43030 + Interview Questions in Mathematics

1.	limx→11+(x−1)21+x2
Answer» ${lim}_{x \to 1} \frac{1 + (x - 1)^{2}}{1 + x^{2}}$

Discussion

2.	If the eccentricity of hyperbola x2−y2 sec2 α =5 is √3 times the eccentricity of the ellipse x2 sec2 α +y2=25,then α=
Answer» $I f t h e e c c e n t r i c i t y o f h y p e r b o l a x^{2} - y^{2} s e c^{2} α = 5 i s \sqrt{3} t i m e s t h e e c c e n t r i c i t y o f t h e e l l i p s e x^{2} s e c^{2} α + y^{2} = 25, t h e n α =$

Discussion

3.	Let A={x:(x−1)(x2−10+21)=0} and B={y:y is a prime factor of 42}, then the number of common elements between A×B and B×A is
Answer» Let $A = {x : (x - 1) (x^{2} - 10 + 21) = 0}$ and $B = {y : y is a prime factor of 42}$ , then the number of common elements between $A \times B$ and $B \times A$ is

Discussion

4.	Which of the following is not a binary operation defined on the set of positive real numbers?
Answer» Which of the following is not a binary operation defined on the set of positive real numbers?

Discussion

5.	If n∑r=1Tr=n8(n+1)(n+2)(n+3) and n∑r=11Tr=n2+3n4p∑k=1k, then p is equal to
Answer» If $n \sum r = 1 T_{r} = \frac{n}{8} (n + 1) (n + 2) (n + 3)$ and $n \sum r = 1 \frac{1}{T_{r}} = \frac{n^{2} + 3 n}{4 p \sum k = 1 k}$ , then $p$ is equal to

Discussion

6.	If the equation of directrix of the parabola y2+4y+4x+2=0 is x=pq, where p and q are co-prime, then the value of pq is
Answer» If the equation of directrix of the parabola $y^{2} + 4 y + 4 x + 2 = 0$ is $x = \frac{p}{q},$ where $p$ and $q$ are co-prime, then the value of $p q$ is

Discussion

7.	If (1)(2020)+(2)(2019)+(3)(2018)+⋯+(2020)(1)=2020×2021×k, then the value of k is
Answer» If $(1) (2020) + (2) (2019) + (3) (2018) + \dots + (2020) (1) = 2020 \times 2021 \times k,$ then the value of $k$ is

Discussion

8.	In the expansion of (1−x−x2+x3)6, the sum of the coefficients of x is
Answer» In the expansion of ${(1 - x - x^{2} + x^{3})}^{6},$ the sum of the coefficients of $x$ is

Discussion

9.	If a and b are positive integers such that N=(a+ib)3−107i is a positive integer, then the value of N is
Answer» If $a$ and $b$ are positive integers such that $N = (a + i b)^{3} - 107 i$ is a positive integer, then the value of $N$ is

Discussion

10.	A real number θ is a root of the equation 7cos2x+4cosx−1=0. If pcos2θ+qcos2θ+r=0 where p,q,r are real constants then r can be
Answer» A real number $θ$ is a root of the equation $7 c o s^{2} x + 4 c o s x - 1 = 0$ . If $p c o s^{2} θ + q c o s^{2} θ + r = 0$ where p,q,r are real constants then r can be

Discussion

11.	The value of sin1∘+sin3∘+sin5∘+sin7∘cos1∘cos2∘sin4∘ is
Answer» The value of $\frac{sin 1^{\circ} + sin 3^{\circ} + sin 5^{\circ} + sin 7^{\circ}}{cos 1^{\circ} cos 2^{\circ} sin 4^{\circ}}$ is

Discussion

12.	The point which does not lie in the half plane 3x−y≤9 is (a) (−3,−2) (b) (1,3)(c) (3,−1) (d) (2,3)
Answer» The point which does not lie in the half plane $3 x - y \leq 9$ is $(a) (- 3, - 2) (b) (1, 3)$ $(c) (3, - 1) (d) (2, 3)$

