Find the amount of Rs 2400 after 3 years,
Question: Find the amount of Rs 2400 after 3 years, when the interest is compounded annually at the rate of 20% per annum. Solution: Given: $\mathrm{P}=\mathrm{Rs} 2,400$ $\mathrm{R}=20 \%$ p. $\mathrm{a}$ $\mathrm{n}=3$ years We know that amount $\mathrm{A}$ at the end of $\mathrm{n}$ years at the rate $\mathrm{R} \%$ per annum when the interest is compounded annually is given by $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$. $\therefore \mathrm{A}=2,400\left(1+\frac...
Read More →Find the amount of Rs 2400 after 3 years,
Question: Find the amount of Rs 2400 after 3 years, when the interest is compounded annually at the rate of 20% per annum. Solution: Given: $\mathrm{P}=\mathrm{Rs} 2,400$ $\mathrm{R}=20 \%$ p. $\mathrm{a}$ $\mathrm{n}=3$ years We know that amount $\mathrm{A}$ at the end of $\mathrm{n}$ years at the rate $\mathrm{R} \%$ per annum when the interest is compounded annually is given by $\mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$. $\therefore \mathrm{A}=2,400\left(1+\frac...
Read More →The least number that is divisible
Question: The least number that is divisible by all the numbers from 1 to 10 (both inclusive) (a) 10 (b) 100 (c) 504 (d) 2520 Solution: (d) Factors of 1 to 10 numbers $1=1$ $2=1 \times 2$ $3=1 \times 3$ $4=1 \times 2 \times 2$ $5=1 \times 5$ $6=1 \times 2 \times 3$ $7=1 \times 7$ $8=1 \times 2 \times 2 \times 2$ $9=1 \times 3 \times 3$ $10=1 \times 2 \times 5$ $\therefore \quad L C M$ of number 1 to $10=L C M(1,2,3,4,5,6,7,8,9,10)$ $=1 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times...
Read More →Solve this
Question: If $A=\left[\begin{array}{cc}2 3 \\ 5 -2\end{array}\right]$ be such that $A^{-1}=k A$, then $k$ equals (a) 19 (b) $1 / 19$ (c) $-19$ (d) $-1 / 19$ Solution: (b) $1 / 19$ $\operatorname{adj} A=\left[\begin{array}{rr}-2 -3 \\ -5 2\end{array}\right]$ $|A|=-19$ $\therefore A^{-1}=\frac{1}{|A|} \operatorname{adj} A$ $\Rightarrow A^{-1}=-\frac{1}{19}\left[\begin{array}{rr}-2 -3 \\ -5 2\end{array}\right]$ Now, $A^{-1}=k A$ $\Rightarrow-\frac{1}{19}\left[\begin{array}{rr}-2 -3 \\ -5 2\end{arra...
Read More →Solve this
Question: If $A=\left[\begin{array}{cc}2 3 \\ 5 -2\end{array}\right]$ be such that $A^{-1}=k A$, then $k$ equals (a) 19 (b) $1 / 19$ (c) $-19$ (d) $-1 / 19$ Solution: (b) $1 / 19$ $\operatorname{adj} A=\left[\begin{array}{rr}-2 -3 \\ -5 2\end{array}\right]$ $|A|=-19$ $\therefore A^{-1}=\frac{1}{|A|} \operatorname{adj} A$ $\Rightarrow A^{-1}=-\frac{1}{19}\left[\begin{array}{rr}-2 -3 \\ -5 2\end{array}\right]$ Now, $A^{-1}=k A$ $\Rightarrow-\frac{1}{19}\left[\begin{array}{rr}-2 -3 \\ -5 2\end{arra...
Read More →66 cubic cm of silver is drawn into a wire 1 mm in diameter.
Question: 66 cubic cm of silver is drawn into a wire 1 mm in diameter. Calculate the length of the wire in metres. Solution: We have, Radius of wire, $r=\frac{1}{2}=0.5 \mathrm{~mm}=0.05 \mathrm{~cm}$ Let the length of the wire be $l$. As, Volume of the wire $=66 \mathrm{~cm}^{3}$ $\Rightarrow \pi r^{2} l=66$ $\Rightarrow \frac{22}{7} \times 0.05 \times 0.05 \times l=66$ $\Rightarrow l=66 \times \frac{7}{22 \times 0.05 \times 0.05}$ $\therefore l=8400 \mathrm{~cm}=84 \mathrm{~m}$ So, the length ...
