Redefine the function f(x) = |x – 2| + |2 + x|,
Question: Redefine the function f(x) = |x 2| + |2 + x|, 3 x 3 Solution: According to the question, function f(x) = |x2| + |2 + x|, 3 x 3 We know that, when x0, |x 2| is (x2), x2 |2 + x| is (2 + x), x2 when x0 |x 2| is (x2), x2 |2 + x| is (2 + x), x2 Given that, f(x) = |x2| + |2 + x|, 3 x 3 It can be rewritten as, $f(x)=\left\{\begin{array}{c}-(x-2)-(2+x),-3 \leq x-2 \\ -(x-2)+(2+x),-2 \leq x2 \\ (x-2)+(2+x), 2 \leq x \leq 3\end{array}\right.$ Or $f(x)=\left\{\begin{array}{c}-x+2-2-x,-3 \leq x-2 ...
Read More →Find the range of the following
Question: Find the range of the following functions given by (i) $f(x)=\frac{3}{2-x^{2}}$ (ii) $f(x)=1-|x-2|$ (iii) $f(x)=|x-3|$ (iv) $f(x)=1+3 \cos 2 x$ Solution: (i) $f(x)=\frac{3}{2-x^{2}}$ According to the question, Let $f(x)=y$, Let $f(x)=y$, $y=\frac{3}{2-x^{2}}$ $\Rightarrow 2-x^{2}=\frac{3}{y}$ $\Rightarrow x^{2}=2-\frac{3}{y}$ But, we know that, $x^{2} \geq 0$ $2-\frac{3}{y} \geq 0$ $\Rightarrow \frac{2 y-3}{y} \geq 0$ ⇒y0 and 2y30 ⇒y0 and 2y3 ⇒y0 andy 3/2 Or f(x)0 andf(x) 3/2 f(x) ( , ...
Read More →The points A(2, 3), B(4, -1) and C(-1, 2)
Question: The points A(2, 3), B(4, -1) and C(-1, 2) are the vertices of ΔABC. Find the length of the perpendicular from C on AB and hence find the area of ΔABC Solution: Given: points A(2, 3), B(4, -1) and C(-1, 2) are the vertices of ΔABC To find : length of the perpendicular from C on AB and the area of ΔABC Formula used: We know that the length of the perpendicular from $(m, n)$ to the line $a x+b y+c=0$ is $D=\frac{|a m+b n+c|}{\sqrt{a^{2}+b^{2}}}$ The equation of the line joining the points...
Read More →Show that
Question: Show that $f(x)=\log \sin x$ is increasing on $(0, \pi / 2)$ and decreasing on $(\pi / 2, \pi)$. Solution: Given:- Function $f(x)=\log \sin x$ Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$. (i) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$ (ii) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equal to $f(x)$ (ii) Find $f^{\prime}(x...
Read More →Find the domain of each of
Question: Find the domain of each of the following functions given by (i) $f(x)=\frac{1}{\sqrt{1-\cos x}}$ (ii) $f(x)=\frac{1}{\sqrt{x+|x|}}$ (iii) $f(x)=x|x|$ (iv) $f(x)=\frac{x^{3}-x+3}{x^{2}-1}$ (v) $f(x)=\frac{3 x}{2 x-8}$ Solution: (i) $f(x)=\frac{1}{\sqrt{1-\cos x}}$ According to the question, We know the value of cos x lies between 1, 1, 1 cos x 1 Multiplying by negative sign, we get Or 1 cos x 1 Adding 1, we get 2 1 cos x 0 (i) Now, $f(x)=\frac{1}{\sqrt{1-\cos x}}$ 1 cosx 0 ⇒cos x 1 Or, ...
Read More →Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function?
Question: Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? Justify. If this is described by the relation, g (x) = x + , then what values should be assigned to and ? Solution: According to the question, g = {(1, 1), (2, 3), (3, 5), (4, 7)}, and is described by relation g (x) = x + Now, given the relation, g = {(1, 1), (2, 3), (3, 5), (4, 7)} g (x) = x + For ordered pair (1,1), g (x) = x + , becomes g (1) = (1) + = 1 ⇒ + = 1 ⇒ = 1 (i) Considering ordered pair (2, 3), g (x) = x + , becomes g (2)...
