Category: Class 11 Mathematics

NCERT Exemplar Class 11 Maths Chapter 11 Conic Sections of NCERT Exemplar Class 11 Maths. Here we have given NCERT Exemplar Class 11 Maths Chapter 11 Conic Sections.

NCERT Exemplar Class 11 Maths Chapter 11 Conic Sections

Short Answer Type Questions

Conic Sections Class 11 Important Questions NCERT

Q1. Find the equation of the circle which touches the both axes in first quadrant and whose radius is a.
Sol:

Q2. Show that the point (x, y) given by  \(x=\frac { 2at }{ 1+{ t }^{ 2 } }   \) and \(y=\frac { 1-{ t }^{ 2 } }{ 1+{ t }^{ 2 } }  \) lies on a circle .

NCERT Exemplar Class 11 Maths Conic Sections

Q3. If a circle passes through the point (0, 0) (a, 0), (0, b) then find the coordinates of its centre.
Sol: We have circle through the point A(0, 0), B(a, 0) and C(0, b).
Clearly triangle is right angled at vertex A.

So, centre of the circle is the mid point of hypotenuse BC which is (a/2, b/2)

Conic Sections Class 11 Exemplar NCERT

Q4. Find the equation of the circle which touches x-axis and whose centre is (1,2).
Sol: Given that, circle with centre (1,2) touches x-axis.
Radius of the circle is, r = 2
So, the equation of the required circle is:
(x – l)² + (y – 2)² = 2²
=>x²-2x + 1 + y²-4y + 4 = 4
=> x² + y² – 2x-4y + 1 = 0

Q5. If the lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangents to a circle, then find the radius of the circle.
Sol: Given lines are 6x – 8y + 8 = 0 and 6x – 8y – 7 = 0.
These parallel lines are tangent to a circle.

Q6. Find the equation of a circle which touches both the axes and the line 3x – 4y + 8 = 0 and lies in the third quadrant.

NCERT Exemplar Class 11 Conic Sections Solutions

Q7. If one end of a diameter of the circle x² + y² -4x -6y + 11 = 0 is (3,4), then find the coordinate of the other end of the diameter.
Sol: Given equation of the circle is:

Q8. Find the equation of the circle having (1, -2) as its centre and passing through 3x +y= 14, 2x + 5y = 18.

Important Questions Of Conic Sections Class 11

Q9. If the line y= √3 x + k touches the circle x² + y² = 16, then find the value of
Sol: Given line is y = √3 x + k and the circle is x² + y² = 16.

Q10. Find the equation of a circle concentric with the circle x² +y² – 6x + 12y + 15 = 0 and has double of its area.

Conic Sections Class 11 Extra Questions NCERT

Q11. If the latus rectum of an ellipse is equal to half of minor axis, then find its eccentricity.

Q12. Given the ellipse with equation 9X² + 25y² = 225, find the eccentricity and foci.

NCERT Exemplar Class 11 Maths Chapter 11 Solutions

Q13. If the eccentricity of an ellipse is 5/8 and the distance between its foci is 10, then find latus rectum of the ellipse.

Class 11 Conic Sections Extra Questions NCERT

Q14. Find the equation of ellipse whose eccentricity is 2/3, latus rectum is 5 and thecentre is (0, 0).

Class 11 Maths Chapter 11 Extra Questions NCERT

Q15. Find the distance between the directrices of the ellipse \(\frac { { x }^{ 2 } }{ 36 } +\quad \frac { { y }^{ 2 } }{ 20 } \quad =\quad 1  \)

Q16. Find the coordinates of a point on the parabola y² = 8x whose focal distance is 4.

Q17. Find the length of the line-segment joining the vertex of the parabola y² = 4ax and a point on the parabola where the line-segment makes an angle 6 to the x-axis.
Sol: Given equation of the parabola isy² = 4ax.
Let the point on the parabola be P(x₁,,y₁).

Q18. If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.

Q19. If the line y = mx + 1 is tangent to the parabola y² = 4x then find the value of m.

Q20. If the distance between the foci of a hyperbola is 16 and its eccentricity is √2, then obtain the equation of the hyperbola.

Q21. Find the eccentricity of the hyperbola 9y² – 4x² =36
Sol: We have the hyperbola:9y² – 4x² = 36

Q22. Find the equation of the hyperbola with eccentricity 3/2 and foci at (±2, 0).

Long Answer Type Questions

Q23. If the lines 2x – 3y = 5 and 3x-4y = 7 are the diameters of a circle of area 154 square units, then obtain the equation of the circle.
Sol: Given that lines 2x – 3y – 5 = 0 and 3x – 4y -1 = 0 are diameters of the circle. Solving these lines we get point of intersection as (1, -1), which is centre of the circle.

Q24. Find the equation of the circle which passes through the points (2, 3) and (4, 5) and the centre lies on the straight line y – 4x + 3 = 0.

Q25. Find the equation of a circle whose centre is (3, -1) and which cuts off a chord of length 6 units on the line 2x — 5y+ 18 = 0.

Sol: Given centre of the circle 0(3, -1)
Chord of the circle is AB.

Q26. Find the equation of a circle of radius 5 which is touching another circle x² + y² – 2x – 4y – 20 = 0 at (5, 5).

Q27. Find the equation of a circle passing through the point (7, 3) having radius 3 units and whose centre lies on the line y = x -1.
Sol: Given that circle passes through the point A(7, 3) and its radius is 3.

