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Mathematics Chapter 9 Sequences And Series

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NCERT Solution For Class 11 Mathematics

Sequences And Series Here is the CBSE Mathematics Chapter 9 for Class 11 students. Summary and detailed explanation of the lesson, including the definitions of difficult words. All of the exercises and questions and answers from the lesson's back end have been completed. NCERT Solutions for Class 11 Mathematics Sequences And Series Chapter 9 NCERT Solutions for Class 11 Mathematics Sequences And Series Chapter 9 The following is a summary in Hindi and English for the academic year 2021-2022. You can save these solutions to your computer or use the Class 11 Mathematics.

Question 1

Mathematics Chapter 9 Sequences And Series

NCERT Solution For Class 11 Mathematics

The x-axis and y-axis taken together determine a plane known as ____________.

The coordinates of points in the XY-plane are of the form __________.

Coordinate planes divide the space into _______ octants.

Let P (-2, 3, -4) is a point in space. Find the co-ordinates of the foot of perpendicular from this point to ZX-plane.

Let P (-2, 3, -4) is a point in space. Find the foot of perpendicular from this point to the z-axis.

Let P (-2, 3, -4) is a point in space. Find the the perpendicular distance of P from XY-plane.

Name the octants in which the following points lie:A (2, 3, 4), B (6, -3, 3), C (2, -1, -6), D (2, 2, -3), E (-1, 3, -6), F (-1, 3, 3), G (-3, -2,5) and H (-1,-2,-5).

Find the perpendicular distance of: (i) A (2, -3, 4) from XY-plane(ii) B (4, -3, 1) from ZX-plane(iii) C (3, -2, -1) from YZ-plane

The co-ordinates of a point A are (2, -3, 6). Write down the co-ordinates of seven points whose absolute values are the same as those of the co-ordinates of the given point.

The co-ordinates of a point P are (a, b, c) where a, b, c > 0. If the perpendicular distance of P from XY-plane is equal to half the sum of its perpendicular distances from ZX-plane and YZ-plane, find a relation between a, b and c.

Find the locus of a point P (x, y, z), where x > y > z > 0, such that its distance from ZX-plane exceeds 1/2 the sum of its distances from XY-plane and YZ-plane by 2.

Find the octant in which the following points lie:A (-3,1, 2), B (-3, 1, -2), C (-3, -1,-2), D (3, -1, -2) and E (3,1, 2).

A point lies on the x-axis at a distance of 5 units towards OX'. What are its co-ordinates?

A point lies on z-axis at a distance of 3 units in the direction OZ. Write its co-ordinates.

A point lies on y-axis and is at a distance of 4 units in the direction OY'. Write its co-ordinates.

A point lies on XY-plane with distances 4 units along OX and 3 units along OY. Write its co-ordinates.

A point lies on YZ-plane with distances 4 units along OY and 5 units along OZ'. Write its co-ordinates.

A point lies on ZX-plane with distances 3 units along OX' and 5 units along OZ'. Write its co-ordinates.

Find the perpendicular distance of point A (-3, 2, -4) from:(a) ZX-plane (b)XY-plane (c) YZ-plane.

The co-ordinates of a point A are (-2, 1, -7). Find the co-ordinates of 7 other points B, C, D, E, F, G and H whose co-ordinates have the same absolute values as those of the co-ordinates of point A.

The co-ordinates of a point H are (-2, 3, -4). Find(i) The co-ordinates of the foot of perpendicular from this point to ZX-plane.(ii) The co-ordinates of the foot of perpendicular from this point to the z-axis.(iii) The length of perpendicular from this point to XY-plane.

The co-ordinates of the foot of perpendicular from point P (a, b, c) to the x-axis are _________.

The co-ordinates of the foot of perpendicular from point (2, -3, 4) to ZX-plane are ____________.

The length of perpendicular from point P (-2, -3, -5) to XY-plane is ____________.

The co-ordinates of a point (a, b, c) on z-axis are of the type __________.

The co-ordinates of a point (a, b, c) on YZ-plane are of the type _________.

The co-ordinates of a point P are (x, y, z), where x, y, z > 0. If the length of perpendicular from it to ZX-plane is 1/2 its perpendicular distance from YZ-plane, then find a relation between x and y.

Find the distance between the following pairs of points:(2, -1, 3) and (-2, -1, 3)

Find the distance between the following pairs of points:(-1, 3, -4) and (1, -3, 4)

Show that the points (-2, 3, 5), (1, 2, 3) and (7, 0, -1) are collinear.

Verify the following : (0, 7, -10), (1, 6, -6) and ( 4, 9, -6) are the vertices of an isosceles triangle.

Show that A (a, b, c), B (b, c, a) and C (c, a, b) are vertices of an equilateral triangle.

