If and are independent events such that and , find
(i)
(ii)
i) $P\left(E_{1} \cap E_{2}\right)$ We know that, when $E_{1}$ and $E_{2}$ are independent, $\begin{array}{l} P\left(E_{1} \cap E_{2}\right)=P\left(E_{1}\right) \times P\left(E_{2}\right) \\ =0.3...
Express each of the following in the form (a + ib):
If and are the two events such that and , show that and are independent events.
We know that, Hence, $P\left(E_{1} \cap E_{2}\right)=P\left(E_{1}\right)+P\left(E_{2}\right)-P\left(E_{1} \cup E_{2}\right)$ $=\frac{1}{4}+\frac{1}{3}-\frac{1}{2}$ $=\frac{1}{12}$ Equation 1 Since...
Express each of the following in the form (a + ib):
Express each of the following in the form (a + ib):
Let and be the events such that and . Find:
(i) , when and are mutually exclusive.
(ii) , when and are independent
(i) We know that, When two events are mutually exclusive $P\left(E_{1} \cap E_{2}\right)=0$ Hence, $P\left(E_{1} \cup E_{2}\right)=P\left(E_{1}\right)+P\left(E_{2}\right)$ $\begin{array}{l}...
Express each of the following in the form (a + ib):
An urn contains 5 white and 8 black balls. Two successive drawings of 3 balls at a time are made such that the balls drawn in the first draw are not replaced before the second draw. Find the probability that the first draw gives 3 white balls and the second draw gives 3 black balls.
Let, success in the first draw be getting 3 white balls. Now, the Probability of success in the first trial is $P_{1}(\text { success })=\frac{5_{c_{3}}}{13_{c_{3}}}=\frac{10}{286}=\frac{5}{143}$...
A bag contains white and 15 black balls. Two balls are drawn in succession without replacement. What is the probability that the first ball is white and the second is black?
Let, success in the first draw be getting a white ball. Now, the Probability of success in the first trial is $\mathrm{P}_{1}(\text { success })=\frac{10}{25}$ Let success in the second draw be...
There is a box containing 30 bulbs, of which 5 are defective. If two bulbs are chosen at random from the box in succession without replacing the first, what is the probability that both the bulbs are chosen are defective?
Let, success :bulb chosen is defective .i.e $\frac{5}{30}$ Now, the Probability of success in the first trial is $P_{1}(\text { success })=\frac{5}{30}$ Probability of success in the second trial...
Express each of the following in the form (a + ib):
A card is drawn from a well-shuffled deck of 52 cards and without replacing this card, a second card is drawn. Find the probability that the first card is a club and the second card is a spade.
Let, success for the first trail be getting a club. Now, the Probability of success in the first trial is $\mathrm{P}_{1} \text { (success) }=\frac{13}{52}$ let, success for the second trail be...
Express each of the following in the form (a + ib):
Express each of the following in the form (a + ib):
Express each of the following in the form (a + ib):
Simplify each of the following and express it in the form (a + ib) : (1 + i)3 – (1 – i)3
Simplify each of the following and express it in the form (a + ib) : (1 + 2i)–3
Simplify each of the following and express it in the form (a + ib) : (2 + i)–2
Two marbles are drawn successively from a box containing 3 black and 4 white marbles. Find the probability that both the marbles are black if the first marble is not replaced before the second draw.
Let, success : marble drawn is black.i.e Number of black marbles/Total number of marbles $=\frac{3}{7}$ Now, the Probability of success in the first trial is $P_{1}(\text { success })=\frac{3}{7}$...
A bag contains 17 tickets, numbered from 1 to 17 . A ticket is drawn, and then another ticket is drawn without replacing the first one. Find the probability that both the tickets may show even numbers.
Let, success : ticket drawn is even.i.e $\frac{8}{17}$ Now, the Probability of success in the first trial is $P_{1}(\text { success })=\frac{8}{17}$ Probability of success in the second trial...
Simplify each of the following and express it in the form (a + ib) :
Two integers are selected at random from integers 1 through 11 . If the sum is even, find the probability that both the numbers selected are odd.
Two integers are selected at random. The first choice has 11 options from the 11 integers, and the second choice has 10 options from the remaining 10 integers. Let $\mathrm{P}(\mathrm{A})$ be the...
In a hostel, of the students read Hindi newspaper, read English newspaper and read both Hindi and English newspapers. A student is selected at random. If he reads English newspaper, what is the probability that he reads Hindi newspaper?
Let $\mathrm{P}(\mathrm{A})$ be the probability of students reading Hindi newspaper. $\therefore P(A)=0.60$ Let $\mathrm{P}(\mathrm{B})$ be the probability of them reading English newspaper....
Simplify each of the following and express it in the form (a + ib) : (4 – 3i)–1
In a hostel, of the students read Hindi newspaper, read English newspaper and read both Hindi and English newspapers. A student is selected at random.
(i) Find the probability that he reads neither Hindi nor English news paper.
(ii) If he reads Hindi newspaper, what is the probability that he reads English newspaper?
Let $\mathrm{P}(\mathrm{A})$ be the probability of students reading Hindi newspaper. $\therefore P(A)=0.60$ Let $\mathrm{P}(\mathrm{B})$ be the probability of them reading English newspaper....
The probability that a certain person will buy a shirt is , the probability that he will buy a coat is and the probability that he will buy a shirt given that he buys a coat is Find the probability that he will buy both a shirt and a coat.
