Maths
Find the slope and the equation of the line passing through the points: (a, b) and ( – a, b)
Find the slope and the equation of the line passing through the points: (5, 3) and ( – 5, – 3)
Find the slope and the equation of the line passing through the points: (i) (3, – 2) and ( – 5, – 7)
Find the equation of the line passing through the point P( – 3, 5) and perpendicular to the line passing through the points A(2, 5) and B( – 3, 6)
Find the equation of the line passing through the point P(4, – 5) and parallel to the line joining the points A(3, 7) and B( – 2, 4).
Find the equation of a line which cuts off intercept 5 on the x – axis and makes an angle of 600 with the positive direction of the x – axis.
Find the equation of a line
Find the equation of a line whose slope is 4 and which passes through the point (5, – 7)
Find the equation of a line which is equidistant from the lines x = – 2 and x = 6.
Answer : For the equation of line equidistant from both lines, we will find point through which line passes and is equidistant from both line. As any point lying on x = - 2 line is ( - 2, 0) and on...
Find the equation of a vertical line passing through the point ( – 5, 6).
Answer : Equation of line parallel to y - axis (vertical) is given by x = constant, as x - coordinate is constant for every point lying on line i.e. 6. So, the required equation of line is given as...
Find the equation of a horizontal line passing through the point (4, – 2).
Answer : Equation of line parallel to x - axis (horizontal) is y = constant, as y - coordinate of every point on the line parallel to x - axis is - 2 i.e. constant. Therefore equation of the line...
Find the equation of a line parallel to the x – axis and having intercept – 3 on the y – axis.
Answer: Equation of line parallel to x - axis is given by y = constant, as x - coordinate of every point on the line parallel to y - axis is - 3 i.e. constant. So, the required equation of line is y...
Find the equation of a line parallel to the x – axis at a distance of
(i) 4 units above it
(ii) 5 units below it
Answer : (i) Equation of line parallel to x - axis is given by y = constant, as the y - coordinate of every point on the line parallel to x - axis is 4,i.e. constant. Now the point lies above x -...
. A(1, 1), B(7, 3) and C(3, 6) are the vertices of a ΞABC. If D is the midpoint of BC and AL β₯ BC, find the slopes of (i) AD and (ii) AL.
Show that the points A(0, 6), B(2, 1) and C(7, 3) are three corners of a square ABCD. Find (i) the slope of the diagonal BD and (ii) the coordinates of the fourth vertex D.
If ΞΈ is the angle between the diagonals of a parallelogram ABCD whose vertices are A(0, 2), B(2,-1), C(4,
If ΞΈ is the angle between the lines joining the points (0, 0) and B(2, 3), and the points C(2, -2) and D(3, 5), show that
If A(1, 2), B(-3, 2) and C(3, 2) be the vertices of a ΞABC, show that
Find the angle between the lines whose slopes are
Find the slope of the line which makes an angle of 300 with the positive direction of the y-axis, measured anticlockwise.
The vertices of a quadrilateral are A(-4, -2), B(2, 6), C(8, 5) and D(9, -7). Using slopes, show that the midpoints of the sides of the quad. ABCD from a parallelogram.
A line passes through the points A(4, -6) and B(-2, -5). Show that the line AB makes an obtuse angle with the x-axis.
If the points A(a, 0), B(0, b) and P(x, y) are collinear, using slopes, prove that
If the three points A(h, k), B(x1, y1) and C(x2, y2) lie on a line then show that (h β x1)(y2 β y1) = (k β y1)(x2 β x).
Using slopes show that the points A(-4, -1), B(-2, -4), C(4, 0) and D(2, 3) taken in order, are the vertices of a rectangle.
Using slopes show that the points A(6, -1), B(5, 0) and C(2, 3) are collinear.
If A(2, -5), B(-2, 5), C(x, 3) and D(1, 1) be four points such that AB and CD are perpendicular to each other, find the value of x.
Show that the line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points (3, -3) and (5, -9).
Show that the line through the points (5, 6) and (2, 3) is parallel to the line through the points (9, -2) and (6, -5)
If the slope of the line joining the points A(x, 2) and B(6, -8) is value of x.
Find the inclination of a line whose slope is
Find the slope of a line whose inclination is
(i) 30Β°
(ii) 120Β°
(iii) 135Β°
(iv) 90Β°
Find the slope of a line whose inclination is (i) 30Β°
(ii) 120Β°
(iii) 135Β°
(iv) 90Β°
In what ratio is the line segment joining the points A(-4, 2) and B(8, 3) divided by the y-axis? Also, find the point of intersection.
Find the ratio in which the x-axis cuts the join of the points A(4, 5) and B(- 10, -2). Also, find the point of intersection.
Find the coordinates of the point which divides the join of A(-5, 11) and B(4, -7) in the ratio 2 : 7.
Find the area of ΞABC, the midpoints of whose sides AB, BC and CA are D(3, -1), E(5, 3) and F(1, -3) respectively.
Find the area of the quadrilateral whose vertices are A(-4, 5), B(0, 7), C(5, -5) and D(-4, -2).
Find the value of k for which the points A(-2, 3), B(1, 2) and C(k, 0) are collinear
Show that the points A(-5, 1), B(5, 5) and C(10, 7) are collinear.
Find the area of ΞABC whose vertices are A(-3, -5), B(5, 2) and C(-9, -3).
If the points A (-2, -1), B(1, 0), C(x, 3) and D(1, y) are the vertices of a parallelogram, find the values of x and y.
Show that the points A(2, -1), B(3, 4), C(-2, 3) and D(-3, -2) are the vertices of a rhombus.
Show that A(1, -2), B(3, 6), C(5, 10) and D(3, 2) are the vertices of a parallelogram.