Show that A (6, 10, 10), B (1, 0, -5) and C (6, -10, 0) are the vertices of a right angled triangle.

Verify the following:(0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of a right angled triangle.

Verify the following:(-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are the vertices of a parallelogram.

Verify the following:Show that A (6, 10, 10), B (1, 0, -5), C (6, -10, 0) and D (11, 0, 15) are the vertices of a rectangle.

Show that points A (1, 2, 3), B (-1, -2, -1), C (2, 3, 2) and D (4, 7, 6) are the vertices of a parallelogram ABCD, but it is not a rectangle.

Show that the points:P (-1,-6, 10), Q (1, -3, 4), R (-5, -1, 1) and S (-7, -4, 7) are the vertices of a rhombus.

Show that the points:A (-2, 6, -2), B (0, 4, -1), C (-2, 3, 1) and D (-4, 5, 0) are the vertices of a square.

Find the co-ordinates of a point equidistant from points A (a, 0, 0), B (0, b, 0), C (0, 0, c) and O (0, 0, 0).

Find the value of x, so that the point (6, 5, -3) is at a distance of 13 units from the point (x, -7, 0).

Find the co-ordinates of a point on the z-axis which is equidistant from points A (3, 2, 1) and B (5, 2, 5).

Find points on x - axis which are at a distance of units from point A (1, 2, 3).

Find the equation of the set of points which are equidistant from the points (1, 2, 3) and B (3, 2, -1).

Find the equation of set of points P the sum of whose distances from A (4, 0, 0) and B (-4, 0, 0) is equal to 10.

If A and B be the points (3, 4, 5) and (-1, 3, - 7) respectively, find the equation of set of points P such that PA2 + PB2 = 2k2. where k is a constant.

Find the equation of the locus of a point P or the equation of set of points P which moves so that its distance from A (1, 2, 3) is four times its distance from the YZ-plane.

Find the distance between the following pairs of points:(a) A (2, 3, 5), B (4, 3, 1) (b) A (-3, 7, 2), B (2, 4, - 1)

Show that the following sets of points are collinear.A (1,2, 3), B (4, 0, 4), C (-2, 4, 2)

Show that the following sets of points are collinear.P (2, - 4, 1), B (4, 4, 3), C (3, 0, 2)

Show that the following sets of points are collinear.A (6, - 7, - 1), B (2, - 3, 1), C (4, - 5, 0)

Show that the following sets of points are collinear.A (1, - 1, 3), B (2, - 4, 5), C(5, - 13, 11).

Show that the following sets of points are collinear.A (3, - 2, 4), B (1, 0, - 2), C (- 1, 2, - 8)

Show that the points A (0, 1, 2), B (2, - 1, 3) and C (1, - 3, 1) are the vertices of an isosceles right angled triangle.

Find the co-ordinates of a point equidistant from four points.O (0, 0, 0), A (2p, 0, 0), B (0, 2q, 0) and C (0, 0, 2r). Find also the distance.

Find the co-ordinates of a point which is equidistant from four points P (1, 0, 0), Q (0, 2, 0), R (0, 0, - 3) and O (0, 0, 0). Find also the distance.

Find the value of k, so that the distance between points (7, - 3, 1) and (4, k, 5) is 13 units.

Show that the points A (0, 7 - 10), B (1, 6, - 3) and C (4, 9, - 6) form the vertices of an isosceles triangle.

Show that the points A (2, 3, 4), B (3, 4, 2) and C (4, 2, 3) form the vertices of an equilateral triangle.

Show that the points A (-1, 2, 1), B (1, -2, 5), C (4, - 7, 8) and D (2, - 3, 4) form the vertices of a parallelogram.

Prove that the four points A (1,3,0), B (- 5, 5, 2), C (- 9, - 1, 2) and D (- 3, - 3, 0) taken in order, form the vertices of a parallelogram. Do they form a rectangle also?

Show that the points A (0, 7, 10), B (- 1, 6, 6), C (- 4, 9, 6) and D (- 3, 10, 10) form the vertices of a rectangle.

Find points on the y-axis which are at a distance of 7 units from the point (-2, 1, 3).

Verify that points A (-1, - 6, 10), B (1, - 3, 4), C (- 5, - 1, 1) and D (- 7, - 4, 7) are the vertices of a rhombus.

Find a point on x-axis which is equidistant from points A (4, - 1, 2) and B (2, 0, 3).

Show that the following four points form the vertices of a square:(a) (0, 4, 1), (2, 3, -1), (4, 5, 0) and (2, 6, 2)(b) (- 2, 6, - 2), (0, 4, - 1), (- 2, 3, 1) and (- 4, 5, 0)

Find the equation of locus of a point which moves so that its distance from point A (- 2, 1, 3) is twice its distance from the XY plane.

