according to ques,
Differentiate w.r.t tan-1 x when x ≠ 0.
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Maths
If then verify that: (kA)¢ = (kA¢).
Answer: According to the given ques
Differentiate x/sinx w.r.t sin x.
according to ques,
If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that
\[x~=~asin2t~(1\text{ }+\text{ }cos2t)~and~y~=~b~cos2t~(1cos2t)\]
If then verify that: (i) (A¢)¢ = A (ii) (AB)¢ = B¢A¢
Answer: According to the ques
If then verify that A2 + A = A (A + I), where I is 3 × 3 unit matrix.
Answer: According to the given matrix, the solution should be
Find dy/dx of each of the functions expressed in parametric: If x = ecos2t and y = esin2t, prove that dy/ dx = – y log x/ x log y.
\[x~=~{{e}^{cos2t}}~and~y~=~{{e}^{sin2t}}\] So, \[cos\text{ }2t\text{ }=\text{ }log\text{ }x\] and \[sin\text{ }2t\text{ }=\text{ }log\text{ }y\]
If verify that A (B + C) = (AB + AC).
Answer: According to the given matrix, the solution should be
Find dy/dx of each of the functions expressed in parametric:
according to ques,
If: = A, find A.
Answer: According to the given matrix, the should be
Solve the matrix:
Answer: According to the given matrix, the solution should be
Find dy/dx of each of the functions expressed in parametric:
according to ques,
Find dy/dx of each of the functions expressed in parametric: x = 3cosq – 2cos3q, y = 3sinq – 2sin3q.
\[x\text{ }=~3cosq\text{ }\text{ }2co{{s}^{3}}q,~y~=\text{ }3sinq\text{ }\text{ }2si{{n}^{3}}q.\]
If (i) (AB) C = A (BC) (ii) A (B + C) = AB + AC.
Answer: According to the given matix, the solution should be
Find dy/dx of each of the functions expressed in parametric:
according to ques,
Give an example of matrices A, B and C such that AB = AC, where A is non-zero matrix, but B ≠ C.
Answer: From (I) and (ii), Thus, \[AB\text{ }=\text{ }AC\]however \[B\text{ }\ne \text{ }C.\]
If A = [3 5], B = [7 3], then find a non-zero matrix C such that AB = AC.
Answer: Given, \[A\text{ }=\text{ }{{\left[ 3\text{ }5 \right]}_{1\times 2}}~\]and \[B\text{ }=\text{ }{{\left[ 7\text{ }3 \right]}_{1\times 2}}\] For \[AC\text{ }=\text{ }BC\] We have order of...
Find dy/dx of each of the functions expressed in parametric:
according to ques,
Differentiate :
according to ques,
If X and Y are 2 x 2 matrices, then solve the following matrix equations for X and Y
Answer: According to the given matix, the solution should be
Differentiate :
according to ques,
Solve for x and y:
Answer: Presently, we have \[2x\text{ }+\text{ }3y\text{ }\text{ }8\text{ }=\text{ }0\text{ }\ldots ..\text{ }\left( 1 \right)~\]and \[x\text{ }+\text{ }5y\text{ }\text{ }11\text{ }=\text{ }0\text{...
Differentiate :
according to ques,
Given Is (AB)’ = B’ A’ ?
Answer: From (I) and (ii), we have \[\left( AB \right)\text{ }=\text{ }B\text{ }A\]
Differentiate :
according to ques,
Differentiate :
according to ques,
Show by an example that for A ≠ 0, B ≠ 0, AB = 0.
Answer: According to the given matrix, the solution should be – Hence Proved
If possible, find BA and AB where
Answer: According to question, the solutions should be
Differentiate :
according to ques,
Differentiate :
according to ques,
If then verify (BA)2 ≠ B2 A2
Answer: The provided matrixs \[A\]has order \[3\text{ }x\text{ }2\]and B has order 2 x 3. Thus, \[BA\] is characterized and will have order 3 x 3. Be that as it may, \[{{A}^{2}}~and\text{...
DIFFERENCIATE: (x + 1)2 + (x + 2)3 + (x + 3)4
ACCORDING TO QUES,
Find A, if:
Answer: On looking at components of the two sides, we have \[4x\text{ }=\text{ }-4\text{ }\Rightarrow x\text{ }=\text{ }-1\] \[4y\text{ }=\text{ }8\text{ }\Rightarrow \text{ }y\text{ }=\text{ }2\]...
DIFFERENCIATE: sinm x . cosn x
ACCORDING TO QUES,
DIFFERENCIATE: (sin x)cos x
ACCORDING TO QUES,
Find the matrix A satisfying the matrix equation:
ANSWER: Presently, \[{{P}^{-1}}~PAQ\text{ }=\text{ }{{P}^{-1~}}I\] Thus, \[IAQ\text{ }=\text{ }{{P}^{-1}}\] \[AQ\text{ }=\text{ }{{P}^{-1}}\] \[AQ{{Q}^{-1}}~=\text{ }{{P}^{-1}}~{{Q}^{-1}}\]...
DIFFERENCIATE:
ACCORDING TO QUES,
DIFFERENCIATE: sin x2 + sin2 x + sin2 (x2)
ACCORDING TO QUES,
DIFFERENCIATE:
ACCORDING TO QUES,
Show that satisfies the equation A2 – 3A – 7I = 0 and hence find A-1.
Answer: According to question, the matrix can be solved as Given, Presently, \[{{A}^{2}}-\text{ }3A\text{ }-\text{ }7I\text{ }=\text{ }0\] Increasing the two sides with\[A-1\], we get...
Differentiate: sin (ax2 + bx + c)
ACCORDING TO QUES,
Differentiate:
ACCORDING TO QUES,
Differentiate:
ACCORDING TO QUES,
Differentiate:
ACCORDING TO QUES,
Differentiate:
ACCORDING TO QUES,
Find the value of x if
Answer: According to given matrix, the solution should be \[[16\text{ }+\text{ }2x\text{ }+\text{ }10(x)\text{ }+\text{ }12\text{ }+\text{ }{{x}^{2}}~+\text{ }4x]\text{ }=\text{ }0\]...
Differentiate:
ACCORDING TO QUES,
A function f : R ® R satisfies the equation f ( x + y) = f (x) f (y) for all x, y ÎR, f (x) ¹ 0. Suppose that the function is differentiable at x = 0 and f ¢ (0) = 2. Prove that f ¢(x) = 2 f (x).
\[f\left( x \right)\text{ }=\text{ }2f\left( x \right).\]
Show that f (x) = |x – 5| is continuous but not differentiable at x = 5.
ACCORDING TO QUES, Therefore, f(x) is not differentiable at \[x\text{ }=\text{ }5.\]
If show that (A + B) (A – B) ≠ A2 – B2
Answer: Thus, from (I) and (ii), \[\left( A\text{ }+\text{ }B \right)\text{ }\left( A\text{ }-\text{ }B \right)\text{ }\ne \text{ }{{A}^{2}}-\text{ }{{B}^{2}}\]
SOLVE:
According to ques, FUNCTION is differentiable at \[x\text{ }=\text{ }2.\]
SOLVE:
NOW ACCORDING TO QUES, AND, HENCE DIFFERENTIABILITY AT \[x\text{ }=\text{ }0.\]
SOLVE:
SO, f(x) is not differentiable at \[x\text{ }=\text{ }2.\]
Find non-zero values of x satisfying the matrix equation:
Answer: Given, On looking at the relating components, we get \[2x\text{ }+\text{ }10x\text{ }=\text{ }48\] \[12x\text{ }=\text{ }48\] In this manner, \[x\text{ }=\text{ }4\] It's additionally seen...
Show that the function f (x) = |sin x + cos x| is continuous at x = p. Examine the differentiability of f, where f is defined by
Given, \[f\left( x \right)\text{ }=\text{ }\left| sin\text{ }x\text{ }+\text{ }cos\text{ }x \right|\text{ }at\text{ }x\text{ }=\text{ }\pi \] Presently, put \[g\left( x \right)\text{ }=\text{...
Find values of a and b if A = B, where
Answer: Given, \[matrix\text{ }A\text{ }=\text{ }matrix\text{ }B\] Then, at that point, their comparing components are equivalent. Thus, we have \[{{a}_{11}}~=\text{ }{{b}_{11}};\]\[a\text{ }+\text{...
Find all points of discontinuity of the function:
Now, if f(t) is discontinuous, then \[2-\text{ }\text{ }x\text{ }=\text{ }0\Rightarrow x\text{ }=\text{ }2\] And, \[2x-\text{ }\text{ }1\text{ }=\text{ }0\Rightarrow x\text{ }=\text{...