Show that A(3, 2), B(0, 5), C(-3, 2) and D(0, -1) are the vertices of a square.
Show that the points A(2, -2), B(8, 4), C(5, 7) and D(-1, 1) are the angular points of a rectangle.
Show that the points A(1, 1), B(-1, -1) and C(-β3, β3) are the vertices of an equilateral triangle each of whose sides is 22 units.
Show that the points A(7, 10), B(-2, 5) and C(3, -4) are the vertices of an isosceles right-angled triangle.
Using the distance formula, show that the points A(3, -2), B(5, 2) and C(8,8) are collinear.
Find a point on the y-axis which is equidistant from A(-4, 3) and B(5, 2).
A is a point on the x-axis with abscissa -8 and B is a point on the y-axis with ordinate 15. Find the distance AB.
Find the distance between the points A(x1, y1) and B(x2, y2), when
(i)AB is parallel to the x-axis
(ii) AB is parallel to the y-axis.
Find a point on the x-axis which is equidistant from the points A(7, 6) and B(- 3, 4).
If a point P(x, y) is equidistant from the points A(6, -1) and B(2, 3), find the relation between x and y.
. Find the distance of the point P(6, -6) from the origin.
Find the distance between the points:
(i) A(2, -3) and B(-6, 3)
(ii) C(-1, -1) and D(8, 11)
(iii) P(-8, -3) and Q(-2, -5)
(iv) R(a + b, a β b) and S(a β b, a + b)
What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin?
Solution: It is given that a cube with 2 units edge, one of whose vertices coincides with the origin and the 3 edges passing through the origin, coincides with the positive direction of the axes...
Choose the correct answer from the given four options indicated against each of the Exercises if the distance between the points $(\mathrm{a}, 0,1)$ and $(0,1,2)$ is $\sqrt{27}$, then the value of $a$ is
(A) 5
(B) $\pm 5$
(C) 5
(D) none of these
Solution: Option(B) $\pm 5$ Explanation: Suppose $P$ be the point whose coordinate is $(a, 0,1)$ and $Q$ represents the point $(0,$, $(1,2) .$ It is given that, $\mathrm{PQ}=\sqrt{27}$ From the...
Choose the correct answer from the given four options indicated against each of the Exercises distance of the point $(3,4,5)$ from the origin $(0,0,0)$ is
(A) $\sqrt{50}$
(B) 3
(C) 4
(D) 5
Solution: Option (A) $\sqrt{50}$ Explanation: Suppose $\mathrm{P}$ be the point whose coordinate is $(3,4,5)$ and $\mathrm{Q}$ represents the origin. From the distance formula it can be written as...
Choose the correct answer from the given four options indicated against each of the Exercises what is the length of foot of perpendicular drawn from the point $P(3,4,5)$ on $y$-axis
(A) $\sqrt{41}$
(B) $\sqrt{34}$
(C) 5
(D) none of these
Solution: Option(B) $\sqrt{34}$ Explanation: As it is known that $y$-axis lies on $x$ y plane and $y z$. Therefore, its distance from $x y$ and $y z$ plane is 0 . $\therefore$ By the basic...
Choose the correct answer from the given four options indicated against each of the Exercises the distance of point P (3, 4, 5) from the y z-plane is
(A) 3 units
(B) 4 units
(C) 5 units
(D) 550
Solution: (A) 3 units Explanation: From basic ideas of three-dimensional geometry, it is known that $x$-coordinate of a point is its distance from $y z$ plane. $\therefore$ The distance of Point $P...
Prove that the points (0, β 1, β 7), (2, 1, β 9) and (6, 5, β 13) are collinear. Find the ratio in which the first point divides the join of the other two.
Solution: It is given that the three points $A(0,-1,-7), B(2,1,-9)$ and $C(6,5,-13)$ are collinear So it can be written as $\begin{array}{l} A...
The mid-point of the sides of a triangle are (1, 5, β 1), (0, 4, β 2) and (2, 3, 4). Find its vertices. Also find the centroid of the triangle.
Solution: It is given that the mid-point of the sides of a triangle are $(1,5,-1),(0,4,-2)$ and $(2,3,$, 4). Suppose the vertices be $A\left(x_{1}, y_{1}, z_{1}\right), B\left(x_{2}, y_{2},...
Show that the three points $A(2,3,4), B(-1,2,-3)$ and $C(-4,1,-10)$ are collinear and find the ratio in which $C$ divides $A B$.
Solution: It is given that the three points are A $(2,3,4), \mathrm{B}(-1,2,-3)$ and $\mathrm{C}(-4,1,-10)$ We need to find collinear points, $\begin{array}{l}...
Let A $(2,2,-3), B(5,6,9)$ and C $(2,7,9)$ be the vertices of a triangle. The internal bisector of the angle $A$ meets $B C$ at the point $D$. Find the coordinates of $D$.
Solution: It is given $A(2,2,-3), B(5,6,9)$ and $C(2,7,9)$ are the vertices of a triangle. And it is also given that the internal bisector of the angle A meets BC at the point D....
If the origin is the centroid of a triangle ABC having vertices A (a, 1, 3), B (β 2, b, β 5) and C (4, 7, c), find the values of a, b, c.
Solution: It is given that the triangle ABC having vertices $A(a, 1,3), B(-2, b,-5)$ and $C(4,7, c)$ and origin is the centroid. The coordinates of the centroid for a triangle is given by the...
Find the coordinate of the points which trisect the line segment joining the points A $(2,1,-3)$ and $\mathrm{B}$ $(5,-8,3)$
Solution: It is given the line segment joining the points are A $(2,1,-3)$ and $B(5,-8,3)$ Now suppose $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ be the points which...
Three vertices of a Parallelogram ABCD are A (1, 2, 3), B (β 1, β 2, β 1) and C (2, 3, 2). Find the fourth vertex D.