Q28. Find the equation of each of the following parabolas.
(i) Directrix, x = 0, focus at (6, 0)
(ii) Vertex at (0,4), focus at (0, 2)
(iii) Focus at (-1, -2), directrix x – 2y + 3 = 0
Sol: We know that the distance of any point on the parabola from its focus and its directrix is same.
(i) Given that, directrix, x = 0 and focus = (6, 0)
So, for any point P(x, y) on the parabola
Distance of P from directrix = Distance of P from focus =>  x² = (x — 6)² + y²
=>         y²– 12x + 36 = 0
(ii) Given that, vertex = (0,4) and focus = (0, 2)
Now distance between the vertex and directrix is same as the distance between the vertex and focus.
Directrix is y – 6 = 0
For any point of P(x, y) on the parabola
Distance of P from directrix = Distance of P from focus

Q29. Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12.

Q30. Find the equation of the set of all points whose distance from (0,4) are 2/3 of their distance from the line y = 9.

Q31. Show that the set of all points such that the difference of their distances from (4, 0)and (-4, 0) is always equal to 2 represent a hyperbola.

True/False Type Questions

Q33. The line x + 3y = 0 is a diameter of the circle x² + y² + 6x + 2y = 0.
Sol: False
Given equation of the circle is x² + y² + 6x + 2y = 0
Centre = (-3, -1)
Clearly, it does not lie on the line x + 3y = 0 as -3 + 3(-l) = -6.
So, this line is not diameter of the circle.

Q34. The shortest distance from the point (2, -7) to the circle x +y² – 1 4jc – lOy- 151 = 0 is equal to 5.
Sol: False

Q35. If the line lx + my = 1 is a tangent to the circle x² + y² = a², then the point (1, m) lies on a circle.

Q36. The point (1,2) lies inside the circle x² + y² – 2x + 6y + 1 = 0.
Sol: False

Q37. The line lx+ my + n = 0 will touch the parabola^² = 4 ax if In = am².
Sol: True




Fill in the Blanks Type Questions

Q41. The equation of the circle having centre at (3, -4) and touching the line 5x + 12y- 12 = 0 is ______.

Q42. The equation of the circle circumscribing the triangle whose sides are the lines y = x + 2, 3y = 4x, 2y = 3x is   _______.
Given equation of line are:

Q43. An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the length of the string and distance between the pins are _____ .

Q44. The equation of the ellipse having foci (0,1), (0, -1) and minor axis of length 1 is ___ .

Q45. The equation of the parabola having focus at (-1, -2) and the directrix x – 2y + 3 = 0 is______ .
Sol: Given that, focus at S(-l, -2) and directrix is x – 2y + 3 = 0

Q46. The equation of the hyperbola with vertices at (0, ±6) and eccentricity 5/3 ________ and its foci are _____    .

Objective Type Questions

Q47. The area of the circle centred at (1,2) and passing through (4, 6) is

Q48. Equation of a circle which passes through (3, 6) and touches the axes is

Q49. Equation of the circle with centre on the j-axis and passing through the origin and the point (2, 3) is

Q50. The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a is

Q51. If the focus of a parabola is (0, -3) and its directrix is y = 3, then its equation is

Q52. If the parabola y² = 4ax passes through the point (3, 2), then the length of its latus rectum is

Q53. If the vertex of the parabola is the point (-3, 0) and the directrix is the line x + 5 = 0, then its equation is
(a) y² = 8(x + 3)
(b) x² = 8(y + 3)
(c) y² = -8(x + 3)
(d) y² = 8(x + 5)

Q54. The equation of the ellipse whose focus is (1, -1), the directrix the line x-y-3 = 0 and eccentricity 1/2 is

Q55. The length of the latus rectum of the ellipse 3x² +y² = 12 is
(a) 4       
(b) 3       
(c) 8       
(d) 4/√3

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We hope the NCERT Exemplar Class 11 Maths Chapter 11 Conic Sections help you. If you have any query regarding NCERT Exemplar Class 11 Maths Chapter 11 Conic Sections, drop a comment below and we will get back to you at the earliest.

NCERT Exemplar Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry are part of NCERT Exemplar Class 11 Maths. Here we have given NCERT Exemplar Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry.

NCERT Exemplar Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry

Short Answer Type Questions

Q1. Locate the following points:

(i) (1,-1, 3),
(ii) (-1,2,4)               
(iii) (-2, -4, -7)
(iv) (-4,2, -5)
Sol: Given, coordinates
(i) (1,-1, 3),
(ii) (-1,2,4)
(iii) (-2, -4, -7)
(iv) (-4,2, -5)

Q2. Name the octant in which each of the following points lies.
(i) (1,2,3)             
(ii) (4,-2, 3)             
(iii) (4,-2,-5)              
(iv)(4,2,-5)
(v) (-4,2,5)           
(vi) (-3,-1,6)           
(vii) (2,-4,-7)
(viii) (-4, 2,-5)

Q3. Let A, B, C be the feet of perpendiculars from a point P on the x, y,z-axes respectively. Find the coordinates of A, B and C in each of the following where the point P is:
(i) (3,4,2)             
(ii) (-5,3,7)              
(iii) (4,-3,-5)
Sol: We know that, on x-axis, y, z = 0, on y-axis, x, z = 0 and on z-axis, x,y = 0. Thus, the feet of perpendiculars from given point P on the axis are as follows.