Given the function f (x) = 1/(x + 2) . Find the points of discontinuity of the composite function y = f (f (x)).
fun wont be define and continuous \[2x\text{ }+\text{ }5\text{ }=\text{ }0\Rightarrow x\text{ }=\text{ }-5/2\] hence,\[x\text{ }=\text{ }-5/2\] is the point of discontinuity.
Find the values of a and b such that the function f defined by following function is a continuous function at x = 4.
So, \[-1\text{ }+\text{ }a\text{ }=\text{ }a\text{ }+\text{ }b\text{ }=\text{ }1\text{ }+\text{ }b\] \[-1\text{ }+\text{ }a\text{ }=\text{ }a\text{ }+\text{ }b\text{ }and\text{ }1\text{ }+\text{...
If A matrix Z such that X + Y + Z is a zero matrix.
Answer: According to the given ques
Prove that the function f defined by following equation remains discontinuous at x = 0, regardless the choice of k
Since the left hand limit is not equal to the right hand limit and both have constant values. so the given function remains discontinuous at \[x\text{ }=\text{ }0.\] ...
If (i) X + Y (ii) 2X – 3Y
Answer: According to the question, the matrix can be solved as
If possible, find the sum of the matrices A and B,
where A = And B = Answer: The given two matrixs \[A\text{ }and\text{ }B\]are of various orders. Two matrices can be added provided that order for both the matrixs is same. So, the sum of matrices...
Find values of a and b if A = B, where
Answer: Given, \[matrix\text{ }A\text{ }=\text{ }matrix\text{ }B\] Then, at that point, their comparing components are equivalent. Thus, we have \[{{a}_{11}}~=\text{ }{{b}_{11}};\]\[a\text{ }+\text{...
Construct a 3 × 2 matrix whose elements are given by aij = ei.xsin jx
Leave A alone a \[3\text{ }x\text{ }2\] matrix With the end goal that, \[{{a}_{ij}}~=\text{ }{{e}^{i.x}}sin\text{ }jx;\]where \[1\text{ }\le \text{ }i\text{ }\le \text{ }3;\text{ }1\text{ }\le...
Find the value of k so that the function f is continuous at the indicated point:
ACCORDING TO QUES,
Find the value of k so that the function f is continuous at the indicated point:
HENCE K IS \[1\]
Construct a2 × 2 matrix where (i) aij = (i – 2j)2/ 2 (ii) aij = |-2i + 3j|
We have, \[A\text{ }=\text{ }{{[{{a}_{ij}}]}_{2\times 2}}\] (I) Such that,\[~{{a}_{ij}}~=\text{ }{{\left( i\text{ }\text{ }2j \right)}^{2}}/\text{ }2;\] \[where\text{ }1\text{ }\le \text{ }i\text{...
Find the value of k so that the function f is continuous at the indicated point:
\[~k\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\]
In the matrix A = , write : Write elements a23, a31, a12
For the given matrix, Elements: \[{{a}_{23}}~=\text{ }{{x}^{2}}~\text{ }y,\text{ }{{a}_{31}}~=\text{ }0,\text{ }{{a}_{12}}~=\text{ }1\]
Find the value of k so that the function f is continuous at the indicated point:
So, \[7\text{ }=\text{ }2k\] \[k\text{ }=\text{ }7/2\text{ }=\text{ }3.5\] SO VALUE OF K IS \[3.5\]
In the matrix A = , write : (i) The order of the matrix A (ii) The number of elements
For the given matrix, (I) The order for the matrix \[A\text{ }is\text{ }3\text{ }x\text{ }3.\] (ii) The number of elements of the matrix \[~=\text{ }3\text{ }x\text{ }3\text{ }=\text{ }9\]
If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?
For a provided matrix of order \[m\text{ }x\text{ }n,\]it has \[mn\]components, where \[m\text{ }and\text{ }n\]are normal numbers. Here we have, \[m\text{ }x\text{ }n\text{ }=\text{ }28\] \[\left(...
Find which of the functions if is continuous or discontinuous at the indicated points: f (x) = |x| + |x – 1| at x = 1
And hence, f(x) is continuous at \[x\text{ }=\text{ }1.\]
Find which of the functions if is continuous or discontinuous at the indicated points:
And hence, f(x) is continuous at \[x\text{ }=\text{ }1.\]
Find which of the functions if is continuous or discontinuous at the indicated points:
And hence, f(x) is discontinuous at \[x\text{ }=\text{ }0.\]
Find which of the functions if is continuous or discontinuous at the indicated points:
And hence, f(x) is continuous at \[x\text{ }=\text{ }0.\]
Find which of the functions if is continuous or discontinuous at the indicated points:
And hence, f(x) is continuous at \[x\text{ }=\text{ }0.\]
If A, B and C are angles of a triangle, then the determinant is equal to (A) 0 (B) -1 (C) 1 (D) None of these
Answer: \[Option\text{ }\left( A \right)\text{ }0\]
The number of distinct real roots of = 0 in the interval -π/4 ≤ x ≤ π/4 is (A) 0 (B) 2 (C) 1 (D) 3
Answer: \[Option\text{ }\left( C \right)\text{ }1\]
The determinant equals (A) abc (b–c) (c – a) (a – b) (B) (b–c) (c – a) (a – b) (C) (a + b + c) (b – c) (c – a) (a – b) (D) None of these
Answer: Option (D) \[None\text{ }of\text{ }these\]
The area of a triangle with vertices (–3, 0), (3, 0) and (0, k) is 9 sq. units. The value of k will be (A) 9 (B) 3 (C) – 9 (D) 6
Option \[\left( B \right)\text{ }3\] According to the question, the area of a triangle with vertices \[({{x}_{1}},\text{ }{{y}_{1}}),\text{ }({{x}_{2}},\text{ }{{y}_{2}})\text{ }and\text{...
The value of determinant (A) a3 + b3 + c3 (B) 3 bc (C) a3 + b3 + c3 – 3abc (D) none of these
ANSWER: Option \[\left( C \right)~{{a}^{3}}~+~{{b}^{3}}~+~{{c}^{3}}~\text{ }3abc\] Given,
If then, value of x is (A) 3 (B) ± 3 (C) ± 6 (D) 6
ANSWER: Option \[~\left( C \right)\text{ }\pm \text{ }6\] On equating the determinants, we get \[2{{x}^{2}}-\text{ }40\text{ }=\text{ }18\text{ }+\text{ }14\] \[2{{x}^{2}}~=\text{ }72\]...
If x + y + z = 0, prove that
Answer: According to the question, the determinant can be solved as
Prove that is divisible by a + b + c and find the quotient.
Answer: According to the question, the determinant can be solved as
If a + b + c ¹ 0 and then prove that a = b = c.
Answer: According to the question, the determinant can be solved as
Given find BA and use this to solve the system of equations y + 2z = 7, x – y = 3, 2x + 3y + 4z = 17.
Answer: According to the question, the determinant can be solved as
Using matrix method, solve the system of equations 3x + 2y – 2z = 3, x + 2y + 3z = 6, 2x – y + z = 2.
Answer: Given equations are: \[3x~+\text{ }2y-\text{ }2z~=\text{ }3\] \[x~+\text{ }2y~+\text{ }3z~=\text{ }6\]and \[2x~-y~+~z~=\text{ }2\] Or, \[AX\text{ }=\text{ }B\]
If A = , find A-1. Using A–1, solve the system of linear equations x – 2y = 10 , 2x – y – z = 8 , –2y + z = 7.
Answer: According to the question, the determinant can be solved as
Find A–1 if and show that A-1 = (A2 – 3I)/ 2.
Answer: According to the question, the determinant can be solved as
Show that the DABC is an isosceles triangle if the determinant
Answer: According to the question, the determinant can be solved as
Show that the points (a + 5, a – 4), (a – 2, a + 3) and (a, a) do not lie on a straight line for any value of a.
Given focuses are \[\left( a\text{ }+\text{ }5,\text{ }a\text{ }\text{ }4 \right),\text{ }\left( a\text{ }\text{ }2,\text{ }a\text{ }+\text{ }3 \right)\]and \[\left( a,\text{ }a \right).\]...
If a1, a2, a3, …, ar are in G.P., then prove that the determinant is independent of r.
Answer: According to the question, the determinant can be solved as So, it is independent of r.
If , then find values of x.
Answer: According to the question, the determinant can be solved as
Find the value of q satisfying
Answer: According to the question, the determinant can be solved as
If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are (x1, y1),(x2, y2), (x3, y3), then
Answer: According to the question, the determinant can be solved as
If A + B + C = 0, then prove that
Answer: According to the question, the determinant can be solved as
Find the area of region bounded by the line x = 2 and the parabola y^2 = 8x
The equation of line x = 2 and parabola y2 = 8x Putting value of x in the other equation, we have \[\begin{array}{*{35}{l}} {{y}^{2}}~=\text{ }8\left( 2 \right) \\ {{y}^{2}}~=\text{ }16 \\...