Discussion

13.	A ladder rests against a wall at an angle α to the horizontal. Its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle β with the horizontal. Show that ab=cos α−cos βsin β−sin α
Answer» A ladder rests against a wall at an angle $α$ to the horizontal. Its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle $β$ with the horizontal. Show that $\frac{a}{b} = \frac{c o s α - c o s β}{s i n β - s i n α}$

Discussion

14.	The sides of two squares are x and y respectively, such that y=x+x2. The rate of change of area of second square with repsect to area of first square is
Answer» The sides of two squares are $x$ and $y$ respectively, such that $y = x + x^{2}$ . The rate of change of area of second square with repsect to area of first square is

Discussion

15.	There are 88 numbers a1,a2,a3,…,a88 and each of them is either equal to −3 or −1. Given that a21+a22+⋯+a288=280, then the value of a41+a42+⋯+a4884−500 is (correct answer + 3, wrong answer 0)
Answer» There are $88$ numbers $a_{1}, a_{2}, a_{3}, \dots, a_{88}$ and each of them is either equal to $- 3$ or $- 1.$ Given that $a_{1}^{2} + a_{2}^{2} + \dots + a_{88}^{2} = 280,$ then the value of $\frac{a_{1}^{4} + a_{2}^{4} + \dots + a_{88}^{4}}{4} - 500$ is (correct answer + 3, wrong answer 0)

Discussion

16.	If S=∞∑n=23n2+1(n2−1)3 then 16S is
Answer» If $S = \infty \sum n = 2 \frac{3 n^{2} + 1}{(n^{2} - 1)^{3}}$ then $16 S$ is

Discussion

17.	Find the equations of the altitudes of a ΔABC whose vertices are A (1, 4), B (−3, 2) and C ((−5, −3).
Answer» Find the equations of the altitudes of a $Δ A B C$ whose vertices are $A (1, 4), B (- 3, 2)$ and $C ((- 5, - 3) .$

Discussion

18.	Given that x,y and b are real numbers and x<y,b>0,then
Answer» Given that x,y and b are real numbers and x<y,b>0,then

Discussion

19.	Three persons A,B,C are to speak at a function along with 5 other persons. If the persons speak in random order, the probability that A speaks before B and B speaks before C is pq, where p,q are co-prime. Then p+q is
Answer» Three persons $A, B, C$ are to speak at a function along with $5$ other persons. If the persons speak in random order, the probability that $A$ speaks before $B$ and $B$ speaks before $C$ is $\frac{p}{q},$ where $p, q$ are co-prime. Then $p + q$ is

Discussion

20.	limk→∞k∑n=1n(e1/k)n(k2+2)(k2+1)2−1
Answer» $lim k \to \infty k \sum n = 1 n \frac{(e^{1 / k})^{n} (k^{2} + 2)}{(k^{2} + 1)^{2} - 1}$

Discussion

21.	If 4 dice are rolled once, the number of ways of getting the sum as 10 is
Answer» If $4$ dice are rolled once, the number of ways of getting the sum as $10$ is

Discussion

22.	Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions from the set A to the set A×B. Then:
Answer» Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A \times B$ . Then:

Discussion

23.	The value of k for which the system of equations: 2x + 3y - 2z = 0; 2x - y + 3z = 0 and 7x + ky - z = 0 has non-trivial solution, is .
Answer» The value of k for which the system of equations: 2x + 3y - 2z = 0; 2x - y + 3z = 0 and 7x + ky - z = 0 has non-trivial solution, is .

