43030 + Interview Questions in Mathematics Page 217 InterviewSolution

10801.	∫a0x dx√a2+x2=
Answer» $\int_{0}^{a} \frac{x d x}{\sqrt{a^{2} + x^{2}}} =$

Discussion

10802.	is equal to
Answer» is equal to

Discussion

10803.	limn→∞1√n2−1+1√n2−4+1√n2−9+.....1√2n−1=
Answer» $lim n \to \infty \frac{1}{\sqrt{n^{2} - 1}} + \frac{1}{\sqrt{n^{2} - 4}} + \frac{1}{\sqrt{n^{2} - 9}} + . . . . . \frac{1}{\sqrt{2 n - 1}} =$

Discussion

10804.	Prove the following identity : ( sin A + sec A )2 + ( cosA + cosecA )2 = ( 1 + secA • cosecA )
Answer» Prove the following identity : ( sin A + sec A )² + ( cosA + cosecA )² = ( 1 + secA • cosecA )

Discussion

10805.	If y=log 1+tan x1-tan x, prove that dydx=sec 2x.
Answer» If $y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}$ , prove that $\frac{d y}{d x} = \sec 2 x$ .

Discussion

10806.	sinx + cos xsin x cos x
Answer» sinx + cos xsin x cos x

Discussion

10807.	If the expansion of 1(1−ax)(1−bx)=a0+a1x+a2x2+⋯+anxn+⋯, then an is (where a≠b,\|ax\|,\|bx\|<1)
Answer» If the expansion of $\frac{1}{(1 - a x) (1 - b x)} = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n} + \dots$ , then $a_{n}$ is (where $a \neq b, \| a x \|, \| b x \| < 1$ )

Discussion

10808.	6. What is the difference between state functions and state variables?
Answer» 6. What is the difference between state functions and state variables?

Discussion

10809.	Let a, x, b be in A.P.; a, y, b in G.P. and a, z, b in H.P. where a and b are distinct positive real numbers. If x = y + 2 and a = 5z, then
Answer» Let a, x, b be in A.P.; a, y, b in G.P. and a, z, b in H.P. where a and b are distinct positive real numbers. If x = y + 2 and a = 5z, then

Discussion

10810.	Area lying in the first quadrant and bounded by the circle x 2 + y 2 = 4 and the lines x = 0 and x = 2 is A. π B. C. D.
Answer» Area lying in the first quadrant and bounded by the circle x 2 + y 2 = 4 and the lines x = 0 and x = 2 is A. π B. C. D.

Discussion

10811.	The number of common tangents to the circles x2+y2−4x−6y−3=0 and x2+y2+2x+2y+1=0 is
Answer» The number of common tangents to the circles $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ and $x^{2} + y^{2} + 2 x + 2 y + 1 = 0$ is

Discussion

10812.	Forrest got a box of chocolate from Jenny. It had n different chocolates. Forrest asked Jenny how many chocolates the box has. Jenny, who is an aspiring data scientist replied, the probability of you getting a KitKat is 119. How many chocolates are there in the box, if each chocolate brand is equally - likely and there is only one chocolate of each brand?___
Answer» Forrest got a box of chocolate from Jenny. It had n different chocolates. Forrest asked Jenny how many chocolates the box has. Jenny, who is an aspiring data scientist replied, the probability of you getting a KitKat is $\frac{1}{19}$ . How many chocolates are there in the box, if each chocolate brand is equally - likely and there is only one chocolate of each brand? ___

Discussion

10813.	The number of non-negative integral values of b for which the origin and point (1,1) lie on the same side of straight line a2x+aby+1=0,∀ a∈R−{0}, is
Answer» The number of non-negative integral values of $b$ for which the origin and point $(1, 1)$ lie on the same side of straight line $a^{2} x + a b y + 1 = 0, \forall a \in R - {0},$ is