Prove that:
Answer: According to the question, the determinant can be solved as
Prove that:
Answer: According to the question, the determinant can be solved as
Prove that:
Answer: According to the question, the determinant can be solved as
Evaluate the following determinant:
Answer: According to the question, the determinant can be solved as Now, expanding along\[R1\], we have \[=\text{ }\left( a\text{ }+\text{ }b\text{ }+\text{ }c \right)\text{ }[1\text{ }x\text{...
Evaluate the following determinant:
Answer: According to the question, the determinant can be solved as
Evaluate the following determinant:
Answer: [Expanding along first column] \[=\text{ }\left( x\text{ }+\text{ }y\text{ }+\text{ }z \right)\text{ }.\text{ }1\left[ 3y\left( 3z\text{ }+\text{ }x \right)\text{ }+\text{ }\left(3z...
Evaluate the following determinant:
Answer: According to the question, the determinant can be solved as
Evaluate the following determinant:
Answer: According to the question, the determinant can be solved as
Evaluate the following determinant:
Answer: \[=\text{ }({{x}^{2}}-\text{ }2x\text{ }+\text{ }2)\text{ }.\text{ }\left( x\text{ }+\text{ }1 \right)\text{ }-\text{ }\left( (x)\text{ }-\text{ }1 \right)\text{ }.\text{ }0\] \[=\text{...
solve the following:
Solution:
Find which of the functions if is continuous or discontinuous at the indicated points:
at \[\mathbf{x}\text{ }=\text{ }\mathbf{4}\] And hence, f(x) is discontinuous at \[x\text{ }=\text{ }4.\]
Find which of the functions if is continuous or discontinuous at the indicated points:
at \[\mathbf{x}\text{ }=\text{ }\mathbf{2}\] And hence, f(x) is continuous at \[x\text{ }=\text{ }2.\]
Find which of the functions if is continuous or discontinuous at the indicated points:
at \[\mathbf{x}\text{ }=\text{ }\mathbf{2}\] And hence, f(x) is discontinuous at \[x\text{ }=\text{ }2.\]
Find which of the functions if is continuous or discontinuous at the indicated points:
at \[\mathbf{x}\text{ }=\text{ }\mathbf{0}\] NOW, HENCE, given function f(x) is discontinuous at \[x\text{ }=\text{ }0.\]
Examine the continuity of the function f (x) = x3 + 2×2 – 1 at x = 1
since, \[y\text{ }=\text{ }f\left( x \right)\] will be continuous at \[x\text{ }=\text{ }a\] if, hence, f(x) is continuous at $$ \[x\text{ }=\text{ }1.\]
AB is a diameter of a circle and C is any point on the circle. Show that the area of Δ ABC is maximum, when it is isosceles.
Let consider AB be the width and C is any point on the circle with range r. \[\angle ACB\text{ }=\text{ }90o\] [angle in the semi-circle is 90o] Let \[AC\text{ }=\text{ }x\] Squaring on both the...
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
We should accept x be to the edge and r be the span of the circle. Surface space of 3D square \[=\text{ }6x2\] Also, surface space of the circle \[=\text{ }4\pi r2\] Presently, their total is...
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also find the maximum volume. Solution:
We should believe x and y to be the length and expansiveness of given square shape ABCD. As per the inquiry, the square shape will be settled with regards to side AD which making a chamber with...
An open box with square base is to be made of a given quantity of card board of area c2. Show that the maximum volume of the box is c3/ 6√3 cubic units.
Leave x alone the length of the side of the square base of the cubical open box and y be its height Along these lines, the surface space of the open box
If the straight-line x cos α + y sin α = p touches the curve x2/a2 + y2/b2 = 1, then prove that a2 cos2 α + b2 sin2 α = p2.
The given bend is \[\mathbf{x2}/\mathbf{a2}\text{ }+\text{ }\mathbf{y2}/\mathbf{b2}\text{ }=\text{ }\mathbf{1}\] and the straight-line \[\mathbf{x}\text{ }\mathbf{cos}\text{ }\mathbf{\alpha }\text{...
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Rs 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
We should consider that the organization expands the yearly membership by \[\mathbf{Rs}\text{ }\mathbf{x}.\] Along these lines, x is the quantity of supporters who end the administrations. ...
Find the points of local maxima, local minima and the points of inflection of the function f (x) = x5 – 5×4 + 5×3 – 1. Also find the corresponding local maximum and local minimum values.
Given, \[\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{x5}\text{ }\text{ }\mathbf{5x4}\text{ }+\text{ }\mathbf{5x3}\text{ }\text{ }\mathbf{1}\] Separating the capacity,...
Prove that f (x) = sin x + √3 cos x has maximum value at x = π/6.
Let ∆ABC be the right-angled triangle in which \[\angle B\text{ }=\text{ }{{90}^{o}}\] Let \[\mathbf{AC}\text{ }=\text{ }\mathbf{x},\text{ }\mathbf{BC}\text{ }=\text{ }\mathbf{y}\] In this way,...
Prove that f (x) = sin x + √3 cos x has maximum value at x = π/6.
ACCORDING TO QUES, HENCE FUNCTION has maximum value at \[x\text{ }=\text{ }\pi /6\] and maximum value is \[2.\]
At what point, the slope of the curve y = – x3 + 3×2 + 9x – 27 is maximum? Also find the maximum slope.
Given, CURVE \[y\text{ }=\text{ }\text{ }x3\text{ }+\text{ }3x2\text{ }+\text{ }9x\text{ }\text{ }27\] Separating the two sides w.r.t. x, we get \[dy/dx\text{ }=\text{ }-\text{ }3x2\text{ }+\text{...
Show that f(x) = tan–1(sin x + cos x) is an increasing function in (0, π/4).
Given, \[f\left( x \right)\text{ }=\text{ }tan1\left( sin\text{ }x\text{ }+\text{ }cos\text{ }x \right)\text{ }in\text{ }\left( 0,\text{ }\pi /4 \right).\] DIFFERENTIATING the two sides w.r.t. x, we...
Show that for a ³ 1, f (x) = √3 sin x – cos x – 2ax + b is decreasing in R.
ACCORDING TO QUES, \[f~\left( x \right)\text{ }=\text{ }\surd 3\text{ }sin~x\text{ }~cos~x\text{ }~2ax\text{ }+\text{ }b,\text{ }a~{}^\text{3}\text{ }1\] differentiating both sides w.r.t. x WE HAVE...
Show that f (x) = 2x + cot-1 x + log [√(1 + x2) – x] is increasing in R.
Given, \[f\text{ }\left( x \right)\text{ }=\text{ }2x\text{ }+\text{ }bed\text{ }1\text{ }x\text{ }+\text{ }log\text{ }\left[ \surd \left( 1\text{ }+\text{ }x2 \right)\text{ }\text{ }x...
Show that the line x/a + y/b = 1, touches the curve y = b . e-x/a at the point where the curve intersects the axis of y.
Given curve condition, \[y\text{ }=\text{ }b\text{ }.\text{ }e-x/an\] and line condition \[x/a\text{ }+\text{ }y/b\text{ }=\text{ }1\] Presently, let the directions of where the curve meets the...
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?
Given, the condition of the bend is \[x2\text{ }+\text{ }y2\text{ }\text{ }2x\text{ }\text{ }4y\text{ }+\text{ }1\text{ }=\text{ }0\text{ }\ldots \text{ }..\text{ }\left( I \right)\] Separating both...
Find the equation of the normal lines to the curve 3×2 – y2 = 8 which are parallel to the line x + 3y = 4
Given curve, \[3x2\text{ }\text{ }y2\text{ }=\text{ }8\] Separating the two sides w.r.t. x, we get \[6x\text{ }\text{ }2y.\text{ }dy/dx\text{ }=\text{ }0\Rightarrow -\text{ }2y\left( dy/dx...
Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2).
Given bend conditions are: \[y2\text{ }=\text{ }4x\text{ }\ldots \text{ }.\text{ }\left( 1 \right)\text{ }and\text{ }x2\text{ }+\text{ }y2\text{ }\text{ }6x\text{ }+\text{ }1\text{ }=\text{ }0\text{...
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
The given bends are \[y\text{ }=\text{ }4\text{ }\text{ }x2\text{ }\ldots \text{ }.\text{ }\left( I \right)\text{ }and\text{ }y\text{ }=\text{ }x2\text{ }\ldots \text{ }\left( ii \right)\] Also, we...
solve the following:
Solution:
Solve the following:
Solution:
Find the co-ordinates of the point on the curve √x + √y = 4 at which tangent is equally inclined to the axes.
ACCORDING TO EQUATION OF CURVE, \[\surd x\text{ }+\text{ }\surd y\text{ }=\text{ }4\] Presently, let (x1, y1) be he required point on the bend SO , \[\surd x1\text{ }+\text{ }\surd y1\text{ }=\text{...