(i) A(3,0,0),5(0,4,0),C(0,0,2)
(ii) A(-5, 0, 0), B(0, 3, 0), C(0, 0, 7)
(iii) A(4, 0, 0), 5(0, -3, 0), C(0,0, -5)

Q4. 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) (-5,3,7)
(iii) (4,-3,-5).
Sol: We know that, on xy-plane z = 0, on yz-plane, x = 0 and on zx-plane, y = 0. Thus, the coordinates of feet of perpendicular on the xy, yz and zx-planes from the given point are as follows:
(i) A(3,4,0), 5(0,4, 5), C(3,0,5)
(ii) A(-5, 3,0), 5(0, 3, 7), C(-5, 0, 7)
(iii) A(4, -3, 0), 5(0, -3, -5), C(4,0, -5)

Q5. How far apart are the points (2,0, 0) and (-3, 0, 0)?
Sol: Given points are A (2, 0, 0) and 5(-3,0, 0).
AB = |2 – (-3)| = 5

Q6. Find the distance from the origin to (6, 6, 7).

Q8. Show that the point ,4(1, -1, 3), 6(2, -4, 5) and (5, -13, 11) are collinear.

Q9. Three consecutive vertices of a parallelogram ABCD are .4(6, -2,4), 6(2,4, -8), C(-2, 2, 4). Find the coordinates of the fourth vertex.

Q10 .Show that the triangle ABC with vertices .4(0,4,1), 6(2,3, -1) and C(4, 5,0) is right angled.
Sol: The vertices of ∆ABC are A(0,4, 1), 5(2, 3, -1) and C(4, 5, 0).

Q11. Find the third vertex of triangle whose centroid is origin and two vertices are (2,4,6) and (0, -2, -5).
Sol: Let the third or unknown vertex of ∆ABC be A(x, y, z).
Other vertices of triangle are 5(2,4, 6) and C(0, -2, -5).
The centroid is G(0, 0, 0).

Q12. Find the centroid of a triangle, the mid-point of whose sides are D (1,2, – 3), E(3,0, l)and F(-l, 1,-4).
Sol: Given that, mid-points of sides of AABC are D(l, 2, -3), E(3, 0, 1) and F(-l, 1,-4).

Q14. Three vertices of a Parallelogram ABCD are A(\, 2, 3), B(-A, -2, -1) and C(2, 3, 2). Find the fourth vertex
Sol: Let the fourth vertex of the parallelogram D(x, y, z).
Mid-point of BD

Q15. Find the coordinate of the points which trisect the line segment joining the points .A(2, 1, -3) and B(5, -8, 3).

Q16. If the origin is the centroid of a triangle ABC having vertices A(a, 1, 3), B(-2, b, -5) and C(4, 7, c), find the values of a, b, c.
Sol: Vertices of AABC are A(a, 1, 3), B(-2, b, -5) and C(4, 7, c).
Also, the centroid is G(0, 0, 0).

Q17. Let A(2, 2, -3), 5(5, 6, 9) and C(2, 7, 9) be the vertices of a triangle. The internal bisector of the angle A meets BC at the point Find the coordinates of D.

Long Answer Type Questions

Q18. Show that the three points A(2, 3, 4), 5(-l, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which Cdivides
Sol: Given points are A(2, 3, 4), B(-1, 2, -3) and C(-4,1,-10)

Q19. The mid-point of the sides of a triangle are (1, 5, -1), (0,4, -2) and (2, 3,4). Find its vertices. Also, find the centroid of the triangle.

Q20. Prove that the points (0, -1, -7), (2, 1, -9) and (6, 5, -13) are collinear. Find the ratio in which the first point divides the join of the other two.
Sol: Given points are 4(0, -1, -7), 8(2, 1, -9) and C(6, 5, -13).

Q21. 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?
Sol: The coordinate of the cube whose edge is 2 units, are:
(2, 0, 0), (2,2, 0), (0, 2, 0), (0, 2,2), (0, 0,2), (2,0, 2), (0, 0, 0) and (2,2, 2)

Objective Type Questions

Q22. The distance of point P(3,4, 5) from the yz-plane is
(a) 3 units
(b) 4 units
(c) 5 units
(d) 550
Sol: (a) Given point is P{3,4, 5).
Distance of P from yz-plane = |x coordinate of P| = 3

Q23. What is the length of foot of perpendicular drawn from the point P(3,4, 5) on y-axis?

Q24. Distance of the point (3,4, 5) from the origin (0, 0, 0) is

Q25. If the distance between the points (a,0,1) and (0,1,2) is √27, then the value of a is
(a)     5                      
(b)     ± 5                   
(c)     -5                    
(d)   none of these

Q26. x-axis is the intersection of two planes
(a) xy and xz                                            
(b) yz and zx
(c) xy and yz                                            
(d) none of these
Sol: (a) We know that, on the xy and xz-planes, the line of intersection is x-axis.

Q27. Equation of Y-axis is considered as
(a) x = 0, y = 0                                         
(b) y = 0, z = 0
(c) z = 0, x = 0                                         
(d) none of these
Sol:(c) On the j-axis, x = 0 and z = 0.

Q28. The point (-2, -3, -4) lies in the
(a) First octant                                        
(b) Seventh octant
(c) Second octant                                   
(d) Eighth octant
Sol: (b) The point (-2, -3, -4) lies in seventh octant.

