If the perpendicular distance of a point
Question: If the perpendicular distance of a point P from the X-axis is 5 units and the foot of the perpendicular lies on the negative direction of X-axis, then the point P has (a)x-coordinate = -5 (b)y-coordinate = 5 only (c)y-coordinate = 5 only (d)y-coordinate = 5 or 5 Solution: (d)We know that, the perpendicular distance of a point from the X-axis gives y-coordinate of that point. Here, foot of perpendicular lies on the negative direction of X-axis, so perpendicular distance can be measure i...
Read More →Solve this
Question: $x-2 y=4$ $-3 x+5 y=-7$ Solution: Given: $\quad x-2 y=4$ $-3 x+5 y=-7$ Using the properties of determinants, we get $\mathrm{D}=\mid \begin{array}{ll}1 -2\end{array}$ $-3 \quad 5 \quad \mid=5-6=-1 \neq 0$ $\mathrm{D}_{1}=\mid 4 \quad-2$ $-7 \quad 5 \mid=20-14=6$ $\mathrm{D}_{2}=\mid 1 \quad 4$ $-3-7 \mid=-7+12=5$ Using Cramer's Rule, we get $\mathrm{x}=\frac{\mathrm{D}_{1}}{\mathrm{D}}=\frac{6}{-1}=-6$ $\mathrm{y}=\frac{\mathrm{D}_{2}}{\mathrm{D}}=\frac{5}{-1}=-5$ $\therefore \mathrm{x...
Read More →An aeroplane is flying at a height of 300 m above the ground.
Question: An aeroplane is flying at a height of 300 m above the ground. Flying at this height the angles of depression from the aeroplane of two points on both banks of a river in opposite directions are 45 ad 60 respectively. Find the width of the river.$[$ Use $\sqrt{3}=1.732]$ Solution: Let CD be the height of the aeroplane above the river at some instant. Suppose A and B be two points on both banks of the river in opposite directions. Height of the aeroplane above the river, CD = 300 mNow,CA...
Read More →The points (- 5, 2) and (2, — 5) lie in the
Question: The points (- 5, 2) and (2, 5) lie in the (a)same quadrant (b)II and III quadrants, respectively (c)II and IV quadrants, respectively (d)IV and II quadrants, respectively Solution: (c)In point (-5,2), x-coordinate is negative and y-coordinate is positive, so it lies in II quadrant and in point (2, 5), x- coordinate is positive and y-coordinate is negative, so it lies in IV quadrant....
Read More →If y-coordinate of a point is zero,
Question: If y-coordinate of a point is zero, then this point always lies (a)in I quadrant (b)in II quadrant (c)on X-axis (d)on Y-axis Solution: (c)If y-coordinate of a point is zero, then this point always lies on X-axis. Because perpendicular distance of the point from X-axis measured along Y-axis is zero....
Read More →Points (1, -1), (2, – 2), (4, – 5) and (-3, – 4)
Question: Points (1, -1), (2, 2), (4, 5) and (-3, 4) (a)lie in II quadrant (b)lie in III quadrant (c)lie in IV quadrant (d)do not lie in the same quadrant Solution: (d)In points (1, -1), (2, -2) and (4, -5) x-coordinate is positive and y-coordinate is negative, So, they all lie in IV quadrant. In point (-3, 4) x- coordinate is negative and y-coordinate is negative. So, it lies in III quadrant So, given points do not lie in the same quadrant....
Read More →Find the quotient when the difference of 985 and 958 is divided by 9.
Question: Find the quotient when the difference of 985 and 958 is divided by 9. Solution: If $\overline{\mathrm{abc}}-\overline{\mathrm{acb}}$ is divided by 9, the quotient is $(\mathrm{b}-\mathrm{c})$. $\therefore$ If $(985-958)$ is divided by 9, quotient $=(8-5)=3$...
Read More →If sum of the number 985 and two other numbers obtained by arranging the digits of 985 in cyclic order is divided by 111, 22 and 37 respectively.
Question: If sum of the number 985 and two other numbers obtained by arranging the digits of 985 in cyclic order is divided by 111, 22 and 37 respectively. Find the quotient in each case. Solution: The sum of $(985+859+598)$ when divided by : (i) 111 $Q$ uotient $=(9+8+5)=22$ (ii) 22 , i. e, $(9+8+5)$ $Q$ uotient $=111$ (iii) $37\left(=\frac{111}{3}\right)$ $Q$ uotient $=3(9+8+5)=66$...