Solve the following:
Solution:
Solve the following:
Solution:
solve the following:
Solution:
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
Given bends are conditions of two circles, \[xy\text{ }=\text{ }4\text{ }\ldots \text{ }..\text{ }\left( I \right)\] and \[x2\text{ }+\text{ }y2\text{ }=\text{ }8\text{ }\ldots \text{ }.\text{...
Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.
It's seen that the given bends are condition of two circles. \[2x\text{ }=\text{ }y2\text{ }\ldots \text{ }..\text{ }\left( 1 \right)\] and \[2xy\text{ }=\text{ }k\text{ }\ldots \text{ }..\text{...
x and y are the sides of two squares such that y = x – x2. Find the rate of change of the area of second square with respect to the area of first square.
How about we consider the space of the primary square \[A1\text{ }=\text{ }x2\] Furthermore, space of the subsequent square be \[A2\text{ }=\text{ }y2\] Presently, \[A1\text{ }=\text{ }x2\text{...
The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.
We should expect x to be the length of the block. In this way, the volume of the solid shape \[V\text{ }=\text{ }x3\text{ }\ldots \text{ }.\text{ }\left( 1 \right)\] Considering that, \[dV/dt\text{...
A swimming pool is to be drained for cleaning. If L represents the number of litres of water in the pool t seconds after the pool has been plugged off to drain and L = 200 (10 – t)2. How fast is the water running out at the end of 5 seconds? What is the average rate at which the water flows out during the first 5 seconds?
Given, \[L\text{ }=\text{ }200\left( 10\text{ }\text{ }t \right)2\] where L addresses the quantity of liters of water in the pool. On separating both the sides w.r.t, t, we get \[dL/dt\text{...
A man, 2m tall, walks at the rate of m/s towards a street light which is m above the ground. At what rate is the tip of his shadow moving? At what rate is the length of the shadow changing when he is m from the base of the light?
Let AB is the stature of streetlamp post and CD is the tallness of the man with the end goal that \[AB\text{ }=\text{ }5\left( 1/3 \right)\text{ }=\text{ }16/3\text{ }m\text{ }and\text{ }CD\text{...
Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3 cm and 3.0005 cm, respectively.
Given, The interior range $$ \[r\text{ }=\text{ }3\text{ }cm\] What's more, outside sweep \[R\text{ }=\text{ }r\text{ }+\text{ }r\text{ }=3.0005\text{ }cm\] \[r\text{ }=\text{ }3.0005\text{...
If a relation R on the set ![\[\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3} \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-df9a10089e417e1bc74e4a43afb3835b_l3.png "Rendered by QuickLaTeX.com")
be defined by ![\[\mathbf{R}\text{ }=\text{ }\left\{ \left( \mathbf{1},\text{ }\mathbf{2} \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7d4fbd5bf6f1080340e0ad730c73e592_l3.png "Rendered by QuickLaTeX.com")
, then R is (A) reflexive (B) transitive (C) symmetric (D) none of these
The correct option is (D) none of these Given R on the set \[\left\{ 1,\text{ }2,\text{ }3 \right\}\]be defined by \[R\text{ }=\text{ }\left\{ \left( 1,\text{ }2 \right) \right\}\] Therefore, its...
The maximum number of equivalence relations on the set ![\[\mathbf{A}\text{ }=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3} \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6b1b0a84bfc5cc886f5c4fd5e453f9a3_l3.png "Rendered by QuickLaTeX.com")
are (A) ![\[1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-64318110f3a374fb786ef713c0b0c575_l3.png "Rendered by QuickLaTeX.com")
(B) ![\[2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b1e8e3db630df9050b1fec3c4066d5e6_l3.png "Rendered by QuickLaTeX.com")
(C) ![\[3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-df8bcc09db12acd0f6bda88b4b8135fb_l3.png "Rendered by QuickLaTeX.com")
(D) ![\[5\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7dc8465919d0838925dbc6c99670258c_l3.png "Rendered by QuickLaTeX.com")
The correct option is (D) \[5\] Given, set \[\mathbf{A}\text{ }=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3} \right\}\] Now, the number of equivalence relations as follows...
Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is (A) symmetric but not transitive (B) transitive but not symmetric (C) neither symmetric nor transitive (D) both symmetric and transitive
The correct option is (B) transitive but not symmetric Given aRb ⇒ a is brother of b. This does not mean b is also a brother of a as b can be a sister of a. Therefore, R is not symmetric. aRb ⇒ a is...
Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is (A) reflexive but not transitive (B) transitive but not symmetric (C) equivalence (D) none of these
The correct option is (C) equivalence Given aRb, if a is congruent to b, ∀ a, b ∈ T. Then, we have aRa ⇒ a is congruent to a; which is always true. So, R is reflexive. Let aRb ⇒ a ~ b b ~ a bRa So,...
Let * be binary operation defined on R by ![\[\mathbf{a}\text{ }*\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{1}\text{ }+\text{ }\mathbf{ab},\forall \mathbf{a},\text{ }\mathbf{b}\in \mathbf{R}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2961f64055cadc8b7cae88fe60a1b709_l3.png "Rendered by QuickLaTeX.com")
. Then the operation * is (i) commutative but not associative (ii) associative but not commutative (iii) neither commutative nor associative (iv) both commutative and associative
(i) Given that * is a binary operation defined on R by \[\mathbf{a}\text{ }*\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{1}\text{ }+\text{ }\mathbf{ab},\forall \mathbf{a},\text{ }\mathbf{b}\in...
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative ![\[\begin{array}{*{35}{l}} \left( \mathbf{i} \right)\text{ }\mathbf{a}\text{ }*\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{a}\text{ }\text{ }\mathbf{b}\forall \mathbf{a},\text{ }\mathbf{b}\in \mathbf{Q}\text{ }\left( \mathbf{ii} \right)\text{ }\mathbf{a}\text{ }*\text{ }\mathbf{b}\text{ }=\text{ }{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}}~\forall \mathbf{a},\text{ }\mathbf{b}\in \mathbf{Q} \\ \left( \mathbf{iii} \right)\text{ }\mathbf{a}\text{ }*\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{a}\text{ }+\text{ }\mathbf{ab}\forall \mathbf{a},\text{ }\mathbf{b}\in \mathbf{Q}\text{ }\left( \mathbf{iv} \right)\text{ }\mathbf{a}\text{ }*\text{ }\mathbf{b}\text{ }=\text{ }{{\left( \mathbf{a}\text{ }\text{ }\mathbf{b} \right)}^{\mathbf{2}}}~\forall \mathbf{a},\text{ }\mathbf{b}\in \mathbf{Q} \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0a1f3a790697fd211e1ec8a051a8dcce_l3.png "Rendered by QuickLaTeX.com")
Given that * is a binary operation defined on Q. (i) \[a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }\text{ }b,\forall a,\text{ }b\in Q\] and \[b\text{ }*\text{ }a\text{ }=\text{ }b\text{ }\text{...
Functions f , g : R → R are defined, respectively, by ![\[\mathbf{f}~\left( \mathbf{x} \right)\text{ }=\text{ }{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{1},~\mathbf{g}~\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{2x}\text{ }\text{ }\mathbf{3}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2dd0a70d016c1b6709d883a89deaa7c8_l3.png "Rendered by QuickLaTeX.com")
, find (i) f o g (ii) g o f (iii) f o f (iv) g o g
Given, \[\mathbf{f}~\left( \mathbf{x} \right)\text{ }=\text{ }{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{1},~\mathbf{g}~\left( \mathbf{x} \right)\text{ }=\text{...
. Let ![\[\mathbf{A}\text{ }=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\ldots \text{ }\mathbf{9} \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4bba703b875f44961cf66438db00c2aa_l3.png "Rendered by QuickLaTeX.com")
and R be the relation in A ×A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in ![\[A\text{ }\times A\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b2d867ff818b6c236e399d822f63f0a7_l3.png "Rendered by QuickLaTeX.com")
. Prove that R is an equivalence relation and also obtain the equivalent class ![\[\left[ \left( \mathbf{2},\text{ }\mathbf{5} \right) \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3affb3b24e311584fc304dd1f72979fc_l3.png "Rendered by QuickLaTeX.com")
.
Given, \[\mathbf{A}\text{ }=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\ldots \text{ }\mathbf{9} \right\}\]and (a, b) R (c, d) if a + d = b + c for \[\left( a,\text{ }b...