Q29. A plane is parallel to yz-plane so it is perpendicular to
(a) x-axis               
(b) y-axis                 
(c) z-axis                 
(d) none of these
Sol: (a) A plane parallel to yz-plane is perpendicular to x-axis.

Q30. The locus of a point for which y = 0, z = 0 is
(a)    equation of x-axis                         
(b)    equation of y-axis
(c)     equation at z-axis                         
(d)    none of these
Sol: (a) We know that, equation of the x-axis is: y = 0, z = 0 So, the locus of the point is equation of x-axis.

Q31. The locus of a point for which x = 0 is
(a)    xy-plane                                          
(b)    yz-plane
(c)     zx-plane                                        
 (d)    none of these
Sol: (b) On the yz-plane, x = 0, hence the locus of the point is yz-plane.

Q32. If a parallelepiped is formed by planes drawn through the points (5,8,10) and (3, 6, 8) parallel to the coordinate planes, then the length of diagonal of the parallelepiped is

Q33. L is the foot of the perpendicular drawn from a point P(3, 4, 5) on the xy-plane. The coordinates of point L are
(a)    (3,0,0)                                              
(b)    (0,4,5)
(c)     (3, 0, 5)                                            
(d)    none of these
Sol: (d) We know that on the xy-plane, z = 0.
Hence, the coordinates of the points L are (3,4, 0).

Q34. L is the foot of the perpendicular drawn from a point (3, 4, 5) on x-axis. The coordinates of L are
(a)    (3,0,0)                                              
(b)    (0,4,0)
(c)     (0, 0, 5)                                            
(d)    none of these
Sol: (a) On the x-axis, y = 0 and z = 0.
Hence, the required coordinates are (3, 0,0).

Fill in the Blanks Type Questions
Q35. The three axes OX, OY, OZ determine______ .
Sol: The three axes OX, OY and OZ determine three coordinate planes.

Q36. The three planes determine a rectangular parallelepiped which has____ of rectangular faces.

Q37. The coordinates of a point are the perpendicular distance from the _____ on the respective axes.
Sol: Given points

Q38. The three coordinate planes divide the space into _________parts.
Sol: Eight

Q39. If a point P lies in yz-plane, then the coordinates of a point on yz-plane is of the form_______.
Sol: We know that, on yz-plane, x = 0.So, the coordinates of the required point are (0, y, z).

Q40. The equation of yz-plane is ______ .
Sol: On yz-plane for any point x-coordinate is zero.
So, yz-plane is locus of point such that x = 0, which is its equation.

Q41. If the point P lies on z-axis, then coordinates of P are of the form_____.
Sol: On the z-axis, x = 0 and y = 0.
So, the required coordinates are of the form (0, 0, z).

Q42. The equation of z-axis, are ______.
Sol: Any point on the z-axis is taken as (0, 0, z).
So, for any point on z-axis, we have x = 0 and y = 0, which together represents its equation.
Q43. A line is parallel to xy-plane if all the points on the line have equal_________.
Sol: A line is parallel to xy-plane if each point P(x, y, z) on it is at same distance from xy-plane.
Distance of point P from xy plane is ‘z’
So, line is parallel to xy-plane if all the points on the line have equal z-coordinate.

Q44. A line is parallel to x-axis if all the points on the line have equal ______.
Sol: A line is parallel to x-axis if each point on it maintains constant distance from y-axis and z-axis.
So, each point has equal y and z-coordinates. .

Q45. x = a represents a plane parallel to .
Sol: Locus of point P(x, y, z) is x = a.
Therefore, each point P has constant x-coordinate.
Now, x is distance of point P from yz-plane.
So, here plane x = a is at constant distance ‘a’ from yz-plane and parallel to _yz-plane.

Q46. The plane parallel to yz-plane is perpendicular to_____ .
Sol: The plane parallel to yz-plane is perpendicular to x-axis.

Q47. The length of the longest piece of a string that can be stretched straight in a  rectangular room whose dimensions are 10, 13 and 8 units are______ .
Sol: Given dimensions are: a = 10, 6=13 andc = 8.
Required length of the string = yja² + b² + c² = ^100 + 169 + 64 = -7333

Q48. If the distance between the points (a, 2,1) and (1,-1,1) is 5, then a_______ .
Sol: Given points are (a, 2,1) and (1,-1,1).

Q49. If the mid-points of the sides of a triangle AB; BC; CA are D(l, 2, – 3), E( 3, 0, 1) and F(-l, 1, -4), then the centroid of the triangle ABC is________ .
Sol: Given that, mid-points of sides of AABC are D( 1, 2, -3), E(3, 0, 1) and F(-l, 1,-4).

Matching Column Type Questions

Q50. Match each item given under the column C₁ to its correct answer given under column C₂.

