43030 + Interview Questions in Mathematics

1.	The graph of a quadratic polynomial f(x)=ax2+bx+c is shown below. Then which of the following option(s) is/are correct ?
Answer» The graph of a quadratic polynomial $f (x) = a x^{2} + b x + c$ is shown below. Then which of the following option(s) is/are correct ?

Discussion

2.	In the following diagram, the shaded part represents
Answer» In the following diagram, the shaded part represents

Discussion

3.	The direction cosines of a line segment AB are −2√17,3√17,−2√17. If AB = √17 and the co-ordinates of A are (3, -6, 10), then the co-ordinates
Answer» The direction cosines of a line segment AB are $\frac{- 2}{\sqrt{17}}, \frac{3}{\sqrt{17}}, \frac{- 2}{\sqrt{17}}$ . If AB = $\sqrt{17}$ and the co-ordinates of A are (3, -6, 10), then the co-ordinates

Discussion

4.	The value of the expression (∫140sin2πx dx)2×⎡⎣(∞∑k=0k7k)−1+13⎤⎦ is
Answer» The value of the expression ${(\int_{0}^{14} {sin}^{2} π x d x)}^{2} \times ⎡ ⎣ {(\infty \sum k = 0 \frac{k}{7^{k}})}^{- 1} + 13 ⎤ ⎦$ is

Discussion

5.	Let f(x)=x−[x]1+x−[x],x ϵ R, where [ x] denotes the greatest integer function. Then, the range of f is
Answer» Let $f (x) = \frac{x - [x]}{1 + x - [x]}, x ϵ R$ , where [ x] denotes the greatest integer function. Then, the range of f is

Discussion

6.	If α and β are the roots of the equation 375x2−25x−2=0, then limn→∞n∑r=1αr+limn→∞n∑r=1βr is equal to :
Answer» If $α$ and $β$ are the roots of the equation $375 x^{2} - 25 x - 2 = 0,$ then $lim n \to \infty n \sum r = 1 α^{r} + lim n \to \infty n \sum r = 1 β^{r}$ is equal to :

Discussion

7.	de-Broglie equation is
Answer» de-Broglie equation is

7.

de-Broglie equation is

Answer»

de-Broglie equation is

Discussion

8.	The equation of the straight line passing through the point (2, –2) and the point of intersection of the lines 5x – y = 9 and x + 6y = 8 is
Answer» The equation of the straight line passing through the point (2, –2) and the point of intersection of the lines 5x – y = 9 and x + 6y = 8 is

Discussion

9.	If cot−1√cosα−tan−1√cosα=x then sin x=
Answer» If $c o t^{- 1} \sqrt{c o s α} - t a n^{- 1} \sqrt{c o s α} = x$ then sin x=

Discussion

10.	Match the following: Given sin x = 25 and x ϵ (0,Π2) (p) cos x(1)52(q) tan x(2)√212(r) cosec x(3)√215(4)2√21
Answer» Match the following: Given sin x = $\frac{2}{5}$ and x $ϵ$ $(0, \frac{Π}{2})$ $\begin{matrix} (p) cos x & (1) \frac{5}{2} (q) tan x & (2) \frac{\sqrt{21}}{2} (r) cosec x & (3) \frac{\sqrt{21}}{5} (4) \frac{2}{\sqrt{21}} \end{matrix}$

10.

Match the following: Given sin x = 25 and x ϵ (0,Π2) (p) cos x(1)52(q) tan x(2)√212(r) cosec x(3)√215(4)2√21

Answer»

Match the following:

Given sin x = $\frac{2}{5}$ and x $ϵ$ $(0, \frac{Π}{2})$

\begin{matrix} (p) cos x & (1) \frac{5}{2} (q) tan x & (2) \frac{\sqrt{21}}{2} (r) cosec x & (3) \frac{\sqrt{21}}{5} (4) \frac{2}{\sqrt{21}} \end{matrix}

Discussion

11.	Question 3 The product of three consecutive positive integers is divisible by 6: Is this statement true or false? Justify your answer.
Answer» Question 3 The product of three consecutive positive integers is divisible by 6: Is this statement true or false? Justify your answer.