Each of the following defines a relation on N: (i) x is greater than y, ![\[\mathbf{x},\text{ }\mathbf{y}\in \mathbf{N}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-53bce6e1c6a20d27eb7981b3f2a0d9cc_l3.png "Rendered by QuickLaTeX.com")
(ii) ![\[\mathbf{x}\text{ }+\text{ }\mathbf{y}\text{ }=\text{ }\mathbf{10},\text{ }\mathbf{x},\text{ }\mathbf{y}\in \mathbf{N}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ac2a04bdc6b7e6a9e6cf59fc647f9a46_l3.png "Rendered by QuickLaTeX.com")
(iii) x y is square of an integer
(iv) ![\[\mathbf{x}\text{ }+\text{ }\mathbf{4y}\text{ }=\text{ }\mathbf{10}\text{ }\mathbf{x},\text{ }\mathbf{y}\in \mathbf{N}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-066b5246906e9f67cf4ce6457424ca50_l3.png "Rendered by QuickLaTeX.com")
. Determine which of the above relations are reflexive, symmetric and transitive.
(i) Given, x is greater than y; \[\mathbf{x},\text{ }\mathbf{y}\in \mathbf{N}\] If \[\left( x,\text{ }x \right)\in R\], then \[x\text{ }>\text{ }x\], which is not true for any \[x\in N\]. Thus, R...
Let ![\[\mathbf{A}\text{ }=\text{ }\left[ \mathbf{1},\text{ }\mathbf{1} \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ec86688849079e7c6493bd34b7abb7a3_l3.png "Rendered by QuickLaTeX.com")
. Then, discuss whether the following functions defined on A are one-one, onto or bijective: (i) ![\[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{x}/\mathbf{2}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-60d315a329816d5d7d74e09a6424ff3b_l3.png "Rendered by QuickLaTeX.com")
(ii) ![\[\mathbf{g}\left( \mathbf{x} \right)\text{ }=\text{ }\left| \mathbf{x} \right|\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7a2feecb05cc88418ae74c021378e4f5_l3.png "Rendered by QuickLaTeX.com")
(iii) ![\[\mathbf{h}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{x}\left| \mathbf{x} \right|\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e4e9b95c341051db94c9983ee99d99dc_l3.png "Rendered by QuickLaTeX.com")
(iv) ![\[\mathbf{k}\left( \mathbf{x} \right)\text{ }=\text{ }{{\mathbf{x}}^{\mathbf{2}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-531f812814bd08b1132646e938b5d74e_l3.png "Rendered by QuickLaTeX.com")
Given, \[A\text{ }=\text{ }\left[ 1,\text{ }1 \right]\] (i) \[f:\text{ }\left[ -1,\text{ }1 \right]\text{ }\to \text{ }\left[ -1,\text{ }1 \right],\text{ }f\text{ }\left( x \right)\text{ }=\text{...
Let ![\[\mathbf{A}\text{ }=\text{ }\mathbf{R}\text{ }\text{ }\left\{ \mathbf{3} \right\},\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{R}\text{ }\text{ }\left\{ \mathbf{1} \right\}.\text{ }\mathbf{Let}\text{ }\mathbf{f}\text{ }:\text{ }\mathbf{A}\text{ }\to \text{ }\mathbf{B}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4826e8120534bd03f6ed6670ce6482c3_l3.png "Rendered by QuickLaTeX.com")
be defined by ![\[\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{x}\text{ }\text{ }\mathbf{2}/\text{ }\mathbf{x}\text{ }\text{ }\mathbf{3}\forall \mathbf{x}\in \mathbf{A}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3be59825d072da5199347321d3e97a56_l3.png "Rendered by QuickLaTeX.com")
. Then show that f is bijective.
According to the question, \[A\text{ }=\text{ }R\text{ }\text{ }\left\{ 3 \right\},\text{ }B\text{ }=\text{ }R\text{ }\text{ }\left\{ 1 \right\}\] And, f : A → B be defined by \[f\text{ }\left( x...
Give an example of a map (i) which is one-one but not onto (ii) which is not one-one but onto (iii) which is neither one-one nor onto.
(i) Let f: N → N, be a mapping defined by \[f\text{ }\left( x \right)\text{ }=\text{ }{{x}^{2}}\] For \[f\text{ }({{x}_{1}})\text{ }=\text{ }f\text{ }({{x}_{2}})\] Then, \[{{x}_{1}}^{2}~=\text{...
Given ![\[\mathbf{A}\text{ }=\text{ }\left\{ \mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4} \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1d0332e8f8738588de84e4d9d4c7882a_l3.png "Rendered by QuickLaTeX.com")
, ![\[\mathbf{B}\text{ }=\text{ }\left\{ \mathbf{2},\text{ }\mathbf{5},\text{ }\mathbf{6},\text{ }\mathbf{7} \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5d67b0b282f80ff369f931d204458d4d_l3.png "Rendered by QuickLaTeX.com")
. Construct an example of each of the following: (a) an injective mapping from A to B (b) a mapping from A to B which is not injective (c) a mapping from B to A.
Given, \[\mathbf{A}\text{ }=\text{ }\left\{ \mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4} \right\}\],\[\mathbf{B}\text{ }=\text{ }\left\{ \mathbf{2},\text{ }\mathbf{5},\text{ }\mathbf{6},\text{...
Let R be relation defined on the set of natural number N as follows: ![\[\mathbf{R}\text{ }=\text{ }\{\left( \mathbf{x},\text{ }\mathbf{y} \right):\text{ }\mathbf{x}\in \mathbf{N},\text{ }\mathbf{y}\in \mathbf{N},\text{ }\mathbf{2x}\text{ }+\text{ }\mathbf{y}\text{ }=\text{ }\mathbf{41}\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-88df5ebd7ad6cdd01200d5d66cf2b418_l3.png "Rendered by QuickLaTeX.com")
. Find the domain and range of the relation R. Also verify whether R is reflexive, symmetric and transitive.
Given function: \[\mathbf{R}\text{ }=\text{ }\{\left( \mathbf{x},\text{ }\mathbf{y} \right):\text{ }\mathbf{x}\in \mathbf{N},\text{ }\mathbf{y}\in \mathbf{N},\text{ }\mathbf{2x}\text{ }+\text{...
Find the approximate value of (1.999)5.
\[\left( 1.999 \right)5\text{ }=\text{ }\left( 2\text{ }\text{ }0.001 \right)5\] Let \[x\text{ }=\text{ }2\text{ }and\text{ }x\text{ }=\text{ }-\text{ }0.001\] Likewise, let \[y\text{ }=\text{ }x5\]...
Find the approximate value of (1.999)5.
question suggests, angle is \[\pi /3.\]
Find an angle q, 0 < q < /2, which increases twice as fast as its sine.
As per the inquiry, we have In this manner, the necessary point is \[\pi /3.\]
The value of ![\[\mathbf{sin}\text{ }(\mathbf{2}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}\left( \mathbf{0}.\mathbf{75} \right))\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-68c7d427817e899bb8b5a68d5aed6574_l3.png "Rendered by QuickLaTeX.com")
is equal to (a) ![\[\mathbf{0}.\mathbf{75}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-446d3f09e70b6cad0293b7ed4c3da967_l3.png "Rendered by QuickLaTeX.com")
(b) ![\[\mathbf{1}.\mathbf{5}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0be31f8506cd6a9fb18570daae72de29_l3.png "Rendered by QuickLaTeX.com")
(c) ![\[\mathbf{0}.\mathbf{96}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f0483be7dacb7fe556620b46cb0154b5_l3.png "Rendered by QuickLaTeX.com")
(d) ![\[\mathbf{sin}\text{ }\mathbf{1}.\mathbf{5}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0b77460703a89e6fbd25aab1d917b285_l3.png "Rendered by QuickLaTeX.com")
The correct option is (c) \[\mathbf{0}.\mathbf{96}\] We have, \[sin\text{ }(2\text{ }ta{{n}^{-1}}\left( 0.75 \right))\]
If ![\[\mathbf{cos}\text{ }(\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{2}/\mathbf{5}\text{ }+\text{ }\mathbf{co}{{\mathbf{s}}^{-\mathbf{1}}}~\mathbf{x})\text{ }=\text{ }\mathbf{0}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1cb547e2a81fe2ddca2b94ec20b7efdb_l3.png "Rendered by QuickLaTeX.com")
, then x is equal to (a) ![\[\mathbf{1}/\mathbf{5}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-698ecc2591e4187b3b31451e4a8412ac_l3.png "Rendered by QuickLaTeX.com")
(b) ![\[\mathbf{2}/\mathbf{5}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5995ed954b2b025ca426324f5a89a8fe_l3.png "Rendered by QuickLaTeX.com")
(c) ![\[\mathbf{0}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fd0385a423a02a2235879aff6bfece5b_l3.png "Rendered by QuickLaTeX.com")
(d)
The correct option is (b) \[\mathbf{2}/\mathbf{5}\] Given, \[cos\text{ }(si{{n}^{-1}}~2/5\text{ }+\text{ }co{{s}^{-1}}~x)\text{ }=\text{ }0\] So, this can be rewritten as \[\begin{array}{*{35}{l}}...