Discussion

24.	If x=g(y)ef(y) is a solution of the differential equation (1+y2)+(x−e−tan−1y)dydx=0, satisfying the condition x=0 at y=0, then the value of f(√2)+g(√2) is
Answer» If $x = g (y) e^{f (y)}$ is a solution of the differential equation $(1 + y^{2}) + (x - e^{- {tan}^{- 1} y}) \frac{d y}{d x} = 0$ , satisfying the condition $x = 0$ at $y = 0$ , then the value of $f (\sqrt{2}) + g (\sqrt{2})$ is

Discussion

25.	Let α,β∈R. If α,β2 be the roots of quadratic equation x2−px+1=0 and α2,β be the roots of quadratic equation x2−qx+8=0, then the value of ′r′ if r8 be arithmetic mean of p and q, is
Answer» Let $α, β \in R$ . If $α, β^{2}$ be the roots of quadratic equation $x^{2} - p x + 1 = 0$ and $α^{2}, β$ be the roots of quadratic equation $x^{2} - q x + 8 = 0$ , then the value of $^{'} r^{'}$ if $\frac{r}{8}$ be arithmetic mean of $p$ and $q$ , is

Discussion

26.	Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16} Determine which of the following sets are functions from X to Y. (i) f1={(1,1),(2,11),(3,1),(4,15)} (ii) f2={(1,1),(2,7),(3,5)} (iii) f3={(1,5),(2,9),(3,1),(4,5),(2,11)}
Answer» Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16} Determine which of the following sets are functions from X to Y. (i) $f_{1} = {(1, 1), (2, 11), (3, 1), (4, 15)}$ (ii) $f_{2} = {(1, 1), (2, 7), (3, 5)}$ (iii) $f_{3} = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}$

Discussion

27.	Find the coordinates of the midpoint of the line segment joining the points (2, 1) and (1, -3).
Answer» Find the coordinates of the midpoint of the line segment joining the points (2, 1) and (1, -3).

Discussion

28.	If α,β,γ are the roots of x3−3x+7=0, then the equation whose roots are α+β−γ,β+γ−α,γ+α−β is
Answer» If $α, β, γ$ are the roots of $x^{3} - 3 x + 7 = 0$ , then the equation whose roots are $α + β - γ, β + γ - α, γ + α - β$ is

Discussion

29.	Let origin is one vertex of an equilateral triangle of side length a units. If other vertex lies on the line x−√3y=0 in the first quadrant, then the co-ordinates of third vertex is/are
Answer» Let origin is one vertex of an equilateral triangle of side length $a$ units. If other vertex lies on the line $x - \sqrt{3} y = 0$ in the first quadrant, then the co-ordinates of third vertex is/are

Discussion

30.	Find the distance of the point (-1, -5, -10) from the point of intersection of the line →r=2^i−^j+2^k+λ(3^i+4^j+2^k) and the plane →r.(^i−^j+^k)=5
Answer» Find the distance of the point (-1, -5, -10) from the point of intersection of the line $\to r = 2^i -^j + 2^k + λ (3^i + 4^j + 2^k)$ and the plane $\to r . (^i -^j +^k) = 5$

Discussion

31.	Pass the necessary Journal entries for the following transactions on the dissolution of the firm of P and Q after the various assets (other than cash) and outside liabilities have been transferred to Realisation Account. (i) Bank loan Rs 12,000 was paid. (ii) Stock worth Rs 16,000 was taken over by partners. (iii) Partner P paid a creditor Rs 4,000. (iv) An asset not appearing in the books of accounts realised Rs 1,200. (v) Expenses of realisation Rs 2,000 were paid by partner Q. (iv) Profit on realisation Rs 36,000 was distributed between P and Q in 5 : 4 ratio.
Answer» Pass the necessary Journal entries for the following transactions on the dissolution of the firm of P and Q after the various assets (other than cash) and outside liabilities have been transferred to Realisation Account. (i) Bank loan Rs 12,000 was paid. (ii) Stock worth Rs 16,000 was taken over by partners. (iii) Partner P paid a creditor Rs 4,000. (iv) An asset not appearing in the books of accounts realised Rs 1,200. (v) Expenses of realisation Rs 2,000 were paid by partner Q. (iv) Profit on realisation Rs 36,000 was distributed between P and Q in 5 : 4 ratio.

31.

