Solve each of the following Cryptarithm:
Question: Solve each of the following Cryptarithm: Solution: $\mathrm{B}+1=8, \mathrm{~B}=7$ $\mathrm{A}+\mathrm{B}=1, \mathrm{~A}+7=1, \mathrm{~A}=4$ So, $\mathrm{A}=4, \mathrm{~B}=7$...
Read More →Solve each of the following Cryptarithm:
Question: Solve each of the following Cryptarithm: Solution: If $1+B=0$ Surely, B $=9$ If $1+\mathrm{A}+1=9$ Surely, $\mathrm{A}=7$...
Read More →Signs of the abscissa and ordinate of
Question: Signs of the abscissa and ordinate of a point in the second quadrant are respectively. (a) $+,+$ (b) $-,-$ (c) $-,+$ (d) $+,-$ Solution: (C)In second quadrant, X-axis is negative and Y-axis is positive. So, sign of abscissa of a point is negative and sign of ordinate of a point is positive...
Read More →Solve each of the following Cryptarithm:
Question: Solve each of the following Cryptarithm: Solution: Two possibilities of A are : (i) If $\mathrm{B}+7 \leq 9, \mathrm{~A}=6$ But clearly, if $\mathrm{A}=6, \mathrm{~B}+7 \geq 9$; it is impossible (ii) If $\mathrm{B}+7 \geq 9, \mathrm{~A}=5$ and $\mathrm{B}+7=5$ Clearly, $\mathrm{B}=8$ $\therefore \mathrm{A}=5, \mathrm{~B}=8$...
Read More →If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.
Question: If the points (3, 2), (x, 2), (8, 8) are collinear, findxusing determinant. Solution: If the points (3, 2), (x, 2) and (8, 8) are collinear, then $\left|\begin{array}{ccc}3 -2 1 \\ x 2 1 \\ 8 8 1\end{array}\right|=0$ $\Delta=\left|\begin{array}{ccc}3 -2 1 \\ x 2 1 \\ 8 8 1\end{array}\right|$ $=\left|\begin{array}{ccc}3 -2 1 \\ x-3 4 0 \\ 8 8 1\end{array}\right| \quad$ [Applying $R_{2} \rightarrow R_{2}-R_{1}$ ] $=\left|\begin{array}{ccc}3 -2 1 \\ x-3 4 0 \\ 5 10 0\end{array}\right| \qu...
Read More →An electrician has to repair an electric fault on a pole of height 4 metres. He needs to reach a point 1 metre below the top of the pole to undertake the repair work.
Question: An electrician has to repair an electric fault on a pole of height 4 metres. He needs to reach a point 1 metre below the top of the pole to undertake the repair work. What should be the length of the ladder that he should use, which when inclined at an angle of 60 to the horizontal would enable him to reach the required position?$[$ Use $\sqrt{3}=$1.73] Solution: Let AC be the pole and BD be the ladder.We have, $\mathrm{AC}=4 \mathrm{~m}, \mathrm{AB}=1 \mathrm{~m}$ and $\angle \mathrm{...
Read More →If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.
Question: If the points (3, 2), (x, 2), (8, 8) are collinear, findxusing determinant. Solution: If the points (3, 2), (x, 2) and (8, 8) are collinear, then $\left|\begin{array}{ccc}3 -2 1 \\ x 2 1 \\ 8 8 1\end{array}\right|=0$ $\Delta=\left|\begin{array}{ccc}3 -2 1 \\ x 2 1 \\ 8 8 1\end{array}\right|$ $=\left|\begin{array}{ccc}3 -2 1 \\ x-3 4 0 \\ 8 8 1\end{array}\right| \quad$ [Applying $R_{2} \rightarrow R_{2}-R_{1}$ ] $=\left|\begin{array}{ccc}3 -2 1 \\ x-3 4 0 \\ 5 10 0\end{array}\right| \qu...