The domain of the function by f(x) = ![\[\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\sqrt{\left( \mathbf{x}\text{ }\text{ }\mathbf{1} \right)}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-dfbf35cc7cb2d872feaf569485ef4904_l3.png "Rendered by QuickLaTeX.com")
is (a) ![\[\left[ \mathbf{1},\text{ }\mathbf{2} \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fcfcb28dd59583bc3e7a020a8cf5f451_l3.png "Rendered by QuickLaTeX.com")
(b) ![\[\left[ -\mathbf{1},\text{ }\mathbf{1} \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b21f7411312e08e9e9c76b1a5369f211_l3.png "Rendered by QuickLaTeX.com")
(c) ![\[\left[ \mathbf{0},\text{ }\mathbf{1} \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4951cedbbb831e919142a21d14151c6a_l3.png "Rendered by QuickLaTeX.com")
(d) none of these
The correct option is (a) \[\left[ \mathbf{1},\text{ }\mathbf{2} \right]\] We know that, \[si{{n}^{-1}}~x\] is defined for \[x\in \left[ -1,\text{ }1 \right]\] So, f(x) =...
The domain of the function ![\[\mathbf{cos}-\mathbf{1}\text{ }\left( \mathbf{2x}-\text{ }\text{ }\mathbf{1} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-99b8e6286b585837ce48c8b01132334e_l3.png "Rendered by QuickLaTeX.com")
is (a)
(b)
(c)
(d) ![\[\left[ \mathbf{0},\text{ }\mathbf{\pi } \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4f3b7c31075d2dae94fe9c4f60b1821a_l3.png "Rendered by QuickLaTeX.com")
The correct option is (a) \[\left[ \mathbf{0},\text{ }\mathbf{1} \right]\] Since, \[cos-1\text{ }x\] is defined for \[x\in \left[ -1,\text{ }1 \right]\] So, f(x) = \[\mathbf{cos}-\mathbf{1}\text{...
The value of ![\[\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{cos}\text{ }\mathbf{33\pi }/\mathbf{5}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-aea705b6330621ecb7f543a1fea098c8_l3.png "Rendered by QuickLaTeX.com")
is (a) ![\[\mathbf{3\pi }/\mathbf{5}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c2d1bda05bc0967141df29b8057cf78a_l3.png "Rendered by QuickLaTeX.com")
(b) ![\[-\mathbf{7\pi }/\mathbf{5}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c3e656e247ec5c6b18a8ecf530d059fe_l3.png "Rendered by QuickLaTeX.com")
(c) ![\[\mathbf{\pi }/\mathbf{10}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-47aaedd99c7dba30799394a11a516fb5_l3.png "Rendered by QuickLaTeX.com")
(d) ![\[-\mathbf{\pi }/\mathbf{10}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4b6678b39bfd2e446cdbc2e73b32692d_l3.png "Rendered by QuickLaTeX.com")
The correct option is (d) \[-\mathbf{\pi }/\mathbf{10}\]
Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45° to each other. If they travel by different roads, find the rate at which they are being separated.
How about we believe P to be any point where the two streets are leaned at a point of \[45o.\] Presently, two men An and B are moving along the streets PA and PB separately with same speed 'V'....
A kite is moving horizontally at a height of 151.5 meters. If the speed of kite is 10 m/s, how fast is the string being let out; when the kite is 250 m away from the boy who is flying the kite? The height of boy is 1.5 m.
Speed of the kite(V) \[=\text{ }10\text{ }m/s\] Leave FD alone the tallness of the kite and AB be the stature of the kite and AB be the tallness of the kid. Presently, let AF \[=\text{ }x\text{ }m\]...
If ![\[\mathbf{3}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{x}\text{ }+\text{ }\mathbf{co}{{\mathbf{t}}^{-\mathbf{1}}}~\mathbf{x}\text{ }=\text{ }\mathbf{\pi }\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f24489a3235ab60a7d1ead918fee879b_l3.png "Rendered by QuickLaTeX.com")
, then x equals (a)
(b) ![\[\mathbf{1}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0313c9365144662794ca80515dcd72ce_l3.png "Rendered by QuickLaTeX.com")
(c) ![\[-\mathbf{1}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-96158281230e100426cf2f540c6264ad_l3.png "Rendered by QuickLaTeX.com")
(d) ½
The correct option is (b) \[\mathbf{1}\] Given, \[\mathbf{3}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{x}\text{ }+\text{ }\mathbf{co}{{\mathbf{t}}^{-\mathbf{1}}}~\mathbf{x}\text{...
Which of the following is the principal value branch of ![\[\mathbf{cose}{{\mathbf{c}}^{-\mathbf{1}}}~\mathbf{x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6d1eb220a0e4116384d174374ab9174c_l3.png "Rendered by QuickLaTeX.com")
? (a) ![\[\left( -\mathbf{\pi }/\mathbf{2},\text{ }\mathbf{\pi }/\mathbf{2} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-04e628daf78abc2132895875d18d553a_l3.png "Rendered by QuickLaTeX.com")
(b) ![\[\left[ \mathbf{0},\text{ }\mathbf{\pi } \right]\text{ }\text{ }\left\{ \mathbf{\pi }/\mathbf{2} \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1095ec138b1c83380bf46ca1d2d9c985_l3.png "Rendered by QuickLaTeX.com")
(c) ![\[\left[ -\mathbf{\pi }/\mathbf{2},\text{ }\mathbf{\pi }/\mathbf{2} \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5354ebba5b1885dfedd7d083862690a1_l3.png "Rendered by QuickLaTeX.com")
(d) ![\[\left[ -\mathbf{\pi }/\mathbf{2},\text{ }\mathbf{\pi }/\mathbf{2} \right]\text{ }\text{ }\left\{ \mathbf{0} \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6730dd1a41f4218ce47d3ee43e4842e8_l3.png "Rendered by QuickLaTeX.com")
The correct option is (d) \[\left[ -\mathbf{\pi }/\mathbf{2},\text{ }\mathbf{\pi }/\mathbf{2} \right]\text{ }\text{ }\left\{ \mathbf{0} \right\}\] According to the principal branch of...
If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius.
We realize that the space of circle, \[A\text{ }=\text{ }\pi r2\] , where r = span of the circle Furthermore, \[edge\text{ }=\text{ }2\pi r\] As indicated by the inquiry, we have Subsequently, it's...
A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate.
Given, a round ball of salt Then, at that point, the volume of ball \[V\text{ }=\text{ }4/3\text{ }\pi r3\] where r = sweep of the ball Presently, as per the inquiry we have \[dV/dt\propto S\] ,...
Which of the following is the principal value branch of ![\[\mathbf{cos}-\mathbf{1}\text{ }\mathbf{x}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d8e6ba03fbd43d222c09e6d7ff5e6405_l3.png "Rendered by QuickLaTeX.com")
? (a)
(b) ![\[\left( \mathbf{0},\text{ }\mathbf{\pi } \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-304afae725f16f5826abffbd7d0e3505_l3.png "Rendered by QuickLaTeX.com")
(c) ![\[\left[ \mathbf{0}.\text{ }\mathbf{\pi } \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-96074c13411cdba9487f8ef3abfc6ff9_l3.png "Rendered by QuickLaTeX.com")
(d)
The correct answer is (c) \[\left[ \mathbf{0}.\text{ }\mathbf{\pi } \right]\] According to the principal value branch \[co{{s}^{-1}}~x\] is \[\left[ 0,\text{ }\pi \right]\].
If ![\[{{\mathbf{a}}_{\mathbf{1}}},\text{ }{{\mathbf{a}}_{\mathbf{2}}},\text{ }{{\mathbf{a}}_{\mathbf{3}}},\text{ }\ldots .,\text{ }{{\mathbf{a}}_{\mathbf{n}}}~\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f0761db9221d8c4a34fb1a319790ab96_l3.png "Rendered by QuickLaTeX.com")
is an arithmetic progression with common difference d, then evaluate the following expression.
Given \[{{\mathbf{a}}_{\mathbf{1}}},\text{ }{{\mathbf{a}}_{\mathbf{2}}},\text{ }{{\mathbf{a}}_{\mathbf{3}}},\text{ }\ldots .,\text{ }{{\mathbf{a}}_{\mathbf{n}}}~\] is an arithmetic progression with...
Show that ![\[\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~(\mathbf{1}/\mathbf{2}\text{ }\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{3}/\mathbf{4})\text{ }=\text{ }\left( \mathbf{4}\text{ }\text{ }\surd \mathbf{7} \right)/\text{ }\mathbf{3}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-dccff6f472609b4472e4046132f40b0b_l3.png "Rendered by QuickLaTeX.com")
and justify why the other value ![\[\left( \mathbf{4}\text{ }+\text{ }\surd \mathbf{7} \right)/\text{ }\mathbf{3}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d3c7d7e4f6ad4b104e1c211168af33a9_l3.png "Rendered by QuickLaTeX.com")
is ignored.
