43030 + Interview Questions in Mathematics

1.	If the angle between two vector A and B is theta the value of product (A×B).A is
Answer» If the angle between two vector A and B is theta the value of product (A×B).A is

Discussion

2.	Let f1:(0,∞)→R and f2:(0,∞)→R be defined by f1(x)=∫x021∏j=1(t−j)jdt, x>0 and f2(x)=98(x−1)50−600(x−1)49+2450, x>0, where, for any positive integer n and real number a1,a2…an, n∏i=1ai denotes the product of a1,a2,...an. Let mi and ni respectively denote the number of points of local minima and the number of points of local maxima of function fi, i=1,2 in the interval (0,∞).The value of 6m1+4n2+8m2n2 is
Answer» Let $f_{1} : (0, \infty) \to R$ and $f_{2} : (0, \infty) \to R$ be defined by $f_{1} (x) = \int_{0}^{x} 21 \prod j = 1 (t - j)^{j} d t,$ $x > 0$ and $f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450,$ $x > 0,$ where, for any positive integer $n$ and real number $a_{1}, a_{2} \dots a_{n},$ $n \prod i = 1 a_{i}$ denotes the product of $a_{1}, a_{2}, . . . a_{n} .$ Let $m_{i}$ and $n_{i}$ respectively denote the number of points of local minima and the number of points of local maxima of function $f_{i}, i = 1, 2$ in the interval $(0, \infty) .$ The value of $6 m_{1} + 4 n_{2} + 8 m_{2} n_{2}$ is

Discussion

3.	Consider the relation 4l2−5m2+6l+1=0, where l,m∈R, then the line lx+my+1=0 touches a fixed circle whose centre and radius of circle are
Answer» Consider the relation $4 l^{2} - 5 m^{2} + 6 l + 1 = 0$ , where $l, m \in R$ , then the line $l x + m y + 1 = 0$ touches a fixed circle whose centre and radius of circle are

Discussion

4.	∫5x4+1dx
Answer» $\int \frac{5}{x^{4} + 1} d x$

Discussion

5.	If x= 7/8-5√2 , then the value of 2x^3-24x^2+ 71x +47.
Answer» If x= 7/8-5√2 , then the value of 2x^3-24x^2+ 71x +47.

Discussion

6.	The slope of normal at any point (x,y) of a curve y=f(x), is given by −2xyx2+y2+1 and curve passes through (1,0).Then, which of the following point(s) can lie on the curve y=f(x)?
Answer» The slope of normal at any point $(x, y)$ of a curve $y = f (x)$ , is given by $\frac{- 2 x y}{x^{2} + y^{2} + 1}$ and curve passes through $(1, 0)$ .Then, which of the following point(s) can lie on the curve $y = f (x) ?$

Discussion

7.	If Sn=sin2πn+sin22πn+…+sin2(n−1)πn, then the value of S100 is
Answer» If $S_{n} = {sin}^{2} \frac{π}{n} + {sin}^{2} \frac{2 π}{n} + \dots + {sin}^{2} \frac{(n - 1) π}{n}$ , then the value of $S_{100}$ is

Discussion

8.	Let P(8,4) be a point on the hyperbola x2a2−y2b2=1. If the normal at point P intersects the x−axis at (12,0), then the value of eccentricity is
Answer» Let $P (8, 4)$ be a point on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ . If the normal at point $P$ intersects the $x -$ axis at $(12, 0)$ , then the value of eccentricity is

Discussion

9.	Determine whether the point (-3, 2) lies inside or outside the triangle whose sides are given by the equations x+y−4=0, 3x−7y+8=0, 4x−y−31=0.
Answer» Determine whether the point (-3, 2) lies inside or outside the triangle whose sides are given by the equations $x + y - 4 = 0$ , $3 x - 7 y + 8 = 0$ , $4 x - y - 31 = 0$ .

Discussion

10.	Find the adjoint of the matrix A=-1-2-221-22-21 and hence show that Aadj A=AI3.
Answer» Find the adjoint of the matrix $A = [\begin{matrix} - 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1 \end{matrix}]$ and hence show that $A (adj A) = \|A\| I_{3}$ .