Let $\mathrm{P}(\mathrm{A})$ be the probability of a certain person buying a shirt. $\therefore \mathrm{P}(\mathrm{A})=0.2$ Let $P(B)$ be the probability of him buying a coat. $\therefore P(B)=0.3$...
The probability that a student selected at random from a class will pass in Hindi is and the probability that he passes in Hindi and English is . What is the probability that he will pass in English if it is known that he has passed in Hindi?
One student is selected at random. Let $\mathrm{P}(\mathrm{A})$ be the probability of students passing in English. Let $\mathrm{P}(\mathrm{B})$ be the probability of students passing in Hindi....
In a class, students study mathematics; study biology and study both mathematics and biology. One student is selected at random. Find the probability that
(i) he studies mathematics if it is known that he studies biology
(ii) he studies biology if it is known that he studies mathematics.
Let $\mathrm{P}(\mathrm{A})$ be the probability of students studying mathematics. $\therefore P(A)=0.40$ Let $\mathrm{P}(\mathrm{B})$ be the probability of students studying biology. $\therefore...
A couple has 2 children. Find the probability that both are boys if it is known that (i) one of the children is a boy, and (ii) the elder child is a boy.
A couple has two children. The sample space $S=\{(B, B),(B, G),(G, B),(G, G)\}$ Let $P(A)$ be the probability of both being boys. (i) Let $P(B)$ be the probability of one of them being a boy. The...
A coin is tossed and then a die is thrown. Find the probability of obtaining a 6, given that a head came up.
A coin is tossed and a die thrown. A coin having two sides have a total outcome of 2 viz. $\{\mathrm{H}, \mathrm{T}\}$ A die has 6 faces and will have a total outcome of 6 i.e. $\{1,2,3,4,5,6\}$ Let...
Two dice were thrown and it is known that the numbers which come up were different. Find the probability that the sum of the two numbers was
Two die having 6 faces each when tossed simultaneously will have a total outcome of $6^{2}=36$ Let $P(A)$ be the probability of getting a sum equal to 5 . Let $P(B)$ be the probability of getting 2...
A die is thrown twice and the sum of the numbers appearing is observed to be What is the conditional probability that the number 5 has appeared at least once?
A die thrown twice will have a total outcome of $6^{2}=36$ Let $P(A)$ be the probability of getting the number 5 at least once. Let $P(B)$ be the probability of getting sum $=8$ The sample space of...
Two unbiased dice are thrown. Find the probability that the sum of the numbers appearing is 8 or greater, if 4 appears on the first die.
Two die having 6 faces each when tossed simultaneously will have a total outcome of $6^{2}=36$ Let $\mathrm{P}(\mathrm{A})$ be the probability of getting a sum greater than $8 .$ Let $P(B)$ be the...
Simplify each of the following and express it in the form (a + ib) :
Three coins are tossed simultaneously. Find the probability that all coins show heads if at least one of the coins shows a head.
When three coins are tossed simultaneously, the total number of outcomes $=2^{3}=8$, and the sample space is given by $\mathrm{S}=\{(\mathrm{H}, \mathrm{H}, \mathrm{H}),(\mathrm{H}, \mathrm{H},...
Simplify each of the following and express it in the form (a + ib) : (–3 + 5i)3
A coin is tossed twice. If the outcome is at most one tail, what is the probability that both head and tail have appeared?
A coin has 2 sides and its sample space $\mathrm{S}=\{\mathrm{H}, \mathrm{T}\}$ The total number of outcomes $=2$ A coin is tossed twice. Let $\mathrm{P}(\mathrm{A})$ be the probability of getting...
A die is rolled. If the outcome is an even number, what is the probability that it is a number greater than
A die has 6 faces and its sample space $S=\{1,2,3,4,5,6\}$ The total number of outcomes $=6$ Let $P(A)$ be the probability of getting an even number. The sample space of $A=\{2,4,6\}$ $\therefore...
Simplify each of the following and express it in the form (a + ib) : (5 – 2i)2
Simplify each of the following and express it in the form (a + ib) :
Simplify each of the following and express it in the form a + ib :
Simplify each of the following and express it in the form a + ib : (3 + 4i) (2 – 3i)
Simplify each of the following and express it in the form a + ib :
Simplify each of the following and express it in the form a + ib : (1 – i)2 (1 + i) – (3 – 4i)2
Simplify each of the following and express it in the form a + ib : (8 – 4i) – (- 3 + 5i)
Simplify each of the following and express it in the form a + ib : (–5 + 6i) – (–2 + i)
Simplify each of the following and express it in the form a + ib :
Simplify each of the following and express it in the form a + ib : 2(3 + 4i) + i(5 – 6i)
Prove that
Prove that i53 + i72 + i93 + i102 = 2i
Prove that (1 + i2 + i4 + i6 + i8 + …. + i20) = 1.
Prove that
Prove that (1 – i)n= 2n for all values of n N
prove that
Prove that
Prove that (1 + i10 + i20 + i30) is a real number.
Prove that
Prove that 6i50 + 5i33 – 2i15 + 6i48 = 7i.