Discussion

12.	Three vertices of a parallelogram ABCD are A(3,−1,2),B(1,2,−4)and C(−1,1,2). Which of the follwing could be the fourth vertex ?
Answer» Three vertices of a parallelogram ABCD are A $(3, - 1, 2), B (1, 2, - 4) a n d C (- 1, 1, 2)$ . Which of the follwing could be the fourth vertex ?

Discussion

13.	Show that the function f:R→R given by f(x)=x3 is injective
Answer» Show that the function $f : R \to R$ given by $f (x) = x^{3}$ is injective

Discussion

14.	The arbitrary constant on which the value of the determinant
Answer» The arbitrary constant on which the value of the determinant

14.

The arbitrary constant on which the value of the determinant

Answer»

The arbitrary constant on which the value of the determinant

Discussion

15.	Solve the following systems of inequations graphically: (i) 2x+y≥8,x+2y≥8,x+y≤6 (ii) 12x+12y≤840,3x+6y≤300,8x+4y≤480 x≥0,y≥0 (iii) x+2y≤40,3x+y≥30,4x+3y≥60,x≥0,y≥0 (iv) 5x+y≥10,2x+2y≥12,x+4y≥12,x≥0,y≥0
Answer» Solve the following systems of inequations graphically: (i) $2 x + y \geq 8, x + 2 y \geq 8, x + y \leq 6$ (ii) $12 x + 12 y \leq 840, 3 x + 6 y \leq 300, 8 x + 4 y \leq 480 x \geq 0, y \geq 0$ (iii) $x + 2 y \leq 40, 3 x + y \geq 30, 4 x + 3 y \geq 60, x \geq 0, y \geq 0$ (iv) $5 x + y \geq 10, 2 x + 2 y \geq 12, x + 4 y \geq 12, x \geq 0, y \geq 0$

15.

Solve the following systems of inequations graphically: (i) 2x+y≥8,x+2y≥8,x+y≤6 (ii) 12x+12y≤840,3x+6y≤300,8x+4y≤480 x≥0,y≥0 (iii) x+2y≤40,3x+y≥30,4x+3y≥60,x≥0,y≥0 (iv) 5x+y≥10,2x+2y≥12,x+4y≥12,x≥0,y≥0

Answer»

Solve the following systems of inequations graphically:

(i) $2 x + y \geq 8, x + 2 y \geq 8, x + y \leq 6$

(ii) $12 x + 12 y \leq 840, 3 x + 6 y \leq 300, 8 x + 4 y \leq 480 x \geq 0, y \geq 0$

(iii) $x + 2 y \leq 40, 3 x + y \geq 30, 4 x + 3 y \geq 60, x \geq 0, y \geq 0$

(iv) $5 x + y \geq 10, 2 x + 2 y \geq 12, x + 4 y \geq 12, x \geq 0, y \geq 0$

Discussion

16.	If ∫3sin x+2cos x3cos x+2sin xdx=ax +b ln(2sinx+3cosx\|+C, then .
Answer» $If \int \frac{3 sin x + 2 cos x}{3 cos x + 2 sin x} d x = a x + b ln (2 sin x + 3 cos x \| + C, then$ .

