}$ be a cofactor of $a_{i j}$ in $A$. The cofactors of element $A$ are given by $C_{11}=d$ $C_{12}=-c$ $C_{21}=-b$ $C_{22}=a$ $\therefore \operatorname{adj} A=\left[\begin{array}{cc}d -c \\ -b a\end{a...
Read More →Solve this
Question: If $A=\left[\begin{array}{cc}1 -3 \\ 2 0\end{array}\right]$, write adj $A$. Solution: $|A|=\left|\begin{array}{cc}1 -3 \\ 2 0\end{array}\right|=6 \neq 0$ $A$ is a non-singular matrix. Therefore, it is invertible. Let $C_{i j}$ be a cofactor of $a_{i j}$ in $A$. The cofactors of element $A$ are given by $C_{11}=0$ $C_{12}=-2$ $C_{21}=3$ $C_{22}=1$ $\therefore \operatorname{adj} A=\left[\begin{array}{cc}0 -2 \\ 3 1\end{array}\right]^{T}=\left[\begin{array}{cc}0 3 \\ -2 1\end{array}\right...
Read More →A mason constructs a wall of dimensions (270 cm × 300 cm × 350 cm) with bricks,
Question: A mason constructs a wall of dimensions $(270 \mathrm{~cm} \times 300 \mathrm{~cm} \times 350 \mathrm{~cm})$ with bricks, each of size $(22.5 \mathrm{~cm} \times 11.25 \mathrm{~cm} \times 8.75 \mathrm{~cm})$ and it is assumed that $\frac{1}{8}$ space is covered by the mortar. Number of bricks used to construct the wall is(a) 11000(b) 11100(c) 11200(d) 11300 Solution: (c) 11200 Volume of wall $=270 \times 300 \times 350 \mathrm{~cm}^{3}$ $\frac{1}{8}$ th of the wall is covered with mort...
Read More →Ashish started the business with an initial investment of Rs 500000.
Question: Ashish started the business with an initial investment of Rs 500000. In the first year he incurred a loss of 4%. However during the second year he earned a profit of 5% which in third year rose to 10%. Calculate the net profit for the entire period of 3 years. Solution: Profit for three year $s=\mathrm{P}\left(1-\frac{\mathrm{R}_{1}}{100}\right)\left(1+\frac{\mathrm{R}_{2}}{100}\right)\left(1+\frac{\mathrm{R}_{3}}{100}\right)$ $\Rightarrow 500,000\left(1-\frac{4}{100}\right)\left(1+\fr...
Read More →Find the inverse of the matrix
Question: Find the inverse of the matrix $\left[\begin{array}{cc}\cot \theta \sin \theta \\ -\sin \theta \cos \theta\end{array}\right]$. Solution: $A=\left[\begin{array}{cc}\cos \theta \sin \theta \\ -\sin \theta \cos \theta\end{array}\right]$ $\therefore|A|=\cos ^{2} \theta+\sin ^{2} \theta=1 \neq 0$ $A$ is a singular matrix. Therefore, it is invertible. Let $C_{i j}$ be a cofactor of $a_{i j}$ in $A$. The cofactors of element $A$ are given by $C_{11}=\cos \theta$ $C_{12}=\sin \theta$ $C_{21}=-...
Read More →Find the zeroes of the following polynomials by factorisation
Question: Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials (i) $4 x^{2}-3 x-1$ Solution: (i) $4 x^{2}-3 x-1$ Solution: Let $f(x)=4 x^{2}-3 x-1$ $=4 x^{2}-4 x+x-1 \quad$ [by splitting the middle term] $=4 x(x-1)+1(x-1)$ $=(x-1)(4 x+1)$ So, the value of $4 x^{2}-3 x-1$ is zero when $x-1=0$ or $4 x+1=0$ i.e., when $x=1$ or $x=-\frac{1}{4}$. So, the zeroes of $4 x^{2}-3 x-1$ are 1 and $-\frac{1}{4...
Read More →The cost of a T.V. set was quoted Rs 17000 at the beginning of 1999.
Question: The cost of a T.V. set was quoted Rs 17000 at the beginning of 1999. In the beginning of 2000 the price was hiked by 5%. Because of decrease in demand the cost was reduced by 4% in the beginning of 2001. What was the cost of the T.V. set in 2001? Solution: Cost of the $\mathrm{TV}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)\left(1-\frac{\mathrm{R}}{100}\right)$ $\Rightarrow 17,000\left(1+\frac{5}{100}\right)\left(1-\frac{4}{100}\right)$ $=17,000(1.05)(0.96)$ $=17,136$ Thus, the cos...
Read More →The value of a refrigerator which was purchased 2 years ago, depreciates at 12% per annum.
Question: The value of a refrigerator which was purchased 2 years ago, depreciates at 12% per annum. If its present value is Rs 9680, for how much was it purchased? Solution: Purchase price $=\mathrm{P}\left(1-\frac{\mathrm{R}}{100}\right)^{-\mathrm{n}}$ $\Rightarrow 9,680\left(1-\frac{12}{100}\right)^{-2}$ $=9,680(0.88)^{-2}$ $=12,500$ Thus, the purchase price of the refrigerator was Rs 12,500 ....
