and 
are two events associated with a random experiment such that 
and 
, find 
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...
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...
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...
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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...
(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{...
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...
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...
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,...
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....
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...
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...
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...
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.
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...
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...
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...
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...
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.”...
Answer: The negation of the statement is: It is false that “The sun is cold.” [Or] “The sun is not cold.”
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]...
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...
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...
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...
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...
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...
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...
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...
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...
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....
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...
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.”
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 ≠...
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.
Answers: (i) Contrapositive: If he is not strong, then he is not a sailor (ii) Contrapositive: If he tries, then he will not 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.
Answers: (i) Contrapositive: If it rains, then it is not cold. (ii) Contrapositive: If it did not snow, then Ravish will not ski.
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.
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.
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...
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...
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...
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) ...
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...
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.
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.
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.
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.
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...
Answer: t: √11 is a rational number. Square root of prime numbers is irrational numbers. The statement is False.
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...
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...
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,...
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...
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...
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...
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...
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]...
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.
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...
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{...
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{...
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{...
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{...
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{...
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...
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{...
(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...
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...
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...
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(...
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...
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...
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{...
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...
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...
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{...
\[\left( \mathbf{i} \right)~15x\text{ }\text{ }18y\text{ }+\text{ }1\text{ }=\text{ }0,\text{ }12x\text{ }+\text{ }10y\text{ }\text{ }3\text{ }=\text{ }0\] and \[6x\text{ }+\text{ }66y\text{ }\text{...
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{...
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...
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 ,...
\[\left( \mathbf{i} \right)~x\text{ }+\text{ }y\text{ }\text{ }4\text{ }=\text{ }0,\text{ }2x\text{ }\text{ }y\text{ }+\text{ }3\text{ }0\] and \[x\text{ }\text{ }3y\text{ }+\text{ }2\text{ }=\text{...
\[\left( \mathbf{i} \right)~2x\text{ }\text{ }y\text{ }+\text{ }3\text{ }=\text{ }0\] And \[x\text{ }+\text{ }y\text{ }\text{ }5\text{ }=\text{ }0\] Given: The equations of the lines are: \[2x\text{...
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)...
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) $486=a 3^{n} / 3$...
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)...
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...
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)...
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 /...
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 /...
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...
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 /...
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 $=a\left(r^{n}-1\right) /(r-1)$...
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...
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 /...
to 5 terms; 
to 
termsSolution: (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...
to 8 terms; 
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...
(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...
to 10 termsSolution: (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 /...
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} /...
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...
\[\left( \mathbf{i} \right)tan\text{ }8x\text{ }-\text{ }tan\text{ }6x\text{ }-\text{ }tan\text{ }2x\text{ }=\text{}tan\text{}8x\text{}tan\text{ }6x\text{ }tan\text{ }2x\] Let us consider LHS:...
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...
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,...
(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{...
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}...
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}...
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,...
(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:...
(i) (ii) Solution: (i) \[=~tan\text{ }A\] \[=\text{ }RHS\] \[\therefore LHS\text{ }=\text{ }RHS\] Hence proved. (ii) \[=~tan\text{ }A\text{ }-\text{ }tan\text{...
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![\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{mx}}-1}{{{b}^{nx}}-1},n\ne 0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-81de84b4d799cb3f3b1de7b7d6a7ec6e_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}+{{a}^{x}}-2}{{{x}^{2}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bd28192aef6ebc1242c534fa2893276d_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log (1+x)}{{{3}^{x}}-1}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e00e3b61a0a9a7d859be0fe9c72bdba9_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{5}^{x}}-1}{\sqrt{4+x}-2}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-41cb7378457296aa54ae8b86894e4511_l3.png "Rendered by QuickLaTeX.com")
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, $I=1 / 4374$...
![\[\underset{x\to \pi }{\mathop{\lim }}\,\frac{\sqrt{2+\cos x}-1}{{{(\pi -x)}^{2}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f988e0b4e2c7d12af081b06d0faff51c_l3.png "Rendered by QuickLaTeX.com")
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 \\...
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}...
![\[\underset{x\to \pi /4}{\mathop{\lim }}\,\frac{2-\cos e{{c}^{2}}x}{1-\cot x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-702be0b0960fec10be6dbf97929960a0_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to \pi /6}{\mathop{\lim }}\,\frac{{{\cot }^{2}}x-3}{\cos ecx-2}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-db6bc2fcdb48a8ca8beaab897e45f9fd_l3.png "Rendered by QuickLaTeX.com")
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 /...
![\[\underset{x\to \pi /4}{\mathop{\lim }}\,\frac{\cos e{{c}^{2}}x-2}{\cot x-1}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e3fa298c7d047ab76c84bb4fb3eb1ada_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to \pi }{\mathop{\lim }}\,\frac{1+\cos x}{{{\tan }^{2}}x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-439129170f630f86d7d620e04d317400_l3.png "Rendered by QuickLaTeX.com")
$$ \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 $\text { (i) }$ &...
![\[\underset{x\to a}{\mathop{\lim }}\,\frac{\cos x-\cos a}{x-a}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-132771a0229411ee89a034a14b5623b9_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to \pi /3}{\mathop{\lim }}\,\frac{\sqrt{1-\cos 6x}}{\sqrt{2}(\pi /3-x)}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9b67b9ea6e09d5781e072c16375d4a39_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{{{\cos }^{2}}x}{1-\sin x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e75045e16cfe3d11edd6e178ea93ec21_l3.png "Rendered by QuickLaTeX.com")
Solution: Using the formula, $\mathrm{T}_{\mathrm{n}}=\mathrm{ar}^{\mathrm{n}-1}$ Given that, $\begin{array}{l} a=0.004 \\ r=t_{2} / t_{1}=(0.02 / 0.004) \\ =5 \\ T_{n}=12.5 \\ n=? \end{array}$...