Discussion

17.	If (a2−a)C2= (a2−a)C4, then a =
Answer» If $^{(a^{2} - a) C_{2}} =^{(a^{2} - a)} C_{4},$ then a =

Discussion

18.	Let f(x) be a differentiable function on [0,8] such that f(1)=6,f(2)=13,f(3)=8,f(4)=−2,f(5)=5,f(6)=15, and f(7)=−13 If the minimum number of roots of the equation f′(x)−f′(x)(f(x))2=0 is λ then λ11 is
Answer» Let $f (x)$ be a differentiable function on $[0, 8]$ such that $f (1) = 6, f (2) = \frac{1}{3}, f (3) = 8, f (4) = - 2, f (5) = 5, f (6) = \frac{1}{5}, and f (7) = - \frac{1}{3}$ If the minimum number of roots of the equation $f^{'} (x) - f^{'} (x) (f (x))^{2} = 0 is λ then \frac{λ}{11}$ is

Discussion

19.	If tangents to the curve y=x44+ax33+ax22+x+1, x ϵ R always lie below the curve, then range of a is
Answer» If tangents to the curve $y = \frac{x^{4}}{4} + \frac{a x^{3}}{3} + \frac{a x^{2}}{2} + x + 1, x ϵ R$ always lie below the curve, then range of $a$ is

Discussion

20.	Let y=y(x) be the solution curve of the differential equation, (y2−x)dydx=1, satisfying y(0)=1. This curve intersects the x-axis at a point whose abscissa is :
Answer» Let $y = y (x)$ be the solution curve of the differential equation, $(y^{2} - x) \frac{d y}{d x} = 1$ , satisfying $y (0) = 1$ . This curve intersects the $x$ -axis at a point whose abscissa is :

Discussion

21.	If the mean deviation about the median of the numbers a,2a,....,50a is 50. Then \|a\| equals
Answer» If the mean deviation about the median of the numbers $a, 2 a, . . . ., 50 a$ is $50$ . Then $\| a \|$ equals

Discussion

22.	Find the centre,eccentricity,foci and directrices of the hyperbola(i) 16x2−9y2+32x+36y−164=0(ii) x2−y2+4x=0(iii) x2−3y2−2x=8
Answer» $Find the centre,eccentricity,foci and directrices of the hyperbola (i) 16 x^{2} - 9 y^{2} + 32 x + 36 y - 164 = 0 (i i) x^{2} - y^{2} + 4 x = 0 (i i i) x^{2} - 3 y^{2} - 2 x = 8$

Discussion

23.	A circle has the same centre as an ellipse and passes through the foci F1 and F2 of the ellipse the two curves intersect in four points. Let P be any point of intersection. If the major axis of the ellipse is 15 and the area of ΔPF1F2 is 26, then the distance between foci is
Answer» A circle has the same centre as an ellipse and passes through the foci $F_{1}$ and $F_{2}$ of the ellipse the two curves intersect in four points. Let P be any point of intersection. If the major axis of the ellipse is 15 and the area of $Δ P F_{1} F_{2}$ is 26, then the distance between foci is

Discussion

24.	The coefficient of x^5 in the expansion of (1+x)21+(1+x)22+.....+(1+x)30 is
Answer» The coefficient of x^5 in the expansion of $(1 + x)^{21} + (1 + x)^{22} + . . . . . + (1 + x)^{30}$ is

24.

The coefficient of x^5 in the expansion of (1+x)21+(1+x)22+.....+(1+x)30 is

Answer»

The coefficient of x^5 in the expansion of $(1 + x)^{21} + (1 + x)^{22} + . . . . . + (1 + x)^{30}$ is

Discussion

25.	Discuss the continuity and differentiability of f(x)=⎧⎨⎩2−x,x<2(2−x)(4−x),2≤x≤44−x,x>4
Answer» Discuss the continuity and differentiability of $f (x) =$ $⎧ ⎨ ⎩ \begin{matrix} 2 - x, & x < 2 (2 - x) (4 - x), & 2 \leq x \leq 4 4 - x, & x > 4 \end{matrix}$

Discussion

26.	If a and b are the roots of x2−3x+p=0 and c, d are the roots x2−12x+q=0, Where a, b, c, d form a G.P. Prove that (q + p) : (q - p) = 17 : 15.
Answer» If a and b are the roots of $x^{2} - 3 x + p = 0$ and c, d are the roots $x^{2} - 12 x + q = 0$ , Where a, b, c, d form a G.P. Prove that (q + p) : (q - p) = 17 : 15.