Read More →The radii of two cylinders are in the ratio of 2 : 3 and their heights are in the ratio of 5 : 3.
Question: The radii of two cylinders are in the ratio of 2 : 3 and their heights are in the ratio of 5 : 3. Find the ratio of their volumes. Solution: Let the radii of the cylinders be $r_{1}$ and $r_{2} ;$ and their heights be $h_{1}$ and $h_{2}$. We have, $r_{1}: r_{2}=2: 3$ or $\frac{r_{1}}{r_{2}}=\frac{2}{3}$ ..............(i) and $h_{1}: h_{2}=5: 3$ or $\frac{h_{1}}{h_{2}}=\frac{5}{3}$ ...........(ii) Now, The ratio of the volumes of the cylinders $=\frac{\text { Volume of the first cylinde...
Read More →If A is a square matrix
Question: If $A$ is a square matrix such that $A^{2}=I$, then $A^{-1}$ is equal to (a) $A+I$ (b) $A$ (c) 0 (d) $2 \mathrm{~A}$ Solution: (b) $A$ Given : $A^{2}=I$ On multiplying both sides by $A^{-1}$, we get $A^{-1} A^{2}=A^{-1} I$ $\Rightarrow A=A^{-1} I$ $\Rightarrow A=A^{-1}$...
Read More →The product of a non-zero rational
Question: The product of a non-zero rational and an irrational number is (a) always irrational (b) always rational (c) rational or irrational (d) one Solution: (a) Product of a non-zero rational and an irrational number is always irrational i.e., $\frac{3}{4} \times \sqrt{2}=\frac{3 \sqrt{2}}{4}$ (irrational)....
Read More →Compute the amount and the compound interest in each of the following by using the formulae when:
Question: Compute the amount and the compound interest in each of the following by using the formulae when: (i) Principal = Rs 3000, Rate = 5%, Time = 2 years (ii) Principal = Rs 3000, Rate = 18%, Time = 2 years (iii) Principal = Rs 5000, Rate = 10 paise per rupee per annum, Time = 2 years (iv) Principal = Rs 2000, Rate = 4 paise per rupee per annum, Time = 3 years (v) Principal = Rs 12800 , Rate $=7 \frac{1}{2} \%$, Time $=3$ years (vi) Principal = Rs 10000, Rate 20% per annum compounded half-y...
Read More →If two positive integers p and q can be expressed’
Question: If two positive integers p and q can be expressed as p=ab2and q = a3b; where 0, b being prime numbers, then LCM (p, q) is equal to (a) ab (b) a2b2 (c) a3b2 (d) a3b3 Solution: (c) Given that, $p=a b^{2}=a \times b \times b$ and $\quad q=a^{3} b=a \times a \times a \times b$ $\therefore \quad$ LCM of $p$ and $q=$ LCM $\left(a b^{2}, a^{3} b\right)=a \times b \times b \times a \times a=a^{3} b^{2}$ [since, LCM is the product of the greatest power of each prime factor involved in the numbe...
Read More →If A and B are invertible matrices, which of the following statement is not correct.
Question: IfAandBare invertible matrices, which of the following statement is not correct. (a) adj $A=|A| A^{-1}$ (b) $\operatorname{det}\left(A^{-1}\right)=(\operatorname{det} A)^{-1}$ (c) $(A+B)^{-1}=A^{-1}+B^{-1}$ (d) $(A B)^{-1}=B^{-1} A^{-1}$ Solution: (c) $(A+B)^{-1}=A^{-1}+B^{-1}$ We have, adj $A=|A| A^{-1}, \operatorname{det}\left(A^{-1}\right)=(\operatorname{det} A)^{-1}$ and $(A B)^{-1}=B^{-1} A^{-1}$ all are the properites of inverse of a matrix....
Read More →If A and B are invertible matrices, which of the following statement is not correct.
