A and B are two events is given P (A′) = 0.5, P (A ∩ B) = 0.3 and P (A ∪ B) = 0.8 Since, P (A′) = 1 – P (A) P (A) = 1 – 0.5 = 0.5 We need to find P (B). By definition of P (A or B) under axiomatic...
If A and B are two events associated with a random experiment such that P (A) = 0.3, P (B) = 0.4 and P (A ∪ B) = 0.5, find P (A ∩ B).
A and B are two events is given P (A) = 0.3, P (B) = 0.5 and P (A ∪ B) = 0.5 We need to find P (A ∩ B). By definition of P (A or B) under axiomatic approach we know, P (A ∪ B) = P (A) + P (B) – P (A...
Which of the following rational numbers is expressible as a non-terminating decimal?
Correct Answer: Option (c) Explanation: 2, 3 and 5 are not the factors of 3219. So, the given rational is in its simplest form. ∴ (23 × 52 × 32) ≠ (2m × 5n) for some integers m, n. This rational...
Classify the following function as injection, surjection or bijection: f: N → N given by
Given f: N → N, given by \[~f\left( x \right)\text{ }=~{{x}^{2}}\] Now we have to check for the given function is injection, surjection and bijection condition. Injection condition: Let x and y be...
Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.(v) On Z+ define * by a * b = a (vi) On R, define * by a * b = a + 4b2
(v) Given on Z+ define * by a * b = a Let \[\begin{array}{*{35}{l}} a,\text{ }b\text{ }\in \text{ }{{Z}^{+}} \\ \Rightarrow \text{ }a\text{ }\in \text{ }{{Z}^{+}} \\ \Rightarrow \text{ }a\text{...
Give three examples of sentences which are not statements. Give reasons for the answers.
Examples: 1. “We won the match!” The sentence “We won the match!” Is an exclamatory sentence. ∴ It is not a statement. 2. Can u get me cup of tea? This sentence is an interrogative sentence. ∴ It is...
Find out which of the following sentences are statements and which are not. Justify your answer. (i) This sentences is a statement. (ii) Is the earth round?
Answers: (i) This sentences is a statement. The statement is not indicating the correct value, so we can say that value contradicts the sense of the sentence. ∴ It is not a statement. (ii) Is the...
Find out which of the following sentences are statements and which are not. Justify your answer. (i) Every rhombus is a square. (ii) x2 + 5|x| + 6 = 0 has no real roots.
Answers: (i) Every rhombus is a square. This sentence is always false, because Rhombuses are not a square. ∴ It is a statement. (ii) x2 + 5|x| + 6 = 0 has no real roots. Let us take, If x>0,...
Find out which of the following sentences are statements and which are not. Justify your answer. (i) Are all circles round? (ii) All triangles have three sides.
Answers: (i) Are all circles round? The given sentence is an interrogative sentence. ∴ It is not a statement. (ii) All triangles have three sides. The given sentence is a true declarative sentence....
Find out which of the following sentences are statements and which are not. Justify your answer. (i) Two non-empty sets have always a non-empty intersection. (ii) The cat pussy is black.
Answers: (i) Two non-empty sets have always a non-empty intersection. This sentence is always false, because there are non-empty sets whose intersection is empty. ∴ It is a statement. (ii) The cat...
Find out which of the following sentences are statements and which are not. Justify your answer. (i) Listen to me, Ravi! (ii) Every set is a finite set.
Answers: (i) Listen to me, Ravi! The sentence “Listen to me, Ravi! “Is an exclamatory sentence. ∴ It is not a statement. (ii) Every set is a finite set. This sentence is always false, because there...
Check whether the following pair of statements is a negation of each other. Give reasons for your answer. (i) a + b = b + a is true for every real number a and b. (ii) There exist real numbers a and b for which a + b = b + a.
Answer: The negation of the statement: p: a + b = b + a is a true for every real number a and b. ~p: There exist real numbers are ‘a’ and ‘b’ for which a+b ≠ b+a. The statement is not the negation...
Write the negation of the following statement: r: There exists a number x such that 0 < x < 1.
Answer: The negation of the statement: r: There exists a number x such that 0 < x < 1. ~r: For every real number x, either x ≤ 0 or x ≥ 1.
Write the negation of the following statements: (i) p: For every positive real number x, the number (x – 1) is also positive. (ii) q: For every real number x, either x > 1 or x < 1.
Answers: (i) The negation of the statement: p: For every positive real number x, the number (x – 1) is also positive. ~p: There exists a positive real number x, such that the number (x – 1) is not...
Are the following pairs of statements are a negation of each other: (i) The number x is not a rational number. The number x is not an irrational number. (ii) The number x is not a rational number. The number x is an irrational number.
Answers: (i) “The number x is an irrational number.” The statement “The number x is not a rational number.” It is a negation of the first statement. (ii) “The number x is an irrational number.” The...
(i) Both the diagonals of a rectangle have the same length. (ii) All policemen are thieves.
Answers: (i) The negation of the statement is: “There is at least one rectangle whose both diagonals do not have the same length.” (ii) The negation of the statement is: “No policemen are...
(i) There is a complex number which is not a real number. (ii) I will not go to school.
Answers: (i) The negation of the statement is: It is false that “complex numbers are not a real number.” [Or] “All complex number are real numbers.” (ii) The negation of the statement is: “I will...
(i) All birds sing. (ii) Some even integers are prime.
Answers: (i) The negation of the statement is: It is false that “All birds sing.” [Or] “All birds do not sing.” (ii) The negation of the statement is: It is false that “even integers are prime.”...
Write the negation of the following statement: The sun is cold.
Answer: The negation of the statement is: It is false that “The sun is cold.” [Or] “The sun is not cold.”
Write the negation of the following statement: (i) Ravish is honest. (ii) The earth is round.
Answers: (i) The negation of the statement is: It is false that “Ravish is honest.” [Or] “Ravish is not honest.” (ii) The negation of the statement is: It is false that “The earth is round.” [Or]...
Write the negation of the following statement: (i) Bangalore is the capital of Karnataka. (ii) It rained on July 4, 2005.
Answers: (i) The negation of the statement is: It is false that “Bangalore is the capital of Karnataka.” [Or] “Bangalore is not the capital of Karnataka.” (ii) The negation of the statement is: It...
Determine whether the following compound statements are true or false: (i) Delhi is in India and 2 + 2 = 5 (ii) Delhi is in England and 2 + 2 = 5
Answers: (i) The components of the compound statement are: P: Delhi is in India. Q: 2 + 2 = 5 Both P and Q are false. The compound statement is False. (ii) The components of the compound statement...
Determine whether the following compound statements are true or false: (i) Delhi is in India and 2 + 2 = 4 (ii) Delhi is in England and 2 + 2 = 4
Answers: (i) The components of the compound statement are: P: Delhi is in India. Q: 2 + 2 = 4 Both P and Q are true. The compound statement is True. (ii) The components of the compound statement...
Write the component statements of the following compound statements and check whether the compound statement is true or false: The sand heats up quickly in the sun and does not cool down fast at night.
Answer: The components of the compound statement are: P: The sand heats up quickly in the sun. Q: The sand does not cool down fast at night. P is false and Q is also false then P and Q both are...
Write the component statements of the following compound statements and check whether the compound statement is true or false: (i) Square of an integer is positive or negative. (ii) x = 2 and x = 3 are the roots of the equation 3×2 – x – 10 = 0.
Answers: (i) The components of the compound statement are: P: Square of an integer is positive. Q: Square of an integer is negative. Both P and Q are true. The compound statement is True. (ii) The...
Write the component statements of the following compound statements and check whether the compound statement is true or false: (i) To enter into a public library children need an identification card from the school or a letter from the school authorities. (ii) All rational numbers are real and all real numbers are not complex.