Solution: It is given that the three consecutive vertices of a parallelogram ABCD are A $(1,2,3), B(-1,$, $-2,-1)$ and $C(2,3,2)$ Suppose the fourth vertex be $D(x, y, z)$. By using midpoint...
The mid-points of the sides of a triangle are (5, 7, 11), (0, 8, 5) and (2, 3, β 1). Find its vertices.
Solution: It is given that the mid-points of the sides of a triangle are $(5,7,11),(0,8,5)$ and $(2,3,-$ 1). Suppose the vertices be $A\left(x_{1}, y_{1}, z_{1}\right), B\left(x_{2}, y_{2},...
Find the centroid of a triangle, the mid-point of whose sides are $D(1,2,-3), E(3,0,1)$ and $F(-1,1,-$ 4).
Solution: It is given that: Mid-points of sides of triangle $\mathrm{DEF}$ are: $\mathrm{D}(1,2,-3), \mathrm{E}(3,0,1)$ and $\mathrm{F}(-1,1,-4)$ Using the geometry of centroid, It is known that the...
Find the third vertex of triangle whose centroid is origin and two vertices are (2, 4, 6) and (0, β2, β5).
Solution: It is given the centroid is origin and two vertices are $(2,4,6)$ and $(0,-2,-5)$ Suppose the third vertex be $(x, y, z)$ The coordinates of the centroid for a triangle is given by the...
Show that the triangle $\mathrm{ABC}$ with vertices $\mathrm{A}(0,4,1), \mathrm{B}(2,3,-1)$ and $\mathrm{C}(4,5,0)$ is right angled.
Solution: The given vertices are $A(0,4,1), B(2,3,-1)$ and $C(4,5,0)$ We need to prove right angled triangle, consider $\begin{array}{l}...
Three consecutive vertices of a parallelogram ABCD are A (6, β 2, 4), B (2, 4, β 8), C (β2, 2, 4). Find the coordinates of the fourth vertex.
Solution: The three consecutive vertices of a parallelogram $A B C D$ are as given $A(6,-2,4), B$ (2, $4,-8), C(-2,2,4)$ Suppose the forth vertex be $D(x, y, z)$ Midpoint of diagonal $A...
Show that the point $A(1,-1,3), B(2,-4,5)$ and $(5,-13,11)$ are collinear.
Solution: The given points are $A(1,-1,3), B(2,-4,5)$ and $(5,-13,11)$. We need to prove collinear, $\begin{array}{l} \mathrm{AB}=\sqrt{(1-2)^{2}+(-1+4)^{2}+(3-5)^{2}}=\sqrt{1+9+4}=\sqrt{14} \\...
Show that if $x^{2}+y^{2}=1$, then the point $\left(x, y \sqrt{1-x^{2}-y^{2}}\right)$ is at a distance 1 unit from the origin.
Solution: It is given that $x^{2}+y^{2}=1 \Rightarrow 1-x^{2}-y^{2}=0$ Distance of the point $\left(x, y \sqrt{1-x^{2}-y^{2}}\right)$ from origin is $=$ $\begin{array}{l}...
Find the distance from the origin to (6, 6, 7).
Solution: The distance from the origin to (6, 6, 7) $=\sqrt{{6^2}+{6^2}+{7^2}}$ $=\sqrt{{36}+{36}+{49}}$ $=\sqrt{121}$ $=11$ units
How far apart are the points (2, 0, 0) and (β3, 0, 0)?
Solution: The points $(2, 0, 0)$ and $(β3, 0, 0)$ are at a distance of $=$ $|2 β (β3)| = 5$ units.
Let A, B, C be the feet of perpendiculars from a point P on the xy, yz and zx planes respectively. Find the coordinates of A, B, C in each of the following where the point P is
(i) (4, β 3, β 5).
Solution: (i) $(4, β 3, β 5):- A (4, β3, 0), B (0, β3, β5), C (4, 0, β5)$
Let A, B, C be the feet of perpendiculars from a point P on the xy, yz and zx planes respectively. Find the coordinates of A, B, C in each of the following where the point P is
(i) (3, 4, 5)
(ii) (β5, 3, 7)
Solution: (i) $(3, 4, 5):- A (3, 4, 0), B (0, 4, 5), C (3, 0, 5)$ (ii) $(β5, 3, 7):- A (β5, 3, 0), B (0, 3, 7), C (β5, 0, 7)$
Let A, B, C be the feet of perpendiculars from a point P on the x, y, z-axis respectively. Find the coordinates of A, B and C in each of the following where the point P is :
(i) (4, β 3, β 5)
Solution: (i) $(4, β 3, β 5):- A (4, 0, 0), B (0, β3, 0), C (0, 0, β5)$
Let A, B, C be the feet of perpendiculars from a point P on the x, y, z-axis respectively. Find the coordinates of A, B and C in each of the following where the point P is :
(i) A = (3, 4, 2)
(ii) (β5, 3, 7)
Solution: (i) $(3, 4, 2):- A (3, 0, 0), B (0, 4, 0), C (0, 0, 2)$ (ii) $(β5, 3, 7):- A (β5, 0, 0), B (0, 3, 0), C (0, 0, 7)$
Name the octant in which each of the following points lies.
(i) (2, β 4, β 7)
(ii) (β 4, 2, β 5).
Solution: (i) $(2, β 4, β 7)$:- 8th Octant, (ii) $(β 4, 2, β 5)$:- 6th Octant.
Name the octant in which each of the following points lies.
(i) (β 4, 2, 5)
(ii) (β3, β1, 6)
Solution: (i) $(β 4, 2, 5)$:- 2nd Octant, (ii) $(β3, β1, 6)$:- 3rd Octant,
Name the octant in which each of the following points lies.