Read More →Solve each of the following Cryptarithms:
Question: Solve each of the following Cryptarithms: Solution: Two possible values of $\mathrm{A}$ are : (i) If $7+\mathrm{B} \leq 9$ $\therefore 3+\mathrm{A}=9$ $\Rightarrow \mathrm{A}=6$ But if $\mathrm{A}=6,7+\mathrm{B}$ must be larger than 9 . Hence, it is impossible. (ii) If $7+B \geq 9$ $\therefore 1+3+\mathrm{A}=9$ $\Rightarrow \mathrm{A}=5$ If $\mathrm{A}=5$ and $7+\mathrm{B}=5, \mathrm{~B}$ must be 8 $\therefore \mathrm{A}=5, \mathrm{~B}=8$...
Read More →Point (-3, 5) lies in the
Question: Point (-3, 5) lies in the (a)first quadrant (b)second quadrant (c)third quadrant (d)fourth quadrant Thinking Process (i)Firstly, check the sign of each coordinate of a point. (ii)If both coordinates x and y has same positive sign i.e., (+, +), then the point lies in first quadrant. (iii)If x-coordinate has negative sign and y-coordinate has positive sign i.e., (-, +), then the point lies in second quadrant. (iv)If both coordinate x and y has negative sing i.e, (-, -), then the point li...
Read More →Which of the following statements are true?
Question: Which of the following statements are true? (i) If a number is divisible by 3, it must be divisible by 9. (ii) If a number is divisible by 9, it must be divisible by 3. (iii) If a number is divisible by 4, it must be divisible by 8. (iv) If a number is divisible by 8, it must be divisible by 4. (v) A number is divisible by 18, if it is divisible by both 3 and 6. (vi) If a number is divisible by both 9 and 10, it must be divisible by 90. (vii) If a number exactly divides the sum of two ...
Read More →If the points (x, −2), (5, 2), (8, 8) are collinear, find x using determinants.
Question: If the points (x, 2), (5, 2), (8, 8) are collinear, findxusing determinants. Solution: If the points (x, 2), (5, 2), (8, 8) are collinear, then $\left|\begin{array}{ccc}x -2 1 \\ 5 2 1 \\ 8 8 1\end{array}\right|=0$ $\Delta=\left|\begin{array}{ccc}x -2 1 \\ 5 2 1 \\ 8 8 1\end{array}\right|$ $\Delta=\left|\begin{array}{ccc}x -2 1 \\ 5-x 4 0 \\ 8 8 1\end{array}\right| \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$ $=\left|\begin{array}{ccc}x -2 1 \\ 5-x 4 0 \\ 8...
Read More →If the points (x, −2), (5, 2), (8, 8) are collinear, find x using determinants.
Question: If the points (x, 2), (5, 2), (8, 8) are collinear, findxusing determinants. Solution: If the points (x, 2), (5, 2), (8, 8) are collinear, then $\left|\begin{array}{ccc}x -2 1 \\ 5 2 1 \\ 8 8 1\end{array}\right|=0$ $\Delta=\left|\begin{array}{ccc}x -2 1 \\ 5 2 1 \\ 8 8 1\end{array}\right|$ $\Delta=\left|\begin{array}{ccc}x -2 1 \\ 5-x 4 0 \\ 8 8 1\end{array}\right| \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$ $=\left|\begin{array}{ccc}x -2 1 \\ 5-x 4 0 \\ 8...
Read More →Given an example of a number which is divisible by
Question: Given an example of a number which is divisible by (i) 2 but not by 4. (ii) 3 but not by 6. (iii) 4 but not by 8. (iv) both 4 and 8 but not by 32. Solution: (i) 10Every number with the structure (4n+ 2) is an example of a number that is divisible by 2 but not by 4.(ii) 15Every number with the structure (6n+ 3) is an example of a number that is divisible by 3 but not by 6.(iii) 28Every number with the structure (8n+ 4) is an example of a number that is divisible by 4 but not by 8.(iv) 8...