Discussion

11.	If y=sin−1(√1+x+√1−x2),x∈(0,1), then dydx is equal to
Answer» If $y = {sin}^{- 1} (\frac{\sqrt{1 + x} + \sqrt{1 - x}}{2}), x \in (0, 1)$ , then $\frac{d y}{d x}$ is equal to

Discussion

12.	Let f: R → R be defined by f(x) such that its inverse exists and given f (1) = 3, f (2) = 7 then find the f−1 (7) . f−1 (3) ? __
Answer» Let f: R $\to$ R be defined by f(x) such that its inverse exists and given f (1) = 3, f (2) = 7 then find the f $^{- 1}$ (7) . f $^{- 1}$ (3) ? __

Discussion

13.	The plane ax+by=0 is rotated through the angle α about it's line of intersection with plane z=0. Then the equation of the plane in the new position is:
Answer» The plane $a x + b y = 0$ is rotated through the angle $α$ about it's line of intersection with plane $z = 0$ . Then the equation of the plane in the new position is:

Discussion

14.	How to write linear equation when two points are given ..let's say (1,0) and (1,4)
Answer» How to write linear equation when two points are given ..let's say (1,0) and (1,4)

Discussion

15.	18. dy/dx = cos (10x + 8y) solve this differential equation
Answer» 18. dy/dx = cos (10x + 8y) solve this differential equation

Discussion

16.	Let X be the set consisting of the first 2018 terms of the arithmetic progression 1,6,11,⋯, and Y be the set consisting of the first 2018 terms of the arithmetic progression 9,16,23,⋯ . Then, the number of elements in the set X∪Y is
Answer» Let $X$ be the set consisting of the first $2018$ terms of the arithmetic progression $1, 6, 11, \dots,$ and $Y$ be the set consisting of the first $2018$ terms of the arithmetic progression $9, 16, 23, \dots$ . Then, the number of elements in the set $X \cup Y$ is

Discussion

17.	Which among the following function is continuous everywhere in its domain but has at least one point where it is not differentiable
Answer» Which among the following function is continuous everywhere in its domain but has at least one point where it is not differentiable

Discussion

18.	If fx=x+k,x<34,x=33x-5,x>3 is continuous at x = 3, then k = _____________.
Answer» If $f (x) = \{\begin{matrix} x + k, & x < 3 \\ 4, & x = 3 \\ 3 x - 5, & x > 3 \end{matrix}$ is continuous at x = 3, then k = _____________.

Discussion

19.	Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph.Let f(x)=sin(πcosx) and g(x)=cos(2πsinx) be two functions defined for x>0. Define the following sets whose elements are written in the increasing order:X={x:f(x)=0}, Y={x:f′(x)=0},Z={x:g(x)=0}, W={x:g′(x)=0},.List−I contains the sets X,Y,Z and W. List−II contains some information regarding these sets. List IList II(I)X(P)⊇{π2,3π2,4π,7π} (II)Y(Q)an arithmetic progression (III)Z(R)NOT an arithmetic progression(IV)Z(S)⊇{π6,7π6,13π6} (T)⊇{π3,2π3,π} (U)⊇{π6,3π4} Q.15 which of the following is the only CORRECT combination?
Answer» Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph. Let $f (x) = sin (π cos x)$ and $g (x) = cos (2 π sin x)$ be two functions defined for $x > 0.$ Define the following sets whose elements are written in the increasing order: $X = {x : f (x) = 0}, Y = {x : f^{'} (x) = 0},$ $Z = {x : g (x) = 0}, W = {x : g^{'} (x) = 0},$ . $L i s t - I$ contains the sets $X, Y, Z$ and $W$ . $L i s t - I I$ contains some information regarding these sets. $\begin{matrix} L i s t I & L i s t I I (I) & X & (P) & \supseteq {\frac{π}{2}, \frac{3 π}{2}, 4 π, 7 π} (I I) & Y & (Q) & an arithmetic progression (I I I) & Z & (R) & NOT an arithmetic progression (I V) & Z & (S) & \supseteq {\frac{π}{6}, \frac{7 π}{6}, \frac{13 π}{6}} (T) & \supseteq {\frac{π}{3}, \frac{2 π}{3}, π} (U) & \supseteq {\frac{π}{6}, \frac{3 π}{4}} \end{matrix}$ Q.15 which of the following is the only CORRECT combination?