Answers: (i) The components of the compound statement are: P: To get into a public library children need an identity card. Q: To get into a public library children need a letter from the school...
For each of the following statements, determine whether an inclusive “OR” o exclusive “OR” is used. Give reasons for your answer. (i) A lady gives birth to a baby boy or a baby girl. (ii) To apply for a driving license, you should have a ration card or a passport.
Answers: (i) “A lady gives birth to a baby boy or a baby girl.” An exclusive “OR” is used because a lady cannot give birth to a baby who is both a boy and a girl. (ii) “To apply for a driving...
For each of the following statements, determine whether an inclusive “OR” o exclusive “OR” is used. Give reasons for your answer. (i) Students can take Hindi or Sanskrit as their third language. (ii) To entry a country, you need a passport or a voter registration card.
Answers: (i) “Students can take Hindi or Sanskrit as their third language.” An exclusive “OR” is used because a student cannot take both Hindi and Sanskrit as the third language. (ii) “To entry a...
Find the component statements of the following compound statements: (i) All rational numbers are real, and all real numbers are complex. (ii) 25 is a multiple of 5 and 8.
Answers: (i) The components of the compound statement are: P: All rational number is real. Q: All real number are complex. (ii) The components of the compound statement are: P: 25 is multiple of 5....
Find the component statements of the following compound statements: (i) The sky is blue, and the grass is green. (ii) The earth is round, or the sun is cold.
Answers: (i) The components of the compound statement are: P: The sky is blue. Q: The grass is green. (ii) The components of the compound statement are: P: The earth is round. Q: The sun is...
Negate each of the following statements: (i) All the students complete their homework. (ii) There exists a number which is equal to its square.
Answers: (i) The negation of the statement is “Some of the students did not complete their homework.” (ii) The negation of the statement is “For every real number x, x2≠x.”
Write the negation of each of the following statements: (i) For every x ∈ N, x + 3 > 10 (ii) There exists x ∈ N, x + 3 = 10
Answers: (i) The negation of the statement is “There exist x ∈ N, such that x + 3 ≥ 10.” (ii) The negation of the statement is “There exist x ∈ N, such that x + 3 ≠...
Determine the Contrapositive of each of the following statement: If x is an integer and x2 is odd, then x is odd.
Answer: The statement: If x is an integer and x2 is odd, then x is odd. Contrapositive: If x is even, then x2 is even.
Determine the Contrapositive of each of the following statements: (i) It is necessary to be strong in order to be a sailor. (ii) Only if he does not tire will he win.
Answers: (i) Contrapositive: If he is not strong, then he is not a sailor (ii) Contrapositive: If he tries, then he will not win.
Determine the Contrapositive of each of the following statements: (i) If x is less than zero, then x is not positive. (ii) If he has courage he will win.
Answers: (i) Contrapositive: If x is positive, then x is not less than zero. (ii) Contrapositive: If he does not win, then he does not have courage.
Determine the Contrapositive of each of the following statements: (i) It never rains when it is cold. (ii) If Ravish skis, then it snowed.
Answers: (i) Contrapositive: If it rains, then it is not cold. (ii) Contrapositive: If it did not snow, then Ravish will not ski.
Determine the Contrapositive of each of the following statements: (i) If she works, she will earn money. (ii) If it snows, then they do not drive the car.
Answers: (i) Contrapositive: If she does not earn money, then she does not work. (ii) Contrapositive: If then they do not drive the car, then there is no snow.
Determine the Contrapositive of each of the following statements: (i) If Mohan is a poet, then he is poor. (ii) Only if Max studies will he pass the test.
Answers: (i) Contrapositive: If Mohan is not poor, then he is not a poet. (ii) Contrapositive: If Max does not study, then he will not pass the test.
Rewrite each of the following statements in the form “p if and only is q.” (i) r: For you to get an A grade, it is necessary and sufficient that you do all the homework you regularly. (ii) s: If a tumbler is half empty, then it is half full, and if a tumbler is half full, then it is half empty.
Answers: (i) In the form “p if and only is q.” You get an A grade if and only if you do all the homework regularly. (ii) In the form “p if and only is q.” A tumbler is half empty if and only if it...
Rewrite each of the following statements in the form “p if and only is q.” (i) p: If you watch television, then your mind is free, and if your mind is free, then you watch television. (ii) q: If a quadrilateral is equiangular, then it is a rectangle, and if a quadrilateral is a rectangle, then it is equiangular.
Answers: (i) In the form “p if and only is q.” You watch television if and only if your mind is free. (ii) In the form “p if and only is q.” A quadrilateral is a rectangle if and only if it is...
State the converse and contrapositive of each of the following statement: If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Answer: Converse: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Contrapositive: If the diagonals of a quadrilateral do not bisect each other, then the...
State the converse and contrapositive of each of the following statements: (i) A positive integer is prime only if it has no divisions other than 1 and itself. (ii) If you live in Delhi, then you have winter clothes.
Answers: (i) Converse: If an integer has no divisor other that 1 and itself, then it is prime. Contrapositive: If an integer has some divisor other than 1 and itself, then it is prime. (ii) ...
State the converse and contrapositive of each of the following statements: (i) If it is hot outside, then you feel thirsty. (ii) I go to a beach whenever it is a sunny day.
Answers: (i) Converse: If you feel thirsty, then it is hot outside. Contrapositive: If you do not feel thirsty, then it is not hot outside. (ii) Converse: If I go to a beach, then it is a sunny...
Write each of the following statements in the form “if p, then q.” (i) Whenever it rains, it is cold. (ii) It never rains when it is cold.
Answers: (i) In the form “if p, then q.” If it rains, then it is cold. (ii) In the form “if p, then q.” If it is cold, then it never rains.
Write each of the following statements in the form “if p, then q.” (i) The game is canceled only if it is raining. (ii) It rains only if it is cold.
Answers: (i) In the form “if p, then q.” If it is raining, then the game is canceled. (ii) In the form “if p, then q.” If it rains, then it is cold.
Write each of the following statements in the form “if p, then q.” (i) It is necessary to have a passport to log on to the server. (ii) It is necessary to be rich in order to be happy.
Answers: (i) In the form “if p, then q.” If you log on the server, then you must have a passport. (ii) In the form “if p, then q.” If he is happy, then he is rich.
Write each of the following statements in the form “if p, then q.” (i) You can access the website only if you pay a subscription fee. (ii) There is traffic jam whenever it rains.
Answers: (i) In the form “if p, then q.” If you access the website, then you pay a subscription fee. (ii) In the form “if p, then q.” If it rains, then there is a traffic jam.
Determine whether the argument used to check the validity of the following statement is correct: p: “If x2 is irrational, then x is rational.” The statement is true because the number x2 = π2 is irrational, therefore x = π is irrational.
Answer: Argument Used: x2 = π2 is irrational So, x = π is irrational. p: “If x2 is irrational, then x is rational.” Consider, An irrational number given by x = √k [k is a rational number] Squaring...
Which of the following statements are true and which are false? In each case give a valid reason for saying so t: √11 is a rational number.
Answer: t: √11 is a rational number. Square root of prime numbers is irrational numbers. The statement is False.
Which of the following statements are true and which are false? In each case give a valid reason for saying so (i) r: Circle is a particular case of an ellipse. (ii) s: If x and y are integers such that x > y, then – x < – y.
Answers: (i) r: Circle is a particular case of an ellipse. A circle can be an ellipse in a particular case when the circle has equal axes. The statement is true. (ii) s: If x and y are integers such...
Which of the following statements are true and which are false? In each case give a valid reason for saying so (i) p: Each radius of a circle is a chord of the circle. (ii) q: The centre of a circle bisect each chord of the circle.
Answers: (i) p: Each radius of a circle is a chord of the circle. The Radius of the circle is not it chord. The statement is False. (ii) q: The centre of a circle bisect each chord of the circle. A...
