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Evaluate [latex] sum_{r=0}^{20} r^2(^{20}C_r)[/latex]

Solution:Answer: 20 ⋅ 21 ⋅ 218... View Article

If [latex]vec{a} . vec{c} =3 , vec{a} = hat{i}+ hat{j}+ hat{k}, vec{b} =hat{j} – hat{k}, vec{a} times vec{c} =vec{b}[/latex], Find [latex][vec{a} ; vec{b} ;; vec{c}][/latex]

Solution:Answer: = -2... View Article

Solve [latex]lim_{n rightarrow infty} frac{1}{n}sum_{r=0}^{2n-1}frac{n^2}{n^2+4r^2}[/latex]

Solution:Answer: [(1/2)] tan-14 - 0 = [(1/2)] tan-14 - 0... View Article

A wire of length 36 units is cut into two parts which are bent respectively to form a square of side x units and a circle of radius of r units. If the sum of areas of the square and the circle so formed is minimum, then find the circumference of the circle.

Solution:Answer: 2π x 18/(4 + π) = 36π/(4 + π)Sum of perimeter of square and circle = 364x + 2πr = 36⇒2x + πr = 18x = (18... View Article

If [latex]arg(frac{z-1}{z+1}) = frac{pi}{4}[/latex], then find equation of circle.

Solution:Answer: √2... View Article

Sum of infinite terms A, AR, AR², AR³,…. is 15. If the sum of the square of these terms is 150, then find the sum of AR², AR⁴, AR⁶, ⋯.

Solution:Answer: ½ A + AR + AR2 + ⋯ = 15⇒A/(1-R) = 15 ⋯(i)Now, A2 + A2 R2 + A2 R4 + ⋯ = 150⇒A2/(1-R2 ) =... View Article

If cosx/(1+sinx) = |tanx|, then find the number of solutions in [0, 2π].

Solution:Answer: x = π/6Case 1: tan x ≥ 0cosx/(1+sinx ) = sinx/cosx⇒cos2x = sin2x+sinx⇒1-sin2x = sin2x + sinx⇒2 sin2x + sin... View Article

Evaluate [latex]int_{frac{-1}{sqrt{2}}}^{frac{1}{sqrt{2}}}[(frac{1-x}{1+x})^2 + (frac{1+x}{1-x})^2 – 2]dx[/latex]

Solution:Answer: 4 ln ½... View Article

P(A) = p, P(B) = 2p, P(Exactly one out of A and B occurs) = 5/9. What will be the maximum value of p?

Solution:Answer: 5/9 Given: P(A) = p and P(B) = 2p P(A)-P(A∩B)+P(B)-P(A∩B) = 5/93p-2P(A∩B) = 5/90 ≤ P(A∩B) ≤ pFor maximum... View Article

In a hospital, 89% has disease A and 98% has disease B, K% is the number of patients who has both, then K can’t be the subset of ____

Solution:Answer: Least k =87, Max k = 89Let total =100n(A)=89, n(B)=98max (n(A),n(B)) ≤ n(AUB) ≤ 10098 ≤ n(A)+n(B)-n(A n B) ≤... View Article

Find value of (5 – e² ) for ellipse x²/8 + y²/4 = 1

Solution:Answer: 9/2x2/8 + y2/4 = 1a2 = 8, b2 = 4e = c/a = √((a2 - b2)/a2 ) = √(1-4/8) = √(1/2)(5 - e2 ) = 5 - 1/2 = 9/2... View Article

If log_e (x+y) = 4xy. Find (d²y)/(dx²) at x = 0.

Solution:Answer: 40loge (x+y) = 4xyor 1/((x + y)) [1 + (dy/dx)] = 4[x (dy/dx) + y]or 1 + (dy/dx) = 4(x + y) [x (dy/dx) + y]⋯(i)If x = 0, then y =... View Article

The sum of the square of the distance of a point from (0, 0), (0, 1), (1, 1), (1, 0) is 18 and its locus is a circle. If d is the diameter of the circle, then find d²

Solution:Answer: 16(OP) 2 + (AP) 2 + (PC) 2 + (PB) 2=18=> x2 + y2 + x2 + (y - 1) 2 + (x - 1) 2 + y2 + (x - 1) 2 + (y - 1) 2 =18=> 4x2 + 4y2... View Article

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Archives

Evaluate [latex] sum_{r=0}^{20} r^2(^{20}C_r)[/latex]

If [latex]vec{a} . vec{c} =3 , vec{a} = hat{i}+ hat{j}+ hat{k}, vec{b} =hat{j} – hat{k}, vec{a} times vec{c} =vec{b}[/latex], Find [latex][vec{a} ; vec{b} ;; vec{c}][/latex]

Solve [latex]lim_{n rightarrow infty} frac{1}{n}sum_{r=0}^{2n-1}frac{n^2}{n^2+4r^2}[/latex]

A wire of length 36 units is cut into two parts which are bent respectively to form a square of side x units and a circle of radius of r units. If the sum of areas of the square and the circle so formed is minimum, then find the circumference of the circle.

If [latex]arg(frac{z-1}{z+1}) = frac{pi}{4}[/latex], then find equation of circle.

Sum of infinite terms A, AR, AR², AR³,…. is 15. If the sum of the square of these terms is 150, then find the sum of AR², AR⁴, AR⁶, ⋯.

If cosx/(1+sinx) = |tanx|, then find the number of solutions in [0, 2π].

Evaluate [latex]int_{frac{-1}{sqrt{2}}}^{frac{1}{sqrt{2}}}[(frac{1-x}{1+x})^2 + (frac{1+x}{1-x})^2 – 2]dx[/latex]

P(A) = p, P(B) = 2p, P(Exactly one out of A and B occurs) = 5/9. What will be the maximum value of p?

In a hospital, 89% has disease A and 98% has disease B, K% is the number of patients who has both, then K can’t be the subset of ____

Find value of (5 – e² ) for ellipse x²/8 + y²/4 = 1

If log_e (x+y) = 4xy. Find (d²y)/(dx²) at x = 0.

The sum of the square of the distance of a point from (0, 0), (0, 1), (1, 1), (1, 0) is 18 and its locus is a circle. If d is the diameter of the circle, then find d²

How many 3 digit numbers by using the digits 0,1,3,4,6,7 can be possible if repetition are allowed?

f(x) = [latex]cos((2 tan^{-1}((sin((cot^{-1}(sqrt{((1-x)/x) )} ) )[/latex] then

Which of the following is the correct sequence of the following conversion?

The ratio of water of crystallization present in Mohr’s salt to that of Potash alum is……..x 10^-1?

In the given equation:

+ ₂(ℎ / = 125^o ) → ?

Choose the correct option for the given statements.

Correct sequence of [Equation] of [Equation] at constant temperature.

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