Discussion

20.	If the angles of the triangle are in the ratio 4:1:1, then the ratio of the longest side to the perimeter is
Answer» If the angles of the triangle are in the ratio $4 : 1 : 1,$ then the ratio of the longest side to the perimeter is

Discussion

21.	Let f(x)={x2k(x2−4)2−xwhen x is an int eger otherwise then limx→2f(x)
Answer» $L e t f (x) = {\begin{matrix} x^{2} & _{\frac{k (x^{2} - 4)}{2 - x}} \end{matrix} when x is an int eger otherwise then {lim}_{x \to 2} f (x)$

Discussion

22.	If f:R→R defined by f(x)=3x−17 be an invertible function then write f−1(x)
Answer» If $f : R \to R$ defined by $f (x) = \frac{3 x - 1}{7}$ be an invertible function then write $f^{- 1} (x)$

Discussion

23.	Consider the four sets A,B,C and D defined asA={a:a=8(log4(x))3+4log√2(x4), x>0}B={b:b=20+13(logx2)2, x>0}C={c∈N:c∈A∩B for same x>0}D={x∈Z:(x−2)2(15x2−56x+17)<0} Then the number of elements in C×D is
Answer» Consider the four sets $A, B, C$ and $D$ defined as $A = {a : a = 8 ({log}_{4} (x))^{3} + 4 {log}_{\sqrt{2}} (x^{4}), x > 0}$ $B = {b : b = 20 + \frac{13}{({log}_{x} 2)^{2}}, x > 0}$ $C = {c \in N : c \in A \cap B for same x > 0}$ $D = {x \in Z : (x - 2)^{2} (15 x^{2} - 56 x + 17) < 0}$ Then the number of elements in $C \times D$ is

Discussion

24.	The statement (p→q)→[(∼p→q)→q] is
Answer» The statement $(p \to q) \to [(\sim p \to q) \to q]$ is

Discussion

25.	If tan Alpha Banten beta be the roots of x square - 3 x + 2 is equal to zero then find cos 2 (alpha + beta)
Answer» If tan Alpha Banten beta be the roots of x square - 3 x + 2 is equal to zero then find cos 2 (alpha + beta)

Discussion

26.	prove that \sqrt5is an irrational number
Answer» prove that \sqrt5is an irrational number

Discussion

27.	Three vectors A, B and C are such that A=B+C and their magnitude are in ratio10:8:6 respectively then angke between A and B
Answer» Three vectors A, B and C are such that A=B+C and their magnitude are in ratio10:8:6 respectively then angke between A and B

Discussion

28.	If a>0 and A,B,C are variable angles of △ABC, such that atanA+2atanB+3atanC=7a, then the minimum integral value of tan2A+tan2B+tan2C=
Answer» If $a > 0$ and $A, B, C$ are variable angles of $△ A B C,$ such that $a tan A + 2 a tan B + 3 a tan C = 7 a,$ then the minimum integral value of ${tan}^{2} A + {tan}^{2} B + {tan}^{2} C =$

Discussion

29.	If l 1 , m 1 , n 1 and l 2 , m 2 , n 2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m 1 n 2 − m 2 n 1 , n 1 l 2 − n 2 l 1 , l 1 m 2 − l 2 m 1 .
Answer» If l 1 , m 1 , n 1 and l 2 , m 2 , n 2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m 1 n 2 − m 2 n 1 , n 1 l 2 − n 2 l 1 , l 1 m 2 − l 2 m 1 .

Discussion

30.	1.Solve 24x < 100, when(i)xis a natural number.(ii) x is an integer.
Answer» 1.Solve 24x < 100, when(i)xis a natural number.(ii) x is an integer.

Discussion

31.	101.If an ant has to move from one corner of a cubical room of a side a to diagonally opposite corner, then the minimum distance covered by ant is
Answer» 101.If an ant has to move from one corner of a cubical room of a side a to diagonally opposite corner, then the minimum distance covered by ant is

Discussion

32.	आमतौर पर मेले मनोरंजन, खरीद फ़रोख्त एवं मेलजोल के लिए होते हैं। वीर कुँवर सिंह ने मेले का उपयोग किस रूप में किया?
Answer» आमतौर पर मेले मनोरंजन, खरीद फ़रोख्त एवं मेलजोल के लिए होते हैं। वीर कुँवर सिंह ने मेले का उपयोग किस रूप में किया?