By giving a counter example, show that the following statement is not true. p: “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.”
Answer: Consider, A triangle ABC with all angles equal. [Each angle of the triangle is equal to 60] ABC is not an obtuse angle triangle. The statement “p: If all the angles of a triangle are equal,...
Show that the following statement is true “The integer n is even if and only if n2 is even”
Answer: Consider, p: Integer n is even q: If n2 is even Let p be true. Let n = 2k Squaring both the sides, n2 = 4k2 n2 = 2.2k2 n2 is an even number. q is true when p is true. The statement is...
Show that the following statement is true by the method of the contrapositive p: “If x is an integer and x2 is odd, then x is also odd.”
Answer: Consider, q: x is an integer and x2 is odd. r: x is an odd integer. p: if q, then r. Let r be false. x is not an odd integer, then x is an even integer x = (2n) for some integer n x2 = 4n2...
Show that the statement p : “If x is a real number such that x3 + x = 0, then x is 0” is true by method of contradiction
Answer: Method of Contradiction: If possible, let p be false. P is not true -p is true -p (p => r) is true q and –r is true x is a real number such that x3+x = 0and x≠ 0 x =0 and x≠0 This is a...
Show that the statement p : “If x is a real number such that x3 + x = 0, then x is 0” is true by (i) Direct method (ii) method of Contrapositive
Answers: (i) Direct Method: Consider, q: x is a real number such that x3 + x=0. r: x is 0. If q, then r. Let q be true. Then, x is a real number such that x3 + x = 0 x is a real number such that...
Check whether the following statement is true or not: (i) p: If x and y are odd integers, then x + y is an even integer. (ii) q : if x, y are integer such that xy is even, then at least one of x and y is an even integer.
Answers: (i) p: If x and y are odd integers, then x + y is an even integer. Conisder, p: x and y are odd integers. q: x + y is an even integer If p, then q. Let p be true. [x and y are odd integers]...
Check the validity of the following statement: r: 60 is a multiple of 3 or 5.
Answer: r: 60 is a multiple of 3 or 5. We know that 60 is a multiple of 3 as well as 5. Hence, the statement is true.
Check the validity of the following statements: (i) p: 100 is a multiple of 4 and 5. (ii) q: 125 is a multiple of 5 and 7.
Answers: (i) p: 100 is a multiple of 4 and 5. We know that 100 is a multiple of 4 as well as 5. Hence, the statement is true. (ii) q: 125 is a multiple of 5 and 7 We know that 125 is a multiple of 5...
Determine whether the point (-3, 2) lies inside or outside the triangle whose sides are given by the equations x + y – 4 = 0, 3x – 7y + 8 = 0, 4x – y – 31 = 0. Solution:
According to ques,: \[x\text{ }+\text{ }y\text{ }\text{ }4\text{ }=\text{ }0,\] \[3x\text{ }\text{ }7y\text{ }+\text{ }8\text{ }=\text{ }0,\] And \[4x\text{ }\text{ }y\text{ }\text{ }31\text{...
Find the values of the parameter a so that the point (a, 2) is an interior point of the triangle formed by the lines x + y – 4 = 0, 3x – 7y – 8 = 0 and 4x – y – 31 = 0.
According to ques,: \[x\text{ }+\text{ }y\text{ }\text{ }4\text{ }=\text{ }0,\] \[3x\text{ }\text{ }7y\text{ }\text{ }8\text{ }=\text{ }0\] and \[4x\text{ }\text{ }y\text{ }\text{ }31\text{ }=\text{...
Find the values of α so that the point P(α 2, α) lies inside or on the triangle formed by the lines x – 5y + 6 = 0, x – 3y + 2 = 0 and x – 2y – 3 = 0.
According to ques,: \[x\text{ }\text{ }5y\text{ }+\text{ }6\text{ }=\text{ }0,\] \[x\text{ }\text{ }3y\text{ }+\text{ }2\text{ }=\text{ }0\] and \[x\text{ }\text{ }2y\text{ }\text{ }3\text{ }=\text{...
if is the angle which the straight line joining the points(x1,y1) and (x2,y2) subtends at the origin , prove that tan = x2y1-x1y2/x1x2+y1y2 and cos = x1y2+y1y2/
to prove: Let us assume A (x1, y1) and B (x2, y2) be the given points and O be the origin. \[Slope\text{ }of\text{ }OA\text{ }=\text{ }{{m}_{1}}~=\text{ }{{y}_{1\times 1}}\] \[Slope\text{ }of\text{...
Find the angle between the line joining the points (2, 0), (0, 3) and the line x + y = 1.
According to ques,: Points (2, 0), (0, 3) and the line x + y = 1. Let us assume A (2, 0), B (0, 3) be the given points. Now, let us find the slopes \[Slope\text{ }of\text{...
Prove that the points (2, -1), (0, 2), (2, 3) and (4, 0) are the coordinates of the vertices of a parallelogram and find the angle between its diagonals.
To prove: The points: \[\left( 2,\text{ }-1 \right),\text{ }\left( 0,\text{ }2 \right),\text{ }\left( 2,\text{ }3 \right)\text{ }and\text{ }\left( 4,\text{ }0 \right)\] are the coordinates of the...
Find the acute angle between the lines 2x – y + 3 = 0 and x + y + 2 = 0.
According to ques,: The equations of the lines are \[2x~-~y\text{ }+\text{ }3\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] \[x\text{ }+\text{ }y\text{ }+\text{ }2\text{...
Find the angles between each of the following pairs of straight lines: (i) 3x + y + 12 = 0 and x + 2y – 1 = 0 (ii) 3x – y + 5 = 0 and x – 3y + 1 = 0
(i) \[3x\text{ }+\text{ }y\text{ }+\text{ }12\text{ }=\text{ }0\text{ }and\text{ }x\text{ }+\text{ }2y\text{ }\text{ }1\text{ }=\text{ }0\] According to ques,: The equations of the lines are...
Find the equation of a line which is perpendicular to the line√3x – y + 5 = 0 and which cuts off an intercept of 4 units with the negative direction of y-axis.
According to ques,: The equation is perpendicular to: \[~\surd 3x\text{ }\text{ }y\text{ }+\text{ }5\text{ }=\text{ }0~\] equation and cuts off an intercept of 4 units with the negative direction of...
Find the equations of the altitudes of a ΔABC whose vertices are A (1, 4), B (-3, 2) and C (-5, -3).
According to ques,: The vertices of ∆ABC are A (1, 4), B (− 3, 2) and C (− 5, − 3). Now let us find the slopes of ∆ABC. \[Slope\text{ }of\text{ }AB\text{ }=\text{ }\left[ \left( 2\text{ }\text{ }4...
Find the equation of the perpendicular bisector of the line joining the points (1, 3) and (3, 1).
According to ques,: A (1, 3) and B (3, 1) be the points joining the perpendicular bisector Let C be the midpoint of AB. hence, coordinates of C \[=\text{ }\left[ \left( 1+3 \right)/2,\text{ }\left(...
Find the equation of a line passing through (3, -2) and perpendicular to the line x – 3y + 5 = 0.
According to ques,: The equation of the line perpendicular to: \[~x~-~3y\text{ }+\text{ }5\text{ }=\text{ }0\] is \[3x\text{ }+\text{ }y\text{ }+~\lambda ~=\text{ }0,\] Where, λ is a constant. It...
Find the equation of a line passing through the point (2, 3) and parallel to the line 3x– 4y + 5 = 0.
Given: The equation of the line parallel to: \[3x~-~4y\text{ }+\text{ }5\text{ }=\text{ }0\] is \[3x\text{ }\text{ }4y\text{ }+~\lambda ~=\text{ }0,\] Where, λ is a constant. It passes through...