(i) (4, β2, β5)
(ii) (4, 2, β5)
Solution: (i) $(4, β2, β5)$:- 8th Octant, (ii) $(4, 2, β5)$:- 5th Octant,
Name the octant in which each of the following points lies.
(i) (1, 2, 3),
(ii) (4, β 2, 3),
Solution: (i) $(1, 2, 3)$:- 1st Octant, (ii) $(4, β 2, 3)$:- 4th Octant,
Locate the following points:
(i) (β 2, β 4, β7)
(ii) (β 4, 2, β 5).
Solution: (i) $(β 2, β 4, β7)$:- 7th octant, (ii) $(β 4, 2, β 5)$:- 6th octant.
Locate the following points:
(i) (1, β 1, 3),
(ii) (β 1, 2, 4)
Solution: (i) $(1, β 1, 3)$:- 4th octant, (ii) $(β 1, 2, 4)$:- 2nd octant,
Find the vector equation of the line passing through the point (1,2,-4) and perpendicular to the two lines: $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$
Solution: The vector eq. of a line passing through a point with position vector $\vec{a}$ and parallel to a vector $\overrightarrow{\mathrm{b}}$ is...
If $\mathrm{l}_{1}, \mathrm{~m}_{1}, \mathrm{n}_{1}$ and $\mathrm{l}_{2}, \mathrm{~m}_{2}, \mathrm{n}_{2}$ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $\left(\mathrm{m}_{1} \mathrm{n}_{2}-\mathrm{m}_{2} \mathrm{n}_{1}\right),\left(\mathrm{n}_{1} \mathrm{l}_{2}-\mathrm{n}_{2} \mathrm{l}_{1}\right),\left(\mathrm{l}_{1} \mathrm{~m}_{2}-\mathrm{l}_{2} \mathrm{~m}_{1}\right)$
Solution: Let's consider $l, m, n$ be the direction cosines of the line perpendicular to each of the given lines. Therefore, $ll_{1}+m m_{1}+n n_{1}=0 \ldots(1)$ And...
In the following cases, find the distance of each of the given points from the corresponding given plane. Point Plane
(a) (0, 0, 0) 3x β 4y + 12 z = 3
(b) (3, -2, 1) 2x β y + 2z + 3 = 0
Solution: (a) The distance of the point $(0,0,0)$ from the plane $3 x-4 y+12=3 \Rightarrow$ $3 x-4 y+12 z-3=0$ is $\begin{array}{l} \frac{\left|a x_{1}+b y_{1}+c...
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
(a) 7x + 5y + 6z + 30 = 0 and 3x β y β 10z + 4 = 0
(b) 2x + y + 3z β 2 = 0 and x β 2y + 5 = 0
Solution: (a) $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ It is given that The eq. of the given planes are $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ Two planes are $\perp$ if the direction ratio of the...
Find the angle between the planes whose vector equations are $\vec{r}:(2 \hat{i}+2 \hat{j}-3 \hat{k})=5, \vec{r} \cdot(3 \hat{i}-3 \hat{j}+5 \hat{k})=3$
Solution: It is given that The eq. of the given planes are $\vec{r}(2 \hat{i}+2 \hat{j}-3 \hat{k})=5 \text { and } \vec{r}(3 \hat{i}-3 \hat{j}+5 \hat{k})=5$ If $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$...
Find the vector equation of the plane passing through the intersection of the planes $\vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})=7, \vec{r} \cdot(2 \hat{i}+5 \hat{j}+3 \hat{k})=9$ and through the point $(2,1,3)$
Solution: Let's consider the vector eq. of the plane passing through the intersection of the planes are $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=7...
Find the intercepts cut off by the plane 2x + y β z = 5.
Solution: It is given that The plane $2 x+y-z=5$ Let us express the equation of the plane in intercept form $x / a+y / b+z / c=1$ Where $a, b, c$ are the intercepts cut-off by the plane at $x, y$...
Find the vector and Cartesian equations of the planes
(a) that passes through the point $(1,0,-2)$ and the normal to the plane is $\hat{i}+\hat{j}-\hat{k}$
(b) that passes through the point $(1,4,6)$ and the normal vector to the plane is $\hat{i}-2 \hat{j}+\hat{k}$
Solution: (a) That passes through the point $(1,0,-2)$ and the normal to the plane is $\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ Let's say that the position vector of the point $(1,0,-2)$...
Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector $3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$
Solution: It is given that, The vector $3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$ Vector equation of the plane with position vector $\overrightarrow{\mathrm{r}}$ is $\vec{r} \cdot...
In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a) z = 2
(b) x + y + z = 1
Solution: (a) $z=2$ It is given that The eq. of the plane, $z=2$ or $0 x+0 y+z=2 \ldots (1) .$ The direction ratio of the normal $(0,0,1)$ Using the formula, $\begin{array}{l}...
Find the shortest distance between the lines whose vector equations are $\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-3 \hat{j}+2 \hat{k})$ and $\vec{r}=4 \hat{i}+5 \hat{j}-6 \hat{k}+\mu(2 \hat{i}+3 \hat{j}+\hat{k})$
Solution: It is known to us that shortest distance between two lines $\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}}$ and $\vec{r}=\overrightarrow{a_{2}}+\mu \overrightarrow{b_{2}}$...
A letter is known to have come either from TATA NAGAR or from CALCUTTA. On the envelope, just two consecutive letter TA are visible. What is the probability that the letter came from TATA NAGAR.
Let E1 be the event that the letter comes from TATA NAGAR, E2 be the event that the letter comes from CALCUTTA And, E3 be the event that on the letter, two consecutive letters TA are visible Now,...