Read More →A cubical ice-cream brick of edge 22 cm is to be distributed among some children by filling ice-cream cones of radius 2 cm and height 7 cm up to the brim.
Question: A cubical ice-cream brick of edge 22 cm is to be distributed among some children by filling ice-cream cones of radius 2 cm and height 7 cm up to the brim. How many children will get the ice-cream cones?(a) 163(b) 263(c) 363(d) 463 Solution: (c) 363The edge of the cubical ice-cream brick =a= 22 cm Volume of the cubical ice-cream brick $=(a)^{3}$ $=(22 \times 22 \times 22) \mathrm{cm}^{3}$ Radius of an ice-cream cone = 2 cmHeight of an ice-cream cone = 7 cm Volume of each ice-cream cone ...
Read More →Find the inverse of the matrix
Question: Find the inverse of the matrix $\left[\begin{array}{cc}3 -2 \\ -7 5\end{array}\right]$. Solution: $|A|=\left|\begin{array}{cc}3 -2 \\ -7 5\end{array}\right|=1 \neq 0$ $A$ is a non-singular matrix. Therefore, it is invertible. Let $C_{i j}$ be a cofactor of $a_{i j}$ in $A$. The cofactors of element $A$ are given by $C_{11}=5$ $C_{12}=7$ $C_{21}=2$ $C_{22}=3$ $\therefore A^{-1}=\frac{1}{|A|}\left[\begin{array}{ll}5 7 \\ 2 3\end{array}\right]^{T}=\left[\begin{array}{ll}5 2 \\ 7 3\end{arr...
Read More →The value of a machine depreciates at the rate of 10% per annum.
Question: The value of a machine depreciates at the rate of 10% per annum. It was purchased 3 years ago. If its present value is Rs 43740, find its purchase price. Solution: Purchase price $=\mathrm{P}\left(1-\frac{\mathrm{R}}{100}\right)^{-\mathrm{n}}$ $\Rightarrow 43,740\left(1-\frac{10}{100}\right)^{-3}$ $=43,740(0.90)^{-3}$ $=60,000$ Thus, the purchase price of the machine was Rs 60,000 ....
Read More →If Cij is the cofactor of the element aij of the matrix
Question: If $C_{i j}$ is the cofactor of the element $a_{i j}$ of the matrix $A=\left[\begin{array}{ccc}2 -3 5 \\ 6 0 4 \\ 1 5 -7\end{array}\right]$, then write the value of $a_{32} C_{32}$. Solution: In the given matrix $A=\left[\begin{array}{ccc}2 -3 5 \\ 6 0 4 \\ 1 5 -7\end{array}\right]$, $C_{32}=(-1)^{3+2}(8-30)=22$ Therefore, $a_{32} C_{32}=5 \times 22=110$. Hence, the value of $a_{32} C_{32}$ is 110 ....
Read More →Mohan purchased a house for Rs 30000 and its value is depreciating at the rate of 25% per year.
Question: Mohan purchased a house for Rs 30000 and its value is depreciating at the rate of 25% per year. Find the value of the house after 3 years. Solution: Value of the house after three years $=\mathrm{P}\left(1-\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $\Rightarrow 30,000\left(1-\frac{25}{100}\right)^{3}$ $=30,000(0.75)^{3}$ $=12,656.25$ Thus, the value of the house after three years will be Rs $12,656.25$....
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:If the radius of the base of a right circular cylinder is halved, keepingthe height the same, then the ratio of the volume of the cylinder thusobtained to the volume of original cylinder is(a) 1 : 2 (b) 2 : 1 (c) 1 : 4 (d) 4 : 1 Solution: Let the base radius and height of the original cylinder be $r$ and $h$, respectively. Also, The radius of the new cylinder, $R=\frac{r}{2}$ and its height, $H=h$. Now, The ratio of the volume of the ...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:If the radius of the base of a right circular cylinder is halved, keepingthe height the same, then the ratio of the volume of the cylinder thusobtained to the volume of original cylinder is(a) 1 : 2 (b) 2 : 1 (c) 1 : 4 (d) 4 : 1 Solution: Let the base radius and height of the original cylinder be $r$ and $h$, respectively. Also, The radius of the new cylinder, $R=\frac{r}{2}$ and its height, $H=h$. Now, The ratio of the volume of the ...
Read More →Let A be a square matrix
Question: Let $A$ be a square matrix such that $A^{2}-A+I=O$, then write $A^{-1}$ interms of $A$. Solution: Given : $A^{2}-A+I=O$ $A^{-1}\left(A^{2}-A+I\right)=A^{-1} O$ (Pre - multiplying both sides because $A^{-1}$ exists) $\left(A^{-1} A^{2}\right)-\left(A^{-1} A\right)+A^{-1} I=O$ $\left(A^{-1} O=O\right)$ $\Rightarrow A-I+A^{-1}=O \quad\left(A^{-1} I=A^{-1}\right)$ $\Rightarrow A^{-1}=I-A$...
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