Prove that 1 + i2 + i4 + i6 = 0
Evaluate:
Evaluate:
Evaluate:
If x and y are acute such that , prove that
Answer: Given: $\sin x=\frac{1}{\sqrt{5}}\,and\,\sin y=\frac{1}{\sqrt{10}}$ Now we will calculate value of cos x and cosy Sin(x + y) = sinx.cosy + cosx.siny
Evaluate:
If θ and Φ lie in the first quadrant such that , find the values of (iii) tan (θ – Φ)
Answer: (iii)We will first find out the Values of tanθ and tanΦ
If θ and Φ lie in the first quadrant such that , find the values of (i) sin (θ – Φ ) (ii) cos (θ – Φ)
Answer: Given: $\sin \theta =\frac{8}{17}\,and\,\cos \phi =\frac{12}{13}$
Prove that:
Answer: Using cos(90° + θ) = - sinθ(I quadrant cosx is positive cosec( - θ) = - cosecθ tan(270° - θ) = tan(180° + 90° - θ) = tan(90° - θ) = cotθ (III quadrant tanx is positive) Similarily sin(270° +...
Prove that:
Answer: Using sin(90° + θ) = cosθ and sin( - θ) = sinθ,tan(90° + θ) = - cotθ Sin(180° + θ) = - sinθ(III quadrant sinx is negative)
Prove that
Answer:
Show that the points A(1, 1, 1), B(-2, 4, 1), C(1, -5, 5) and D(2, 2, 5) are the vertices of a square.
Answer: (x1,y1,z1) = (1, 1, 1) (x2,y2,z2) = (-2, 4, 1) (x3,y3,z3) = (1, 5, 5) (x4,y4,z4) = (2, 2, 5) $\begin{array}{l} Length AB = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} -...
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.
Answer: (x1,y1,z1) = (0, 1, 2) (x2,y2,z2) = (2, -1, 3) (x3,y3,z3) = (1, -3, 1) $\begin{array}{l} Length AB = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} \\ =>...
Show that the points A(4, 6, -5), B(0, 2, 3) and C(-4, -4, -1) from the vertices of an isosceles triangle.
Answer: (x1,y1,z1) = (4, 6, -3) (x2,y2,z2) = (0, 2, 3) (x3,y3,z3) = (-4, -4, -1) $\begin{array}{l} Length AB = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} \\ =>...
Show that the points A(1, -1, -5), B(3, 1,3) and C(9, 1, -3) are the vertices of an equilateral triangle.
Answer: (x1,y1,z1) = (1, -1, -5) (x2,y2,z2) = (3, 1,3) (x3,y3,z3) = (9, 1, -3) $\begin{array}{l} Length AB = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} \\ =>...
Find the distance between the points : (i) R(1, -3, 4) and S(4, -2, -3) (ii) C(9, -12, -8) and the origin
Answers: (i) R(1, -3, 4) and S(4, -2, -3) (x1,y1,z1) = (1, -3, 4) (x2,y2,z2) = (4, -2, -3) $\begin{array}{l} D = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} \\ D =...
Find the distance between the points : (i) A(5, 1, 2) and B(4, 6, -1) (ii) P(1, -1, 3) and Q(2, 3, -5)
Answers: (i) A(5, 1, 2) and B(4, 6, -1) (x1,y1,z1) = (5, 1, 2) (x2,y2,z2)= (4, 6, -1) $\begin{array}{l} D = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} \\ D = \sqrt...
In which octant does each of the given points lie? (i) (-1, -6, 5) (ii) (4, 6, 8)
Answers: (i) (-1, -6, 5) lies in octant III (ii) (4, 6, 8) lies in octant...
In which octant does each of the given points lie? (i) (-6, 5, -1) (ii) (4, -3, -2)
Answers: (i) (-6, 5, -1) lies in octant VI (ii) (4, -3, -2) lies in octant...
In which octant does each of the given points lie? (i) (-4, -1, -6) (ii) (2, 3, -4)
Answers: (i) (-4, -1, -6) lies in octant VII (ii) (2, 3, -4) lies in octant...
In which plane does the point (4, -3, 0) lie?
The x, y, z coordinates of the point are 4, -3, 0. As the distance of point along the z-axis is 0, the plane in which the point lies is the xy-plane.
If a point lies on yz-plane then what is its x-coordinate?
The x-coordinate is the distance of a point from the origin parallel or along the x- axis. To measure the x coordinate, you must move either to the left of the origin or to its right. In case of a...
If a point lies on the z-axis, then find its x-coordinate and y-coordinate.
The X and y coordinates of a point are its distance from the origin along or parallel to the horizontal x-axis and y-axis. To measure the x and y coordinates, you must move either to the left of the...
Prove that:
Answer:
Find the equation of the parabola with vertex at the origin, passing through the point P(5, 2) and symmetric with respect to the y-axis.
Answer: The equation of a parabola with vertex at the origin and symmetric about the y-axis is x2 = 4ay The point P(5,2) passes through above parabola, 52 =...
Find the equation of the parabola with vertex at the origin and focus F(0, 5).
Answer: Vertex : A (0,0) Focus F(0,5) is of the form F(0,a) Vertex A(0,0) and Focus F(0,a), The equation of parabola is x2 = 4ay a = 5 The equation of parabola is...
Find the equation of the parabola with focus F(0, -3) and directrix y = 3.
Answer: Given, The equation of directrix is, y = 3 y - 3 = 0 Above equation is of the form, y - a = 0 Focus of the parabola F(0,-3) is of the form F(0,-a) a = 3...
Find the equation of the parabola with focus F(4, 0) and directrix x = -4.
Answer: Given, Equation of directrix, x = -4 x + 4 = 0 The equation is of the form, x + a = 0 Focus of the parabola F(4,0) is of the form F(a,0) a = 4 For...
Find the equation of the parabola with vertex at the origin and focus at F(-2, 0).