Show that the straight lines L1 = (b + c)x + ay + 1 = 0, L2 = (c + a)x + by + 1 = 0 and L3 = (a + b)x + cy + 1 = 0 are concurrent.
Given: \[{{L}_{1}}~=\text{ }\left( b\text{ }+\text{ }c \right)x\text{ }+\text{ }ay\text{ }+\text{ }1\text{ }=\text{ }0\] \[{{L}_{2}}~=\text{ }\left( c\text{ }+\text{ }a \right)x\text{ }+\text{...
If the lines p1x + q1y = 1, p2x + q2y = 1 and p3x + q3y = 1 be concurrent, show that the points (p1, q1), (p2, q2) and (p3, q3) are collinear.
Given: \[{{p}_{1}}x\text{ }+\text{ }{{q}_{1}}y\text{ }=\text{ }1\] \[{{p}_{2}}x\text{ }+\text{ }{{q}_{2}}y\text{ }=\text{ }1\] and , \[{{p}_{3}}x\text{ }+\text{ }{{q}_{3}}y\text{ }=\text{ }1\] The...
Find the conditions that the straight lines y = m1x + c1, y = m2x + c2 and y = m3x + c3 may meet in a point.
Given: \[{{m}_{1}}x\text{ }\text{ }y\text{ }+\text{ }{{c}_{1}}~=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] \[{{m}_{2}}x\text{ }\text{ }y\text{ }+\text{ }{{c}_{2}}~=\text{ }0\text{ }\ldots...
For what value of λ are the three lines 2x – 5y + 3 = 0, 5x – 9y + λ = 0 and x – 2y + 1 = 0 concurrent?
Given: \[2x~-~5y\text{ }+\text{ }3\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] \[5x~-~9y\text{ }+~\lambda ~=\text{ }0\text{ }\ldots \text{ }\left( 2 \right)\] And , \[x~-~2y\text{...
Prove that the following sets of three lines are concurrent: (i) 15x – 18y + 1 = 0, 12x + 10y – 3 = 0 and 6x + 66y – 11 = 0 (ii) 3x – 5y – 11 = 0, 5x + 3y – 7 = 0 and x + 2y = 0
\[\left( \mathbf{i} \right)~15x\text{ }\text{ }18y\text{ }+\text{ }1\text{ }=\text{ }0,\text{ }12x\text{ }+\text{ }10y\text{ }\text{ }3\text{ }=\text{ }0\] and \[6x\text{ }+\text{ }66y\text{ }\text{...
Prove that the lines y = √3x + 1, y = 4 and y = -√3x + 2 form an equilateral triangle.
Given: \[y\text{ }=\text{ }\surd 3x\text{ }+\text{ }1\ldots \ldots \text{ }\left( 1 \right)\] \[y\text{ }=\text{ }4\text{ }\ldots \ldots .\text{ }\left( 2 \right)\] and \[y\text{ }=\text{ }\text{...
Find the equations of the medians of a triangle, the equations of whose sides are: 3x + 2y + 6 = 0, 2x – 5y + 4 = 0 and x – 3y – 6 = 0
Given: \[3x\text{ }+\text{ }2y\text{ }+\text{ }6\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] \[2x\text{ }-\text{ }5y\text{ }+\text{ }4\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 2...
Find the area of the triangle formed by the lines y = m1x + c1, y = m2x + c2 and x = 0
Given: \[y\text{ }=\text{ }{{m}_{1}}x\text{ }+\text{ }{{c}_{1}}~\ldots \text{ }\left( 1 \right)\] \[y\text{ }=\text{ }{{m}_{2}}x\text{ }+\text{ }{{c}_{2}}~\ldots \text{ }\left( 2 \right)\] and ,...
Find the coordinates of the vertices of a triangle, the equations of whose sides are: (i) x + y – 4 = 0, 2x – y + 3 0 and x – 3y + 2 = 0 (ii) y (t1 + t2) = 2x + 2at1t2, y (t2 + t3) = 2x + 2at2t3 and, y(t3 + t1) = 2x + 2at1t3.
\[\left( \mathbf{i} \right)~x\text{ }+\text{ }y\text{ }\text{ }4\text{ }=\text{ }0,\text{ }2x\text{ }\text{ }y\text{ }+\text{ }3\text{ }0\] and \[x\text{ }\text{ }3y\text{ }+\text{ }2\text{ }=\text{...
Find the point of intersection of the following pairs of lines: (i) 2x – y + 3 = 0 and x + y – 5 = 0 (ii) bx + ay = ab and ax + by = ab
\[\left( \mathbf{i} \right)~2x\text{ }\text{ }y\text{ }+\text{ }3\text{ }=\text{ }0\] And \[x\text{ }+\text{ }y\text{ }\text{ }5\text{ }=\text{ }0\] Given: The equations of the lines are: \[2x\text{...
The ratio of the sum of the first three terms is to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
Solution: Given that, The sum of G.P of 3 terms is 125 Using the formula, The sum of GP for $n$ terms $=a\left(r^{n}-1\right) /(r-1)$ $125=\mathrm{a}\left(\mathrm{r}^{\mathrm{n}}-1\right)...
The common ratio of a G.P. is 3, and the last term is 486. If the sum of these terms be 728, find the first term.
Solution: Given that, The sum of GP $= 728$ Where, $r = 3, a = ?$ Firstly, $T_{n}=a r^{n-1}$ $486=a 3^{n-1}$ $486=a 3^{n} / 3$ $486(3)=a 3^{n}$ $1458=a 3^{n} \ldots .$ Eq. (i) $486=a 3^{n} / 3$...
The sum of n terms of the G.P. 3, 6, 12, … is 381. Find the value of n.
Solution: Given that, The sum of GP $=381$ Where, $a=3, r=6 / 3=2, n=?$ Using the formula, The sum of GP for $n$ terms $=a\left(r^{n}-1\right) /(r-1)$ $\begin{array}{l} 381=3\left(2^{n}-1\right)...
How many terms of the sequence √3, 3, 3√3,… must be taken to make the sum 39+ 13√3 ?
Solution: Given that, The sum of GP $=39+13 \sqrt{3}$ Where, $a=\sqrt{3}, r=3 / \sqrt{3}=\sqrt{3}, n=?$ Using the formula, The sum of GP for $n$ terms $=a\left(r^{n}-1\right) /(r-1)$ $39+13...
How many terms of the series 2 + 6 + 18 + …. Must be taken to make the sum equal to 728?
Solution: Given that, The sum of GP $=728$ Where, $a=2, r=6 / 2=3, n=?$ Using the formula, The sum of GP for $n$ terms $=a\left(r^{n}-1\right) /(r-1)$ $\begin{array}{l} 728=2\left(3^{n}-1\right)...
How many terms of the G.P. 3, 3/2, ¾, … Be taken together to make 3069/512 ?
Solution: Given that, The sum of G.P $=3069 / 512$ Where, $a=3, r=(3 / 2) / 3=1 / 2, n=?$ Using the formula, The sum of GP for $n$ terms $=a\left(r^{n}-1\right) /(r-1)$ $\begin{array}{l} 3069 /...
Find the sum of the following series:
(i) 0.6 + 0.66 + 0.666 + …. to n terms.
Solution: (i) $0.6+0.66+0.666+\ldots$ to $n$ terms. Let,s take 6 as a common term therefore we obtain, $6(0.1+0.11+0.111+\ldots n$ terms $)$ Now multiplying and dividing by 9 we obtain, $6 /...
Find the sum of the following series:
(i) 9 + 99 + 999 + … to n terms.
(ii) 0.5 + 0.55 + 0.555 + …. to n terms
Solution: (i) $9+99+999+\ldots$ to $n$ terms. We can write the given terms as $\begin{array}{l} (10-1)+(100-1)+(1000-1)+\ldots+n \text { terms } \\ \left(10+10^{2}+10^{3}+\ldots n \text { terms...