Pass the necessary Journal entries for the following transactions on the dissolution of the firm of P and Q after the various assets (other than cash) and outside liabilities have been transferred to Realisation Account. (i) Bank loan Rs 12,000 was paid. (ii) Stock worth Rs 16,000 was taken over by partners. (iii) Partner P paid a creditor Rs 4,000. (iv) An asset not appearing in the books of accounts realised Rs 1,200. (v) Expenses of realisation Rs 2,000 were paid by partner Q. (iv) Profit on realisation Rs 36,000 was distributed between P and Q in 5 : 4 ratio.

Answer»

Pass the necessary Journal entries for the following transactions on the dissolution of the firm of P and Q after the various assets (other than cash) and outside liabilities have been transferred to Realisation Account.

(i) Bank loan Rs 12,000 was paid.

(ii) Stock worth Rs 16,000 was taken over by partners.

(iii) Partner P paid a creditor Rs 4,000.

(iv) An asset not appearing in the books of accounts realised Rs 1,200.

(v) Expenses of realisation Rs 2,000 were paid by partner Q.

(iv) Profit on realisation Rs 36,000 was distributed between P and Q in 5 : 4 ratio.

Discussion

32.	In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?
Answer» In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

Discussion

33.	Differentation of a function, containing ex multipled by a constant is? f(x)=5ex
Answer» Differentation of a function, containing e^x multipled by a constant is? f(x)=5e^x

Discussion

34.	Differentiate the following functions with respect to x. cos (sin x).
Answer» Differentiate the following functions with respect to x. cos (sin x).

34.

Differentiate the following functions with respect to x. cos (sin x).

Answer»

Differentiate the following functions with respect to x.

cos (sin x).

Discussion

35.	If the point P(4,−2) is one end of the focal chord PQ of the parabola y2=x, then the slope of the tangent at Q is
Answer» If the point $P (4, - 2)$ is one end of the focal chord $P Q$ of the parabola $y^{2} = x$ , then the slope of the tangent at $Q$ is

Discussion

36.	Vasundhra Ltd., a Company manufacturing high quality sanitary equipments decided to appoint instructors to impart free training to the youth of the city so that they may get suitable employment in various industrial units. Following information is derived from the books of the Company: Particulars31.3.201731.3.2016Revenue from Operations12,00,00010,00,000Cost of Materials Consumed6,60,0006,00,000Employee Benefit Expenses1.20.0001,20,000Other Expenses96,00050,000 You are required to : (a) Prepare a Common Size Statement of Profit & Loss, and (b) Identify the value involved.
Answer» Vasundhra Ltd., a Company manufacturing high quality sanitary equipments decided to appoint instructors to impart free training to the youth of the city so that they may get suitable employment in various industrial units. Following information is derived from the books of the Company: $\begin{matrix} Particulars & 31.3.2017 & 31.3.2016 Revenue from Operations & 12, 00, 000 & 10, 00, 000 Cost of Materials Consumed & 6, 60, 000 & 6, 00, 000 Employee Benefit Expenses & 1.20.000 & 1, 20, 000 Other Expenses & 96, 000 & 50, 000 \end{matrix}$ You are required to : (a) Prepare a Common Size Statement of Profit & Loss, and (b) Identify the value involved.

36.

Vasundhra Ltd., a Company manufacturing high quality sanitary equipments decided to appoint instructors to impart free training to the youth of the city so that they may get suitable employment in various industrial units. Following information is derived from the books of the Company: Particulars31.3.201731.3.2016Revenue from Operations12,00,00010,00,000Cost of Materials Consumed6,60,0006,00,000Employee Benefit Expenses1.20.0001,20,000Other Expenses96,00050,000 You are required to : (a) Prepare a Common Size Statement of Profit & Loss, and (b) Identify the value involved.