Discussion

27.	A man arranges to pay off a debt of Rs: 3600 by 40 annual instalments which are in AP. When 30 of the installments are paid, he dies leaving one third of the debt unpaid. The value of the 8th instalment is
Answer» A man arranges to pay off a debt of Rs: 3600 by 40 annual instalments which are in AP. When 30 of the installments are paid, he dies leaving one third of the debt unpaid. The value of the $8^{t h}$ instalment is

27.

A man arranges to pay off a debt of Rs: 3600 by 40 annual instalments which are in AP. When 30 of the installments are paid, he dies leaving one third of the debt unpaid. The value of the 8th instalment is

Answer»

A man arranges to pay off a debt of Rs: 3600 by 40 annual instalments which are in AP. When 30 of the installments are paid, he dies leaving one third of the debt unpaid. The value of the $8^{t h}$ instalment is

Discussion

28.	The absolute value of π/2∫0(xcosx+1)esinx dxπ/2∫0(xsinx−1)ecosx dx is equal to
Answer» The absolute value of $\frac{π / 2 \int 0 (x cos x + 1) e^{sin x} d x}{π / 2 \int 0 (x sin x - 1) e^{cos x} d x}$ is equal to

Discussion

29.	Write the set builder form A = {-1, 1}.
Answer» Write the set builder form A = {-1, 1}.

Discussion

30.	If a,x are real numbers and \|a\|<1,\|x\|<1, then 1+(1+a)x+(1+a+a2)x2......∞ is
Answer» If $a, x$ are real numbers and $\| a \| < 1, \| x \| < 1,$ then $1 + (1 + a) x + (1 + a + a^{2}) x^{2} . . . . . . \infty$ is

Discussion

31.	If θ be the angle subtended at the focus by the chord which is normal at the point (λ,λ),λ≠0 to the parabola y2=4x, then the equation of the line making angle θ with positive x−axis and passing through (1,2) is
Answer» If $θ$ be the angle subtended at the focus by the chord which is normal at the point $(λ, λ), λ \neq 0$ to the parabola $y^{2} = 4 x,$ then the equation of the line making angle $θ$ with positive $x -$ axis and passing through $(1, 2)$ is

Discussion

32.	If x2+y2=a2, then ∫a0√1+(dydx)2dx=
Answer» If $x^{2} + y^{2} = a^{2}$ , then $\int_{0}^{a} \sqrt{1 + {(\frac{d y}{d x})}^{2}} d x =$

Discussion

33.	The equation of plane containing intersecting lines x+33=y1=z−22 and x−34=y−22=z−63 is _______
Answer» The equation of plane containing intersecting lines $\frac{x + 3}{3} = \frac{y}{1} = \frac{z - 2}{2}$ and $\frac{x - 3}{4} = \frac{y - 2}{2} = \frac{z - 6}{3}$ is _______

Discussion

34.	Let f(x)=(x3+(a−1)x2+(b−a)x−b)\|x2+8x+6\|, where a,b∈R. If f(x) is derivable in (−∞,∞), then the value of (a+b4) is
Answer» Let $f (x) = (x^{3} + (a - 1) x^{2} + (b - a) x - b) \| x^{2} + 8 x + 6 \|,$ where $a, b \in R$ . If $f (x)$ is derivable in $(- \infty, \infty)$ , then the value of $(\frac{a + b}{4})$ is

Discussion

35.	SinA+cosecA=2; Find the value of sin²A + cosec²A.
Answer» SinA+cosecA=2; Find the value of sin²A + cosec²A.

35.

SinA+cosecA=2; Find the value of sin²A + cosec²A.

Answer»

SinA+cosecA=2;

Find the value of sin²A + cosec²A.