Column C,		Column C2
(a)	In xy-plane	(i)	1st octant
(b)	Point (2, 3,4) lies in the	(ii)	vz-plane
(c)	Locus of the points having x coordinate 0 is	(iii)	z-coordinate is zero
(d)	A line is parallel to x-axis if and only	(iv)	z-axis                      .
(e)	If x = 0, y = 0 taken together will represent the	(v)	plane parallel to xy-plane
(f)	z = c represent the plane	(vi)	if all the points on the line have equal y and z-coordinates.
(g)	Planes x = a, y = b represent the line	(vii)	from the point on the respective axis.
00	Coordinates of a point are the distances from the origin to the feet of perpendiculars	(viii)	parallel to z-axis
(i)	A ball is the solid region in the space	(ix)	disc
G)	Region in the plane enclosed by a circle is known as a	00	sphere

 

Sol: (a) In xy-plane, z-coordinate is zero.
(b) The point (2, 3,4) lies in 1st octant.
(c) Locus of the points having x-coordinate zero is yz-plane.
(d) A line is parallel to x-axis if and only if all the points on the line have equal y and z-coordinates.
(e)x = 0, y = 0 represent z-axis
(f) z = c represents the plane parallel to xy-plane.

(g) The plane x = a is parallel to yz-plane.
Plane y = b is parallel to xz-plane.
So,    planes x = a and y = b is line of intersection of these planes.
Now, line of intersection of yz-plane and xz-plane is z-axis.
So, line of intersection of planes x = a andy = b is line parallel to z-axis.
(h) Coordinates of a point are the distances from the origin to the feet of perpendicular from the point on the respective axis.
(i) A ball is the solid region in the space enclosed by a sphere.
(j) The region in the plane enclosed by a circle is known as a disc.
Hence, the correct matches are:
(a) – (iii), (b) – (i), (c) – (ii), (d) – (vi), (e) – (iv),
(f) – (v), (g) – (viii), (h) – (vii), (i) – (x), (j) – (ix),

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We hope the NCERT Exemplar Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry help you. If you have any query regarding NCERT Exemplar Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry, drop a comment below and we will get back to you at the earliest.

NCERT Exemplar Class 11 Maths Chapter 15 Statistics are part of NCERT Exemplar Class 11 Maths. Here we have given NCERT Exemplar Class 11 Maths Chapter 15 Statistics.

NCERT Exemplar Class 11 Maths Chapter 15 Statistics

Short Answer Type Questions

Q1. Find the mean deviation about the mean of the distribution:

Size	20	21	22	23	24
Frequency	6	4	5	1	4

Q2. Find the mean deviation about the median of the following distribution:

Marks obtained	10	11	12	14	15
Number of students	2	3	8	3	4

 

Q3. Calculate the mean deviation about the mean of the set of first n natural numbers when n is an odd number.
Sol: Consider first natural number when n is an odd i.e., 1, 2, 3,4,… , n [odd].

Q4. Calculate the mean deviation about the mean of the set of first n natural numbers when n is an even number.
Sol: Consider first n natural number, when n is even i.e., 1, 2, 3,4..n.

Q5. Find the standard deviation of the first n natural numbers.

Q6. The mean and standard deviation of some data for the time taken to complete . a test are calculated with the following results:
Number of observations = 25, mean = 18.2 seconds, standard deviation = 3.25 s.

Q8. Two sets each of 20 observations, have the same standard derivation 5. The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.

 

Q9. The frequency distribution:

X	A	2A	3 A	4A	5 A	6A
f	2	1	1	1	1	1

where A is a positive integer, has a variance of 160. Determine the value of A.

Q10. For the frequency distribution:

X	2	3	4	5	6	7
f	4	9	16	14	11	6

Find the standard deviation.

Q11. There are 60 students in a class. The following is the frequency distribution of the marks obtained by the students in a test:

Marks	0	i	2	3	4	5
Frequency	x – 2	X	x2	(x+1)2	2x	x + 1

where x is a positive integer. Determine the mean and standard deviation of the marks.

Q12. The mean life of a sample of 60 bulbs was 650 hours and the standard deviation was 8 hours. A second sample of 80 bulbs has a mean life of 660 hours and standard deviation 7 hours. Find the overall standard deviation.

Q13. Mean and standard deviation of 100 items are 50 and 4, respectively. Then find the sum of all the item and the sum of the squares of the items.

Q14. If for a distribution Σ (x -5)= 3,Σ (x -5)²= 43 and the total number of item is 18, find the mean and standard deviation.
Sol: Given, n = 18, Σ (x – 5) = 3 and Σ (x – 5)² = 43

Q15. Find the mean and variance of the frequency distribution given below:

Long Answer Type Questions
Q16. Calculate the mean deviation about the mean for the following frequency distribution:

Class interval	0-4	4-8	8-12	12-16	16-20
Frequency	4	6	8	5	2

Q17. Calculate the mean deviation from the median of the following data

Class interval	0 – 6	6 – 12	12 -18	18 -24	24 -30
Frequency	4	5	3	6	2

Q18. Determine the mean and standard deviation for the following distribution:

Marks	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Frequency	1	6	6	8	8	2	2	3	0	2	1	0	0	0	1

Q19. The weights of coffee in 70 jars are shown in the following table:

Weight (in grams)	Frequency
200-201	13
201-202	27
202 – 203	18
203-204	10
204-205	1
205-206	1

Determine variance and standard deviation of the above distribution.

Q20. Determine mean and standard deviation of first n terms of an A.P. whose first term is a and common difference is d.

Q21. Following are the marks obtained, out of 100, by two students Ravi and Hashinain 10 tests.

Ravi	25	50	45	30	70	42	36	48	35	60
Hashina	10	70	50	20	95	55	42	60	48	80

Who is more intelligent and who is more consistent?

Q22. Mean and standard deviation of 100 observations were found to be 40 and 10,respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.