Read More →Find values of k, if area of triangle is 4 square units whose vertices are
Question: Find values ofk, if area of triangle is 4 square units whose vertices are (i) (k, 0), (4, 0), (0, 2)(ii) (2, 0), (0, 4), (0,k) Solution: (i) If the area of a triangle with vertices $(k, 0),(4,0)$ and $(0,2)$ is 4 square units, then $\Delta=\frac{1}{2} \mid k \quad 0 \quad 1$ $4 \quad 0 \quad 1$ $0 \quad 2 \quad 1 \mid$ $=\frac{1}{2}\{(2) \times \mid \mathrm{k} \quad 1$ $4 \quad 1 \mid\} \quad$ [Expanding along $\mathrm{C}_{2}$ ] $=(\mathrm{k}-4)$ Since area is always $+$ ve, we take it...
Read More →A point both of whose coordinates are negative will lie in
Question: A point both of whose coordinates are negative will lie in (a)I quadrant (b)II quadrant (c)III quadrant (d)IV quadrant Solution: (c)A point both of whose coordinates are negative will lie in III quadrant because, in III quadrant x-coordinate and y-coordinate both are negative....
Read More →The point at which the two coordinate axes meet is called the
Question: The point at which the two coordinate axes meet is called the (a)abscissa (b)ordinate (c)origin (d)quadrant Solution: (c)The point at which the two coordinate axes meet is called the origin....
Read More →Without performing actual computations,
Question: Without performing actual computations, find the quotient when 94 49 is divided by (i) 9 (ii) 5 Solution: (i) We know that when $\overline{\mathrm{ab}}-\overline{\mathrm{ba}}$ is divided by 9 , the quotient is $\mathrm{a}-\mathrm{b}$. Therefore, when $(94-49)$ is divided by 9 , the quotient is $(9-4=5)$. (ii) We know that when $\overline{\mathrm{ab}}-\overline{\mathrm{ba}}$ is divided by $(\mathrm{a}-\mathrm{b})$, the quotient is 9 . Therefore, when $(94-49)$ is divided by $(9-4=5)$, t...
Read More →A man observes a car from the top of a tower, which is moving towards the tower with a uniform speed.
Question: A man observes a car from the top of a tower, which is moving towards the tower with a uniform speed. If the angle of depression of the car changes from 30 to 45 in 12 minutes, find the time taken by the car now to reach the tower. Solution: Suppose AB be the tower of heighthmeters. Let C be the initial position of the car and let after 12 minutes the car be at D. It is given that the angles of depression at C and D are 30 and 45 respectively.Let the speed of the car bevmeter per minut...
Read More →Without performing actual addition and division write the quotient when the sum of 69 and 96 is divided by
Question: Without performing actual addition and division write the quotient when the sum of 69 and 96 is divided by (i) 11 (ii) 15 Solution: (i) Clearly, 69 and 96 are two numbers such that one can be obtained be reversing the digits of the other. Therefore, when the sum of 69 and 96 is divided by 11, we get 15 (sum of the digits) as quotient.(ii) Clearly, 69 and 96 are two numbers such that one can be obtained be reversing the digits of the other. Therefore, when the sum of 69 and 96 is divide...
Read More →Ordinate of all points on the X-axis is
Question: Ordinate of all points on the X-axis is (a)0 (b)1 (c) 1 (d)anynumber Solution: (a) Ordinate of all points on the X-axis is zero.Because ordinate (or y-coordinate) of a point is perpendicular distance of this point from the X-axis measured along the Y-axis. If point lies on X- axis, then the perpendicular distance of point from X- axis will be zero, so ordinate will be zero....
Read More →Show that the Cryptarithm
Question: Show that the Cryptarithm $4 \times \overline{A B}=\overline{C A B}$ does not have any solution. Solution: 0 is the only unit digit number, which gives the same 0 at the unit digit when multiplied by 4 . So, the possible value of $\mathrm{B}$ is 0 . Similarly, for A also, 0 is the only possible digit. But then $A, B$ and $C$ will all be 0 . And if $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ become 0, these numbers cannot be of two $-$ digit or three $-$ digit. Therefore, both will become...
Read More →Abscissa of all the points on the X-axis is
Question: Abscissa of all the points on the X-axis is (a)0 (b)1 (c)2 (d)any number Solution: (d)Abscissa of all the points on the X-axis is any number because X-axis is a number line which contains many real numbers on it....
Read More →Using determinants, find the equation of the line joining the points
Question: Using determinants, find the equation of the line joining the points (i) $(1,2)$ and $(3,6)$ (ii) $(3,1)$ and $(9,3)$ Solution: (i) Given: $A=(1,2)$ and $B=(3,6)$ Let the point $P$ be $(x, y) .$ So, Area of triangle $A B P=0$ $\Rightarrow \Delta=\frac{1}{2}\left|\begin{array}{lll}1 2 1 \\ 3 6 1 \\ x y 1\end{array}\right|=0$ $\Rightarrow 1(6-y)-2(3-x)+1(3 y-6 x)=0$ $\Rightarrow 6-y-6+2 x+3 y-6 x=0$ $\Rightarrow 2 y-4 x=0$ $\Rightarrow y=2 x$ (ii) Given: $A=(3,1)$ and $B=(9,3)$ Let the p...