Discussion

36.	If two adjacent vertices of a regular hexagon are (1,2) and (2,1), then equation of the circumcircle of the hexagon is
Answer» If two adjacent vertices of a regular hexagon are $(1, 2)$ and $(2, 1),$ then equation of the circumcircle of the hexagon is

Discussion

37.	Evaluate the definite integrals. ∫π40sin x+cos x9+16sin 2xdx.
Answer» Evaluate the definite integrals. $\int_{0}^{\frac{π}{4}} \frac{s i n x + c o s x}{9 + 16 s i n 2 x} d x .$

Discussion

38.	Prove that (sin 7x +sin 5x)+(sin 9x+ sin 3x)(cos 7x+ cos 5x)+(cos 9x + cos 3x)=tan 6x
Answer» Prove that $\frac{(sin 7x +sin 5x) + (sin 9x+ sin 3x)}{(cos 7x+ cos 5x) + (cos 9x + cos 3x)} = tan 6x$

38.

Prove that (sin 7x +sin 5x)+(sin 9x+ sin 3x)(cos 7x+ cos 5x)+(cos 9x + cos 3x)=tan 6x

Answer»

Prove that

$\frac{(sin 7x +sin 5x) + (sin 9x+ sin 3x)}{(cos 7x+ cos 5x) + (cos 9x + cos 3x)} = tan 6x$

Discussion

39.	Show that semi-vertical angle of a cone of maximum volume and given slant height is cos−1[1√3].
Answer» Show that semi-vertical angle of a cone of maximum volume and given slant height is ${cos}^{- 1} [\frac{1}{\sqrt{3}}] .$

Discussion

40.	If the shortest distance(in units) between the parabolas y2=4x and y2=2x−6 is d units, then the value of d2 is
Answer» If the shortest distance(in units) between the parabolas $y^{2} = 4 x$ and $y^{2} = 2 x - 6$ is $d$ units, then the value of $d^{2}$ is

Discussion

41.	5 girls are there. 4 are fair complexioned 2 are rich. If a be the event selecting a rich girl and b be the event selecting a fair complexioned girl. Find P(a/b).
Answer» 5 girls are there. 4 are fair complexioned 2 are rich. If a be the event selecting a rich girl and b be the event selecting a fair complexioned girl. Find P(a/b).

Discussion

42.	For the matrices A and B, verify that (AB)' =B'A', where A=⎡⎢⎣1−43⎤⎥⎦,B=[−1 2 1]
Answer» For the matrices A and B, verify that (AB)' =B'A', where $A = ⎡ ⎢ ⎣ \begin{matrix} 1 - 4 3 \end{matrix} ⎤ ⎥ ⎦, B = [- 121]$

Discussion

43.	The centre of regular polygon of n sides is located at z=0 and one of its vertices is z1. If z2 is vertex adjacent to z1, then z2=
Answer» The centre of regular polygon of $n$ sides is located at $z = 0$ and one of its vertices is $z_{1}$ . If $z_{2}$ is vertex adjacent to $z_{1}$ , then $z_{2} =$

Discussion

44.	If the lines 2x−3y=5 and 3x−4y=7 are the diameters of a circle of area 154 square units, then obtain the equation of the circle.
Answer» If the lines $2 x - 3 y = 5$ and $3 x - 4 y = 7$ are the diameters of a circle of area 154 square units, then obtain the equation of the circle.

Discussion

45.	Find the shortest distance between lines →r=(4^i−^j)+λ(^i+2^j−3^k) and →r=(^i−^j+2^k)+μ(2^i+4^j−5^k)
Answer» Find the shortest distance between lines $\to r = (4^i -^j) + λ (^i + 2^j - 3^k)$ and $\to r = (^i -^j + 2^k) + μ (2^i + 4^j - 5^k)$

Discussion

46.	What is the value of npn?
Answer» What is the value of ⁿp_n?

Discussion

47.	A couple has two children. Find the probability that both children are males, if it is known that atleast one of the children is male. Find the probability that both children are females if it is known that the elder child is a female.
Answer» A couple has two children. Find the probability that both children are males, if it is known that atleast one of the children is male. Find the probability that both children are females if it is known that the elder child is a female.

47.