Answer»

Vasundhra Ltd., a Company manufacturing high quality sanitary equipments decided to appoint instructors to impart free training to the youth of the city so that they may get suitable employment in various industrial units. Following information is derived from the books of the Company:

$\begin{matrix} Particulars & 31.3.2017 & 31.3.2016 Revenue from Operations & 12, 00, 000 & 10, 00, 000 Cost of Materials Consumed & 6, 60, 000 & 6, 00, 000 Employee Benefit Expenses & 1.20.000 & 1, 20, 000 Other Expenses & 96, 000 & 50, 000 \end{matrix}$

You are required to :

(a) Prepare a Common Size Statement of Profit & Loss, and

(b) Identify the value involved.

Discussion

37.	If x=1+a+a^2+a^3....to infinity; and y=1+b+b^2+b^3+...to infinity Then prove that 1+ab+a^2b^2+...to infinity=xy/(x+y-1)
Answer» If x=1+a+a^2+a^3....to infinity; and y=1+b+b^2+b^3+...to infinity Then prove that 1+ab+a^2b^2+...to infinity=xy/(x+y-1)

37.

If x=1+a+a^2+a^3....to infinity; and y=1+b+b^2+b^3+...to infinity Then prove that 1+ab+a^2b^2+...to infinity=xy/(x+y-1)

Answer»

If x=1+a+a^2+a^3....to infinity;

and y=1+b+b^2+b^3+...to infinity

Then prove that 1+ab+a^2b^2+...to infinity=xy/(x+y-1)

Discussion

38.	Evaluate limx→0(x+1)5−1x ___
Answer» Evaluate $lim x \to 0 \frac{(x + 1)^{5} - 1}{x}$ ___

Discussion

39.	What is the range of x2+x+1x2+x−1
Answer» What is the range of $\frac{x^{2} + x + 1}{x^{2} + x - 1}$

Discussion

40.	Let y=f(x) be a real-valued differentiable function on the set of all real numbers R such that f(1)=1. If f(x) satisfies xf′(x)=x2+f(x)−2, then the area enclosed by y=f(x) with x-axis between ordinates x=0 and x=3 is
Answer» Let $y = f (x)$ be a real-valued differentiable function on the set of all real numbers $R$ such that $f (1) = 1$ . If $f (x)$ satisfies $x f^{'} (x) = x^{2} + f (x) - 2$ , then the area enclosed by $y = f (x)$ with $x$ -axis between ordinates $x = 0$ and $x = 3$ is

Discussion

41.	If tanA=ntanB and sinA=msinB ,prove that cos2A =m2 -1/n2 - 1
Answer» If tanA=ntanB and sinA=msinB ,prove that cos²A =m² -1/n² - 1

Discussion

42.	If \|z\|=1 then value of \|¯¯¯z+z−1\|2 is
Answer» If $\| z \| = 1$ then value of $\| ¯ ¯ ¯ z + z^{- 1} \|^{2}$ is

Discussion

43.	When Rolle's Theorem is verified for f(x) on [a, b] then there exists c such that
Answer» When Rolle's Theorem is verified for f(x) on [a, b] then there exists c such that

Discussion

44.	A spherical balloon is being inflated at the rate of 35 cc/min. The rate of increase in surface area (in sq.cm/min) of the balloon when its diameter is 14 cm, is
Answer» A spherical balloon is being inflated at the rate of 35 cc/min. The rate of increase in surface area (in sq.cm/min) of the balloon when its diameter is 14 cm, is

Discussion

45.	If f(x)=⎧⎪⎪⎪⎨⎪⎪⎪⎩sin (a+1)x+2sin xx,x<02,x=0√1+bx−1x,x>0 is continuous at x=0, then find the values of a and b.
Answer» If $f (x) = ⎧ ⎪⎪⎪ ⎨ ⎪⎪⎪ ⎩ \begin{matrix} \frac{sin (a + 1) x + 2 sin x}{x} & , & x < 0 2 & , & x = 0 \frac{\sqrt{1 + b x} - 1}{x} & , & x > 0 \end{matrix}$ is continuous at $x = 0$ , then find the values of $a$ and $b$ .