Answer : Vertex : A (0,0) focus F(-2,0) is of the form F(-a,0) For Vertex A(0,0) and Focus F(-a,0), The equation of parabola is y2 = - 4ax a = 2 The equation of...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
Answer: Given, 3x2 = -16y x2 = -16/3 y Comparing the given equation with parabola having an equation, x2 = 4ay 4a = 16/2 a = 4/3 Focus: F(0, -a) = F(0, -4/3) Vertex: A(0,...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
Answer: Given, x2 = -18y Comparing given equation with parabola having equation, x2 = -4ay 4a = 18 a = 9/2 Focus: F(0, -a) =F(0, -9/2) Vertex: A(0, 0) =A(0, 0) Equation...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
Answer: Given, x2 = - 8y Comparing given equation with parabola having equation, x2 = - 4ay 4a = 8 a = 2 Focus : F(0,-a) = F(0,-2) Vertex : A(0,0) = A(0,0) Equation of...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
Answer: Given, 3x2 = 8y x2 = 8/3 y Comparing the given equation with parabola having an equation, x2 = 4ay 4a = 8/3 a = 2/3 Focus : F(0, a) = F(0, 2/3) Vertex :...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
Answer: Given, x2 = 10y Comparing given equation with parabola having equation, x2 = 4ay 4a = 10 a = 5 Focus : F(0,a) = F(0,2.5) Vertex : A(0,0) = A(0,0) Equation of the...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
Answer: Given, x2 = 16y Comparing given equation with parabola having equation, x2 = 4ay 4a = 16 a = 4 Focus : F(0,a) = F(0,4) Vertex : A(0,0) = A(0,0) Equation of the...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
Answer: Given, 5y2 = -16x y2 = -16/5 x Comparing the given equation with parabola having an equation, y2 = - 4ax 4???? = 16/5 ???? = 4/5 Focus : F(-a,0) = F(-4/5, 0)...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
Answer: Given, y2 = -6x Comparing given equation with parabola having equation, y2 = - 4ax 4a = 6 a = 2/3 Focus: F(-a, 0) = F(-2/3, 0) Vertex: A(0, 0) =A(0, 0)...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :
Answer: Given, y2 = -8x Comparing given equation with parabola having equation, y2 = - 4ax 4a = 8 a = 2 Focus : F(-a,0) = F(-2,0) Vertex : A(0,0) = A(0,0) Equation...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola:
Answer: Given, 3y2 = 8x y2 = 8/3 x Comparing the given equation with parabola having equation, y2 = 4ax 4a = 8/3 a = 2/3 Focus : F(a, 0) = F(2/3, 0) Vertex :...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola:
Answer: Given, y2 = 10x Comparing given equation with parabola having equation, y2 = 4ax 4a = 10 a =2.5 Focus : F(a,0) = F(2.5,0) Vertex : A(0,0) = A(0,0) Equation of the...
Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola:
Answer: Given, y2= 12x Comparing given equation with parabola having equation, y2 = 4ax 4a = 12 a = 3 Focus : F(a,0) = F(3,0) Vertex : A(0,0) = A(0,0) Equation of...
Prove that:
Answer:
Prove that:
Answer: (II quadrant tanx negative) - tan45° = - 1
Find the equation of the parabola, which is symmetric about the y-axis and passes through the point P(2, -3).
Answer: The equation of a parabola with vertex at the origin and symmetric about the y-axis is x2 = 4ay The point P(2,-3) passes through above parabola, 22...
Prove that
Answer: (ii)cot105° - tan105° = cot(180° - 75°) - tan(180° - 75°) (II quadrant tanx is negative and cotx as well) = - cot75° - ( - tan75°) = tan75° - cot75°
Prove that: tan15° + cot15° = 4
Answer: (iii) tan15° + cot15° = First, we will calculate tan15°,
Prove that:
Answer: (i) sin75° = sin(90° - 15°) .…….(using sin(A - B) = sinAcosB - cosAsinB) = sin90°cos15° - cos90°sin15° = 1.cos15° - 0.sin15° = cos15° Cos15° = cos(45° - 30°) …………(using cos(A - B) = cosAcosB...
Prove that:
Answer:
Prove that:
(i) cos(n + 2)x.cos(n + 1)x + sin(n + 2)x.sin(n + 1)x = cos x Answer: (i) cos(n + 2)x.cos(n + 1)x + sin(n + 2)x.sin(n + 1)x = sin((n + 2)x + (n + 1)x)(using cos(A - B) = cosAcosB + sinAsinB) =...
If ( – 1, 3) and (????, β) are the extremities of the diameter of the circle x2 + y2 – 6x + 5y – 7 = 0, find the coordinates (????, β).
Answer: Given, x2 + y2 – 6x + 5y – 7 = 0 Centre (3, -5/2) ( - 1, 3) & (????, β) are the 2 extremities of the diameter Using mid - point formula, $\begin{array}{l} \frac{{\alpha - 1}}{2} = 3\\...
Show that the quadrilateral formed by the straight lines x – y = 0, 3x + 2y = 5, x – y = 10 and 2x + 3y = 0 is cyclic and hence find the equation of the circle.
Answer: Slope of CD = 1 AB||CD Slope of BD = AC = - 1 AC||B They form a rectangle with all sides = 900 The quadrilateral is cyclic as sum of opposite angles...
Find the equation of the circle circumscribing the triangle formed by the lines x + y = 6, 2x + y = 4 and x + 2y = 5.
Answer: The required circle equation is, Using Laplace Expansion, 27(x2 + y2) - 459x - 513y + 1350 = 0 x2 + y2 - 17x - 19 + 50 =...
Find the equation of a circle passing through the origin and intercepting lengths a and b on the axes.