Find the equation of the line in vector and in Cartesian form that passes through the point with position vector $2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$. is in the direction
Solution: Given: Vector equation of a line that passes through a given point whose position vector is $\vec{a}$ and parallel to a given vector $\vec{b}$ is $\vec{r}=\vec{a}+\lambda \vec{b}$ Let,...
Show that the three lines with direction cosines $\frac{12}{13}, \frac{-3}{13}, \frac{-4}{13} ; \frac{4}{13}, \frac{12}{13}, \frac{3}{13} ; \frac{3}{13}, \frac{-4}{13}, \frac{12}{13}$ Are mutually perpendicular.
Solution: Consider the direction cosines of $L_{1}, L_{2}$ and $L_{3}$ be $l_{1}, m_{1}, n_{1} ; l_{2}, m_{2}, n_{2}$ and $l_{3}, m_{3}, n_{3}$. It is known that If $\mathrm{f}_{1}, \mathrm{~m}_{1},...
Write the negation of the following simple statements
(i) All similar triangles are congruent.
(ii) Area of a circle is same as the perimeter of the circle.
(i) "Not p" is the negation of the assertion p. The negation of p is represented by $\sim p$. The truth value of $\sim p$ is the opposite of the truth value of p. The negation of the statement is...
Write the negation of the following simple statements
(i) 2 is not a prime number.
(ii) Every real number is an irrational number.
(i) "Not p" is the negation of the assertion p. The negation of p is represented by $\sim p$. The truth value of $\sim p$ is the opposite of the truth value of p. The negation of the statement is β2...
Write the negation of the following simple statements
(i) The number 17 is prime.
(ii) 2 + 7 = 6.
(i) "Not p" is the negation of the statement p. The negation of p is represented by the "$\sim p$." The truth value of $\sim p$ is the opposite of the truth value of p. The negation of the statement...
Find the component statements of the following compound statements.
(i) β7 is a rational number or an irrational number.
(ii) 0 is less than every positive integer and every negative integer.
(i) A compound statement is made up of two or more statements (Components). As a result, the components of the given statement 7are a rational or irrational number, respectively. p: β7is a rational...
Find the component statements of the following compound statements.
(i) Number 7 is prime and odd.
(ii) Chennai is in India and is the capital of Tamil Nadu.
(i) A compound statement is made up of two or more statements (Components). As a result, the elements of the provided statement "Number 7 is prime and odd" are as follows: p: The number 7 is prime....
Which of the following sentences are statements? Justify
(i) Where is your bag?
(ii) Every square is a rectangle.
(i) If a statement is true or false but not both, it is a declarative sentence. "Where is your bag?" is a questionΒ in this context. As a result, it is not a statement. (ii) Every square is a...
Which of the following sentences are statements? Justify
(i) A triangle has three sides.
(ii) 0 is a complex number.
(i) A statement is a declarative sentence if it is either true or false but not both. Hence, it is a true statement (ii) If a statement is true or false but not both, it is a declarative sentence....
Find the term independent of $x$ in the expansion of $\left(3 x-\frac{2}{x^{2}}\right)^{15}$
Given function is $\left(3 x-\frac{2}{x^{2}}\right)^{15}$ We know from the standard formula of $T_{r+1}$ we get, $\mathrm{T}_{r+1}={ }^{15} C_{r}(3 x)^{15-r}\left(\frac{-2}{x^{2}}\right)^{r}={...
Find the equation of a circle which touches both the axes and the line 3x β 4y + 8 = 0 and lies in the third quadrant.
The equation of the given circle is \[{{x}^{2}}~+\text{ }{{y}^{2}}~+\text{ }4x\text{ }+\text{ }4y\text{ }+\text{ }4\text{ }=\text{ }0.\]
If the lines 3x β 4y + 4 = 0 and 6x β 8y β 7 = 0 are tangents to a circle, then find the radius of the circle.
Given lines are 6x β 8y + 8 = 0 and 6x β 8y β 7 = 0. Distance d between two parallel lines y = mx + c1Β and y = mx + c2Β is given by d = |C1βC2|/β(A2Β + B2Β ) These parallel lines are tangent to a...
Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is (A) symmetric but not transitive (B) transitive but not symmetric (C) neither symmetric nor transitive (D) both symmetric and transitive
The correct option is (B) transitive but not symmetric Given aRb β a is brother ofΒ b. This does not meanΒ bΒ is also a brother of a asΒ bΒ can be a sister ofΒ a. Therefore, R is not symmetric. aRb β a is...
. Let \[\mathbf{A}\text{ }=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\ldots \text{ }\mathbf{9} \right\}\] and R be the relation in A ΓA defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in \[A\text{ }\times A\]. Prove that R is an equivalence relation and also obtain the equivalent class \[\left[ \left( \mathbf{2},\text{ }\mathbf{5} \right) \right]\].
Given, \[\mathbf{A}\text{ }=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\ldots \text{ }\mathbf{9} \right\}\]and (a, b) R (c, d) if a + d = b + c for \[\left( a,\text{ }b...
Let R be relation defined on the set of natural number N as follows: \[\mathbf{R}\text{ }=\text{ }\{\left( \mathbf{x},\text{ }\mathbf{y} \right):\text{ }\mathbf{x}\in \mathbf{N},\text{ }\mathbf{y}\in \mathbf{N},\text{ }\mathbf{2x}\text{ }+\text{ }\mathbf{y}\text{ }=\text{ }\mathbf{41}\}\]. Find the domain and range of the relation R. Also verify whether R is reflexive, symmetric and transitive.
Given function: \[\mathbf{R}\text{ }=\text{ }\{\left( \mathbf{x},\text{ }\mathbf{y} \right):\text{ }\mathbf{x}\in \mathbf{N},\text{ }\mathbf{y}\in \mathbf{N},\text{ }\mathbf{2x}\text{ }+\text{...