Q23. While calculating the mean and variance of 10 readings, a student wrongly used the reading 52 for the correct reading 25. He obtained the mean and variance as 45 and 16 respectively. Find the correct mean and the variance.

Objective Type Questions
Q24. The mean deviation of the data 3,10, 10,4, 7, 10, 5 from the mean is (a) 2 (b) 2.57 (c) 3 (d) 3.75
Sol: (b) Given, observations are 3, 10, 10, 4, 7, 10 and 5.

Q26. When tested, the lives (in hours) of 5 bulbs were noted as follows: 1357, 1090, 1666, 1494, 1623 The mean deviations (in hours) from their mean is (a) 178 (b) 179 (c) 220 (d) 356

Q27. Following are the marks obtained by 9 students in a mathematics test:
50, 69,20, 33, 53, 39,40, 65, 59 The mean deviation from the median is:
(a) 9 (b) 10.5 (c) 12.67 (d) 14.76
Sol: (c) Since, marks obtained by 9 students in Mathematics are 50,69,20,33,53, 39,40, 65 and 59.
Rewrite the given data in ascending order.
20, 33, 39,40, 50, 53, 59, 65, 69,
Here, n = 9 [odd]

Q30. The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations is
(a) 50000 (b) 250000 (c) 252500 (d) 255000

Q31. Let a, b, c, d, e be the observations with mean m and standard deviation V. The standard deviation of the observations a + k,b + k,c + k,d+k,e + k is

Q34. Standard deviations for first 10 natural numbers is
(a) 5.5 (b) 3.87 (c) 2.97 (d) 2.87

Q35. Consider the numbers 1,2, 3,4, 5, 6, 7, 8,9,10. If 1 is added to each number, the variance of the numbers so obtained is
(a) 6.5 (b) 2.87 (c) 3.87 (d) 8.25
Sol: (d) Given numbers are 1, 2, 3,4, 5, 6, 7, 8, 9 and 10
If 1 is added to each number, then observations will be 2, 3,4, 5, 6,7, 8, 9, 10 and 11.

Q36. Consider the first 10 positive integers. If we multiply each number by -1 and then add 1 to each number, the variance of the numbers so obtained is (a) 8.25 (b) 6.5 (c) 3.87 (d) 2.87
Sol: (a) Since, the first 10 positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.
On multiplying each number by -1, we get
-1, -2, -3, -4, -5, -6, -7, -8, -9, -10 On adding 1 in each number, we get
0, -1, -2, -3, -4, -5, -6, -7, -8, -9

Q38. Coefficient of variation of two distributions are 50 and 60, and their arithmetic means are 30 and 25 respectively. Difference of their standard deviation is
(a) 0 (b) 1 (c) 1.5 (d) 2.5

Q39. The standard deviation of some temperature data in °C is 5. If the data were converted into °F, the variance would be
(a) 81 (b) 57 (c) 36 (d) 25

Fill in the Blanks

Q43. The standard deviation of a data is _____ of any change in origin, but is ________ on the change of scale.
Sol: The standard deviation of a data is independent of any change in origin but is dependent of charge of scale.
Q44. The sum of the squares of the deviations of the values of the variable is ________ when taken about their arithmetic mean.
Sol: The sum of the squares of the deviations of the values of the variable is minimum when taken about their arithmetic mean.
Q45. The mean deviation of the data is ________ when measured from the median.
Sol: The mean deviation of the data is least when measured from the median.
Q46. The standard deviation is________ to the mean deviation taken from the arithmetic mean.
Sol: The standard deviation is greater than or equal to the mean deviation taken from the arithmetic mean.

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We hope the NCERT Exemplar Class 11 Maths Chapter 15 Statistics help you. If you have any query regarding NCERT Exemplar Class 11 Maths Chapter 15 Statistics, drop a comment below and we will get back to you at the earliest.

NCERT Exemplar Class 11 Maths Chapter 16 Probability are part of NCERT Exemplar Class 11 Maths. Here we have given NCERT Exemplar Class 11 Maths Chapter 16 Probability.

NCERT Exemplar Class 11 Maths Chapter 16 Probability

Short Answer Type Questions 

Q1. If the letters of the word ALGORITHM are arranged at random in a row what is the probability the letters GOR must remain together as a unit?
Sol: We have word ALGORITHM Number of letters = 9

Q2. Six new employees, two of whom are married to each other, are to be assigned six desks that are lined up in a row. If the assignment of employees to desks is made randomly, what is the probability that the married couple will have nonadjacent desks?
Sol: Six employees can be arranged in 6! ways.
n(S) = 6!
Two adjacent desks for married couple can be selected in 5 ways viz.,(l, 2), (2, 3), (3,4), (4, 5), (5,6).
This couple can be arranged in the two desks in 2! ways.
Other four persons can be arranged in 4! ways.
So, number of ways in which married couple occupy adjacent desks
= 5×2! x4! =2×5!
So, number of ways in which married couple occupy non-adjacent desks = 6! – 2 x 5! = 4 x 5! = n(E)