Read More →Find the values of x for which the functions
Question: Find the values of x for which the functions f (x) = 3x2 1 and g (x) = 3 + x are equal Solution: According to the question, f and g functions defined by f (x) = 3x2 1 and g (x) = 3 + x For what real numbers x, f (x) = g (x) To satisfy the condition f(x) = g(x), Should also satisfy, 3x2 1 = 3 + x ⇒3x2 x 3 1 = 0 ⇒3x2 x 4 = 0 Splitting the middle term, We get, ⇒3x2+ 3x 4x4 = 0 ⇒3x(x + 1) 4(x + 1) = 0 ⇒(3x 4)(x + 1) = 0 ⇒3x 4 = 0 or x + 1 = 0 ⇒3x = 4 or x = 1 ⇒ x = 4/3, 1 Hence, forx = 4/3...
Read More →Express the following functions
Question: Express the following functions as set of ordered pairs and determine their range. f: XR, f (x) = x3 + 1, where X = {1, 0, 3, 9, 7} Solution: According to the question, A function f: XR, f (x) = x3+ 1, where X = {1, 0, 3, 9, 7} Domain = f is a function such that the first elements of all the ordered pair belong to the set X = {1, 0, 3, 9, 7}. The second element of all the ordered pair are such that they satisfy the condition f (x) = x3+ 1 When x = 1, f (x) = x3+ 1 f ( 1) = ( 1)3+ 1 = 1...
Read More →If f and g are two real valued functions
Question: If f and g are two real valued functions defined as f (x) = 2x + 1, g (x) = x2+ 1, then find. (i) f + g (ii) f g (iii) fg (iv)f/g Solution: According to the question, f and g be real valued functions defined as f (x) = 2x + 1, g (x) = x2+ 1, (i) f + g ⇒f + g = f(x) + g(x) = 2x + 1 + x2+ 1 = x2+ 2x + 2 (ii) f g ⇒f g = f(x) g(x) = 2x + 1 (x2+ 1) = 2x x2 (iii) fg ⇒fg = f(x) g(x) = (2x + 1)( x2+ 1) = 2x(x2) + 2x(1) + 1(x2) + 1(1) = 2x3+ 2x + x2+ 1 = 2x3+ x2+ 2x + 1 (iv) f/g f/g = f(x)/g(x)...
Read More →Let f and g be real functions defined by
Question: Let f and g be real functions defined by f (x) = 2x + 1 and g (x) = 4x 7. (a) For what real numbers x, f (x) = g (x)? (b) For what real numbers x, f (x) g (x)? Solution: According to the question, f and g be real functions defined by f(x) = 2x + 1 and g(x) = 4x 7 (a) For what real numbers x, f (x) = g (x) To satisfy the condition f(x) = g(x), Should also satisfy, 2x + 1 = 4x7 ⇒7 + 1 = 4x2x ⇒8 = 2x Or, 2x = 8 ⇒x = 4 Hence, we get, For x = 4, f (x) = g (x) (b) For what real numbers x, f ...
Read More →If f and g are real functions defined by
Question: If f and g are real functions defined by f (x) = x2+ 7 and g (x) = 3x + 5, find each of the following (a) f (3) + g ( 5) (b) f() g(14) (c) f ( 2) + g ( 1) (d) f (t) f ( 2) (e) (f(t) f(5))/ (t 5), if t 5 Solution: According to the question, f and g are real functions such that f (x) = x2+ 7 and g (x) = 3x + 5 (a) f (3) + g ( 5) f (x) = x2+ 7 Substituting x = 3 in f(x), we get f (3) = 32+ 7 = 9 + 7 = 16 (i) And, g (x) = 3x + 5 Substituting x = 5 in g(x), we get g (5) = 3(5) + 5 = 15 + 5 ...
Read More →Is the given relation a function?
Question: Is the given relation a function? Give reasons for your answer. (i) h = {(4, 6), (3, 9), ( 11, 6), (3, 11)} (ii) f = {(x, x) | x is a real number} (iii) g = n, (1/n) |n is a positive integer (iv) s = {(n, n2) | n is a positive integer} (v) t = {(x, 3) | x is a real number. Solution: (i) According to the question, h = {(4, 6), (3, 9), ( 11, 6), (3, 11)} Therefore, element 3 has two images, namely, 9 and 11. A relation is said to be function if every element of one set has one and only o...