. If A = \[\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4}\text{ } \right\}\],define relations on A which have properties of being: (a) reflexive, transitive but not symmetric (b) symmetric but neither reflexive nor transitive (c) reflexive, symmetric and transitive.
According to the question, \[A\text{ }=\text{ }\left\{ 1,\text{ }2,\text{ }3 \right\}\]. (i) Let \[{{R}_{1}}~=\text{ }\left\{ \left( 1,\text{ }1 \right),\text{ }\left( 1,\text{ }2 \right),\text{...
The domain of the function by f(x) = \[\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\sqrt{\left( \mathbf{x}\text{ }\text{ }\mathbf{1} \right)}\] is (a) \[\left[ \mathbf{1},\text{ }\mathbf{2} \right]\] (b) \[\left[ -\mathbf{1},\text{ }\mathbf{1} \right]\] (c) \[\left[ \mathbf{0},\text{ }\mathbf{1} \right]\] (d) none of these
The correct option is Β (a) \[\left[ \mathbf{1},\text{ }\mathbf{2} \right]\] We know that, \[si{{n}^{-1}}~x\] is defined for \[x\in \left[ -1,\text{ }1 \right]\] So, f(x) =...
Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{\sin 2x + 3x}}{{2x + \tan 3x}}$.
We are given, $\mathop {\lim }\limits_{x \to 0} \frac{{\sin 2x + 3x}}{{2x + \tan 3x}}$. Multiplying and dividing the numerator by $2x$ then, $\mathop {\lim }\limits_{x \to 0} \frac{{\sin 2x +...
Evaluate $\mathop {\lim }\limits_{x \to \frac{\pi }{6}} \frac{{\sqrt 3 \sin x – \cos x}}{{x – \frac{\pi }{6}}}$.
We are given, $\mathop {\lim }\limits_{x \to \frac{\pi }{6}} \frac{{\sqrt 3 \sin x - \cos x}}{{x - \frac{\pi }{6}}}$ Simplifying the numerator, $\sqrt 3 {\sin ^^\circ } - \cos x = 2\left(...
Evaluate $\mathop {\lim }\limits_{x \to \frac{\pi }{4}} \frac{{\sin x – \cos x}}{{x – \frac{\pi }{4}}}$.
We are given, $\mathop {\lim }\limits_{x \to \frac{\pi }{4}} \frac{{\sin x - \cos x}}{{x - \frac{\pi }{4}}}$ $\sin x - \cos x = \sqrt 2 \left( {\frac{{\sin x}}{{\sqrt 2 }} - \frac{{\cos x}}{{\sqrt 2...
Evaluate $\mathop {\lim }\limits_{x \to \frac{\pi }{3}} \frac{{\sqrt {1 – \cos 6x} }}{{\sqrt 2 \frac{\pi }{3} – x}}$.
We are given, $\mathop {\lim }\limits_{x \to \frac{\pi }{3}} \frac{{\sqrt {1 - \cos 6x} }}{{\sqrt 2 \frac{\pi }{3} - x}}$. Using the identity, $\cos 6x = 1 - 2{\sin ^2}3x$ to get, $\mathop {\lim...
Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{1 – \cos mx}}{{1 – \cos nx}}$.
We are given, $\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos mx}}{{1 - \cos nx}}$ Using the identities, $\cos mx = 1 - 2{\sin ^2}\frac{{mx}}{2}$ and $\operatorname{cosn} x = 1 - 2{\sin...
Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{1 – \cos 2x}}{{{x^2}}}$.
We are given, $\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos 2x}}{{{x^2}}}$. Using the identity, $\cos 2x = 1 - 2{\sin ^2}x$ to get, $\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos...
Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{{{\sin }^2}2x}}{{{{\sin }^2}4x}}$.
Β We are given, $\mathop {\lim }\limits_{x \to 0} \frac{{{{\sin }^2}2x}}{{{{\sin }^2}4x}}$ To simplify we have to multiply and divide the numerator and denominator by $\frac{{4{x^2}}}{{16{x^2}}}$ to...
Find $n$ if $\mathop {\lim }\limits_{x \to 2} \frac{{{x^n} – {2^n}}}{{x – 2}} = 80,n \in {\mathbf{N}}$.
We are given, $\mathop {\lim }\limits_{x \to 2} \frac{{{x^n} - {2^n}}}{{x - 2}} = 80,n \in {\mathbf{N}}$ Using the formula, $\mathop {\lim }\limits_{x \to a} \frac{{{x^n} - {a^n}}}{{x - a}} = n{a^{n...
Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {1 + {x^3}}Β – \sqrt {1 – {x^3}} }}{{{x^2}}}$.
We are given, $\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {1 + {x^3}}Β - \sqrt {1 - {x^3}} }}{{{x^2}}}$ Rationalize the numerator to get, $ = \mathop {\lim }\limits_{{\text{x}} \to 0}...
Evaluate $\mathop {\lim }\limits_{x \to 1} \frac{{{x^7} – 2{x^5} + 1}}{{{x^3} – 3{x^2} + 2}}$.
We solve the given limit by using L. Hospitalβs rule which is, If $\mathop {\lim }\limits_{{\text{x}} \to {\text{a}}} \frac{{{\text{f}}({\text{x}})}}{{{\text{g}}({\text{x}})}} = \frac{0}{0}$. Then,...
Evaluate $\mathop {\lim }\limits_{x \to 2} \frac{{{x^2} – 4}}{{\sqrt {3x – 2}Β – \sqrt {x + 2} }}$.
We are given, $\mathop {\lim }\limits_{x \to 2} \frac{{{x^2} - 4}}{{\sqrt {3x - 2}Β - \sqrt {x + 2} }}$ Rationalize the denominator to get, $ = \mathop {\lim }\limits_{x \to 2} \frac{{{x^2} -...