A couple has two children. Find the probability that both children are males, if it is known that atleast one of the children is male. Find the probability that both children are females if it is known that the elder child is a female.

Answer»

A couple has two children.

Find the probability that both children are males, if it is known that atleast one of the children is male.

Find the probability that both children are females if it is known that the elder child is a female.

Discussion

48.	Find the domain and range of the function of defined by: f(x)= 1/\|x\|-x
Answer» Find the domain and range of the function of defined by: f(x)= 1/\|x\|-x

48.

Find the domain and range of the function of defined by: f(x)= 1/|x|-x

Answer»

Find the domain and range of the function of defined by:

f(x)= 1/|x|-x

Discussion

49.	The line 3x+4y=√7 touches the ellipse 3x2+4y2=1 at the point
Answer» The line $3 x + 4 y = \sqrt{7}$ touches the ellipse $3 x^{2} + 4 y^{2} = 1$ at the point

Discussion

50.	The product of two natural numbers is 17. Then, the sum of the reciprocals of their squares is:
Answer» The product of two natural numbers is 17. Then, the sum of the reciprocals of their squares is:

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

The graph of a quadratic polynomial f(x)=ax2+bx+c is shown below. Then which of the following option(s) is/are correct ?

In the following diagram, the shaded part represents

The direction cosines of a line segment AB are −2√17,3√17,−2√17. If AB = √17 and the co-ordinates of A are (3, -6, 10), then the co-ordinates

The value of the expression (∫140sin2πx dx)2×⎡⎣(∞∑k=0k7k)−1+13⎤⎦ is

Let f(x)=x−[x]1+x−[x],x ϵ R, where [ x] denotes the greatest integer function. Then, the range of f is

If α and β are the roots of the equation 375x2−25x−2=0, then limn→∞n∑r=1αr+limn→∞n∑r=1βr is equal to :

de-Broglie equation is

The equation of the straight line passing through the point (2, –2) and the point of intersection of the lines 5x – y = 9 and x + 6y = 8 is

If cot−1√cosα−tan−1√cosα=x then sin x=

Match the following: Given sin x = 25 and x ϵ (0,Π2) (p) cos x(1)52(q) tan x(2)√212(r) cosec x(3)√215(4)2√21

Question 3 The product of three consecutive positive integers is divisible by 6: Is this statement true or false? Justify your answer.

Three vertices of a parallelogram ABCD are A(3,−1,2),B(1,2,−4)and C(−1,1,2). Which of the follwing could be the fourth vertex ?

Show that the function f:R→R given by f(x)=x3 is injective

The arbitrary constant on which the value of the determinant

Solve the following systems of inequations graphically: (i) 2x+y≥8,x+2y≥8,x+y≤6 (ii) 12x+12y≤840,3x+6y≤300,8x+4y≤480 x≥0,y≥0 (iii) x+2y≤40,3x+y≥30,4x+3y≥60,x≥0,y≥0 (iv) 5x+y≥10,2x+2y≥12,x+4y≥12,x≥0,y≥0

If ∫3sin x+2cos x3cos x+2sin xdx=ax +b ln(2sinx+3cosx|+C, then .

If (a2−a)C2= (a2−a)C4, then a =

Let f(x) be a differentiable function on [0,8] such that f(1)=6,f(2)=13,f(3)=8,f(4)=−2,f(5)=5,f(6)=15, and f(7)=−13 If the minimum number of roots of the equation f′(x)−f′(x)(f(x))2=0 is λ then λ11 is

If tangents to the curve y=x44+ax33+ax22+x+1, x ϵ R always lie below the curve, then range of a is

Let y=y(x) be the solution curve of the differential equation, (y2−x)dydx=1, satisfying y(0)=1. This curve intersects the x-axis at a point whose abscissa is :

If the mean deviation about the median of the numbers a,2a,....,50a is 50. Then |a| equals

Find the centre,eccentricity,foci and directrices of the hyperbola(i) 16x2−9y2+32x+36y−164=0(ii) x2−y2+4x=0(iii) x2−3y2−2x=8