Discussion

10814.	The maximum value of cos2(π3−x)−cos2(π3+x) is
Answer» The maximum value of $c o s^{2} (\frac{π}{3} - x) - c o s^{2} (\frac{π}{3} + x)$ is

Discussion

10815.	Solve forx:9x^2-6b^2x-(a^4-b^4)=13.
Answer» Solve forx:9x^2-6b^2x-(a^4-b^4)=13.

Discussion

10816.	Find the values of k for which the quadratic equation (k+4)x2+(k+1)x+1=0 has equal roots. Also find these roots.
Answer» Find the values of k for which the quadratic equation $(k + 4) x^{2} + (k + 1) x + 1 = 0$ has equal roots. Also find these roots.

Discussion

10817.	If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length ( in cm) of their common chord is :
Answer» If the angle of intersection at a point where the two circles with radii $5 c m$ and $12 c m$ intersect is $90 °,$ then the length $(in c m)$ of their common chord is :

Discussion

10818.	51. How to proof that (4,4) , (3,5) , (-1,-1) is a right angle triangle
Answer» 51. How to proof that (4,4) , (3,5) , (-1,-1) is a right angle triangle

Discussion

10819.	Find the amplitude of the complex number sin(6π/5) + i (1- cos(6π/5))
Answer» Find the amplitude of the complex number sin(6π/5) + i (1- cos(6π/5))

Discussion

10820.	The area of the triangle formed by any tangent to the hyperbola x2a2−y2b2=1 with its asymptotes is
Answer» The area of the triangle formed by any tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ with its asymptotes is

Discussion

10821.	Q1. was incomplete due typographic mistake. Corrected one is:- Solve: \sqrt{2x-6}+\sqrt{x+4}=5
Answer» Q1. was incomplete due typographic mistake. Corrected one is:- Solve: \sqrt{2x-6}+\sqrt{x+4}=5

Discussion

10822.	Tangent to a non-linear curve y=f(x) at any point P intersects x−axis and y−axis at A and B respectively. If normal to the curve y=f(x) at P intersects y−axis at C such that AC=BC and f(2)=3, then the equation of curve is
Answer» Tangent to a non-linear curve $y = f (x)$ at any point $P$ intersects $x -$ axis and $y -$ axis at $A$ and $B$ respectively. If normal to the curve $y = f (x)$ at $P$ intersects $y -$ axis at $C$ such that $A C = B C$ and $f (2) = 3,$ then the equation of curve is

Discussion

10823.	Show that the closed right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
Answer» Show that the closed right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

Discussion

10824.	The solution set for 2cos2θ+sinθ≤2,where π2≤θ≤3π2, is
Answer» The solution set for $2 {cos}^{2} θ + sin θ \leq 2,$ where $\frac{π}{2} \leq θ \leq \frac{3 π}{2}$ , is

Discussion

10825.	If the vertex of the conic y2−4y=4x−4a always lies between the straight lines x+y=3 and 2x+2y−1=0 then
Answer» If the vertex of the conic $y^{2} - 4 y = 4 x - 4 a$ always lies between the straight lines $x + y = 3$ and $2 x + 2 y - 1 = 0$ then

Discussion

10826.	Let x,y be positive real numbers and m,n positive integers. The maximum value of the expression xmyn(1+x2m)(1+y2n) is :
Answer» Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $\frac{x^{m} y^{n}}{(1 + x^{2 m}) (1 + y^{2 n})}$ is :

Discussion

10827.	If sinA=45 and cosB=513, where 0<A, B<π2, find the values of the following: (i) sin(A+B) (ii) cos(A+B) (iii) sin(A-B) (iv) cos(A-B)
Answer» If $s i n A = \frac{4}{5}$ and $c o s B = \frac{5}{13}$ , where $0 < A$ , $B < \frac{π}{2}$ , find the values of the following: (i) sin(A+B) (ii) cos(A+B) (iii) sin(A-B) (iv) cos(A-B)