Answer: AD = b units and AE = a units. D(0, b), E(a, 0) and A(0, 0) lies on the circle. C is the centre. The general equation of a circle: (x - h)2 + (y - k)2 = r2...
Find the equation of the circle which passes through the points A(1, 1) and B(2, 2) and whose radius is 1. Show that there are two such circles.
Answer: The general equation of a circle is, (x - h)2 + (y - k)2 = r2 …(i) (h, k) is the centre and r is the radius. Putting A(1, 1) in (i) (1 - h)2 + (1 - k)2 = 12 ...
Prove that the centres of the three circles , and are collinear.
Answer: Given, x2 + y2 – 4x – 6y – 12 = 0 Centre ( - g1, - f1) = (2, 3) x2 + y2 + 2x + 4y – 5 = 0 Centre ( - g2, - f2) = ( - 1, - 2) x2 + y2 – 10x – 16y + 7 = 0 Centre ( - g3, - f3) = (5, 8) ...
Find the equation of the circle concentric with the circle and of double its area.
Answer: Two or more circles are said to be concentric If they have the same centre and different radii. Given, x2 + y2 - 6x + 12y + 15 = 0 Radius r = The...
Find the equation of the circle concentric with the circle and which touches the y-axis.
Answer: The general equation of the circle is, x2 + y2 + 2gx + 2fy + c = 0 Radius, r = $\begin{array}{l} r = \sqrt {{{(2)}^2} + {{(3)}^2} - ( - 3)} \\ r =...
Find the equation of the circle which passes through the points (1, 3) and (2, – 1), and has its centre on the line 2x + y – 4 = 0.
Answer: The equation of a circle: x2 + y2 + 2gx + 2fy + c = 0…(i) Putting (1, 3) & (2, - 1) in (i) 2g + 6f + c = - 10..(ii) 4g - 2f + c = - 5..(iii) The centre lies on the given straight line, (...
Prove that (i) sin(50° + θ)cos(20° + θ) – cos(50° + θ)sin(20° + θ) = 1/2 (ii) cos(70° + θ)cos(10° + θ) + sin(70° + θ)sin(10° + θ) = 1/2
Answer: (i) We have: sin(50° + θ)cos(20° + θ) - cos(50° + θ)sin(20° + θ) = sin(50° + θ - (20° + θ))(using sin(A - B) = sinAcosB - cosAsinB) = sin(50° + θ - 20° - θ) = sin30° = 1/2 (ii) We have:...
Show that the points A(1, 0), B(2, – 7), c(8, 1) and D(9, – 6) all lie on the same circle. Find the equation of this circle, its centre and radius.
Answer: The general equation of a circle: (x - h)2 + (y - k)2 = r2 …(i) (h, k) is the centre and r is the radius. Putting (1, 0) in (i) (1 - h)2 + (0 - k)2 = r2 h2 + k2 +...
Find the equation of the circle concentric with the circle and passing through the point P(5, 4).
Answer: Two or more circles are said to be concentric If they have the same centre and different radii. Given, x2 + y2 + 4x + 6y + 11 = 0 The concentric circle...
Find the equation of the circle which is circumscribed about the triangle whose vertices are A( – 2, 3), b(5, 2) and C(6, – 1). Find the centre and radius of this circle.
Answer: The general equation of a circle: (x - h)2 + (y - k)2 = r2 ...(i) (h, k) is the centre r is the radius Putting A( - 2, 3), B(5, 2) and c(6, - 1) in the equation, h2 + k2 + 4h - 6k + 13 = r2...
Find the equation of the circle passing through the points (20, 3), (19, 8) and (2, – 9) Also, find the centre and radius.
Answer: The required circle equation, Using Laplace Expansion, 102(x2 + y2) - 1428x - 612y - 11322 = 0 x2 + y2 - 14x -6y - 11 = 0 The equation with centre = (7, 3) Radius...
Find the equation of the circle passing through the points (i) (0, 0), (5, 0) and (3, 3) (ii) (1, 2), (3, – 4) and (5, – 6). Also, find the centre and radius
Answers: (i) The required circle equation, Using Laplace Expansion, 15(x2 + y2) - 75x - 15y = 0 x2 + y2 - 5x - y =0 The equation with centre = (2.5, 0.5) Radius = (ii) The...
Show that the equation does not represent a circle.
Answer: Radius = The radius is negative which is not possible x2 + y2 - 3x + 3y + 10 = 0 does not represent a circle.
Show that the equation represents a point circle. Also, find its centre.
Answer: The general equation of a circle is, x2 + y2 + 2gx + 2fy + c = 0 c, g, f are constants. x2 + y2 + 2x + 10y + 26 = 0 The equation represents a circle with 2g = 2 ⇒g = 1, 2f = 10 ⇒f = 5 and c...
Show that the equation represents a circle. Find its centre and radius.
Answer: The general equation of a conic is, ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 a, b, c, f, g, h are constants For a circle, a = b and h = 0. The equation is, x2 + y2 + 2gx + 2fy + c = 0 ...
Prove that (v) cos130°cos40° + sin130°sin40° = 0
(v) cos130°cos40° + sin130°sin40° = cos(130° - 40°) (using cos(A - B) = cosAcosB + sinAsinB) = cos90° = 0
Show that the equation represents a circle. Find its centre and radius.
Answer: The general equation of a conic is, ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 a, b, c, f, g, h are constants For a circle, a = b and h = 0. The equation is, x2 + y2 + 2gx + 2fy + c = 0 x2 + y2 +...