Read More →From the top of a vertical tower, the angles of depression of two cars in the same straight line with the base of the tower, at an instant are found to be 45° and 60°.
Question: From the top of a vertical tower, the angles of depression of two cars in the same straight line with the base of the tower, at an instant are found to be 45 and 60. If the cars are 100 m apart and are on the same side of the tower, find the height of the tower. Solution: Let OP be the tower and points A and B be the positions of the cars.We have, $\mathrm{AB}=100 \mathrm{~m}, \angle \mathrm{OAP}=60^{\circ}$ and $\angle \mathrm{OBP}=45^{\circ}$ Let $\mathrm{OP}=h$ In $\Delta \mathrm{AO...
Read More →Prove the following
Question: Prove that (a +b +c)3-a3-b3 c3=3(a +b)(b +c)(c +a). Solution: To prove, $(a+b+c)^{3}-a^{3}-b^{3}-c^{3}=3(a+b)(b+c)(c+a)$ $\mathrm{LHS}=\left[(a+b+c)^{3}-a^{3}\right]-\left(b^{3}+c^{3}\right)$ $=(a+b+c-a)\left[(a+b+c)^{2}+a^{2}+a(a+b+c)\right]$ $-\left[(b+c)\left(b^{2}+c^{2}-b c\right)\right]$ [using identity, $a^{3}+b^{3}=(a+b)\left(a^{2}+b^{2}-a b\right)$ and $\left.a^{3}-b^{3}=(a-b)\left(a^{2}+b^{2}+a b\right)\right]$ $=(b+c)\left[a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a+a^{2}+a^{2}+a b+a...
Read More →Without performing actual division,
Question: Without performing actual division, find the remainder when 928174653 is divided by 11. Solution: $928174653=$ A multiple of $11+$ (Sum of its digits at odd places - Sum of its digits at even places) $928174653=$ A multiple of $11+\{(9+8+7+6+3)-(2+1+4+5)\}$ $928174653=$ A multiple of $11+(33-12)$ $928174653=$ A multiple of $11+21$ $928174653=$ A multiple of $11+(11 \times 1+10)$ $928174653=$ A multiple of $11+10$ Therefore, the remainder is 10...
Read More →Find the remainder, without performing actual division,
Question: Find the remainder, without performing actual division, when 798 is divided by 11. Solution: $798=$ A multiple of $11+$ (Sum of its digits at odd places $-$ Sum of its digits at even places) $798=$ A multiple of $11+(7+8-9)$ $798=$ A multiple of $11+(15-9)$ $798=$ A multiple of $11+6$ Therefore, the remainder is 6 ....
Read More →If a+b+c= 5 and ab+bc+ca =10,
Question: If a+b+c= 5 and ab+bc+ca =10, then prove that a3+b3+c3 3abc = -25. Solution: To prove, $a^{3}+b^{3}+c^{3}-3 a b c=-25$ Given, $a+b+c=5, a b+b c+c a=10$ $\because$ $(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2(a b+b c+c a)$ $\therefore$ $(5)^{2}=a^{2}+b^{2}+c^{2}+2(10)$ $\Rightarrow \quad 25=a^{2}+b^{2}+c^{2}+20$ $\Rightarrow \quad a^{2}+b^{2}+c^{2}=25-20$ $\Rightarrow \quad a^{2}+b^{2}+c^{2}=5$ $\mathrm{LHS}=\mathrm{a}^{3}+b^{3}+c^{3}-3 a b c$ $=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)$ $=...