We have, \[ta{{n}^{-1}}\left( 1/2\text{ }si{{n}^{-1}}~3/4 \right)\] Let, \[~{\scriptscriptstyle 1\!/\!{ }_2}\text{ }si{{n}^{-1}}~{\scriptscriptstyle 3\!/\!{ }_4}\text{ }=\text{ }\theta \] or,...
Find the value of ![\[\mathbf{4}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{1}/\mathbf{5}\text{ }\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{1}/\mathbf{239}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1c868672679244103bf531dc883f771b_l3.png "Rendered by QuickLaTeX.com")
\[\begin{array}{*{35}{l}} 4\text{ }ta{{n}^{-1}}~1/5\text{ }\text{ }ta{{n}^{-1}}~1/239 \\ =\text{ }2\text{ }(ta{{n}^{-1}}~1/5)\text{ }\text{ }ta{{n}^{-1}}~1/239 \\ \end{array}\] \[=\text{ }2\text{...
Prove that ![\[\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{1}/\mathbf{4}\text{ }+\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{2}/\mathbf{9}\text{ }=\text{ }\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{1}/\surd \mathbf{5}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-17821387054a6129ddeeec9c8d470549_l3.png "Rendered by QuickLaTeX.com")
Taking the LHS, \[ta{{n}^{-1}}~1/4\text{ }+\text{ }ta{{n}^{-1}}~2/9\] = RHS – Hence Proved
Show that ![\[\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{5}/\mathbf{13}\text{ }+\text{ }\mathbf{co}{{\mathbf{s}}^{-\mathbf{1}}}~\mathbf{3}/\mathbf{5}\text{ }=\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{63}/\mathbf{16}\text{ }\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9f862e2ddc16f49b0d8cb191b1265cde_l3.png "Rendered by QuickLaTeX.com")
Solution: Here, \[si{{n}^{-1}}~5/13\text{ }=\text{ }ta{{n}^{-1}}~5/12\] And, \[co{{s}^{-1}}~3/5\text{ }=\text{ }ta{{n}^{-1}}~4/3\] Taking the L.H.S, we have Thus, L.H.S = R.H.S – Hence Proved...
Prove that ![\[\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{8}/\mathbf{17}\text{ }+\text{ }\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{3}/\mathbf{5}\text{ }=\text{ }\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{77}/\mathbf{85}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4a29de9d312057185353faf3423c5473_l3.png "Rendered by QuickLaTeX.com")
Taking the L.H.S, = \[\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{8}/\mathbf{17}\text{ }+\text{ }\mathbf{si}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{3}/\mathbf{5}\] = tan-1 8/15 + tan-1 3/4 – Hence...
. Find the simplified form of ![\[\mathbf{co}{{\mathbf{s}}^{-\mathbf{1}}}~\left[ \mathbf{3}/\mathbf{5}\text{ }\mathbf{cos}\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{4}/\mathbf{5}\text{ }\mathbf{sin}\text{ }\mathbf{x} \right],\text{ }\mathbf{x}\in \left[ -\mathbf{3\pi }/\mathbf{4},\text{ }\mathbf{\pi }/\mathbf{4} \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d850eeacf1c4cdcc74e478df322d0565_l3.png "Rendered by QuickLaTeX.com")
.
We have, \[\mathbf{co}{{\mathbf{s}}^{-\mathbf{1}}}~\left[ \mathbf{3}/\mathbf{5}\text{ }\mathbf{cos}\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{4}/\mathbf{5}\text{ }\mathbf{sin}\text{ }\mathbf{x}...
Prove that
Taking L.H.S, = R.H.S – Hence Proved
Solve the equation ![\[\mathbf{cos}\text{ }(\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}~}}\mathbf{x})\text{ }=\text{ }\mathbf{sin}\text{ }(\mathbf{co}{{\mathbf{t}}^{-\mathbf{1}~}}\mathbf{3}/\mathbf{4})\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6cd380c08ad8ec5642d29a05211ae34a_l3.png "Rendered by QuickLaTeX.com")
.
Given equation, \[\mathbf{cos}\text{ }(\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}~}}\mathbf{x})\text{ }=\text{ }\mathbf{sin}\text{ }(\mathbf{co}{{\mathbf{t}}^{-\mathbf{1}~}}\mathbf{3}/\mathbf{4})\]...
Show that ![\[\mathbf{cos}\text{ }\left( \mathbf{2}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{1}/\mathbf{7} \right)\text{ }=\text{ }\mathbf{sin}\text{ }(\mathbf{4}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{1}/\mathbf{3}).\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c138d201ed7e18941451df2a61b7d942_l3.png "Rendered by QuickLaTeX.com")
Taking L.H.S, we have Thus, L.H.S = R.H.S – Hence proved
If ![\[\mathbf{2}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}\left( \mathbf{cos}\text{ }\mathbf{\theta } \right)\text{ }=\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\left( \mathbf{2}\text{ }\mathbf{cosec}\text{ }\mathbf{\theta } \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d2c7880428f4dda9d218ce3ce1285562_l3.png "Rendered by QuickLaTeX.com")
,then show that ![\[\mathbf{\theta }\text{ }=\text{ }\mathbf{\pi }/\mathbf{4}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8f04b641eff03fb75d5c846a8eb665ad_l3.png "Rendered by QuickLaTeX.com")
.
Given, \[\mathbf{2}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}\left( \mathbf{cos}\text{ }\mathbf{\theta } \right)\text{ }=\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\left( \mathbf{2}\text{...
Find the value of the expression ![\[\mathbf{sin}\text{ }(\mathbf{2}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{1}/\mathbf{3})\text{ }+\text{ }\mathbf{cos}\text{ }(\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{2}\surd \mathbf{2})\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5540e2f416552b974588e8588acb6daf_l3.png "Rendered by QuickLaTeX.com")
.
Given expression, \[\mathbf{sin}\text{ }(\mathbf{2}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\mathbf{1}/\mathbf{3})\text{ }+\text{ }\mathbf{cos}\text{...
Find the real solution of the equation : 
Given equation, Hence, the real solutions of the given trigonometric equation are \[0\] and \[-1\].
Show that ![\[\mathbf{2}\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}\left( -\mathbf{3} \right)\text{ }=\text{ }\text{ }\mathbf{\pi }/\mathbf{2}\text{ }+\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}\left( -\mathbf{4}/\mathbf{3} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1bf1fa38e67e8dd0a1d86ec247fb01c5_l3.png "Rendered by QuickLaTeX.com")
Taking L.H.S = \[2\text{ }ta{{n}^{-1}}\left( -3 \right)\text{ }=\text{ }-2\text{ }ta{{n}^{-1}}~3\text{ }(\because ta{{n}^{-1}}~\left( -x \right)\text{ }=\text{ }\text{ }ta{{n}^{-1}}~x)\text{ }\in...
Find the value of ![\[\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}~}}\left( \mathbf{tan}\text{ }\mathbf{2\pi }/\mathbf{3} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d9347e59b88d99bd217f83bd865f0408_l3.png "Rendered by QuickLaTeX.com")
.
We know that, \[ta{{n}^{-1}}~tan\text{ }x\text{ }=\text{ }x,\text{ }x\in \left( -\pi /2,\text{ }\pi /2 \right)\]
Find the value of ![\[\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\left( -\mathbf{1}/\surd \mathbf{3} \right)\text{ }+\text{ }\mathbf{co}{{\mathbf{t}}^{-\mathbf{1}}}\left( \mathbf{1}/\surd \mathbf{3} \right)\text{ }+\text{ }\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}\left( \mathbf{sin}\text{ }\left( -\mathbf{\pi }/\mathbf{2} \right) \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-648eb67f374bfb1b0ea85a535aa93285_l3.png "Rendered by QuickLaTeX.com")
According to the question, \[\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\left( -\mathbf{1}/\surd \mathbf{3} \right)\text{ }+\text{ }\mathbf{co}{{\mathbf{t}}^{-\mathbf{1}}}\left( \mathbf{1}/\surd...