Find the sum of the following series:
(i) 5 + 55 + 555 + … to n terms.
(ii) 7 + 77 + 777 + … to n terms.
Solution: (i) $5+55+555+\ldots$ to $n$ terms. Let's take 5 as a common term therefore we obtain, $5[1+11+111+\ldots \mathrm{n}$ terms $]$ Now multiplying and dividing by 9 we obtain, $5 /...
Evaluate the following:
(i)
Solution: (i) $\sum_{n=2}^{10} 4^{n}$ $=4^{2}+4^{3}+4^{4}+\ldots+4^{10}$ Where, $a=4^{2}=16, r=4^{3} / 4^{2}=4, n=9$ Using the formula, The sum of GP for $n$ terms $=a\left(r^{n}-1\right) /(r-1)$...
Evaluate the following:
(i)
(ii)
Solution: (i) $\sum_{n=1}^{11}\left(2+3^{n}\right)$ $\begin{array}{l} =\left(2+3^{1}\right)+\left(2+3^{2}\right)+\left(2+3^{3}\right)+\ldots+\left(2+3^{11}\right) \\ =2 \times...
Find the sum of the following geometric series:
(i) to terms;
Solution: (i) $3 / 5+4 / 5^{2}+3 / 5^{3}+4 / 5^{4}+\ldots$ to $2 \mathrm{n}$ terms; We can write the series as: $3\left(1 / 5+1 / 5^{3}+1 / 5^{5}+\ldots\right.$ to $n$ terms $)+4\left(1 / 5^{2}+1 /...
Find the sum of the following geometric series:
(i) to 5 terms;
(ii) to terms
Solution: (i) $2 / 9-1 / 3+1 / 2-3 / 4+\ldots$ to 5 terms; Given that $\begin{array}{l} a=2 / 9 \\ r=t_{2} / t_{1}=(-1 / 3) /(2 / 9)=-3 / 2 \\ n=5 \end{array}$ Using the formula, The sum of GP for...
Find the sum of the following geometric series:
(i) to 8 terms;
(ii) to 8 terms;
Solution: (i) $0.15+0.015+0.0015+\ldots$ to 8 terms Given that $\begin{array}{l} a=0.15 \\ r=t_{2} / t_{1}=0.015 / 0.15=0.1=1 / 10 \\ n=8 \end{array}$ Using the formula, The sum of GP for $n$ terms...
Prove that : (i) tan 36o + tan 9o + tan 36o tan 9o = 1 (ii) tan 13x – tan 9x – tan 4x = tan 13x tan 9x tan 4x
(i) \[tan\text{ }{{36}^{o}}~+\text{ }tan\text{ }{{9}^{o}}~+\text{ }tan\text{ }{{36}^{o}}~tan\text{ }{{9}^{o}}~=~1\] We know \[36{}^\circ \text{ }+\text{ }9{}^\circ \text{ }=\text{ }45{}^\circ \] So...
Find the sum of the following geometric progressions:
(i) to 10 terms
Solution: (i) $4,2,1,1 / 2 \ldots$ to 10 terms It is known that, the sum of GP for $n$ terms $=a\left(r^{n}-1\right) /(r-1)$ Given that, $\mathrm{a}=4, \mathrm{r}=\mathrm{t}_{2} / \mathrm{t}_{1}=2 /...
Find the sum of the following geometric progressions:
(i)
(ii) to
Solution: (i) $1,-1 / 2,1 / 4,-1 / 8, \ldots$ It is known that, the sum of $\mathrm{GP}$ for infinity $=\mathrm{a} /(1-\mathrm{r})$ Given that, $\mathrm{a}=1, \mathrm{r}=\mathrm{t}_{2} /...
Find the sum of the following geometric progressions:
(i) 2, 6, 18, … to 7 terms
(ii) 1, 3, 9, 27, … to 8 terms
Solution: (i) $2,6,18, \ldots$ to 7 terms It is known that, the sum of GP for $n$ terms $=a\left(r^{n}-1\right) /(r-1)$ Given that, $a=2, r=t_{2} / t_{1}=6 / 2=3, n=7$ Substitute the values in...
Prove that: (i) tan 8x – tan 6x – tan 2x = tan 8x tan 6x tan 2x (ii) tan π/12 + tan π/6 + tan π/12 tan π/6 = 1
\[\left( \mathbf{i} \right)tan\text{ }8x\text{ }-\text{ }tan\text{ }6x\text{ }-\text{ }tan\text{ }2x\text{ }=\text{}tan\text{}8x\text{}tan\text{ }6x\text{ }tan\text{ }2x\] Let us consider LHS:...
The sum of the first three terms of a G.P. is 39 / 10, and their product is 1 . Find the common ratio and the terms.
Solution: Let the three numbers be $a / r, a, ar$ According to the question $\mathrm{a} / \mathrm{r}+\mathrm{a}+\mathrm{ar}=39 / 10 \ldots$ eq. (1) $\mathrm{a} / \mathrm{r} \times \mathrm{a} \times...
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is 871 / 2. Find them.
Solution: Let the three numbers be $a/r$, $a$, $ar$ According to the question $\mathrm{a} / \mathrm{r} \times \mathrm{a} \times \mathrm{ar}=125 \ldots$ eq.(1) From eq.(1) we obtain,...
Prove that: (i) cos^2 A + cos^2 B – 2 cos A cos B cos (A +B) = sin^2 (A + B)
(ii) Solution: (i) \[co{{s}^{2}}A+co{{s}^{2}}B - 2cosAcosBcos\left(A+B \right)=si{{n}^{2}}\left(A+B\right)\] Let us consider LHS: \[co{{s}^{2}}A\text{ }+\text{...
The sum of first three terms of a G.P. is 13 / 12, and their product is -1 . Find the G.P.
Solution: Let the three numbers be $a/r$, $a$, $ar$ According to the question $\mathrm{a} / \mathrm{r}+\mathrm{a}+\mathrm{ar}=13 / 12 \ldots$ eq. (1) $\mathrm{a} / \mathrm{r} \times \mathrm{a}...
Find three number in G.P. whose sum is 38 and their product is 1728.
Solution: Let the three numbers be $\mathrm{a} / \mathrm{r}, \mathrm{a}$, $ar$ According to the question, $\mathrm{a} / \mathrm{r}+\mathrm{a}+\mathrm{ar}=38 \ldots$ eq.(1) $\mathrm{a} / \mathrm{r}...
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Solution: According to the question $\mathrm{a} / \mathrm{r}+\mathrm{a}+\mathrm{ar}=65 \ldots$ equation (1) $a / r \times a \times a r=3375 \ldots$ equation (2) From eq.(2) we obtain,...
Prove that: (i)sin2 B = sin2 A + sin^2 (A-B) – 2sin A cos B sin (A – B)
(ii) Solution: (i) \[si{{n}^{2}}B=si{{n}^{2}}A+si{{n}^{2}}\left( A-B \right)2sin\text{}A\text{}cosB\text{}sin\left(A\text{}-\text{}B\right)\] Let us consider RHS:...
Prove that:
(i) (ii) Solution: (i) \[=~tan\text{ }A\] \[=\text{ }RHS\] \[\therefore LHS\text{ }=\text{ }RHS\] Hence proved. (ii) \[=~tan\text{ }A\text{ }-\text{ }tan\text{...
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Find the 4th term from the end of the G.P. ½, 1/6, 1/18, 1/54, … , 1/4374
Solution: $n^{th}$ term from the end is given by: $a_{n}=I(1 / r)^{n-1}$ where, $I$ is the last term, $r$ is the common ratio, $n$ is the $n^{th}$ term Given that, last term, $I=1 / 4374$...