Find the equation of locus of a point so that it is equidistant from the points A (1, 3, - 1) and B(4, -1, 7).

Given points A (- 2, 0, 4), B (3, 2, - 1). Find the equation of locus of a point P so that PA2 - PB2 = 20.

Find the coordinates of the point which divides the line segment formed by joining the points (- 2, 3, 5) and (1, -4, 6) in the ratio 2 : 3 internally.

Find the co-ordinates of the point which divides the line segment formed by joining the points (- 2, 3, 5) and (1, -4, 6) in the ratio of 2 : 3 externally.

Given that P (3, 2, -4), Q (5, 4, -6) and R (9, 8, -10) are collinear. Find the ratio in which Q divides PR.

Using section formula, prove that the three points (-4, 6, 10), (2, 4, 6) and (14, 0, - 2) are collinear. Also find the ratio in which point B divides the join of A and C.

Find the co-ordinates of the points of trisection of the line segment joining the points P (3, 2, - 1) and Q (1, 2, 5).

Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8).

Find the ratio in which the join of P (2, 1, -1) and Q (3, 2, 4) is divided by XY-plane.

A point R with x-coordinate 4 lies on the line joining the points P (2, - 3, 4) and Q (8, 0, 10). Find the co-ordinates of point R.

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and C (6, 0, 0).

Name the octants in which the following points lie:
A (2, 3, 4), B (6, -3, 3), C (2, -1, -6), D (2, 2, -3), E (-1, 3, -6), F (-1, 3, 3), G (-3, -2,5) and H (-1,-2,-5).

Find the perpendicular distance of:
(i) A (2, -3, 4) from XY-plane
(ii) B (4, -3, 1) from ZX-plane
(iii) C (3, -2, -1) from YZ-plane

Find the locus of a point P (x, y, z), where x > y > z > 0, such that its distance from ZX-plane exceeds ¹/₂ the sum of its distances from XY-plane and YZ-plane by 2.

Find the octant in which the following points lie:
A (-3,1, 2), B (-3, 1, -2), C (-3, -1,-2), D (3, -1, -2) and E (3,1, 2).

Find the perpendicular distance of point A (-3, 2, -4) from:
(a) ZX-plane (b)XY-plane (c) YZ-plane.

The co-ordinates of a point H are (-2, 3, -4). Find
(i) The co-ordinates of the foot of perpendicular from this point to ZX-plane.
(ii) The co-ordinates of the foot of perpendicular from this point to the z-axis.
(iii) The length of perpendicular from this point to XY-plane.

The co-ordinates of a point P are (x, y, z), where x, y, z > 0. If the length of perpendicular from it to ZX-plane is ¹/₂ its perpendicular distance from YZ-plane, then find a relation between x and y.

Find the distance between the following pairs of points:
(2, -1, 3) and (-2, -1, 3)

Find the distance between the following pairs of points:
(-1, 3, -4) and (1, -3, 4)

Verify the following:
(0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of a right angled triangle.

Verify the following:
(-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are the vertices of a parallelogram.

Verify the following:
Show that A (6, 10, 10), B (1, 0, -5), C (6, -10, 0) and D (11, 0, 15) are the vertices of a rectangle.

Show that the points:
P (-1,-6, 10), Q (1, -3, 4), R (-5, -1, 1) and S (-7, -4, 7) are the vertices of a rhombus.

Show that the points:
A (-2, 6, -2), B (0, 4, -1), C (-2, 3, 1) and D (-4, 5, 0) are the vertices of a square.

Find points on x - axis which are at a distance of $square root of 29$ units from point A (1, 2, 3).

If A and B be the points (3, 4, 5) and (-1, 3, - 7) respectively, find the equation of set of points P such that PA² + PB² = 2k². where k is a constant.

Find the distance between the following pairs of points:
(a) A (2, 3, 5), B (4, 3, 1)
(b) A (-3, 7, 2), B (2, 4, - 1)

Show that the following sets of points are collinear.
A (1,2, 3), B (4, 0, 4), C (-2, 4, 2)

Show that the following sets of points are collinear.
P (2, - 4, 1), B (4, 4, 3), C (3, 0, 2)

Show that the following sets of points are collinear.
A (6, - 7, - 1), B (2, - 3, 1), C (4, - 5, 0)

Show that the following sets of points are collinear.
A (1, - 1, 3), B (2, - 4, 5), C(5, - 13, 11).

Show that the following sets of points are collinear.
A (3, - 2, 4), B (1, 0, - 2), C (- 1, 2, - 8)

Find the co-ordinates of a point equidistant from four points.
O (0, 0, 0), A (2p, 0, 0), B (0, 2q, 0) and C (0, 0, 2r). Find also the distance.