Read More →Show that the length of the perpendicular from the point
Question: Show that the length of the perpendicular from the point (7, 0) to the line 5x + 12y 9 = 0 is double the length of perpendicular to it from the point (2, 1) Solution: Given: Points (7,0) and (2,1) , line 5x + 12y 9 = 0 To Prove : length of the perpendicular from the point $(7,0)$ to the line $5 x+12 y-9=0$ is double the length of perpendicular to it from the point $(2,1)$ Formula used: We know that the length of the perpendicular from $(\mathrm{m}, \mathrm{n})$ to the line $\mathrm{ax}...
Read More →If R3 = {(x, |x| ) |x is a real number} is a relation.
Question: If R3= {(x, |x| ) |x is a real number} is a relation. Then find domain and range of R3. Solution: According to the question, R3= {(x, |x|) |x is a real number} is a relation Domain of R3consists of all the first elements of all the ordered pairs of R3, i.e., x, It is also given that x is a real number, So, Domain of R3= R Range of R contains all the second elements of all the ordered pairs of R3, i.e., |x| It is also given that x is a real number, So, |x| = |R| ⇒|x|0, i.e., |x| has all...
Read More →Show that
Question: Show that $f(x)=\log \sin x$ is increasing on $(0, \pi / 2)$ and decreasing on $(\pi / 2, \pi)$. Solution: Given:- Function $f(x)=\log \sin x$ Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$. (i) If $f^{\prime}(x)0$ for $a l l x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$ (ii) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equal to $f(x)$ (ii) Find $f^{\prime}...
Read More →If R2 = {(x, y) | x and y are integers
Question: If R2= {(x, y) | x and y are integers and x2+ y2= 64} is a relation. Then find R2. Solution: We have, R2= {(x, y) |x and y are integers and x2+ y2 64} So, we get, x2= 0 and y2= 64 or x2= 64 and y2= 0 x = 0 and y = 8 or x = 8 and y = 0 Therefore, R2 = {(0, 8), (0, 8), (8,0), (8,0)}...
Read More →If R1 = {(x, y) | y = 2x + 7, where x ∈ R and – 5 ≤ x ≤ 5}
Question: If R1= {(x, y) | y = 2x + 7, where xRand 5 x 5} is a relation. Then find the domain and Range of R1. Solution: According to the question, R1= {(x, y) | y = 2x + 7, where xRand 5 x 5} is a relation The domain of R1consists of all the first elements of all the ordered pairs of R1, i.e., x, It is also given 5 x 5. Therefore, Domain of R1= [5, 5] The range of R contains all the second elements of all the ordered pairs of R1, i.e., y It is also given y = 2x + 7 Now x[5,5] Multiply LHS and R...
Read More →Given R = {(x, y) : x, y ∈ W, x2 + y2 = 25}.
Question: Given R = {(x, y) : x, yW, x2+ y2= 25}. Find the domain and Range of R. Solution: According to the question, R = {(x, y) : x, yW, x2+ y2= 25} R = {(0,5), (3,4), (4, 3), (5,0)} The domain of R consists of all the first elements of all the ordered pairs of R. Domain of R = {0, 3, 4, 5} The range of R contains all the second elements of all the ordered pairs of R. Range of R = {5, 4, 3, 0}...
Read More →Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}.
Question: Given A = {1, 2, 3, 4, 5}, S = {(x, y) : xA, yA}. Find the ordered pairs which satisfy the conditions given below: (i) x + y = 5 (ii) x + y 5 (iii) x + y 8 Solution: According to the question, A = {1, 2, 3, 4, 5}, S = {(x, y) : xA, yA} (i) x + y = 5 So, we find the ordered pair such that x + y = 5, where x and y belongs to set A = {1, 2, 3, 4, 5}, 1 + 1 = 25 1 + 2 = 35 1 + 3 = 45 1 + 4 = 5⇒the ordered pair is (1, 4) 1 + 5 = 65 2 + 1 = 35 2 + 2 = 45 2 + 3 = 5⇒the ordered pair is (2, 3) ...