Discussion

46.	If the vertex and focus of a parabola are (3,6) and (4,5) then the equation of its directrix is
Answer» If the vertex and focus of a parabola are (3,6) and (4,5) then the equation of its directrix is

Discussion

47.	The equation of the curve which passes through the point (2a,a) and for which the sum of the Cartesian sub tangent and the abscissa is equal to the constant a is
Answer» The equation of the curve which passes through the point (2a,a) and for which the sum of the Cartesian sub tangent and the abscissa is equal to the constant a is

Discussion

48.	The complex number z (Re(z)≠2) satisfies the equation z2=4z+\|z\|2−16\|z\|3, then the value of \|z\|4 is
Answer» The complex number $z (R e (z) \neq 2)$ satisfies the equation $z^{2} = 4 z + \| z \|^{2} - \frac{16}{\| z \|^{3}},$ then the value of $\| z \|^{4}$ is

Discussion

49.	Using eular's formula prove that cosπ11+cos3π11+co5π11+cos7π11+cos9π11=12sinx
Answer» Using eular's formula prove that $\frac{c o s π}{11} + \frac{c o s 3 π}{11} + \frac{c o 5 π}{11} + \frac{c o s 7 π}{11} + \frac{c o s 9 π}{11} = \frac{1}{2} s i n x$

Discussion

50.	If a1−2x.b1+2x=a4+x.b4−x; a,b>0 & a≠b, then the value of x is
Answer» If $a^{1 - 2 x} . b^{1 + 2 x} = a^{4 + x} . b^{4 - x}; a, b > 0 & a \neq b$ , then the value of $x$ is

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

limx→11+(x−1)21+x2

If the eccentricity of hyperbola x2−y2 sec2 α =5 is √3 times the eccentricity of the ellipse x2 sec2 α +y2=25,then α=

Let A={x:(x−1)(x2−10+21)=0} and B={y:y is a prime factor of 42}, then the number of common elements between A×B and B×A is

Which of the following is not a binary operation defined on the set of positive real numbers?

If n∑r=1Tr=n8(n+1)(n+2)(n+3) and n∑r=11Tr=n2+3n4p∑k=1k, then p is equal to

If the equation of directrix of the parabola y2+4y+4x+2=0 is x=pq, where p and q are co-prime, then the value of pq is

If (1)(2020)+(2)(2019)+(3)(2018)+⋯+(2020)(1)=2020×2021×k, then the value of k is

In the expansion of (1−x−x2+x3)6, the sum of the coefficients of x is

If a and b are positive integers such that N=(a+ib)3−107i is a positive integer, then the value of N is

A real number θ is a root of the equation 7cos2x+4cosx−1=0. If pcos2θ+qcos2θ+r=0 where p,q,r are real constants then r can be

The value of sin1∘+sin3∘+sin5∘+sin7∘cos1∘cos2∘sin4∘ is

The point which does not lie in the half plane 3x−y≤9 is (a) (−3,−2) (b) (1,3)(c) (3,−1) (d) (2,3)

A ladder rests against a wall at an angle α to the horizontal. Its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle β with the horizontal. Show that ab=cos α−cos βsin β−sin α

The sides of two squares are x and y respectively, such that y=x+x2. The rate of change of area of second square with repsect to area of first square is

There are 88 numbers a1,a2,a3,…,a88 and each of them is either equal to −3 or −1. Given that a21+a22+⋯+a288=280, then the value of a41+a42+⋯+a4884−500 is (correct answer + 3, wrong answer 0)

If S=∞∑n=23n2+1(n2−1)3 then 16S is

Find the equations of the altitudes of a ΔABC whose vertices are A (1, 4), B (−3, 2) and C ((−5, −3).

Given that x,y and b are real numbers and x&lt;y,b&gt;0,then

Three persons A,B,C are to speak at a function along with 5 other persons. If the persons speak in random order, the probability that A speaks before B and B speaks before C is pq, where p,q are co-prime. Then p+q is

limk→∞k∑n=1n(e1/k)n(k2+2)(k2+1)2−1

If 4 dice are rolled once, the number of ways of getting the sum as 10 is

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions from the set A to the set A×B. Then:

The value of k for which the system of equations: 2x + 3y - 2z = 0; 2x - y + 3z = 0 and 7x + ky - z = 0 has non-trivial solution, is .