Show that the equation represents a circle. Find its centre and radius.
Answer: The general equation of a conic is, ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 a, b, c, f, g, h are constants For a circle, a = b and h = 0. The equation is, x2 + y2 + 2gx + 2fy + c = 0 x2 + y2 –...
Prove that
Answer: (iii) cos75°cos15° + sin75°sin15° = cos(75° - 15°) (using cos(A - B) = cosAcosB + sinAsinB) = cos60° = 1/2 (iv) sin40°cos20° + cos40°sin20° = sin(40° + 20°) (using sin(A + B) = sinAcosB +...
Prove that
Answer: (i) sin80°cos20° - cos80°sin20° = sin(80° - 20°) (using sin(A - B) = sinAcosB - cosAsinB) = sin60° = $\frac{\sqrt{3}}{2}$ (ii)cos45°cos15° - sin45°sin15° = cos(45° + 15°) (using cos(A + B) =...
Find the values of all trigonometric functions of 135 deg
Answer:
Find the value of (ix) cos (495֯ )
(ix)cos495° = cos(360° + 135°) …………(using cos(360° + x) = cosx) = cos135° = cos(180° - 45°) ………….(using cos(180° - x) = - cosx) = - cos45° = - 1//2
Find the value of (vii) cot ( – 315֯ ) (viii) sin ( – 1230֯ )
Answer: vii) $ \cot \left( -{{315}^{\circ }} \right)=\frac{1}{\tan \left( -{{315}^{\circ }} \right)} $ $ \Rightarrow \frac{1}{-\tan \left( {{315}^{\circ }} \right)}=\frac{1}{-\tan \left(...
Find the value of (v) cosec ( – 690֯ ) (vi) tan (225֯ )
Answer:
Find the value of (iii) tan ( – 120֯ ) (iv) sec ( – 420֯ )
Answer: (iii) tan( - 120°) = - tan12 …….(tan( - x) = tanx) = - tan(180° - 60°) ……. (in II quadrant tanx is negative) = - ( - tan60°) = tan60°
Find the value of (i) cos 840֯(ii) sin 870֯
Answer: (i) Cos840° = Cos(2.360° + 120°) …………(using Cos(2ϖ + x) = Cosx) = Cos(120°) = Cos(180° - 60°) = - Cos60° ……………(using Cos(ϖ - x) = - Cosx) = - 1/2 (ii) sin870° = sin(2.360° + 150°)...
Find the probability that a leap year selected at random w ill contain 53 Sundays.
All kings, queens, and aces are removed from a pack o f 52 cards. The remaining cards are well-shuffled and then a card Is drawn from I t Find the probability that the drawn card Is
(I) a black face card,
(II)a red face card.
All red face cards are removed from a pack o f playing cards. The remaining cards are well-shuffled and then a card Is drawn at random from them. Find the probability that the drawn card Is
(I) a red card,
(II) a face card,
(III)a card of clubs.
What Is the probability that an ordinary year has 53 Mondays?
A card Is drawn at random from a well-shuffled pack o f 52 cards. Find the probability that the card was drawn Is neither a red card nor a queen.
5 cards the ten, Jack, queen, king and ace o f diamonds are well shuffled with their faces downward. One card Is then picked up at random. (a) What Is the probability that the drawn card Is the queen? (b) If the queen Is drawn and put aside and a second card Is drawn, find the probability that the second card Is (I) an ace, (II) a queen.
A letter Is chosen at random from the letter o f the word ‘ASSOCIATION’. Find the probability that the chosen letter Is a (I) vowel (II) consonant (III) S
Two dice are rolled once. Find the probability o f getting such numbers on 2 dice whose product Is a perfect square.
A die Is rolled twice. Find the probability that _9_ _ 3 12 4
(I) 5 w ill not come up either time,
(II) 5 w ill come up exactly one time,
(III) 5 w ill come up both the times.
A group consists o f 12 persons, o f which 3 are extremely patient, other 6 are extremely honest and rest are extremely kind. A person from the group Is selected at random. Assuming that each person Is equally likely to be selected, find the probability o f selecting a person who Is
(I) extremely patient,
(II) extremely kind o r honest. Which o f the above values did you prefer more?
A carton consists o f 100 shirts o f which 88 are good and 8 have minor defects. Rohlt, a trader, w ill only accept the shirts which are good. But, Kamal, and another trader w ill only reject the shirts which have major defects. 1 shirt Is drawn at random from the carton. What Is the probability that It Is acceptable to
(I)Rohlt,
(II) Kamal?
A Jar contains 54 marbles, each o f which some are blue, some are green and some are white. The probability o f selecting a blue marble at random Is and the probability o f selecting a green marble at random Is | . How many white marbles does the Jar contain?
The sides of a rectangle are given by the equations x = – 2, x = 4, y = – 2 and y = 5. Find the equation of the circle drawn on the diagonal of this rectangle as its diameter.
Answer: The intersection points in clockwise fashion are:( - 2, 5), (4, 5), (4, - 2), ( - 2, -2). The equation of a circle passing through the coordinates of the end points of diameters is (x - x1)...
A Jar contains 24 marbles. Some o f these are green others are blue. If a marble Is drawn at random from the Jar, the probability that It Is green Is | . Find the number o f blue marbles In the Jar.
Find the equation of the circle, the coordinates of the end points of one of whose diameters are A(p, q) and B(r, s)
Answer: The equation of a circle passing through the coordinates of the end points of diameters is (x - x1) (x - x2) + (y - y1)(y - y2) = 0 Substituting the values:(x1, y1) = (p, q) & (x2, y2) =...