Read More →Find the values of k for which the length
Question: Find the values of k for which the length of the perpendicular from the point (4, 1) on the line 3x 4y + k = 0 is 2 units Solution: Given: Point (4,1) , line 3x 4y + k = 0 and length of perpendicular is 2 units To find: The values of k Formula used: We know that the length of the perpendicular from (m,n) to the line ax + by + c = 0 is given by, $D=\frac{|a m+b n+c|}{\sqrt{a^{2}+b^{2}}}$ The equation of the line is $3 x-4 y+k=0$ Here $m=4$ and $n=1, a=3, b=-4, c=k$ and $D=2$ units $D=\f...
Read More →In each of the following cases,
Question: In each of the following cases, find a and b. (i) (2a + b, a b) = (8, 3) (ii) (a/4 , a 2b) = (0, 6 + b) Solution: (i) According to the question, (2a + b, a b) = (8, 3) Given the ordered pairs are equal, so corresponding elements will be equal. Hence, 2a + b = 8 and ab = 3 Now ab = 3 ⇒a = 3 + b Substituting the value of a in the equation 2a + b = 8, We get, 2(3 + b) + b = 8 ⇒6 + 2b + b = 8 ⇒3b = 86 = 2 ⇒ b = 2/3 Substituting the value of b in equation (ab = 3), We get, ⇒ a 2/3 = 3 ⇒ a =...
Read More →If A = {x : x ∈ W, x < 2}, B = {x : x ∈ N, 1 < x < 5},
Question: If A = {x : xW, x 2}, B = {x : xN, 1 x 5}, C = {3, 5} find (i) A (BC) (ii) A (B C) Solution: According to the question, A = {x: xW, x 2}, B = {x : xN, 1 x 5} C = {3, 5};Wis the set of whole numbers A = {x: xW, x 2} = {0, 1} B = {x : xN, 1 x 5} = {2, 3, 4} (i) (BC) = {2, 3, 4} {3, 5} (BC) = {3} A (BC) = {0, 1} {3} = {(0, 3), (1, 3)} Hence, the Cartesian product = {(0, 3), (1, 3)} (ii) (BC) = {2, 3, 4} {3, 5} (BC) = {2, 3, 4, 5} A (BC) = {0, 1} {2, 3, 4, 5} = {(0, 2), (0, 3), (0, 4), (0,...
Read More →Prove that the product of the lengths of perpendiculars drawn from the points
Question: Prove that the product of the lengths of perpendiculars drawn from the points $\mathrm{A}\left(\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}, 0\right)$ and $\mathrm{B}\left(-\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}, 0\right)$ to the line $\frac{\mathrm{x}}{\mathrm{a}} \cos \theta+\frac{\mathrm{y}}{\mathrm{b}} \sin \theta=1$, is $\mathrm{b}^{2}$ Solution: Given: Point $\mathrm{A}\left(\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}, 0\right), \mathrm{B}\left(-\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}, 0\right)$ and...
Read More →Show that f(x)=sin x is increasing
Question: Show that $f(x)=\sin x$ is increasing on $(0, \pi / 2)$ and decreasing on $(\pi / 2, \pi)$ and neither increasing nor decreasing in $(0, \pi)$. Solution: Given:- Function $f(x)=\sin x$ Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$. (i) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$ (ii) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put ...
Read More →If P = {x : x < 3, x ∈ N}, Q = {x : x ≤ 2, x ∈ W}.
Question: If P = {x : x 3, x N}, Q = {x : x 2, x W}. Find (P Q) (P Q), where W is the set of whole numbers. Solution: According to the question, P = {x: x 3, xN}, Q = {x : x 2, xW} whereWis the set of whole numbers P = {1, 2} Q = {0, 1, 2} Now (PQ) = {1, 2}{0, 1, 2} = {0, 1, 2} And, (PQ) = {1, 2}{0, 1, 2} = {1, 2} We need to find the Cartesian product of (PQ) = {0, 1, 2} and (PQ) = {1, 2} So, (PQ) (PQ) = {0, 1, 2} {1, 2} = {(0, 1), (0, 2), (1, 1), (1, 2), (2, 1), (2, 2)} Hence, the Cartesian pro...
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