Discussion

33.	Show that the set of all points such that the difference of their distances from (4, 0) and (− 4,0) is always equal to 2 represents a hyperbola.
Answer» Show that the set of all points such that the difference of their distances from (4, 0) and (− 4,0) is always equal to 2 represents a hyperbola.

Discussion

34.	Let f(x)=x3+1x3,x≠0. If the intervals in which f(x) increases are (−∞,a] and [b,∞) then min(b - a) is equal to
Answer» Let $f (x) = x^{3} + \frac{1}{x^{3}}, x \neq 0$ . If the intervals in which f(x) increases are $(- \infty, a]$ and $[b, \infty)$ then min(b - a) is equal to

Discussion

35.	If f(x)=f(x−2)+f(x−3) for x=3,4,5,⋯ and f(0)=0,f(1)=1,f(2)=2, then the value of f(8) is
Answer» If $f (x) = f (x - 2) + f (x - 3) for x = 3, 4, 5, \dots$ and $f (0) = 0, f (1) = 1, f (2) = 2,$ then the value of $f (8)$ is

Discussion

36.	find the domain and range of f(x)=(x-2) / (x-3)
Answer» find the domain and range of f(x)=(x-2) / (x-3)

Discussion

37.	Differentiation of x into Mod x is equal to
Answer» Differentiation of x into Mod x is equal to

Discussion

38.	If 0<θ,ϕ<π2,x=∞∑n=0cos2nθ,y=∞∑n=0sin2nϕ and z=∞∑n=0cos2nθ⋅sin2nϕ then :
Answer» If $0 < θ, ϕ < \frac{π}{2}, x = \infty \sum n = 0 {cos}^{2 n} θ, y = \infty \sum n = 0 {sin}^{2 n} ϕ$ and $z = \infty \sum n = 0 {cos}^{2 n} θ \cdot {sin}^{2 n} ϕ$ then :

Discussion

39.	What are constants in Cauchy equation
Answer» What are constants in Cauchy equation

Discussion

40.	A function f (x) = 1 + 1x is defined on the closed interval [1, 3]. A point in the interval, where the function satisfies the mean value theorem, is ______________.
Answer» A function f (x) = 1 + $\frac{1}{x}$ is defined on the closed interval [1, 3]. A point in the interval, where the function satisfies the mean value theorem, is ______________.

Discussion

41.	How B=meu knot(1 + x) HCan be written as B=meuHWhere, x is susceptibility constant
Answer» How B=meu knot(1 + x) H Can be written as B=meuH Where, x is susceptibility constant

Discussion

42.	The value of π∫0sin2kxsinxdx, where k∈I, is
Answer» The value of $π \int 0 \frac{sin 2 k x}{sin x} d x$ , where $k \in I$ , is

Discussion

43.	Prove that:sin22π5-sin2-π3=5-18
Answer» Prove that: $\sin^{2} \frac{2 π}{5} - \sin^{2 -} \frac{π}{3} = \frac{\sqrt{5} - 1}{8}$

Discussion

44.	Let f(x)=20x+49x2+1∫0(xy+x2y2)f(y)dy where x and y are variables independent of each other. If 1∫0xf(x)dx=40k3, then the value of k is
Answer» Let $f (x) = 20 x + 49 x^{2} + 1 \int 0 (x y + x^{2} y^{2}) f (y) d y$ where $x$ and $y$ are variables independent of each other. If $1 \int 0 x f (x) d x = \frac{40 k}{3},$ then the value of $k$ is

Discussion

45.	√yx+√xy=2⇒dydx=
Answer» $\sqrt{\frac{y}{x}} + \sqrt{\frac{x}{y}} = 2 \Rightarrow \frac{d y}{d x} =$

Discussion

46.	∫1√e2x+4ex+1dx is equal to
Answer» $\int \frac{1}{\sqrt{e^{2 x} + 4 e^{x} + 1}} d x$ is equal to