Show that the following four points form the vertices of a square:
(a) (0, 4, 1), (2, 3, -1), (4, 5, 0) and (2, 6, 2)
(b) (- 2, 6, - 2), (0, 4, - 1), (- 2, 3, 1) and (- 4, 5, 0)

Given points A (- 2, 0, 4), B (3, 2, - 1). Find the equation of locus of a point P so that PA² - PB² = 20.

Find the co-ordinates of a point R which divides the join of P ((x₁,y₁,z₁) and Q (x₂, y₂, z₂) externally in the ratio 1 : 2. Verify that P is mid-point of QR.

Using section formula, show that points A (2, -3, 4), B (-1, 2, 1) and $straight C space open parentheses 0 comma space 1 third comma space 2 close parentheses$ are collinear.

Find the octant in which following points lie:
(i) (1, 4, -2) (ii) (2, 1, 4) (iii) (-2, -1, 5) (iv) (1, -3, -1) (v) (2, -4, 3)

Using distance formula, show that the following sets of points are collinear.
(i) (-1, 4, -2), (2, -2, 1), (0, 2, -1) (ii) (3, -5, 1), (-1, 0, 8), (7, - 10. - 6).

Using section formula, show that the following sets of points are collinear:
(i) (4, 5, -5), (0, -11, 3), (2, -3, -1) (ii) (-2, 3, 5), (1, 2, 3), (7, 0, -1).

Find the co-ordinates of a point on y-axis which is at distance of $5 square root of 2$ units from point P(3, -2, 5).

A (1, 2, -1) and B (0, -3, 4) are two points. Find the locus of a point P so that PA² - PB² = 27.

Show that point A (5, 4, 2), B (-1, -2, 4) and $straight C space open parentheses 13 over 5 comma space 8 over 5 comma space 14 over 5 close parentheses$ are collinear. Find also the ratio in which C divides the join of A and B.

The number of common tangent to the circles x²+y²-4x-6y-12=0 and x²+y²+6x+18y+26 = 0 is

1

2

3

4

The angle between the lines whose direction cosines satisfy the equations l +m+n=0 and l² = m²+n² is

π/3

π/4

π/6

π/2

If PS is the median of the triangle with vertices P(2,1), Q(6,-1) and R (7,3), then equation of the line passing through (1,-1) and parallel to PS is

4x-7y - 11 =0

2x+9y+7=0

4x+7y+3 = 0

2x-9y-11 =0

Let a,b,c and d be non-zero numbers. If the point of intersection of the lines 4ax +2ay +c= 0 and 5bx +2by +d = 0 lies in the fourth quadrant and is equidistant from the two axes, then

2bc-3ad =0

2bc+3ad =0

2ad-3bc =0

3bc+2ad=0

The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as (0, 1) (1, 1) and (1, 0) is

$2 plus square root of 2$
$2 minus square root of 2$
$1 plus square root of 2$
$1 minus square root of 2$

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector.Then the maximum area (in sq. m) of the flower-bed, is

30

12.5

10

25

The sum to infinity of the series $1 space plus space 2 over 3 space plus space 6 over 3 squared space plus space 10 over 3 cubed space plus space 14 over 3 to the power of 4 space plus.... space is$

2

3

4

6

If p and q are positive real numbers such that p² + q² = 1, then the maximum value of (p + q) is

2

1/2

$fraction numerator 1 over denominator square root of 2 end fraction$
$square root of 2$

If the coefficients of rth, (r+ 1)th and (r + 2)th terms in the binomial expansion of (1 + y)^m are in A.P., then m and r satisfy the equation

m² – m(4r – 1) + 4r² – 2 = 0

m² – m(4r+1) + 4r² + 2 = 0

m² – m(4r + 1) + 4r² – 2 = 0

m² – m(4r – 1) + 4r² + 2 = 0

If the letters of word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

601

600

602

603

If in a triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sin B, sin C are in

G.P.

A.P.

Arithmetic − Geometric Progression

H.P.

If non-zero numbers a, b, c are in H.P., then the straight line $straight x over straight a space plus straight y over straight b space plus 1 over straight c space equals space 0$ always passes through a fixed point. That point is

(-1, 2)

(-1, -2)

(1, -2)

(1, 1/2)

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

x² + 18x +16 = 0

x²-18x-16 = 0

x²+18x-16 =0

x²-18x +16 =0

Let T be the rth term of an A.P. whose first term is a and the common difference is d. If for r some positive integers m, n, m ≠ n, T_m = 1/n and T_n = 1/m, then a-b equals

0

1

1/mn

$1 over straight m plus 1 over straight n$