Question: IfAandBare invertible matrices, which of the following statement is not correct. (a) adj $A=|A| A^{-1}$ (b) $\operatorname{det}\left(A^{-1}\right)=(\operatorname{det} A)^{-1}$ (c) $(A+B)^{-1}=A^{-1}+B^{-1}$ (d) $(A B)^{-1}=B^{-1} A^{-1}$ Solution: (c) $(A+B)^{-1}=A^{-1}+B^{-1}$ We have, adj $A=|A| A^{-1}, \operatorname{det}\left(A^{-1}\right)=(\operatorname{det} A)^{-1}$ and $(A B)^{-1}=B^{-1} A^{-1}$ all are the properites of inverse of a matrix....
Read More →If two positive integers a and b are written
Question: If two positive integers a and b are written as a = x3y2and b = xy3, wfiere x, y are prime numbers, then HCF (a, b) is (a) xy (b) xy2 (c)x3y3 (d) xy2 Solution: (b) Given that, $\quad a=x^{3} y^{2}=x \times x \times x \times y \times y$ and $\quad b=x y^{3}=x \times y \times y \times y$ $\therefore$ HCF of $a$ and $b \quad=\operatorname{HCF}\left(x^{3} y^{2}, x y^{3}\right)=x \times y \times y=x y^{2}$ [since, HCF is the product of the smallest power of each common prim facter involved ...
Read More →The ratio between the radius of the base and the height of a cylinder is 2 : 3.
Question: The ratio between the radius of the base and the height of a cylinder is2 : 3. If the volume of the cylinder is 12936 cm3, then find the radius of the base of the cylinder. Solution: Let the radius of the base and the height of the cylinder be $r$ and $h$, respectively. We have, $r: h=2: 3$ i. e. $\frac{r}{h}=\frac{2}{3}$ or $h=\frac{3 r}{2} \quad \ldots$ (i) As, Volume of the cylinder $=12936 \mathrm{~cm}^{3}$ $\Rightarrow \pi r^{2} h=12936$ $\Rightarrow \frac{22}{7} \times r^{2} \tim...
Read More →Solve this
Question: If $A^{2}-A+I=0$, then the inverse of $A$ is (a) $A^{-2}$ (b) $A+1$ (c) $I-A$ (d) $A-1$ Solution: (c) $I-A$ Given : $A^{2}-A+I=O$ $A^{-1}\left(A^{2}-A+I\right)=A^{-1} O \quad$ [multiplying both sides by $A^{-1}$ ] $\Rightarrow\left(A^{-1} A^{2}\right)-\left(A^{-1} A\right)+A^{-1} I=O \quad\left[\because A^{-1} O=O\right]$ $\Rightarrow A-I+A^{-1}=O \quad\left[\because A^{-1} I=A^{-1}\right]$ $\Rightarrow A^{-1}=I-A$...
Read More →The largest number which divides
Question: The largest number which divides70 and 125, leaving remainders respectively, is (a) 13 (b) 65 (c) 875 (d) 1750 Solution: (a) Since, 5 and 8 are the remainders of 70 and 125, respectively. Thus, after subtracting these remainders from the numbers, we have the numbers 65 = (70-5), 117 = (125 8), which is divisible by the required number. Now, required number = HCF of 65,117 [for the largest number] For this, $\quad 117=65 \times 1+52 \quad[\because$ dividend $=$ divisor $\times$ quotient...
Read More →If A is a matrix of order 3
Question: If $A$ is a matrix of order 3 and $|A|=8$, then $|\operatorname{adj} A|=$ (a) 1 (b) 2 (c) $2^{3}$ (d) $2^{6}$ Solution: (d) $2^{6}$ $|a d j A|=|A|^{n-1}$ $=8^{2}$ $=2^{6}$...