Evaluate the following limits:
Which term of the progression 18, -12, 8, … is 512/729 ?
Solution: Using the formula, $\begin{array}{l} \mathrm{T}_{n}=\mathrm{ar}^{\mathrm{n}-1} \\ \mathrm{a}=18 \\ \mathrm{r}=\mathrm{t}_{2} / \mathrm{t}_{1}=(-12 / 18) \\ =-2 / 3 \\...
Which term of the G.P.:
(i) √3, 3, 3√3, … is 729 ?
(ii) 1/3, 1/9, 1/27… is 1/19683 ?
Solution: (i) $\sqrt{3}, 3,3 \sqrt{3}, \ldots$ is $729 ?$ Using the formula, $\begin{array}{l} \mathrm{T}_{\mathrm{n}}=\mathrm{ar}^{\mathrm{n}-1} \\ \mathrm{a}=\sqrt{3} \\ \mathrm{r}=\mathrm{t}_{2}...
Evaluate the following limits:
Evaluate the following limits:
Which term of the G.P.:
(i) √2, 1/√2, 1/2√2, 1/4√2, … is 1/512√2 ?
(ii) 2, 2√2, 4, … is 128 ?
Solution: (i) $\sqrt{2}, 1 / \sqrt{2}, 1 / 2 \sqrt{2}, 1 / 4 \sqrt{2}, \ldots$ is $1 / 512 \sqrt{2} ?$ Using the formula, $\begin{array}{l} T_{n}=a r^{n-1} \\ a=\sqrt{2} \\ r=t_{2} / t_{1}=(1 /...
Evaluate the following limits:
Evaluate the following limits:
Fill in the blanks in the following table:
$$ \begin{tabular}{|l|l|l|l|l|} \hline & $\mathrm{P}(\mathrm{A})$ & $\mathrm{P}(\mathrm{B})$ & $\mathrm{P}(\mathrm{A} \cap \mathrm{B})$ & $\mathrm{P}(\mathrm{AUB})$ \\ \hline $\text { (i) }$ &...
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Which term of the progression 0.004, 0.02, 0.1, …. is 12.5?
Solution: Using the formula, $\mathrm{T}_{\mathrm{n}}=\mathrm{ar}^{\mathrm{n}-1}$ Given that, $\begin{array}{l} a=0.004 \\ r=t_{2} / t_{1}=(0.02 / 0.004) \\ =5 \\ T_{n}=12.5 \\ n=? \end{array}$...
Evaluate the following limits:
Evaluate the following limits:
Find the term from the end of the G.P. .
Solution: The $n^{th}$ term from the end is given by: $a_{n}=I(1 / r)^{n-1}$ where, $I$ is the last term, $r$ is the common ratio, $n$ is the nth term Given that, last term, $\mid=162$...
A and B are two events such that P (A) = 0.54, P (B) = 0.69 and P (A ∩ B) = 0.35. Find
(i) P (A ∩ B′)
(ii) P (A′ ∩ B)
A and B are two events is given P (A) = 0.54, P (B) = 0.69 and P (A ∩ B) = 0.35 By definition of P (A or B) under axiomatic approach we can write, P (A ∪ B) = P (A) + P (B) – P (A ∩ B) (i) P (A ∩...
Find:
(i) term of the G.P.
(ii) the term of the G.P.
Solution: (i) nth term of the G.P. $\sqrt{3}, 1 / \sqrt{3}, 1 / 3 \sqrt{3}, \ldots$ It is known that, $t_{1}=a=\sqrt{3}, r=t_{2} / t_{1}=(1 / \sqrt{3}) / \sqrt{3}=1 /(\sqrt{3} \times \sqrt{3})=1 /...
A and B are two events such that P (A) = 0.54, P (B) = 0.69 and P (A ∩ B) = 0.35. Find
(i) P (A ∪ B)
(ii) P (A′ ∩ B′)
A and B are two events is given P (A) = 0.54, P (B) = 0.69 and P (A ∩ B) = 0.35 By definition of P (A or B) under axiomatic approach we can write, P (A ∪ B) = P (A) + P (B) – P (A ∩ B) (i) P (A ∪ B)...
Find:
(i) the term of the G.P.
(ii) the term of the G.P.
Solution: (i) the $8^{\text {th }}$ term of the G.P., $0.3,0.06,0.012, \ldots$ It is known that, $\mathrm{t}_{1}=\mathrm{a}=0.3, \mathrm{r}=\mathrm{t}_{2} / \mathrm{t}_{1}=0.06 / 0.3=0.2$ Using the...
If A and B be mutually exclusive events associated with a random experiment such that P (A) = 0.4 and P (B) = 0.5, then find:
(i) P (A′ ∩ B)
(ii) P (A ∩ B′)
A and B are two mutually exclusive events is given to us. P (A) = 0.4 and P (B) = 0.5 By definition of mutually exclusive events we can write, P (A ∪ B) = P (A) + P (B) (i) P (A′ ∩ B) P (only B) = P...
If A and B be mutually exclusive events associated with a random experiment such that P (A) = 0.4 and P (B) = 0.5, then find:
(i) P(A ∪ B)
(ii) P (A′ ∩ B′)
A and B are two mutually exclusive events is given to us. P (A) = 0.4 and P (B) = 0.5 By definition of mutually exclusive events we can write, P (A ∪ B) = P (A) + P (B) (i) P (A ∪ B) = P (A) + P (B)...
Find:
(i) the ninth term of the G.P.
(ii) the term of the G.P.
Solution: (i) the ninth term of the G.P. $1,4,16,64, \ldots$ It is known that, $t_{1}=a=1, r=t_{2} / t_{1}=4 / 1=4$ Using the formula. $\begin{array}{l}...
Show that the sequence defined by is a G.P.
Solution: Given that, $a_{n}=2 / 3^{n}$ Consider $\mathrm{n}=1,2,3,4, \ldots$ since $\mathrm{n}$ is a natural number. Therefore, $\begin{array}{l} a_{1}=2 / 3 \\ a_{2}=2 / 3^{2}=2 / 9 \\ a_{3}=2 /...
Show that each one of the following progressions is a G.P. Also, find the common ratio in each case:
(i)
(ii)
Solution: (i) a, $3 \mathrm{a}^{2} / 4,9 \mathrm{a}^{3} / 16, \ldots$ Let $a=a, b=3 a^{2} / 4, c=9 a^{3} / 16$ In Geometric Progression, $\begin{array}{l} b^{2}=a c \\ \left(3 a^{2} / 4\right)^{2}=9...
Show that each one of the following progressions is a G.P. Also, find the common ratio in each case:
(i)
(ii)
Solution: (i) $4,-2,1,-1 / 2, \ldots$ Let $a=4, b=-2, c=1$ In $\mathrm{GP}$ $\begin{array}{l} b^{2}=a c \\ (-2)^{2}=4(1) \\ 4=4 \end{array}$ Therefore, the Common ratio $=r=-2 / 4=-1 / 2$ (ii) $-2 /...
A farmer buys a used tractor for ₹ 12000. He pays ₹ 6000 cash and agrees to pay the balance in annual instalments of ₹ 500 plus 12% interest on the unpaid amount. How much the tractor cost him?
Solution: We have to find the total cost of the tractor if he buys it in installments. Total price $=$ ₹ 12000 Paid amount $=$ ₹ 6000 Unpaid amount $=$ ₹ $12000-6000=$ ₹ 6000 He pays remaining ₹...
A piece of equipment cost a certain factory 600,000 . If it depreciates in value the first, the next year, the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?
Solution: Given that a piece of equipment cost a certain factory is ₹ 600,000 We have to find the value of the equipment at the end of 10 years. The price of equipment depreciates $15 \%, 13.5 \%,...