![\[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{\sin 2x}{\cos x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c679c96299b68e3df8ad9b89e09e25b5_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to \pi /2}{\mathop{\lim }}\,\left( \frac{\pi }{2}-x \right)\tan x\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-49fba0ad396d020e8cd41091c84df09d_l3.png "Rendered by QuickLaTeX.com")
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 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 ∩...
term of the G.P. 
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 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)...
term of the G.P. 
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...
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...
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)...
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}...
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 /...
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...
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 /...
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 ₹...
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 \%,...
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...
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...
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...
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...
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 /...
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...
![\[\underset{x\to 0}{\mathop{\lim }}\,\frac{3\sin x-4{{\sin }^{3}}x}{x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8b76b1ccb4f2ce12f40f876ac9ccbe6f_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x\cos x}{3x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ff3045952f6e022e7fb2d2f360d5db86_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{2}}}{\sin {{x}^{2}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-62afe3c2e0d0065ce01cc49c854b22ba_l3.png "Rendered by QuickLaTeX.com")
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...
![\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin {{x}^{0}}}{x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0d12f4d9b324b33dfee6e28be924ac74_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\operatorname{Sin}3x}{5x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f05a5ac1a2e8f1af6c4280947d81df8f_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{x+1}-\sqrt{x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-48d065dcace8865bd61446291179535f_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{{{x}^{2}}+cx}-x\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6d5bc7c8d90b21fb8c14acee41dd4054_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{5{{x}^{3}}-6}{\sqrt{9+4{{x}^{6}}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c78af11744a51b8685ee5330084fc1a0_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{3{{x}^{3}}-4{{x}^{2}}+6x-1}{2{{x}^{3}}+{{x}^{2}}-5x+7}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-68959d78b61b2862b9c11984a02ff27a_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{(3x-1)(4x-2)}{(x+8)(x-1)}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d13b99bb29edba434423f82e98d6fdd7_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{5/7}}-{{a}^{5/7}}}{{{x}^{2/7}}-{{a}^{2/7}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a22f764139a0a1786a261b77691d481c_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{2/7}}-{{a}^{2/7}}}{x-a}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-35c4aab7e9c86096fb2e9cc178c5b816_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to a}{\mathop{\lim }}\,\frac{{{(1+x)}^{6}}-1}{{{(1+x)}^{2}}-1}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-34440c86a9873a59fe5f3e74bbc893f8_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to a}{\mathop{\lim }}\,\frac{{{(x+2)}^{3/2}}-{{(a+2)}^{3/2}}}{x-a}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-80b46d0c4b2f9d2192f5b64b6b989419_l3.png "Rendered by QuickLaTeX.com")
![\[\underset{x\to a}{\mathop{\lim }}\,\frac{{{(x+2)}^{5/2}}-{{(a+2)}^{5/2}}}{x-a}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ec895b139a66da8c64ed301ae1f43672_l3.png "Rendered by QuickLaTeX.com")
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 \[7x\text{ }+\text{ }24y\text{ }=\text{ }50.\]...
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...
Given: the equation is: \[~x/a\text{ }+\text{ }y/b\text{ }=\text{ }1~\] As , General equation of line \[~y\text{ }=\text{ }mx\text{ }+\text{ }c.\] \[bx\text{ }+\text{ }ay\text{ }=\text{ }ab\]...
\[\left( \mathbf{i} \right)~x\text{ }+\text{ }\surd 3y\text{ }\text{ }4\text{ }=\text{ }0\] \[x\text{ }+\text{ }\surd 3y\text{ }=\text{ }4\] The normal form of the given line, \[where\text{ }p\text{...
(iii) Given: \[\surd 3x\text{ }+\text{ }y\text{ }+\text{ }2\text{ }=\text{ }0~\] \[-\surd 3x\text{ }\text{ }y\text{ }=\text{ }2\]
(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 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:
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...
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...
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...
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...
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{...
Given: \[p\text{ }=\text{ }3,\text{ }\alpha \text{ }=\text{ }ta{{n}^{-1}}~\left( 5/12 \right)\] hence , \[tan\text{ }\alpha \text{ }=\text{ }5/12\] \[sin\text{ }\alpha \text{ }=\text{ }5/13\]...
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...
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{...
\[\left( \mathbf{i} \right)~p\text{ }=\text{ }5,\text{ }\alpha \text{ }=\text{ }60{}^\circ \] Given: \[p\text{ }=\text{ }5,\text{ }\alpha \text{ }=\text{ }60{}^\circ \] The equation of the line in...
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{...
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...
(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{...
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{...
(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...
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...
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{...
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) ...
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. 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...
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] \\...
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...
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(...
\[A\text{ }\left( -1,\text{ }6 \right),\text{ }B\text{ }\left( -3,\text{ }-9 \right)\text{ }and\text{ }C\text{ }\left( 5,\text{ }-8 \right)\] be the coordinates of the given triangle. Let: D, E, and...
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) +...
\[~\left( \mathbf{i} \right)~\left( 1,\text{ }4 \right),\text{ }\left( 2,\text{ }-3 \right)\text{ }and\text{ }\left( -1,\text{ }-2 \right)\] Given: Points A (1, 4), B (2, -3) and C (-1, -2). Let ,...
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...
(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 +...
(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:...