Read More →Using determinants, find the value of k so that
Question: Using determinants, find the value of $k$ so that the points $(k, 2-2 k),(-k+1,2 k)$ and $(-4-k, 6-2 k)$ may be collinear. Solution: If the points $(k, 2-2 k),(-k+1,2 k)$ and $(-4-k, 6-2 k)$ are collinear, then $\Delta=\left|\begin{array}{ccc}k 2-2 k 1 \\ -k+1 2 k 1 \\ -4-k 6-2 k 1\end{array}\right|=0$ $\Rightarrow\left|\begin{array}{ccc}k 2-2 k 1 \\ -2 k+1 4 k-2 0 \\ -4-k 6-2 k 1\end{array}\right|=0 \quad$ [Applying $R_{2} \rightarrow R_{2}-R_{1}$ ] $\Rightarrow\left|\begin{array}{ccc...
Read More →Find the remainder when 51439786 is divided by 3.
Question: Find the remainder when 51439786 is divided by 3. Do this without performing actual division. Solution: Sum of the digits of the number $51439786=5+1+4+3+9+7+8+6=43$ The remainder of 51439786 , when divided by 3 , is the same as the remainder when the sum of the digits is divided 3 . When 43 is divided by 3, remainder is 1 . Therefore, when 51439786 is divided by 3 , remainder will be 1 ....
Read More →Find the remainder when 981547 is divided by 5.
Question: Find the remainder when 981547 is divided by 5. Do this without doing actual division. Solution: If a natural number is divided by 5 , it has the same remainder when its unit digit is divided by 5 . Here, the unit digit of 981547 is 7 . When 7 is divided by 5 , remainder is 2 . Therefore, remainder will be 2 when 981547 is divided by 5 ....
Read More →If x denotes the digit at hundreds place of the number
Question: If $x$ denotes the digit at hundreds place of the number $\overline{67 x 19}$ such that the number is divisible by $11 .$ Find all possible values of $x$. Solution: A number is divisible by 11 , if the difference of the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11 . Sum of digits at odd places - Sum of digits at even places $=(6+\mathrm{x}+9)-(7+1)$ $=(15+\mathrm{x})-8=\mathrm{x}+7$ $\therefore \mathrm{x}+7=11$ $\Rightarrow \m...
Read More →Using determinants, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).
Question: Using determinants, find the area of the triangle with vertices (3, 5), (3, 6), (7, 2). Solution: Given:Vertices of triangle: ( 3, 5), (3, 6) and (7, 2) Area of the triangle $=\Delta=\frac{1}{2}\left|\begin{array}{ccc}-3 5 1 \\ 3 -6 1 \\ 7 2 1\end{array}\right|$ $=\frac{1}{2}\left|\begin{array}{ccc}-3 5 1 \\ 6 -11 0 \\ 7 2 1\end{array}\right| \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$ $=\frac{1}{2}\left|\begin{array}{ccc}-3 5 1 \\ 6 -11 0 \\ 10 -3 0\end{a...
Read More →Find all possible values of x .
Question: If $\overline{98215 x 2}$ is a number with $x$ as its tens digit such that is is divisible by 4 . Find all possible values of $x$. Solution: A natural number is divisible by 4 if the number formed by its digits in units and tens places is divisible by $4 .$ $\therefore \overline{98215 x 2}$ will be divisible by 4 if $\overline{x 2}$ is divisible by 4 . $\therefore \overline{x 2}=10 x+2$ $x$ is a digit; therefore possible values of $x$ are $0,1,2,3 \ldots 9$. $\overline{x 2}=2,12,22,32,...
Read More →Using determinants, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).
Question: Using determinants, find the area of the triangle with vertices (3, 5), (3, 6), (7, 2). Solution: Given:Vertices of triangle: ( 3, 5), (3, 6) and (7, 2) Area of the triangle $=\Delta=\frac{1}{2}\left|\begin{array}{ccc}-3 5 1 \\ 3 -6 1 \\ 7 2 1\end{array}\right|$ $=\frac{1}{2}\left|\begin{array}{ccc}-3 5 1 \\ 6 -11 0 \\ 7 2 1\end{array}\right| \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$ $=\frac{1}{2}\left|\begin{array}{ccc}-3 5 1 \\ 6 -11 0 \\ 10 -3 0\end{a...
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