A man is employed to count ₹ 10710 . He counts at the rate of 180 per minute for half an hour. After this he counts at the rate of ₹3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
Solution: Given that the amount to be counted is ₹ 10710 We have to find the time taken by man to count the entire amount. He counts the amount at the rate of ₹ 180 per minute for 30 minutes. Amount...
There are 25 trees at equal distances of 5 meters in a line with a well, the distance of well from the nearest tree being 10 meters. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.
Solution: It is given that total number of trees are 25 and the distance between two adjacent trees are 5 meters To find the total distance the gardener will cover. As given the gardener is coming...
A manufacturer of the radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find
(і) the product in the 10th year.
Solution: Given that, In the third and seventh year 600 and 700 radio sets units are produced, respectively. $a_3 = 600$ and $a_7 = 700$ (i) The product in the $10^{\text {th }}$ year. Find the...
A manufacturer of the radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find
(і) the production in the first year
(іі) the total product in the 7 years and
Solution: Given that, In the third and seventh year 600 and 700 radio sets units are produced, respectively. $a_3 = 600$ and $a_7 = 700$ (і) The production in the first year Find the production in...
A man arranges to pay off a debt of ₹ 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the instalment.
Solution: As per the question: There are 40 annual instalments that form an arithmetic series. Let '$a$' be the first instalment $S_{40}=3600, n=40$ Using the formula, $\begin{array}{l} S_{n}=n /...
A man saves ₹ 32 during the first year, ₹ 36 in the second year and in this way he increases his savings by ₹ 4 every year. Find in what time his saving will be ₹ 200.
Solution: As per the question: Savings for the first year is ₹ 32 Savings for the second year is ₹ 36 Every year he increases his savings by ₹ 4. Therefore, A.P. will be $32,36,40, \ldots \ldots...
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
A man saved ₹ 16500 in ten years. In each year after the first he saved ₹ 100/- more than he did in the preceding year. How much did he saved in the first year?
Solution: As per the question: A man saved ₹$16500$ in ten years Let be his savings in the first year be ₹ $x$ Every year his savings increased by ₹ 100. Therefore, A.P will be $x$, $100 + x$, $200...
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Evaluate the following limits:
Show that the origin is equidistant from the lines 4x + 3y + 10 = 0; 5x – 12y + 26 = 0 and 7x + 24y = 50.
Given: The lines: \[4x\text{ }+\text{ }3y\text{ }+\text{ }10\text{ }=\text{ }0;\text{ }5x\text{ }\text{ }12y\text{ }+\text{ }26\text{ }=\text{ }0\] And \[7x\text{ }+\text{ }24y\text{ }=\text{ }50.\]...
Reduce the lines 3x – 4y + 4 = 0 and 2x + 4y – 5 = 0 to the normal form and hence find which line is nearer to the origin.
Given: The normal forms of the lines: \[3x\text{ }-\text{ }4y\text{ }+\text{ }4\text{ }=\text{ }0\] And \[~2x\text{ }+\text{ }4y\text{ }-\text{ }5\text{ }=\text{ }0.\] To find, in given normal form...
Put the equation x/a + y/b = 1 the slope intercept form and find its slope and y – intercept.
Given: the equation is: \[~x/a\text{ }+\text{ }y/b\text{ }=\text{ }1~\] As , General equation of line \[~y\text{ }=\text{ }mx\text{ }+\text{ }c.\] \[bx\text{ }+\text{ }ay\text{ }=\text{ }ab\]...
Reduce the following equations to the normal form and find p and α in each case: (i) x + √3y – 4 = 0 (ii) x + y + √2 = 0
\[\left( \mathbf{i} \right)~x\text{ }+\text{ }\surd 3y\text{ }\text{ }4\text{ }=\text{ }0\] \[x\text{ }+\text{ }\surd 3y\text{ }=\text{ }4\] The normal form of the given line, \[where\text{ }p\text{...
Reduce the equation √3x + y + 2 = 0 to: (iii) The normal form and find p and α.
(iii) Given: \[\surd 3x\text{ }+\text{ }y\text{ }+\text{ }2\text{ }=\text{ }0~\] \[-\surd 3x\text{ }\text{ }y\text{ }=\text{ }2\]
Reduce the equation √3x + y + 2 = 0 to: (i) slope – intercept form and find slope and y – intercept; (ii) Intercept form and find intercept on the axes
(i) Given: \[\surd 3x\text{ }+\text{ }y\text{ }+\text{ }2\text{ }=\text{ }0~\] \[y\text{ }=\text{ }\text{ }\surd 3x\text{ }\text{ }2\] following is the slope intercept form of the given line....
The straight line through P(x1, y1) inclined at an angle θ with the x–axis meets the line ax + by + c = 0 in Q. Find the length of PQ.
The equation of the line that passes through P(x1, y1) and makes an angle of θ with the x–axis. To find the length of PQ. By using the formula, The equation of the line is given by:
A line a drawn through A (4, – 1) parallel to the line 3x – 4y + 1 = 0. Find the coordinates of the two points on this line which are at a distance of 5 units from A.
Given: \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\text{ }=\text{ }A\text{ }\left( 4,\text{ }-1 \right)\] Let us find Coordinates of the two points on this line which are at a distance of 5 units...
A straight line drawn through the point A (2, 1) making an angle π/4 with positive x–axis intersects another line x + 2y + 1 = 0 in the point B. Find length AB.
Given: \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\text{ }=\text{ }A\text{ }\left( 2,\text{ }1 \right),\] \[\theta ~=~\pi /4\text{ }=\text{ }45{}^\circ \] Let us find the length AB. By using the...
If the straight line through the point P(3, 4) makes an angle π/6 with the x–axis and meets the line 12x + 5y + 10 = 0 at Q, find the length PQ.
Given: \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\text{ }=\text{ }A\text{ }\left( 3,\text{ }4 \right),\] \[~\theta ~=\text{ }\pi /6\text{ }=\text{ }30{}^\circ \] To find the length PQ. By using...
A line passes through a point A (1, 2) and makes an angle of 600 with the x–axis and intercepts the line x + y = 6 at the point P. Find AP.
Given: \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\text{ }=\text{ }A\text{ }\left( 1,\text{ }2 \right),\text{ }\theta ~=\text{ }60{}^\circ \] To find the distance AP, On using the formula, The...
Find the equation of the straight line on which the length of the perpendicular from the origin is 2 and the perpendicular makes an angle α with x–axis such that sin α = 1/3.
Given: \[p\text{ }=\text{ }2,\text{ }sin\text{ }\alpha \text{ }=\text{ }1/3\] As , \[~cos\text{ }\alpha \text{ }=\text{ }\surd \left( 1\text{ }\text{ }si{{n}^{2}}~\alpha \right)\] \[=\text{...
Find the equation of the straight line at a distance of 3 units from the origin such that the perpendicular from the origin to the line makes an angle α given by tan α = 5/12 with the positive direction of x–axis.
Given: \[p\text{ }=\text{ }3,\text{ }\alpha \text{ }=\text{ }ta{{n}^{-1}}~\left( 5/12 \right)\] hence , \[tan\text{ }\alpha \text{ }=\text{ }5/12\] \[sin\text{ }\alpha \text{ }=\text{ }5/13\]...
Find the equation of the line whose perpendicular distance from the origin is 4 units and the angle which the normal makes with the positive direction of x–axis is 15°.
Given: \[p\text{ }=\text{ }4,\text{ }\alpha \text{ }=\text{ }15{}^\circ \] The equation of the line in normal form is given by as, \[~cos\text{ }15{}^\circ ~=\text{ }cos\text{ }\left( 45{}^\circ...
Find the equation of the line on which the length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x–axis is 30°.