A circle has the same centre as an ellipse and passes through the foci F1 and F2 of the ellipse the two curves intersect in four points. Let P be any point of intersection. If the major axis of the ellipse is 15 and the area of ΔPF1F2 is 26, then the distance between foci is

The coefficient of x^5 in the expansion of (1+x)21+(1+x)22+.....+(1+x)30 is

Discuss the continuity and differentiability of f(x)=⎧⎨⎩2−x,x&lt;2(2−x)(4−x),2≤x≤44−x,x&gt;4

If a and b are the roots of x2−3x+p=0 and c, d are the roots x2−12x+q=0, Where a, b, c, d form a G.P. Prove that (q + p) : (q - p) = 17 : 15.

A man arranges to pay off a debt of Rs: 3600 by 40 annual instalments which are in AP. When 30 of the installments are paid, he dies leaving one third of the debt unpaid. The value of the 8th instalment is

The absolute value of π/2∫0(xcosx+1)esinx dxπ/2∫0(xsinx−1)ecosx dx is equal to

Write the set builder form A = {-1, 1}.

If a,x are real numbers and |a|&lt;1,|x|&lt;1, then 1+(1+a)x+(1+a+a2)x2......∞ is

If θ be the angle subtended at the focus by the chord which is normal at the point (λ,λ),λ≠0 to the parabola y2=4x, then the equation of the line making angle θ with positive x−axis and passing through (1,2) is

If x2+y2=a2, then ∫a0√1+(dydx)2dx=

The equation of plane containing intersecting lines x+33=y1=z−22 and x−34=y−22=z−63 is _______

Let f(x)=(x3+(a−1)x2+(b−a)x−b)|x2+8x+6|, where a,b∈R. If f(x) is derivable in (−∞,∞), then the value of (a+b4) is

SinA+cosecA=2; Find the value of sin²A + cosec²A.

If two adjacent vertices of a regular hexagon are (1,2) and (2,1), then equation of the circumcircle of the hexagon is

Evaluate the definite integrals. ∫π40sin x+cos x9+16sin 2xdx.

Prove that (sin 7x +sin 5x)+(sin 9x+ sin 3x)(cos 7x+ cos 5x)+(cos 9x + cos 3x)=tan 6x

Show that semi-vertical angle of a cone of maximum volume and given slant height is cos−1[1√3].

If the shortest distance(in units) between the parabolas y2=4x and y2=2x−6 is d units, then the value of d2 is

5 girls are there. 4 are fair complexioned 2 are rich. If a be the event selecting a rich girl and b be the event selecting a fair complexioned girl. Find P(a/b).

For the matrices A and B, verify that (AB)' =B'A', where A=⎡⎢⎣1−43⎤⎥⎦,B=[−1 2 1]

The centre of regular polygon of n sides is located at z=0 and one of its vertices is z1. If z2 is vertex adjacent to z1, then z2=

If the lines 2x−3y=5 and 3x−4y=7 are the diameters of a circle of area 154 square units, then obtain the equation of the circle.

Find the shortest distance between lines →r=(4^i−^j)+λ(^i+2^j−3^k) and →r=(^i−^j+2^k)+μ(2^i+4^j−5^k)

What is the value of npn?

A couple has two children. Find the probability that both children are males, if it is known that atleast one of the children is male. Find the probability that both children are females if it is known that the elder child is a female.

Find the domain and range of the function of defined by: f(x)= 1/|x|-x

The line 3x+4y=√7 touches the ellipse 3x2+4y2=1 at the point

The product of two natural numbers is 17. Then, the sum of the reciprocals of their squares is:

Discuss the continuity and differentiability of f(x)=⎧⎨⎩2−x,x<2(2−x)(4−x),2≤x≤44−x,x>4

If a,x are real numbers and |a|<1,|x|<1, then 1+(1+a)x+(1+a+a2)x2......∞ is