Discussion

10828.	limx→2(9∑n=1xn(n+1)x2+2(2n+1)x+4) is equal to:
Answer» $lim x \to 2 (9 \sum n = 1 \frac{x}{n (n + 1) x^{2} + 2 (2 n + 1) x + 4})$ is equal to:

Discussion

10829.	Two sets A & B such that A⊆B & B⊆A, then
Answer» Two sets $A & B$ such that $A \subseteq B & B \subseteq A,$ then

Discussion

10830.	The roots of the equation 3x−5+2xx−3=5 are (where x≠3,5)
Answer» The roots of the equation $\frac{3}{x - 5} + \frac{2 x}{x - 3} = 5$ are (where $x \neq 3, 5$ )

Discussion

10831.	limx→2√x−2+√x−√2√x2−4is equal to
Answer» $l i m x \to 2 \frac{\sqrt{x - 2} + \sqrt{x} - \sqrt{2}}{\sqrt{x^{2} - 4}} is equal to$

Discussion

10832.	The triangle formed by the tangent to the parabola y2=4x at the point whose abscissa lies in the interval [a2, 4a2], the ordinate and the X-axis, has greatest area equal to
Answer» The triangle formed by the tangent to the parabola y²=4x at the point whose abscissa lies in the interval [a², 4a²], the ordinate and the X-axis, has greatest area equal to

Discussion

10833.	If [x] denotes the greatest integer function less than or equal to x, then [ ( 1 + 0.0001)10000] equals to?
Answer» If [x] denotes the greatest integer function less than or equal to x, then [ ( 1 + 0.0001)¹⁰⁰⁰⁰] equals to?

Discussion

10834.	Find n if n−1P3:nP4=1:9.
Answer» Find $n$ if $^{n - 1} P_{3} :^{n} P_{4} = 1 : 9.$

Discussion

10835.	A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8 th set of letter is mailed.
Answer» A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8 th set of letter is mailed.

Discussion

10836.	If xϕ(x)=x∫5(3t2−2ϕ′(t))dt, x>−2, and ϕ(0)=4, then ϕ(2) is
Answer» If $x ϕ (x) = x \int 5 (3 t^{2} - 2 ϕ^{'} (t)) d t, x > - 2,$ and $ϕ (0) = 4,$ then $ϕ (2)$ is

Discussion

10837.	Let A={x1,x2,x3,x4,x5,x6} and f:A→A. The number of bijective functions such that f(xi)=xi for exactly four of the xi's is (i=1 to 6)
Answer» Let $A = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}}$ and $f : A \to A$ . The number of bijective functions such that $f (x_{i}) = x_{i}$ for exactly four of the $x_{i}$ 's is $(i = 1 to 6)$

Discussion

10838.	Which of the following is/are true?(where C is constant of integration)
Answer» Which of the following is/are true? (where $C$ is constant of integration)

Discussion

10839.	If ∫1sinxt2.f(t)dt=1−sinx,∀xϵ(0,π2) then the value of f(1√3) is
Answer» If $\int_{s i n x}^{1} t^{2} . f (t) d t = 1 - s i n x, \forall x ϵ (0, \frac{π}{2})$ then the value of $f (\frac{1}{\sqrt{3}})$ is

Discussion

10840.	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 x−y+z=5 is
Answer» 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 $x - y + z = 5$ is

Discussion

10841.	Let f(x)=x6+2x4+x3+2x+3,x∈R. Then the natural number n for which limx→1xnf(1)−f(x)x−1=44 is
Answer» Let $f (x) = x^{6} + 2 x^{4} + x^{3} + 2 x + 3, x \in R .$ Then the natural number $n$ for which $lim x \to 1 \frac{x^{n} f (1) - f (x)}{x - 1} = 44$ is

Discussion

10842.	If y = f(x) represents a straight line passing through origin and not passing through any of the points with integral Co-ordinates in the co-ordinate plane. Then the number of such continuous functions on ‘R’ is ( it is known that straight line represents a function)
Answer» If y = f(x) represents a straight line passing through origin and not passing through any of the points with integral Co-ordinates in the co-ordinate plane. Then the number of such continuous functions on ‘R’ is ( it is known that straight line represents a function)

Discussion

10843.	What is the sum of 11 terms of A.P. whose middle term is 30?
Answer» What is the sum of 11 terms of A.P. whose middle term is 30?