Find the equation of the circle, the coordinates of the end points of one of whose diameters are A( – 2, – 3) and B( – 3, 5)
Answer: The equation of a circle passing through the coordinates of the end points of diameters is (x - x1) (x - x2) + (y - y1)(y - y2) = 0 Substituting the values:(x1, y1) = ( - 2, - 3) & (x2,...
Find the equation of the circle, the coordinates of the end points of one of whose diameters are A(5, – 3) and B(2, – 4)
Answer: The equation of a circle passing through the coordinates of the end points of diameters is (x - x1) (x - x2) + (y - y1)(y - y2) = 0 Substituting the values:(x1, y1) = (5, - 3) & (x2, y2)...
Find the equation of the circle, the coordinates of the end points of one of whose diameters are A(3, 2) and B(2, 5)
Answer: The equation of a circle passing through the coordinates of the end points of diameters is (x - x1) (x - x2) + (y - y1)(y - y2) = 0 Substituting the values (x1, y1) = (3, 2) & (x2, y2) =...
If two diameters of a circle lie along the lines x – y = 9 and x – 2y = 7, and the area of the circle is 38.5 sq cm, find the equation of the circle.
Answer : The point of intersection of two diameters is the centre of the circle. The point of intersection of two diameters x – y = 9 and x – 2y = 7 is (11, 2). ∴...
A bag contains 18 balls out o f which x balls are red.
(I)If one ball Is drawn at random from the bag, what Is the probability that It Is not red?
(II) If two more red balls are put In the bag, the probability o f drawing a red ball w ill be | times the probability o f drawing a red ball In the firs t case. Find the value o f x.
Find the equation of the circle passing through the point ( – 1, – 3) and having its centre at the point of intersection of the lines x – 2y = 4 and 2x + 5y + 1 = 0.
Answer: The intersection of the lines: x – 2y = 4 and 2x + 5y + 1 = 0. is (2, - 1) The general form of the equation of a circle is (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is...
Find the equation of the circle whose centre is (2, – 3) and which passes through the intersection of the lines 3x + 2y = 11 and 2x + 3y = 4.
Answer: The intersection of the lines: 3x + 2y = 11 and 2x + 3y = 4 Is (5, - 2) The general form of the equation of a circle is (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the...
Find the equation of the circle of radius 5 cm, whose centre lies on the y – axis and which passes through the point (3, 2).
Answer: The general form of the equation of a circle is (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. The centre lies on Y - axis, ∴ it’s X - coordinate...
Find the equation of the circle whose centre is (2, – 5) and which passes through the point (3, 2).
Answer: The general form of the equation of a circle is (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. (h, k) = (2, - 5) For determining the equation of...
Find the centre and radius of each of the following circles :
Answer: The general form of the equation of a circle is (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. Comparing the given equation of circle with...
Find the centre and radius of each of the following circles :
Answer: The general form of the equation of a circle is (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. Comparing the given equation of circle with...
Find the centre and radius of each of the following circles :
Answer: The general form of the equation of a circle is (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. Comparing the given equation of circle with...
Find the centre and radius of each of the following circles :
Answer: The general form of the equation of a circle is (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. Comparing the given equation of circle with...
Find the equation of a circle with Centre at the origin and radius 4
Answer: The general form of the equation of a circle is: (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. Substituting the centre and radius of the circle...
Find the equation of a circle with Centre ( – a, – b) and radius
Answer: The general form of the equation of a circle is (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. Substituting the centre and radius of the circle...
Find the equation of a circle with Centre (a cos ????, a sin ????) and radius a
Answer: The general form of the equation of a circle is: (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. Substituting the centre and radius of the circle...
Find the equation of a circle with Centre (a, a) and radius √2
Answer: The general form of the equation of a circle is (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. Substituting the centre and radius of the circle...
The probability o f selecting a red ball at random from a Jar that contains only red, blue and orange balls Is j . The probability of selecting a blue ball at random from the same Jar Is j .If the Jar contains 10 orange balls, find the total number o f balls In the Jar.
Find the equation of a circle with Centre ( – 3, – 2) and radius 6
Answer: The general form of the equation of a circle is: (x - h)2 + (y - k)2 = r2 (h, k) is the center of the circle. r is the radius of the circle. Substituting the center and radius of the circle...
Find the equation of a circle with Centre (2, 4) and radius 5
Answer: The general form of the equation of a circle is: (x - h)2 + (y - k)2 = r2 (h, k) is the centre of the circle. r is the radius of the circle. Substituting the centre and radius of the circle...
A piggy bank contains hundred 50-p coins, seventy Rs. 1 coin, fifty Rs. 2 coins and th irty Rs. 5 coins. If It Is equally likely that one of the coins will fall out when the blank Is turned upside down, what Is the probability that the coin(I) will bea R s. 1 coin? (II) will not be a Rs. 5 coin (III) will be 50-p or a Rs. 2 coin?
A box contains 80 discs, which are numbered from 1 to 80. If one disc Is drawn at random from the box, find the probability that It bears a perfect square number.
Tickets numbered 2 ,3 ,4 , 5……………100,101 are placed In a box and mix thoroughly. One ticket Is drawn at random from the box. Find the probability that the number on the ticket Is(III) a number which Is a perfect square (Iv) a prime number less than 40.