Discussion

47.	Statements:(A) No A is B.(B) Some C are A.(C) All B are D.Conclusions:(I) Some D are C(II) Some A are not COptions:a)Both I and II followb)Only conclusion II followsc)Only conclusion I followsd)Neither I nor II follow
Answer» Statements: (A) No A is B. (B) Some C are A. (C) All B are D. Conclusions: (I) Some D are C (II) Some A are not C Options: a)Both I and II follow b)Only conclusion II follows c)Only conclusion I follows d)Neither I nor II follow

Discussion

48.	If y=[x]4+{x}+{x}[x]−3, then dydx at x=π is(Where [⋅] denotes the greatest integer function and {⋅} denotes the fractional part function)
Answer» If $y = [x]^{4 + {x}} + {x}^{[x] - 3},$ then $\frac{d y}{d x}$ at $x = π$ is $($ Where $[\cdot]$ denotes the greatest integer function and ${\cdot}$ denotes the fractional part function $)$

Discussion

49.	Let S be the sum of all solutions (in radians) of the equation sin4θ+cos4θ−sinθcosθ=0 in [0,4π]. Then 8Sπ is equal to
Answer» Let $S$ be the sum of all solutions (in radians) of the equation ${sin}^{4} θ + {cos}^{4} θ - sin θ cos θ = 0$ in $[0, 4 π]$ . Then $\frac{8 S}{π}$ is equal to

Discussion

50.	The range of a for which the equation x2+ax−4=0 has its smaller root in the interval (−1,2) is
Answer» The range of $a$ for which the equation $x^{2} + a x - 4 = 0$ has its smaller root in the interval $(- 1, 2)$ is

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

If the angle between two vector A and B is theta the value of product (A×B).A is

Consider the relation 4l2−5m2+6l+1=0, where l,m∈R, then the line lx+my+1=0 touches a fixed circle whose centre and radius of circle are

∫5x4+1dx

If x= 7/8-5√2 , then the value of 2x^3-24x^2+ 71x +47.

The slope of normal at any point (x,y) of a curve y=f(x), is given by −2xyx2+y2+1 and curve passes through (1,0).Then, which of the following point(s) can lie on the curve y=f(x)?

If Sn=sin2πn+sin22πn+…+sin2(n−1)πn, then the value of S100 is

Let P(8,4) be a point on the hyperbola x2a2−y2b2=1. If the normal at point P intersects the x−axis at (12,0), then the value of eccentricity is

Determine whether the point (-3, 2) lies inside or outside the triangle whose sides are given by the equations x+y−4=0, 3x−7y+8=0, 4x−y−31=0.

Find the adjoint of the matrix A=-1-2-221-22-21 and hence show that Aadj A=AI3.

If y=sin−1(√1+x+√1−x2),x∈(0,1), then dydx is equal to

Let f: R → R be defined by f(x) such that its inverse exists and given f (1) = 3, f (2) = 7 then find the f−1 (7) . f−1 (3) ? __

The plane ax+by=0 is rotated through the angle α about it's line of intersection with plane z=0. Then the equation of the plane in the new position is:

How to write linear equation when two points are given ..let's say (1,0) and (1,4)

18. dy/dx = cos (10x + 8y) solve this differential equation

Let X be the set consisting of the first 2018 terms of the arithmetic progression 1,6,11,⋯, and Y be the set consisting of the first 2018 terms of the arithmetic progression 9,16,23,⋯ . Then, the number of elements in the set X∪Y is

Which among the following function is continuous everywhere in its domain but has at least one point where it is not differentiable

If fx=x+k,x&lt;34,x=33x-5,x&gt;3 is continuous at x = 3, then k = _____________.