Given: \[p\text{ }=\text{ }4,\text{ }\alpha \text{ }=\text{ }30{}^\circ \] The equation of the line in normal form is given by On using the formula, \[x\text{ }cos~\alpha \text{ }+\text{ }y\text{...
Find the equation of a line for which (i) p = 5, α = 60° (ii) p = 4, α = 150°
\[\left( \mathbf{i} \right)~p\text{ }=\text{ }5,\text{ }\alpha \text{ }=\text{ }60{}^\circ \] Given: \[p\text{ }=\text{ }5,\text{ }\alpha \text{ }=\text{ }60{}^\circ \] The equation of the line in...
Find the equation to the straight line which cuts off equal positive intercepts on the axes and their product is 25.
Given: \[a\text{ }=\text{ }b\text{ }and\text{ }ab\text{ }=\text{ }25\] to find the equation of the line which cutoff intercepts on the axes. \[\therefore ~{{a}^{2}}~=\text{ }25\] \[a\text{ }=\text{...
For what values of a and b the intercepts cut off on the coordinate axes by the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x – 3y + 6 = 0 on the axes.
Given: Intercepts cut off on the coordinate axes by the line \[~ax\text{ }+\text{ }by\text{ }+8\text{ }=\text{ }0\text{ }\ldots \ldots \text{ }\left( i \right)\] And are equal in length but opposite...
Find the equation to the straight line which passes through the point (5, 6) and has intercepts on the axes (i) Equal in magnitude and both positive (ii) Equal in magnitude but opposite in sign
(i) Equal in magnitude and both positive Given: \[a\text{ }=\text{ }b\] to find the equation of line cutoff intercepts from the axes. On using the formula, The equation of the line is: \[x/a\text{...
Find the equation of the straight line which passes through (1, -2) and cuts off equal intercepts on the axes.
Given: A line passing through (1, -2) Let the equation of the line cutting equal intercepts at coordinates of length ‘a’ is By using the formula, The equation of the line is: \[~x/a\text{ }+\text{...
Find the equation to the straight line (i) cutting off intercepts 3 and 2 from the axes. (ii) cutting off intercepts -5 and 6 from the axes.
(i) Cutting off intercepts 3 and 2 from the axes. Given: \[a\text{ }=\text{ }3,\] \[~b\text{ }=\text{ }2\] to find the equation of line cutoff intercepts from the axes. on using the formula, The...
Find the equation of the circle which passes through the origin and cuts off intercepts a and b respectively from x and y – axes.
Since the circle has intercept ‘a’ from x – axis, the circle must pass through (a, 0) and (-a, 0) as it already passes through the origin. Also,since the circle has intercept ‘b’ from x – axis, the...
Find the equation of the circle passing through the origin and the points where the line 3x + 4y = 12 meets the axes of coordinates.
The line 3x + 4y = 12 The value of x is 0 on meeting the y – axis. So, \[\begin{array}{*{35}{l}} 3\left( 0 \right)\text{ }+\text{ }4y\text{ }=\text{ }12 \\ 4y\text{ }=\text{ }12 \\ y\text{...
Find the equation of the circle circumscribing the rectangle whose sides are x – 3y = 4, 3x + y = 22, x – 3y = 14 and 3x + y = 62.
The sides \[\begin{array}{*{35}{l}} x\text{ }-\text{ }3y\text{ }=\text{ }4\text{ }\ldots .\text{ }\left( 1 \right) \\ 3x\text{ }+\text{ }y\text{ }=\text{ }22\text{ }\ldots \text{ }\left( 2 \right) ...
Find the equation of the side BC of the triangle ABC whose vertices are A (-1, -2), B (0, 1) and C (2, 0) respectively. Also, find the equation of the median through A (-1, -2).
Given: The vertices of triangle ABC are: \[A\text{ }\left( -1,\text{ }-2 \right),\text{ }B\left( 0,\text{ }1 \right)\text{ }and\text{ }C\left( 2,\text{ }0 \right).\] Let us find the equation of...
The sides of a squares are x = 6, x = 9, y = 3 and y = 6. Find the equation of a circle drawn on the diagonal of the square as its diameter.
The sides of a squares are x = 6, x = 9, y = 3 and y = 6. assuming A, B, C, D be the vertices of the square. we get, the coordinates as: A = (6, 3) B = (9, 3) C = (9, 6) D = (6, 6) the equation of...
Find the equation of the circle the end points of whose diameter are the centres of the circles x^2 + y^2 + 6x – 14y – 1 = 0 and x^2 + y^2 – 4x + 10y – 2 = 0.
x2 + y2 + 6x – 14y – 1 = 0…. (1) So the centre \[\begin{array}{*{35}{l}} =\text{ }\left[ \left( -6/2 \right),\text{ }-\left( -14/2 \right) \right] \\ =\text{ }\left[ -3,\text{ }7 \right] \\...
Find the equations to the diagonals of the rectangle the equations of whose sides are x = a, x = a’, y = b and y = b’.
Given: The rectangle formed by the lines, \[x\text{ }=\text{ }a,\text{ }x\text{ }=\text{ }a,\text{ }y\text{ }=\text{ }b\text{ }and\text{ }y\text{ }=\text{ }b\] It is clear that, the vertices of the...
Find the equation of the circle, the end points of whose diameter are (2, -3) and (-2, 4). Find its centre and radius.
The diameters (2, -3) and (-2, 4). By using the formula, Centre = (-a, -b) \[\begin{array}{*{35}{l}} =\text{ }\left[ \left( 2-2 \right)/2,\text{ }\left( -3+4 \right)/2 \right] \\ =\text{ }\left(...
Find the equations of the medians of a triangle, the coordinates of whose vertices are (-1, 6), (-3,-9) and (5, -8).
\[A\text{ }\left( -1,\text{ }6 \right),\text{ }B\text{ }\left( -3,\text{ }-9 \right)\text{ }and\text{ }C\text{ }\left( 5,\text{ }-8 \right)\] be the coordinates of the given triangle. Let: D, E, and...
Verify that the area of the triangle with vertices (2, 3), (5, 7) and (-3 -1) remains invariant under the translation of axes when the origin is shifted to the point (-1, 3).
Solution: According to the question, the points are (2, 3), (5, 7), and (-3, -1). The area of triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is as follows: = ½ [x1(y2 – y3) + x2(y3 -y1) +...
Find the equations to the sides of the triangles the coordinates of whose angular points are respectively: (i) (1, 4), (2, -3) and (-1, -2) (ii) (0, 1), (2, 0) and (-1, -2)
\[~\left( \mathbf{i} \right)~\left( 1,\text{ }4 \right),\text{ }\left( 2,\text{ }-3 \right)\text{ }and\text{ }\left( -1,\text{ }-2 \right)\] Given: Points A (1, 4), B (2, -3) and C (-1, -2). Let ,...
At what point the origin be shifted so that the equation x2 + xy – 3x + 2 = 0 does not contain any first-degree term and constant term?
Solution: We have the equation x2 + xy – 3x + 2 = 0 The origin has been relocated from (0, 0) to (p, q) As a result, any arbitrary point (x, y) is changed to (x + p, y + q). Therefore, the new...
Find what the following equations become when the origin is shifted to the point (1, 1)?
(iii) xy – x – y + 1 = 0 (iv) xy – y2 – x + y = 0 Solution: (iii) xy – x – y + 1 = 0 We will replace the value of x by x + 1 and y by y + 1 Then, (x + 1) (y + 1) – (x + 1) – (y + 1) + 1 = 0 xy + x +...
Find what the following equations become when the origin is shifted to the point (1, 1)?
(i) x2 + xy – 3x – y + 2 = 0 (ii) x2 – y2 – 2x + 2y = 0 Solution: (i) x2 + xy – 3x – y + 2 = 0 We will first substitute the value of x by x + 1 and y by y + 1. Then, te above-given ewuation becomes:...