If x=g(y)ef(y) is a solution of the differential equation (1+y2)+(x−e−tan−1y)dydx=0, satisfying the condition x=0 at y=0, then the value of f(√2)+g(√2) is

Let α,β∈R. If α,β2 be the roots of quadratic equation x2−px+1=0 and α2,β be the roots of quadratic equation x2−qx+8=0, then the value of ′r′ if r8 be arithmetic mean of p and q, is

Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16} Determine which of the following sets are functions from X to Y. (i) f1={(1,1),(2,11),(3,1),(4,15)} (ii) f2={(1,1),(2,7),(3,5)} (iii) f3={(1,5),(2,9),(3,1),(4,5),(2,11)}

Find the coordinates of the midpoint of the line segment joining the points (2, 1) and (1, -3).

If α,β,γ are the roots of x3−3x+7=0, then the equation whose roots are α+β−γ,β+γ−α,γ+α−β is

Let origin is one vertex of an equilateral triangle of side length a units. If other vertex lies on the line x−√3y=0 in the first quadrant, then the co-ordinates of third vertex is/are

Find the distance of the point (-1, -5, -10) from the point of intersection of the line →r=2^i−^j+2^k+λ(3^i+4^j+2^k) and the plane →r.(^i−^j+^k)=5

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

Differentation of a function, containing e​​​​​​x multipled by a constant is? f(x)=5ex

Differentiate the following functions with respect to x. cos (sin x).

If the point P(4,−2) is one end of the focal chord PQ of the parabola y2=x, then the slope of the tangent at Q is

If x=1+a+a^2+a^3....to infinity; and y=1+b+b^2+b^3+...to infinity Then prove that 1+ab+a^2b^2+...to infinity=xy/(x+y-1)

Evaluate limx→0(x+1)5−1x ___

What is the range of x2+x+1x2+x−1

Let y=f(x) be a real-valued differentiable function on the set of all real numbers R such that f(1)=1. If f(x) satisfies xf′(x)=x2+f(x)−2, then the area enclosed by y=f(x) with x-axis between ordinates x=0 and x=3 is

If tanA=ntanB and sinA=msinB ,prove that cos2A =m2 -1/n2 - 1

If |z|=1 then value of |¯¯¯z+z−1|2 is

When Rolle's Theorem is verified for f(x) on [a, b] then there exists c such that

A spherical balloon is being inflated at the rate of 35 cc/min. The rate of increase in surface area (in sq.cm/min) of the balloon when its diameter is 14 cm, is

If f(x)=⎧⎪⎪⎪⎨⎪⎪⎪⎩sin (a+1)x+2sin xx,x&lt;02,x=0√1+bx−1x,x&gt;0 is continuous at x=0, then find the values of a and b.

If the vertex and focus of a parabola are (3,6) and (4,5) then the equation of its directrix is

The equation of the curve which passes through the point (2a,a) and for which the sum of the Cartesian sub tangent and the abscissa is equal to the constant a is

The complex number z (Re(z)≠2) satisfies the equation z2=4z+|z|2−16|z|3, then the value of |z|4 is

Using eular's formula prove that cosπ11+cos3π11+co5π11+cos7π11+cos9π11=12sinx

If a1−2x.b1+2x=a4+x.b4−x; a,b&gt;0 &amp; a≠b, then the value of x is

Given that x,y and b are real numbers and x<y,b>0,then

Differentation of a function, containing ex multipled by a constant is? f(x)=5ex

If f(x)=⎧⎪⎪⎪⎨⎪⎪⎪⎩sin (a+1)x+2sin xx,x<02,x=0√1+bx−1x,x>0 is continuous at x=0, then find the values of a and b.

If a1−2x.b1+2x=a4+x.b4−x; a,b>0 & a≠b, then the value of x is