If the angles of the triangle are in the ratio 4:1:1, then the ratio of the longest side to the perimeter is

Let f(x)={x2k(x2−4)2−xwhen x is an int eger otherwise then limx→2f(x)

If f:R→R defined by f(x)=3x−17 be an invertible function then write f−1(x)

Consider the four sets A,B,C and D defined asA={a:a=8(log4(x))3+4log√2(x4), x&gt;0}B={b:b=20+13(logx2)2, x&gt;0}C={c∈N:c∈A∩B for same x&gt;0}D={x∈Z:(x−2)2(15x2−56x+17)&lt;0} Then the number of elements in C×D is

The statement (p→q)→[(∼p→q)→q] is

If tan Alpha Banten beta be the roots of x square - 3 x + 2 is equal to zero then find cos 2 (alpha + beta)

prove that \sqrt5is an irrational number

Three vectors A, B and C are such that A=B+C and their magnitude are in ratio10:8:6 respectively then angke between A and B

If a&gt;0 and A,B,C are variable angles of △ABC, such that atanA+2atanB+3atanC=7a, then the minimum integral value of tan2A+tan2B+tan2C=

If l 1 , m 1 , n 1 and l 2 , m 2 , n 2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m 1 n 2 − m 2 n 1 , n 1 l 2 − n 2 l 1 , l 1 m 2 ­− l 2 m 1 .

1.Solve 24x < 100, when(i)xis a natural number.(ii) x is an integer.

101.If an ant has to move from one corner of a cubical room of a side a to diagonally opposite corner, then the minimum distance covered by ant is

आमतौर पर मेले मनोरंजन, खरीद फ़रोख्त एवं मेलजोल के लिए होते हैं। वीर कुँवर सिंह ने मेले का उपयोग किस रूप में किया?

Show that the set of all points such that the difference of their distances from (4, 0) and (− 4,0) is always equal to 2 represents a hyperbola.

Let f(x)=x3+1x3,x≠0. If the intervals in which f(x) increases are (−∞,a] and [b,∞) then min(b - a) is equal to

If f(x)=f(x−2)+f(x−3) for x=3,4,5,⋯ and f(0)=0,f(1)=1,f(2)=2, then the value of f(8) is

find the domain and range of f(x)=(x-2) / (x-3)

Differentiation of x into Mod x is equal to

If 0&lt;θ,ϕ&lt;π2,x=∞∑n=0cos2nθ,y=∞∑n=0sin2nϕ and z=∞∑n=0cos2nθ⋅sin2nϕ then :

What are constants in Cauchy equation

A function f (x) = 1 + 1x is defined on the closed interval [1, 3]. A point in the interval, where the function satisfies the mean value theorem, is ______________.

How B=meu knot(1 + x) HCan be written as B=meuHWhere, x is susceptibility constant

The value of π∫0sin2kxsinxdx, where k∈I, is

Prove that:sin22π5-sin2-π3=5-18

Let f(x)=20x+49x2+1∫0(xy+x2y2)f(y)dy where x and y are variables independent of each other. If 1∫0xf(x)dx=40k3, then the value of k is

√yx+√xy=2⇒dydx=

∫1√e2x+4ex+1dx is equal to

Statements:(A) No A is B.(B) Some C are A.(C) All B are D.Conclusions:(I) Some D are C(II) Some A are not COptions:a)Both I and II followb)Only conclusion II followsc)Only conclusion I followsd)Neither I nor II follow

If y=[x]4+{x}+{x}[x]−3, then dydx at x=π is(Where [⋅] denotes the greatest integer function and {⋅} denotes the fractional part function)

Let S be the sum of all solutions (in radians) of the equation sin4θ+cos4θ−sinθcosθ=0 in [0,4π]. Then 8Sπ is equal to

The range of a for which the equation x2+ax−4=0 has its smaller root in the interval (−1,2) is

If fx=x+k,x<34,x=33x-5,x>3 is continuous at x = 3, then k = _____________.

Consider the four sets A,B,C and D defined asA={a:a=8(log4(x))3+4log√2(x4), x>0}B={b:b=20+13(logx2)2, x>0}C={c∈N:c∈A∩B for same x>0}D={x∈Z:(x−2)2(15x2−56x+17)<0} Then the number of elements in C×D is

If a>0 and A,B,C are variable angles of △ABC, such that atanA+2atanB+3atanC=7a, then the minimum integral value of tan2A+tan2B+tan2C=

If l 1 , m 1 , n 1 and l 2 , m 2 , n 2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m 1 n 2 − m 2 n 1 , n 1 l 2 − n 2 l 1 , l 1 m 2 − l 2 m 1 .

If 0<θ,ϕ<π2,x=∞∑n=0cos2nθ,y=∞∑n=0sin2nϕ and z=∞∑n=0cos2nθ⋅sin2nϕ then :