Q3. Suppose an integer from 1 through 1000 is chosen at random, find the probability that the integer is a multiple of 2 or a multiple of 9.
Sol: We have integers 1,2, 3,…1000
We have integers 1,2, 3,…1000
n(S) = 1000
Number of integers which are multiple of 2 = 500 Let the number of integers which are multiple of 9 be n.
nth term = 999 =>   9 + (n -1)9 = 999
=>             9 + 9n – 9 = 999
=>          n = 111
From 1 to 1000, the number of multiples of 9 is 111.
The multiple of 2 and 9 both are 18, 36,…, 990.
Let m be the number of terms in above series.
.’.               mth term = 990
=>             18 + (m- 1)18 = 990
=>             18+18m-18 = 990
=>        m = 55
Number of multiples of 2 or 9 = 500 +111-55 = 556 = n(E)

Q4. An experiment consists of rolling a die until a 2 appears.
(i) How many elements of the sample space correspond to the event that the 2 appears on the Ath roll of the die?
(ii) How many elements of the sample space correspond to the event that the 2 appears not later than the Ath roll of the die?
Sol: Number of outcomes when die is thrown is ‘6’.
(i) If 2 appears on the Ath roll of the die.
So, first (k -1) roll have 5 outcomes each and Kth roll results 2
Number of outcomes = 5^k-1

(ii) If we consider that 2 appears not later than K th roll of the die, then 2 comes before Ath roll.
If 2 appears in first roll, number of ways = 1 If 2 appears in second roll, number of ways
= 5 x 1 (as first roll does not result in 2)
If 2 appears in third roll, number of ways
= 5 x 5 x 1 (as first two rolls do not result in 2)
Similarly if 2 appears in (k – l)th roll, number of ways = [5x5x5… (k- 1) times] x 1 = 5^k-1 Possible outcomes if 2 appears before kth roll = 1 +5 + 5² + 5³+ … +5^k-l

Q5. A die is loaded in such a way that each odd number is twice as likely to occur as each even number. Find P(G), where G is the event that a number greater than 3 occurs on a single roll of the die.
Sol: If is given that 2 x Probability of even number = Probability of odd

Q6. In a large metropolitan area, the probabilities are .87, .36, .30 that a family (randomly chosen for a sample survey) owns a colour television set, a black and white television set, or both kinds of sets. What is the probability that a family owns either anyone or both kinds of sets?
Sol: Let C be the even that family own colour television set and B be the event that family owns a black and white television set It is given that,
P(C) = 0.87, P{B) = 0.36 and P(C∩B) = 0.30 We have to find probability that a family owns either anyone or both kind of sets i.e., P(B ∪ C)
Now, P(B∪C) = P(B) + P(C)-P(C∩ B)
= 0.87 + 0.36-0.30= 0.93

Q7. If A and B are mutually exclusive events, P(A) =35 and P(B) = 0.45, find
(a) P(A’)
(b) P(B’)
(c) P(A∪ B)
(d) P(A∩ B)
(e) P(A∪ B’)                                           
(f) P(A’∩B’)

Q8. A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated as very complex, complex, routine, simple or very simple are respectively, 0.15,0.20, 0.31, 0.26, .08. Find the probabilities that a particular surgery will be rated
(a) complex or very complex;
(b) neither very complex nor very simple;
(c) routine or complex
(d) routine or simple

Q9. Four candidates A, B, C, ZJhave applied for the assignment to coach a school cricket team. If A is twice as likely to be selected as B, and B and C are given about the same chance of being selected, while C is twice as likely to be selected as D, what are the probabilities that
(a) C will be selected? (b) A will not be selected?
Sol: It is given that A is twice as likely to be selected as B.
P(A) = 2P(B)
B and C are given about the same chance of being selected.
P(B) = P(C)
C is twice as likely to be selected as D.
P(C) = 2 P(D)

Q10. One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month. Consequently the sample space consists of four elementary outcomes S = {John promoted, Rita promoted, Aslam promoted, Gurpreet promoted}. You are told that the chances of John’s promotion is same as that of Gurpreet, Rita’s chances of promotion are twice as likely as Johns. Aslam’s chances are four times that of John.
(a) Determine
P(John promoted)
P(Rita promoted)
P(Aslam promoted)
P(Gurpreet promoted)
(b) If A = {John promoted or Gurpreet promoted}, find P(A).
Sol: Let Event: J = John promoted
R = Rita promoted
A = Aslam promoted
G = Gurpreet promoted
Given sample space, S = {John promoted, Rita promoted, Aslam promoted, Gurpreet promoted}
i.e. S={J,R,A,G)
It is given that, chances of John’s promotion is same as that of Gurpreet.
P(J) = P(G)
Rita’s chances of promotion are twice as likely as John.
P(R) = 2P(J)
And Aslam’s chances of promotion are four times that of John.
P(A) = 4P(J)
Now, P(J) + P(R) + P(A) + P(G) = 1 => P(J) + 2P(J) + 4P(J) + P(J) = 1
=> 8P(J) = 1
P(J) = 1/8

Q11. The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, P(A∩B) = .07).

Q12. One urn contains two black balls (labelled Bx and B2) and one white ball. A second urn contains one black ball and two white balls (labelled W₁ and W₂). Suppose the following experiment is performed. One of the two urns is chosen at random. Next a ball is randomly chosen from the urn. Then a second ball is chosen at random from the same urn without replacing the first ball.
(a) Write the sample space showing all possible outcomes
(b) What is the probability that two black balls are chosen?
(c) What is the probability that two balls of opposite colour are chosen?
Sol:It is given that one of the two urn is chosen, then a ball is randomly chosen
from the urn, then a second ball is chosen at random from the same urn without replacing the first ball.