Evaluate $\mathop {\lim }\limits_{x \to 1} \frac{{{x^4} – \sqrt x }}{{\sqrt x – 1}}$.
We are given, $\mathop {\lim }\limits_{x \to 1} \frac{{{x^4} - \sqrt x }}{{\sqrt xΒ - 1}}$ Rationalize the denominator to get, $\mathop {\lim }\limits_{x \to 1} \frac{{{x^4} - \sqrt x }}{{\sqrt xΒ -...
Evaluate $\mathop {\lim }\limits_{x \to a} \frac{{{{(2 + x)}^{\frac{5}{2}}} – {{(a + 2)}^{\frac{5}{2}}}}}{{x – a}}$.
We are given, $\mathop {\lim }\limits_{x \to a} \frac{{{{(2 + x)}^{\frac{5}{2}}} - {{(a + 2)}^{\frac{5}{2}}}}}{{x - a}}$ $ = \mathop {\lim }\limits_{x \to a} \frac{{{{(2 + x)}^{\frac{5}{2}}} - {{(a...
Evaluate $\mathop {\lim }\limits_{x \to \frac{1}{2}} \frac{{4{x^2} – 1}}{{2x – 1}}$.
We are given, $\mathop {\lim }\limits_{x \to \frac{1}{2}} \frac{{4{x^2} - 1}}{{2x - 1}}$ $ = \mathop {\lim }\limits_{x \to \frac{1}{2}} \frac{{{{(2x)}^2} - 1}}{{2x - 1}}$ Simplifying to get, $ =...
Evaluate: $\mathop {\lim }\limits_{x \to 3} \frac{{{x^2} – 9}}{{x – 3}}$.
We are given, $\mathop {\lim }\limits_{x \to 3} \frac{{{x^2} - 9}}{{x - 3}}$ $ = \mathop {\lim }\limits_{x \to 3} \frac{{(x - 3)(x + 3)}}{{x - 3}}$ Applying the limits to get, $ = \mathop {\lim...
A straight line moves so that the sum of the reciprocals of its intercepts made on axes is constant. Show that the line passes through a fixed point.
If A = {x : x β W, x < 2}, B = {x : x β N, 1 < x < 5}, C = {3, 5} find(i) A Γ (B β© C)(ii) A Γ (B βͺ C)
Given, A = {x: x β W, x < 2}, B = {x : x βN, 1 < x < 5} C = {3, 5}; W is the set of whole numbers A = {x: x β W, x < 2} = {0, 1} B = {x : x βN, 1 < x < 5} = {2, 3, 4} (i) (Bβ©C) =...
Let A = {β1, 2, 3} and B = {1, 3}. Determine(i) B Γ B(ii) A Γ A
Given, A = {β1, 2, 3} and B = {1, 3} (i) B Γ B {1, 3} Γ{1, 3} So, B Γ B = {(1, 1), (1, 3), (3, 1), (3, 3)} As a result, the Cartesian product is {(1, 1), (1, 3), (3, 1), (3, 3)} (ii) A Γ A {β1, 2,...
Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universals set (s) for all the three sets A, B and C(i) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}(ii) {1, 2, 3, 4, 5, 6, 7, 8}
(i) A β {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B β {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} C β {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} As a result, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set...
Which of the following are examples of the null set(i) Set of odd natural numbers divisible by 2(ii) Set of even prime numbers
(i) Because odd numbers are not divisible by two, a set of odd natural numbers divisible by two is a null set. (ii) Because 2 is an even prime number, the set of even prime numbers is not a null...
Which of the following sets are finite or infinite(i) The set of months of a year(ii) 1, 2, 3 β¦
(i) Because there are 12 items in the set of months of a year, it is a finite set. (ii) Because it contains an unlimited amount of natural numbers, 1, 2, 3,... is an infinite set.
State whether the following set is finite or infinite: The set of circles passing through the origin (0, 0)
Because an infinite number of circles can pass through the origin (0, 0), the set of circles travelling through the origin is endless.
In the following, state whether A = B or not:(i) A = {2, 4, 6, 8, 10}; B = {x: x is positive even integer and x β€ 10}(ii) A = {x: x is a multiple of 10}; B = {10, 15, 20, 25, 30 β¦}
(i) A = {2, 4, 6, 8, 10}; B = {x: x is a positive even integer and x β€ 10} = {2, 4, 6, 8, 10} So, A = B (ii) A = {x: x is a multiple of 10} B = {10, 15, 20, 25, 30 β¦} We know that 15 β B but 15 β A....
From the sets given below, select equal sets:A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14}, D = {3, 1, 4, 2}, E = {β1, 1}, F = {0, a}, G = {1, β1}, H = {0, 1}
A = {2, 4, 8, 12}; B = {1, 2, 3, 4}; C = {4, 8, 12, 14} D = {3, 1, 4, 2}; E = {β1, 1}; F = {0, a} G = {1, β1}; H = {0, 1} We know, 8 β A, 8 β B, 8 β D, 8 β E, 8 β F, 8 β G, 8 β H A β B, A β D, A β ...
Solve for x, the inequalities in
Solution: Hence, \[\begin{array}{*{35}{l}} 1\text{ }\le \text{ }y\text{ }<\text{ }2Β \\ \Rightarrow ~1\text{ }\le \text{ }\left| x-\text{ }\text{ }2 \right|\text{ }<\text{ }2Β \\ \end{array}\]...
If arg (z β 1) = arg (z + 3i), then find x β 1 : y. where z = x + iy
Let \[z\text{ }=\text{ }x\text{ }+\text{ }iy\] Given that, \[arg\text{ }\left( z-\text{ }\text{ }1 \right)\text{ }=\text{ }arg\text{ }\left( z\text{ }+\text{ }3i \right)\] \[\Rightarrow ~arg\text{...