Discussion

10844.	Write down a unit vector in XY- plane, making an angle of 30∘ in anticlockwise direction with the positive direction of X-axis.
Answer» Write down a unit vector in XY- plane, making an angle of $30^{\circ}$ in anticlockwise direction with the positive direction of X-axis.

Discussion

10845.	The solution set the given set of equations will be x+y+z=6 x+2y+3z=10 x+2y+z=1
Answer» The solution set the given set of equations will be $x + y + z = 6$ $x + 2 y + 3 z = 10$ $x + 2 y + z = 1$

10845.

The solution set the given set of equations will be x+y+z=6 x+2y+3z=10 x+2y+z=1

Answer»

The solution set the given set of equations will be

$x + y + z = 6$

$x + 2 y + 3 z = 10$

$x + 2 y + z = 1$

Discussion

10846.	If direction cosines of a vector of magnitude 3 are 23,−93,23 and a > 0 then vector is____________
Answer» If direction cosines of a vector of magnitude 3 are $\frac{2}{3}, - \frac{9}{3}, \frac{2}{3}$ and a > 0 then vector is____________

Discussion

10847.	Which among the following graph(s) is/are even function?
Answer» Which among the following graph(s) is/are even function?

Discussion

10848.	Prove that: cos22x−cos26x=sin4xsin8x
Answer» Prove that: ${cos}^{2} 2 x - {cos}^{2} 6 x = sin 4 x sin 8 x$

Discussion

10849.	For what value of is the function defined by continuous at x = 0? What about continuity at x = 1?
Answer» For what value of is the function defined by continuous at x = 0? What about continuity at x = 1?

Discussion

10850.	Show that the function defined by is a continuous function.
Answer» Show that the function defined by is a continuous function.

Discussion

Explore topic-wise InterviewSolutions in .

∫a0x dx√a2+x2=

is equal to

limn→∞1√n2−1+1√n2−4+1√n2−9+.....1√2n−1=

Prove the following identity : ( sin A + sec A )2 + ( cosA + cosecA )2 = ( 1 + secA • cosecA )

If y=log 1+tan x1-tan x, prove that dydx=sec 2x.

sinx + cos xsin x cos x

If the expansion of 1(1−ax)(1−bx)=a0+a1x+a2x2+⋯+anxn+⋯, then an is (where a≠b,|ax|,|bx|&lt;1)

6. What is the difference between state functions and state variables?

Let a, x, b be in A.P.; a, y, b in G.P. and a, z, b in H.P. where a and b are distinct positive real numbers. If x = y + 2 and a = 5z, then

Area lying in the first quadrant and bounded by the circle x 2 + y 2 = 4 and the lines x = 0 and x = 2 is A. π B. C. D.

The number of common tangents to the circles x2+y2−4x−6y−3=0 and x2+y2+2x+2y+1=0 is

The number of non-negative integral values of b for which the origin and point (1,1) lie on the same side of straight line a2x+aby+1=0,∀ a∈R−{0}, is

The maximum value of cos2(π3−x)−cos2(π3+x) is

Solve forx:9x^2-6b^2x-(a^4-b^4)=13.

Find the values of k for which the quadratic equation (k+4)x2+(k+1)x+1=0 has equal roots. Also find these roots.