Q13. A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the Probability that
(a) All the three balls are white
(b) All the three balls are red
(c) One ball is red and two balls are white

Q15. A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.
Sol: Number of cards = 52 .-.            n(S) = 52
4 king + 13 heart + 26 red – 13 – 2 = 28 = n{E)
.’.             Required probability = 28/52 = 7/13

Q17. Determine the probability p, for each of the following events.
(a) An odd number appears in a single toss of a fair die.
(b) At least one head appears in two tosses of a fair coin.
(c) The sum of 6 appears in a single toss of a pair of fair dice.
Sol: (a) When a die is thrown the possible outcomes are
S = {1, 2, 3,4, 5, 6} out of which 1, 3, 5 are odd,

Objective type Questions

Q18. In a non-leap year, the probability of having 53 Tuesdays or 53 Wednesdays is

Q19. Three numbers are chosen from 1 to 20. Find the probability that they are not consecutive

Q20. While shuffling a pack of 52 playing cards, 2 are accidentally dropped. Find the probability that the missing cards to be of different colours

Q21. Seven persons are to be seated in a row. The probability that two particular persons sit next to each other is

Q22. Without repetition of the numbers, four digit numbers are formed with the numbers 0, 2, 3, 5. The probability of such a number divisible by 5 is

Q23. If A and B are mutually exclusive events, then

Q24. If P(A ∪B) = P(A n B) for any two events A and B, then
(a) P(A) = P(B) (b) P (A) > P (B)
(c) P(A ) < P(B) (d) none of these
Sol: (a) We have, P(A ∪ B) = P(A n B)
P(A) + P(B) – P(A ∩ B) = P(A ∩ B)

Q25. If 6 boys and 6 girls sit in a row at random. The probability that all the girls sit together is

Q26. A single letter is selected at random from the word ‘PROBABILITY’. The probability that it is a vowel is

Q27. If the probabilities for A to fail in an examination is 0.2 and that for B is 0.3, then the probability that either A or B fails is

Q29. If M and N are any two events, tlie probability that at least one of them occurs is                            .
(a) P(M) + P(N) – 2 P(M ∩N)         
(b) P(M) + P(N) – P(M ∩ N)
(c) P(M) + P(N) + P(M ∩ N)
(d) P(M) + P(N) + 2P(M∩N)
Sol: (B) If M and N are any two events.
.-. P(M ∪N) = P(M) + P(N) – P(M ∩ N) .

True/False Type Questions

Q30. The probability that a person visiting a zoo will see the giraffe is 0.72, the probability that he will see the bears is 0.84 and the probability that he will see both is 0.52.
Sol: False
P(to see giraffe or bear) = P (giraffe) + P (bear) – P(giraffe and bear)

=0.72 + 0.84-0.52= 1.04
which is not possible.

Q31. The probability that a student will pass his examination is 0.73, the probability of the student getting a compartment is 0.13, and the probability that the student will either pass or get compartment is 0.96.
Sol: False
Let A = Student will pass examination
B = Student will getting compartment
P(A) = 0.73, P(B) = 0.13 and P(A or B) = 0.96
P(A or B) = P(A) + P(B) = 0.73 + 0.13 = 0.86
But P(A or B) = 0.96
Hence, given statement is false.

Q32.The probabilities that a typist will make 0, 1,2, 3, 4, 5 or more mistakes in typing a report are, respectively, 0.12, 0.25, 0.36, 0.14, 0.08, 0.11.
Sol: False
Sum of these probabilities must be equal to 1.
P(0) + P( 1) + P( 2) + P(3) + P{ 4) + P(5)
= 0.12 + 0.25+0.36 + 0.14 + 0.08 + 0.11 = 1.06 which is greater than 1,
So, it is false statement.

Q33. If A and B are two candidates seeking admission in an engineering College. The probability that A is selected is .5 and the probability that both A and B are selected is at most .3. Is it possible that the probability of B getting selected is 0.7?

Sol. False
Given that, P(A ) = 0.5, P(A ∩B)< 0.3
Now, P(A) x P(B) ≤ 0.3
=>0.5 x P(B) ≤0.3
=> P(B) ≤0.6
Hence, it is false statement

Q34. The probability of intersection of two events A and B is always less than or equal to those favourable to the event
Sol: True
We know that A ∩ B ⊂ A
P(A ∩ B) ≤ P(A)
Hence, it is a true statement.

Q35. The probability of an occurrence of event A is .7 and that of the occurrence of event B is .3 and the probability of occurrence of both is .4.
Sol: False
A ∩B⊆ A, B
P(A ∩B) ≤ P(A), P(B)
But given that P(B) = 0.3 and P(A ∩B) = 0.4, which is not possible.

Q36. The sum of probabilities of two students getting distinction in their final examinations is 1.2.
Sol: True
Probability of each student getting distinction in their final examination is less than or equal to 1, sum of the probabilities of two may be 1.2.
Hence, it is true statement.

Fill in the Blanks Type Questions

Q37. The probability that the home team will win an upcoming football game is 0.77, the probability that it will tie the game is 0.08, and the probability that it will lose the game is ______.
Sol: P(losing) = 1 – (0.77 + 0.08) = 0.15

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We hope the NCERT Exemplar Class 11 Maths Chapter 16 Probability help you. If you have any query regarding NCERT Exemplar Class 11 Maths Chapter 16 Probability, drop a comment below and we will get back to you at the earliest.