If the real part of ( zΜ + 2)/ ( zΜ β 1) is 4, then show that the locus of the point representing z in the complex plane is a circle.
Let z = x + iy Now, \[\Rightarrow ~{{x}^{2}}~+\text{ }x\text{ }\text{ }-2\text{ }+\text{ }{{y}^{2}}~=\text{ }4\text{ }\left( {{x}^{2}}~\text{ }-2x\text{ }+\text{ }1\text{ }+\text{ }{{y}^{2}}...
Prove 2 + 4 + 6 + β¦+ 2n = n^2 + n for all natural numbers n.
As indicated by the inquiry, \[P\left( n \right)\text{ }is\text{ }2\text{ }+\text{ }4\text{ }+\text{ }6\text{ }+\text{ }\ldots \text{ }+\text{ }2n\text{ }=\text{ }n^2\text{ }+\text{...
Prove 2n < (n + 2)! for all natural number n.
As indicated by the inquiry, \[P\left( n \right)\text{ }is\text{ }2n\text{ }<\text{ }\left( n\text{ }+\text{ }2 \right)!\] In this way, subbing various qualities for n, we get, \[P\left( 0...
Prove n(n^2 + 5) is divisible by 6, for each natural number n.
As per the inquiry, \[P\left( n \right)\text{ }=\text{ }n\left( n^2\text{ }+\text{ }5 \right)\] is distinguishable by 6. Along these lines, subbing various qualities for n, we get, \[P\left( 0...
Prove For any natural number n, x^n β y^n is divisible by x β y, where x integers with x β y.
As indicated by the inquiry, \[P\left( n \right)\text{ }=\text{ }xn\text{ }\text{ }yn\] is detachable by \[x\text{ }\text{ }y,\text{ }x\] whole numbers with\[x\text{ }\ne \text{ }y\] . In this way,...
Prove each of the statements in 4n β 1 is divisible by 3, for each natural number n.
As per the inquiry, \[P\left( n \right)\text{ }=\text{ }4n\text{ }\text{ }1\] is separable by 3. Thus, subbing various qualities for n, we get, \[P\left( 0 \right)\text{ }=\text{ }40\text{...
List all the elements of the following sets: (i) E = {$x: x$ is a month of a year not having 31 days} (ii) F = {$x: x$ is a consonant in the English alphabet which proceeds k}.
Solution: (i) E = {$x: x$ is a month of a year not having 31 days} As a result the elements are E = {February, April, June, September, November} (ii) F = {$x: x$ is a consonant in the English...
Write the following sets in roster form: (i) A = {$x: x$ is an integer and $β3 < x < 7$}. (ii) B = {$x: x$ is a natural number less than 6}.
Solution: (i) A = {$x: x$ is an integer and $β3 < x < 7$} β2, β1, 0, 1, 2, 3, 4, 5, and 6 are the only elements of the given set. As a result, the provided set can be written in the roster...
If in two circles, arcs of the same length subtend angles 60Β° and 75Β° at the centre, find the ratio of their radii
7. The following table shows the ages of the patients admitted in a hospital during a year:
Ages (in years):5 β 1515 β 2525 β 3535 β 4545 β 5555 β 65No of students:6112123145 Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.Β ...
6. Calculate the missing frequency from the following distribution, it being given that the median of the distribution is \[\mathbf{24}.\] \[\]
Solution: Let the unknown frequency be taken as x, Itβs given that Median \[=\text{ }24\] Then, median class = \[20\text{ }\text{ }30;\text{ }L\text{ }=\text{ }20,~h\text{ }=\text{ }30\text{...
3. Following is the distribution of I.Q of \[\mathbf{100}\] students. Find the median I.Q.
Solution: Here, we have \[N\text{ }=\text{ }100,\] \[So,\text{ }N/2\text{ }=\text{ }100/\text{ }2\text{ }=\text{ }50\] The cumulative frequency just greater than \[N/\text{ }2\text{ }is\text{...
Solve for$x:2x+7\ge 5x-14$, where x is a positive prime number.
From the question it is given that, $2x+7\ge 5x-14$ So, by transposing we get, $5x-2x\le 14+7$ $3x\le 21$ $x\le 21/3$ $x\le 7$ As per the condition given in the question, x is a positive prime...
If \[p=\{x:-3
\[PQ=\left\{ -2,1,0,1,\text{ }2,3,4,5,6,7 \right\}\left\{ -7,-6,-5,-4,-3,-2,-1,0,1,2 \right\}\] \[=\left\{ 3,4,5,6,7 \right\}\]
If \[p=\{x:-3
As per the condition given in the question, \[p=\{x:-3<x\le 7,x\in R\}\] So, \[P=\{-2,-1,0,1,2,3,4,5,6,7\}\] Then, \[Q=\{x:-7\le x<3,x\in R\}\] \[Q=\left\{ -7,-6,-5,-4,-3,-2,-1,0,1,2...
Solve for$x:x/4+3\le x/3+4$, where x is a negative odd number.
From the question it is given that, $x/4+3\le x/3+4$ So, by transposing we get, $x/4-x/3\le 4-3$ $(3x-4x)/12\le 1$ $-x\le 12$ $x\ge -12$ As per the condition given in the question, x is a negative...
Solve for $x:5x-14<18-3x,x\in w$
An equation between two variables that gives a straight line when plotted on a graph. From the question it is given that, $5x+3x<18-3x$ So, by transposing we get, $5x+3x<18+14$ $8x<32$...
Solve for $x:3-2x\ge x-12,X\in N$
From the question it is given that, $3-2x\ge x-12$ So, by transposing we get, $2x+x\le 12+3$ $3x\le 15$ $3x\le 15$ $x\le 15/3$ $x\le 5$ As per the condition given in the question, x β W. Therefore,...