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length ( in cm) of their common chord is :

51. How to proof that (4,4) , (3,5) , (-1,-1) is a right angle triangle

Find the amplitude of the complex number sin(6π/5) + i (1- cos(6π/5))

The area of the triangle formed by any tangent to the hyperbola x2a2−y2b2=1 with its asymptotes is

Q1. was incomplete due typographic mistake. Corrected one is:- Solve: \sqrt{2x-6}+\sqrt{x+4}=5

Tangent to a non-linear curve y=f(x) at any point P intersects x−axis and y−axis at A and B respectively. If normal to the curve y=f(x) at P intersects y−axis at C such that AC=BC and f(2)=3, then the equation of curve is

Show that the closed right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

The solution set for 2cos2θ+sinθ≤2,where π2≤θ≤3π2, is

If the vertex of the conic y2−4y=4x−4a always lies between the straight lines x+y=3 and 2x+2y−1=0 then

Let x,y be positive real numbers and m,n positive integers. The maximum value of the expression xmyn(1+x2m)(1+y2n) is :

If sinA=45 and cosB=513, where 0&lt;A, B&lt;π2, find the values of the following: (i) sin(A+B) (ii) cos(A+B) (iii) sin(A-B) (iv) cos(A-B)

limx→2(9∑n=1xn(n+1)x2+2(2n+1)x+4) is equal to:

Two sets A &amp; B such that A⊆B &amp; B⊆A, then

The roots of the equation 3x−5+2xx−3=5 are (where x≠3,5)

limx→2√x−2+√x−√2√x2−4is equal to

The triangle formed by the tangent to the parabola y2=4x at the point whose abscissa lies in the interval [a2, 4a2], the ordinate and the X-axis, has greatest area equal to

If [x] denotes the greatest integer function less than or equal to x, then [ ( 1 + 0.0001)10000] equals to?

Find n if n−1P3:nP4=1:9.

If xϕ(x)=x∫5(3t2−2ϕ′(t))dt, x&gt;−2, and ϕ(0)=4, then ϕ(2) is

Let A={x1,x2,x3,x4,x5,x6} and f:A→A. The number of bijective functions such that f(xi)=xi for exactly four of the xi's is (i=1 to 6)

Which of the following is/are true?(where C is constant of integration)

If ∫1sinxt2.f(t)dt=1−sinx,∀xϵ(0,π2) then the value of f(1√3) is

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 x−y+z=5 is

Let f(x)=x6+2x4+x3+2x+3,x∈R. Then the natural number n for which limx→1xnf(1)−f(x)x−1=44 is

If y = f(x) represents a straight line passing through origin and not passing through any of the points with integral Co-ordinates in the co-ordinate plane. Then the number of such continuous functions on ‘R’ is ( it is known that straight line represents a function)

What is the sum of 11 terms of A.P. whose middle term is 30?

Write down a unit vector in XY- plane, making an angle of 30∘ in anticlockwise direction with the positive direction of X-axis.

The solution set the given set of equations will be x+y+z=6 x+2y+3z=10 x+2y+z=1

If direction cosines of a vector of magnitude 3 are 23,−93,23 and a &gt; 0 then vector is____________

Which among the following graph(s) is/are even function?

Prove that: cos22x−cos26x=sin4xsin8x

For what value of is the function defined by continuous at x = 0? What about continuity at x = 1?

Show that the function defined by is a continuous function.

If the expansion of 1(1−ax)(1−bx)=a0+a1x+a2x2+⋯+anxn+⋯, then an is (where a≠b,|ax|,|bx|<1)

If sinA=45 and cosB=513, where 0<A, B<π2, find the values of the following: (i) sin(A+B) (ii) cos(A+B) (iii) sin(A-B) (iv) cos(A-B)

Two sets A & B such that A⊆B & B⊆A, then

If xϕ(x)=x∫5(3t2−2ϕ′(t))dt, x>−2, and ϕ(0)=4, then ϕ(2) is

If direction cosines of a vector of magnitude 3 are 23,−93,23 and a > 0 then vector is____________