Read More →Using determinants, find the equation of the line joining the points
Question: Using determinants, find the equation of the line joining the points (i) $(1,2)$ and $(3,6)$ (ii) $(3,1)$ and $(9,3)$ Solution: (i) Given: $A=(1,2)$ and $B=(3,6)$ Let the point $P$ be $(x, y) .$ So, Area of triangle $A B P=0$ $\Rightarrow \Delta=\frac{1}{2}\left|\begin{array}{lll}1 2 1 \\ 3 6 1 \\ x y 1\end{array}\right|=0$ $\Rightarrow 1(6-y)-2(3-x)+1(3 y-6 x)=0$ $\Rightarrow 6-y-6+2 x+3 y-6 x=0$ $\Rightarrow 2 y-4 x=0$ $\Rightarrow y=2 x$ (ii) Given: $A=(3,1)$ and $B=(9,3)$ Let the p...
Read More →Solve each of the following Cryptarithm:
Question: Solve each of the following Cryptarithm: Solution: If $\mathrm{A}+\mathrm{B}=8, \mathrm{~A}+\mathrm{B} \geq 9$ is possible only if $\mathrm{A}=\mathrm{B}=9$ But from $7+\mathrm{B}=\mathrm{A}, \mathrm{A}=\mathrm{B}=9$ is impossible Surely, $\mathrm{A}+\mathrm{B}=8, \mathrm{~A}+\mathrm{B} \leq 9$ So, $\mathrm{A}+7=9$, Surely $\mathrm{A}=2$ $7+\mathrm{B}=\mathrm{A}, 7+\mathrm{B}=2, \mathrm{~B}=5$ So, $\mathrm{A}=2, \mathrm{~B}=5$...
Read More →Using determinants, find the equation of the line joining the points
Question: Using determinants, find the equation of the line joining the points (i) $(1,2)$ and $(3,6)$ (ii) $(3,1)$ and $(9,3)$ Solution: (i) Given: $A=(1,2)$ and $B=(3,6)$ Let the point $P$ be $(x, y) .$ So, Area of triangle $A B P=0$ $\Rightarrow \Delta=\frac{1}{2}\left|\begin{array}{lll}1 2 1 \\ 3 6 1 \\ x y 1\end{array}\right|=0$ $\Rightarrow 1(6-y)-2(3-x)+1(3 y-6 x)=0$ $\Rightarrow 6-y-6+2 x+3 y-6 x=0$ $\Rightarrow 2 y-4 x=0$ $\Rightarrow y=2 x$ (ii) Given: $A=(3,1)$ and $B=(9,3)$ Let the p...
Read More →Point (- 10,0) lies
Question: Point (- 10,0) lies (a)on the negative direction of the X-axis (b)on the negative direction of the Y-axis (c)in the third quadrant (d)in the fourth quadrant Solution: (a)In point (-10, 0) y-coordinate is zero, so it lies on X-axis and its x-coordinate is negative, so the point (-10, 0) lies on the X-axis in the negative direction....
Read More →From the top of a building AB, 60 m high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30° and 60°, respectively.
Question: From the top of a building AB, 60 m high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30 and 60, respectively. Find (i) the horizontal distance between AB and CD,(ii) the height of the lamp post,(iii) the difference between the heights of the building and the lamp post. Solution: We have, $\mathrm{AB}=60 \mathrm{~m}, \angle \mathrm{ACE}=30^{\circ}$ and $\angle \mathrm{ADB}=60^{\circ}$ Let $\mathrm{BD}=\mathrm{CE}=x$ and $\mathrm{CD}=\mat...
Read More →Solve each of the following Cryptarithm:
Question: Solve each of the following Cryptarithm: Solution: $\mathrm{A}+\mathrm{B}=9$ as the sum of two digits can never be 19 $2+\mathrm{A}=0, \mathrm{~A}$ must be 8 $\mathrm{A}+\mathrm{B}=9,8+\mathrm{B}=9, \mathrm{~B}=1$ So, $\mathrm{A}=8, \mathrm{~B}=1$...
Read More →Prove the following
Question: Point (0, 7) lies (a)on the X-axis (b)in the second quadrant (c)on the Y-axis (d)in the fourth quadrant Thinking Process (i)Firstly, check whether any coordinate of point is zero or not. (a)If x-coordinate is zero and y-coordinate is non-zero, then the point lies on Y-axis. (b)If y-coordinate is zero and x-coordinate is non-zero, then the point lies on X- axis. (c)If x-coordinate and y- coordinate are zero, then the point lies on origin (or on both the axes). (d)If none of the coordina...
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