Tickets numbered 2 ,3 ,4 , 5……………100,101 are placed In a box and mix thoroughly. One ticket Is drawn at random from the box. Find the probability that the number on the ticket Is
(I)an even number
(II)a number less than 16
Cards marked with numbers 1,3, 5……………..101 are placed In a bag and mixed thoroughly. A card Is drawn at random from the bag. Find the probability that the number on the drawn card Is
(I)less than 19,
(II) a prime number less than 20.
A box contains cards bearing numbers 6 to 70. If one card Is drawn at random from the box, find the probability that It bearsA box contains cards bearing numbers 6 to 70. If one card Is drawn at random from the box, find the probability that It bearsA box contains cards bearing numbers 6 to 70. If one card Is drawn at random from the box, find the probability that It bears
(III) an odd number less than 30,
(Iv) a composite number between 50 and 70.
Prove that:
Answer: Taking LHS: We know that: Putting the values, we get = 4 = RHS ∴ LHS = RHS
A box contains cards bearing numbers 6 to 70. If one card Is drawn at random from the box, find the probability that It bears
(I) a 1 digit number,
(II)a number divisible by 5,
Cards bearing numbers 1,3, 5……………..35 are kept In a bag. A card Is drawn at random from the bag. Find the probability o f getting a card bearing
(I)a prime number less than 15,
(II) a number divisible by 3 and 5.
Prove that
Answer: = RHS ∴ LHS = RHS
Card numbered 1 to 30 are put In a bag. A card Is drawn at random from the bag. Find the probability that the number on the drawn card Is
(I) not divisible by 3,
(II)a prime number greater than 7,
(III)not a perfect square number.
Prove that
Answer: Taking LHS, we have: = tan2 π/3 + 2 cos2 π/4 + 3 sec2 π/6 + 4 cos2 π/2 Putting π = 180° = tan2 60° + 2 cos2 45° + 3 sec2 30° + 4 cos2 90°Now, we know that, = 3 + 1 + 4 = 8 = RHS ∴ LHS =...
A box contains cards numbered 3, 5 , 7 , 9 ……..35,37. A card Is drawn at random from the box. Find the probability that the number on the card Is a prime number.
Find the value of cos (-2220 )
Answer: To find: Value of cos 2220° We have, cos (-2220 ) = cos 2220° [∵ cos(-θ) = cos θ] = cos [2160 + 60°] = cos [360° × 6 + 60°] = cos 60° [Clearly, 2220° is in I Quadrant and the multiple of...
Find the value of cosec (-750 )
Answer: To find: Value of cosec (-750°) We have, cosec (-750°) = - cosec(750°) [∵ cosec(-θ) = -cosec θ] = - cosec [90° × 8 + 30°] Clearly, 405° is in I Quadrant and the multiple of 90° is even = -...
A card Is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card was drawn Is
(III)either a king or queen
(Iv) neither a king nor the queen.
Find the value of cot (585 )
Answer: We have, cot (585°) = cot [90° × 6 + 45°] = cot 45° [Clearly, 585° is in III Quadrant and the multiple of 90° is even] = 1 [∵ cot 45° = 1]
Find the value of tan (-300 )
Answer: To find: Value of tan (-300°) We have, tan (-300°) = - tan (300°) [∵ tan(-θ) = -tan θ] = - tan [90° × 3 + 30°] Clearly, 300° is in IV Quadrant and the multiple of 90° is odd = - cot 30° =...
A card Is drawn at random from a well-shuffled deck o f playing cards. Find the probability that the card was drawn Is
(I)a card of a spade or an Ace
(II)a red king
One card Is drawn from a well-shuffled deck of 52 cards. Find the probability o f getting(v) a Jack o f hearts (vl) a spade.
Find the value of sec (-1470 )
Answer: To find: Value of sec (-1470°) We have, sec (-1470°) = sec (1470°) [∵ sec(-θ) = sec θ] = sec [90° × 16 + 30°] Clearly, 1470° is in I Quadrant and the multiple of 90° is even = sec 30° $\sec...
One card Is drawn from a well-shuffled deck of 52 cards. Find the probability o f getting
(III)a red face card
(Iv) a queen o f black suit
Find the value of sin 405°
Answer: To find: Value of sin 405° We have, sin 405° = sin [90° × 4 + 45°] = sin 45° [Clearly, 405° is in I Quadrant and the multiple of 90° is even] $\sin {{405}^{\circ...
Find the value of
Answer: We have: $ co\sec \left( -\frac{41\pi }{4} \right) $ We know that: $ co\sec \left( -\frac{41\pi }{4} \right)=-co\sec \left( \frac{41\pi }{4} \right) $ [∵ cosec(-θ) = -cosec θ] Putting π =...
Find the value of .
Answer: We have $\sec \left( -\frac{25\pi }{3} \right)$ We know that: $\sec \left( -\frac{25\pi }{3} \right)=\sec \left( \frac{25\pi }{3} \right)$ [∵ sec(-θ) = sec θ] Putting π = 180° = sec[25 ×...
Find the value of
Answer: We have: = cot (13 × 45°) = cot (585°) = cot [90° × 6 + 45°] = cot 45° [Clearly, 585° is in III Quadrant and the multiple of 90° is even] = 1 [∵ cot 45° = 1]
Find the general solution of each of the following equations: cot x + tan x = 2 cosec x
Answer: Given, cot x + tan x = 2 cosec x cos2x + sin2x = 2 sinx cosx cosec x 1 = sin 2x cosec x
Find the general solution of each of the following equations: sin x tan x – 1 = tan x – sin x
Answer: Given, sin x tan x – 1 = tan x – sin x sin x(tan x + 1) = tan x + 1