Prove that ![\[\mathbf{cot}\text{ }(\mathbf{\pi }/\mathbf{4}\text{ }\text{ }\mathbf{-2}\text{ }\mathbf{co}{{\mathbf{t}}^{-\mathbf{1}}}~\mathbf{3})\text{ }=\text{ }\mathbf{7}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-158939f2ec8b756711a45b4e1f072354_l3.png "Rendered by QuickLaTeX.com")
According to the question, L.H.S = R.H.S – Hence Proved
Evaluate ![\[\mathbf{cos}[\mathbf{co}{{\mathbf{s}}^{-\mathbf{1}}}\left( -\surd \mathbf{3}/\mathbf{2} \right)\text{ }+\text{ }\mathbf{\pi }/\mathbf{6}]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-129f4b73ad8a452af907b31ca9d311f8_l3.png "Rendered by QuickLaTeX.com")
Find the value of ![\[\mathbf{ta}{{\mathbf{n}}^{-\mathbf{1}}}~\left[ \mathbf{tan}\text{ }\left( \mathbf{5\pi }/\mathbf{6} \right) \right]\text{ }+\text{ }\mathbf{co}{{\mathbf{s}}^{-\mathbf{1}}}~\left[ \mathbf{cos}\text{ }\left( \mathbf{13\pi }/\mathbf{6} \right) \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5cebef4c3e42c84da0e476a637feb0b1_l3.png "Rendered by QuickLaTeX.com")
We know that, \[ta{{n}^{-1}}~tan\text{ }x\text{ }=\text{ }x,\text{ }x\in \left( -\pi /2,\text{ }\pi /2 \right)\] And, here \[ta{{n}^{-1}}~tan\text{ }\left( 5\pi /6 \right)\text{ }=\text{ }5\pi...
Let n be a fixed positive integer. Define a relation R in Z as follows: ![\[\forall \mathbf{a},\text{ }\mathbf{b}\in \mathbf{Z}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ad8e1c13fb20f78579c0c903ffbb8c13_l3.png "Rendered by QuickLaTeX.com")
, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation.
Given \[\forall \mathbf{a},\text{ }\mathbf{b}\in \mathbf{Z}\], aRb if and only if a – b is divisible by n. Now, for \[aRa\Rightarrow \left( a\text{ }\text{ }a \right)\]is divisible by n, which is...
Let ![\[\mathbf{f}:\text{ }\mathbf{R}\text{ }\to \text{ }\mathbf{R}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-dd6ed4993146108e6223c8db2541e3f7_l3.png "Rendered by QuickLaTeX.com")
be the function defined by ![\[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{1}/\left( \mathbf{2}\text{ }\text{ }\mathbf{cos}\text{ }\mathbf{x} \right)\forall \mathbf{x}\in \mathbf{R}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-84cddfb14d8332eb9da79af0df936940_l3.png "Rendered by QuickLaTeX.com")
. Then, find the range of f.
According to the question, \[\begin{array}{*{35}{l}} f\left( x \right)\text{ }=\text{ }1/\left( 2\text{ }\text{ }cos\text{ }x \right)\forall x\in R \\ Let\text{ }y\text{ }=\text{ }1/\left( 2\text{...
If functions ![\[\mathbf{f}:\text{ }\mathbf{A}\text{ }\to \text{ }\mathbf{B}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a65b4f87417f32d09792148a75d456c6_l3.png "Rendered by QuickLaTeX.com")
and ![\[\mathbf{g}:\text{ }\mathbf{B}\text{ }\to \text{ }\mathbf{A}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1fe8f0a4ce8e1b942287832dc9c329ac_l3.png "Rendered by QuickLaTeX.com")
satisfy ![\[\mathbf{g}\text{ }\mathbf{o}\text{ }\mathbf{f}\text{ }=\text{ }{{\mathbf{I}}_{\mathbf{A}}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2822f57295d364ff6b54fcd63ddbadb5_l3.png "Rendered by QuickLaTeX.com")
, then show that f is one-one and g is onto.
Given, \[\mathbf{f}:\text{ }\mathbf{A}\text{ }\to \text{ }\mathbf{B}\]and \[\mathbf{g}:\text{ }\mathbf{B}\text{ }\to \text{ }\mathbf{A}\] satisfy \[\mathbf{g}\text{ }\mathbf{o}\text{...
Let ![\[\mathbf{X}\text{ }=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3} \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-00a236a7e355a7d670dcb8476df3ab87_l3.png "Rendered by QuickLaTeX.com")
and ![\[\mathbf{Y}\text{ }=\text{ }\left\{ \mathbf{4},\text{ }\mathbf{5} \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-213326489477be96ad3cd80e2d540705_l3.png "Rendered by QuickLaTeX.com")
. Find whether the following subsets of ![\[\mathbf{X}\text{ }\times \text{ }\mathbf{Y}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f014b536a97f4ce864b7dfedf28891d9_l3.png "Rendered by QuickLaTeX.com")
are functions from X to Y or not. (i) ![\[\mathbf{f}\text{ }=\text{ }\left\{ \left( \mathbf{1},\text{ }\mathbf{4} \right),\text{ }\left( \mathbf{1},\text{ }\mathbf{5} \right),\text{ }\left( \mathbf{2},\text{ }\mathbf{4} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{5} \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6691f233314673e77e22007af106b401_l3.png "Rendered by QuickLaTeX.com")
(ii) ![\[\mathbf{g}\text{ }=\text{ }\left\{ \left( \mathbf{1},\text{ }\mathbf{4} \right),\text{ }\left( \mathbf{2},\text{ }\mathbf{4} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{4} \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5042e575b4ccda6cf2e48eaa227a27b8_l3.png "Rendered by QuickLaTeX.com")
(iii) ![\[\mathbf{h}\text{ }=\text{ }\left\{ \left( \mathbf{1},\mathbf{4} \right),\text{ }\left( \mathbf{2},\text{ }\mathbf{5} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{5} \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8b578f4fddc68e8e0e586ca65111a21a_l3.png "Rendered by QuickLaTeX.com")
(iv) ![\[\mathbf{k}\text{ }=\text{ }\left\{ \left( \mathbf{1},\mathbf{4} \right),\text{ }\left( \mathbf{2},\text{ }\mathbf{5} \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-345e65558e5e27c048ec049113afadcd_l3.png "Rendered by QuickLaTeX.com")
.
Solution: From the question, \[\mathbf{X}\text{ }=\text{ }\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3} \right\}\]and \[\mathbf{Y}\text{ }=\text{ }\left\{ \mathbf{4},\text{ }\mathbf{5}...
. Let the function
be defined by f (x) = cos x, ![\[\forall \mathbf{x}\in \mathbf{R}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-93681e8b3dafd096e20967b3836271fa_l3.png "Rendered by QuickLaTeX.com")
. Show that f is neither one-one nor onto.
We have, \[f:\text{ }R\text{ }\to \text{ }R\], \[f\left( x \right)\text{ }=\text{ }cos\text{ }x\] Now, \[\begin{array}{*{35}{l}} f\text{ }({{x}_{1}})\text{ }=\text{ }f\text{ }({{x}_{2}}) \\...
Let C be the set of complex numbers. Prove that the mapping ![\[\mathbf{f}:\text{ }\mathbf{C}\text{ }\to \text{ }\mathbf{R}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b91d0e5e712c414bc7e31733b22d22a9_l3.png "Rendered by QuickLaTeX.com")
given by ![\[f\left( \mathbf{z} \right)\text{ }=\text{ }\left| \mathbf{z} \right|\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f825948db38f9f56374d6b1effa2f56c_l3.png "Rendered by QuickLaTeX.com")
, ![\[\forall \mathbf{z}\in \mathbf{C}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a74567bb0e077aca3e2b1f592c075bd3_l3.png "Rendered by QuickLaTeX.com")
, is neither one-one nor onto.
According to the question, \[\mathbf{f}:\text{ }\mathbf{C}\text{ }\to \text{ }\mathbf{R}\] such that \[f\left( \mathbf{z} \right)\text{ }=\text{ }\left| \mathbf{z} \right|\], \[\forall \mathbf{z}\in...
If the mappings f and g are given by ![\[\mathbf{f}\text{ }=\text{ }\left\{ \left( \mathbf{1},\text{ }\mathbf{2} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{5} \right),\text{ }\left( \mathbf{4},\text{ }\mathbf{1} \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-58479171ac85089e2b7c869f6a70f385_l3.png "Rendered by QuickLaTeX.com")
and ![\[\mathbf{g}\text{ }=\text{ }\left\{ \left( \mathbf{2},\text{ }\mathbf{3} \right),\text{ }\left( \mathbf{5},\text{ }\mathbf{1} \right),\text{ }\left( \mathbf{1},\text{ }\mathbf{3} \right) \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6bae4d20c2ddb3f71bbf7e1016b46c25_l3.png "Rendered by QuickLaTeX.com")
, write ![\[\mathbf{f}\text{ }\mathbf{o}\text{ }\mathbf{g}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-079b2a9780de7bb51c6a828ea8162b7e_l3.png "Rendered by QuickLaTeX.com")
.
From the question, \[\mathbf{f}\text{ }=\text{ }\left\{ \left( \mathbf{1},\text{ }\mathbf{2} \right),\text{ }\left( \mathbf{3},\text{ }\mathbf{5} \right),\text{ }\left( \mathbf{4},\text{ }\mathbf{1}...