Read More →The volume of a right circular cylinder with its height equal to the radius is
Question: The volume of a right circular cylinder with its height equal to the radius is $25 \frac{1}{7} \mathrm{~cm}^{3}$. Find the height of the cylinder. Solution: We have, Height $=$ Base radius i. e. $h=r$ As, Volume of the cylinder $=25 \frac{1}{7} \mathrm{~cm}^{3}$ $\Rightarrow \pi r^{2} h=\frac{176}{7}$ $\Rightarrow \frac{22}{7} \times h^{2} \times h=\frac{176}{7}$ $\Rightarrow h^{3}=\frac{176 \times 7}{7 \times 22}$ $\Rightarrow h^{3}=8$ $\Rightarrow h=\sqrt[3]{8}$ $\therefore h=2 \math...
Read More →If the HCF of 65 and 117 is
Question: If the HCF of 65 and 117 is expressible in the form 65m -117, then the value of m is (a) 4 (b) 2 (c) 1 (d) 3 Solution: (b) By Euclid's division algorithm, $b=a q+r, 0 \leq ra \quad[\because$ dividend $=$ divisor $\times$ quotient $+$ remainder $]$ $\Rightarrow \quad 117=65 \times 1+52$ $\Rightarrow \quad 65=52 \times 1+13$ $\Rightarrow \quad 52=13 \times 4+0$ $\therefore \quad \operatorname{HCF}(65,117)=13$ $\ldots$ (i) Also, given that, $\operatorname{HCF}(65,117)=65 m-117$ .....(ii) ...
Read More →If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is
Question: Ifdis the determinant of a square matrixAof ordern, then the determinant of its adjoint is (a) $d^{n}$ (b) $d^{n-1}$ (c) $d^{n+1}$ (d) $d$ Solution: (b) $d^{n-1}$ We know, $|\operatorname{adj} A|=|A|^{n-1}$ $\Rightarrow|\operatorname{adj} A|=d^{n-1}$...
Read More →If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is
Question: Ifdis the determinant of a square matrixAof ordern, then the determinant of its adjoint is (a) $d^{n}$ (b) $d^{n-1}$ (c) $d^{n+1}$ (d) $d$ Solution: (b) $d^{n-1}$ We know, $|\operatorname{adj} A|=|A|^{n-1}$ $\Rightarrow|\operatorname{adj} A|=d^{n-1}$...
Read More →The volumes of two cubes are in the ratio 8 : 27.
Question: The volumes of two cubes are in the ratio 8 : 27. Find the ratio of their surface areas. Solution: Let the edges of the cubes be $a$ and $b$. As, $\frac{\text { Volume of the first cube }}{\text { Volume of the second cube }}=\frac{8}{27}$ $\Rightarrow \frac{a^{3}}{b^{3}}=\frac{8}{27}$ $\Rightarrow \frac{a}{b}=\sqrt[3]{\frac{8}{27}}$ $\Rightarrow \frac{a}{b}=\frac{2}{3}$ Now, The ratio of the surface areas of the cubes $=\frac{\text { Surface area of the first cube }}{\text { Surface a...
Read More →List price of a washing machine is Rs 9000.
Question: List price of a washing machine is Rs 9000. If the dealer allows a discount of 5% on the cash payment, how much money will a customer pay to the dealer in cash, if the rate of VAT is 10%? Solution: List price of the washing machine $=$ Rs. 9000 Discount allowed $=5 \%$ Discount $=5 \%$ of Rs. 9000 $=\frac{5}{100} \times 9000=$ Rs. 450 So, the $c$ ost of the washing machine $=$ List price $-$ Discount $=$ Rs. $(9000-450)$ $=$ Rs. 8550 VAT $=10 \%$ of Rs. 8550 $=\frac{10}{100} \times 855...
Read More →solve the matrix
Question: The matrix $\left[\begin{array}{ccc}5 10 3 \\ -2 -4 6 \\ -1 -2 b\end{array}\right]$ is a singular matrix, if the value of $b$ is (a) $-3$ (b) 3 (c) 0 (d) non-existent Solution: (d) non-existent For any singular matrix, the value of the determinant is 0 . Here, $A=\left[\begin{array}{ccc}5 10 3 \\ -2 -4 6 \\ -1 -2 b\end{array}\right]$ $|A|=5(-4 b+12)-10(-2 b+6)+3(4-4)=0$ $\Rightarrow-20 b+60+20 b-12=0$ Hence, $b$ is non-existent....
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