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Suppose f(x) is a polynomial of degree four, having critical points at (-1,0,1). If T = {xεR f(x) = f(0) }, then the sum of squares of all the elements of T is:

a) 6b) 2c) 8d) 4Solution:f’(x) = k(x+1)x(x-1)f’(x) = k[ x3-x]Integrating both sidesf(x) = k[(x4/4)- (x2/2)]+cf(0) = cf(x) = f(0) ⇒ k[(x4/4)-... View Article

Let a, b, cεR be such that a²+b²+c² = 1, if a cos θ = b cos(θ+2π/3) = c cos(θ+4π/3), where θ = π/9, then the angle between the vectors [latex]ahat{i}+bhat{j}+chat{k}[/latex] and [latex]bhat{i}+chat{j}+ahat{k}[/latex] is:

a) π/2b) 2π/3c) π/9d) 0Solution:cos α = = (ab+bc+ca)/a2+b2+c2= (ab+bc+ca)/1cos α = ab+bc+cacos α =... View Article

Let A be a 3×3 matrix such that adj A is given and B = adj(adj A)…..

Let A be a 3×3 matrix such that adj A = ⇒ |adj A| = 9⇒ |A2| = 9⇒ |A| = 3 = |λ| and |A| = -3 = |λ|det (adj (adj A))... View Article

If x³dy+xy dx = x²dy+2y dx; y(2) = e and x>1, then y(4) is equal to

a) √e/2 b) (3/2)√e c) (1/2)+√e d) (3/2)+√e Solution: (x3-x2)dy = (2-x) ydx , , , p = 1, q = 2, r = -1 ln|y| = -ln x+(2/x)+ln... View Article

If ((1+i)/(1-i))^m/2 = ((1+i)/(i-1))^n/3 = 1, (m, n∈N) then the greatest common divisor of the least values of m and n is

[(1+i)(1+i)/(1+i)(1-i)]m/2 = [(1+i)(-1-i)/(-1+i)(-1-i)]n/3 = 1 (i = √-1)Solving LHS of above equation, we get(2i/2)m/2 = 1⇒ m = 8Solving... View Article

Let A matrix is given and A4 = [aij]. If a11 = 109, then a22 is equal to:

A = a11 ⇒ x4+3x2+1 = 109x4 +3x2-108 = 0⇒ (x2+12)(x2-9) = 0x = ±3a22 = x2+1 = 10Answer: 10... View Article

The value of [latex]0.16^{log_{2.5}} (\frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}}+…\infty)[/latex] is equal to:

(1/3)+(1/32)+(1/33)+….∞ = 1/3(1-1/3) = 1/2 log2.5(1/2) ⇒ log5/2 (1/2) 0.16 = 16/100 = 4/25 = (2/5)2 ⇒ ⇒ = 4 Answer: 4... View Article

The diameter of the circle, whose centre lies on the line x+y = 2 in the first quadrant and which touches both the lines x = 3 and y = 2, is ……

p = rfor y = 2r = |(2-α-2)/1| = |α|for x = 3r = |(α-3)/1| = |α-3|| α | = |α - 3 |⇒ α2 =... View Article

If [latex]\lim_{x\to0}\left \{ \frac{1}{x^{8}}\left ( 1-\cos \frac{x^{2}}{2} -\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2}\cos \frac{x^{2}}{4}\right ) \right \}= 2^{-k}[/latex] then the value of k is

If then the value of k is Solution: ⇒ 1/256 = 2-k ⇒ 2-8 = 2-k ⇒ k = 8Answer: 8... View Article

Let [t] denote the greatest integer ≤t. If for some λ ∈ R -{0, 1}, [latex]lim_{xto0}left | frac{1-x+left | x right |}{lambda -x+[x]} right |= L[/latex]. Then L is equal to

a) 0b) 2c) 1/2d) 1Solution: = 1/(-1)Since |λ| = |λ-1|, [-h] = -1λ2 = λ2-2λ+1⇒ λ= 1/2L = 2Answer: b... View Article

The lines [latex]vec{r}= (hat{i}-hat{j})+l(2hat{i}+hat{k})[/latex] and [latex]vec{r}= (2hat{i}-hat{j})+m(hat{i}+hat{j}-hat{k})[/latex]

a) do not intersect for any values of l and mb) intersect when l = 1 and m = 2c) intersect when l = 2 and m = 1/2d) intersect for all values of l... View Article

If α and β are the roots of the equation x²+px+2 = 0 and (1/α) and (1/β) are the roots of the equation 2x²+2qx+1 = 0, then (α-1/α)(β-1/β)(α+1/α)(β+1/β) is equal to

a) (9/4)(9+p2)b) (9/4)(9+q2)c) (9/4)(9-p2)d) (9/4)(9-q2)Solution:α+β = -pαβ = 2(1/α)+(1/β) = -q1/αβ =... View Article

The area (in sq. units) of the region {(x,y):0 ≤y ≤x²+1, 0 ≤y ≤x+1, 1/2 ≤x≤2} is

a) 23/16b) 79/16c) 23/6d) 79/24Solution:A= = (1/3)+1-[(1/24)+(1/2)]+[2+2-(3/2)]= (4/3)-(13/24)+(5/2)= (19+60)/24= 79/24Answer: d... View Article

The solution curve of the differential equation, (1+e^-x)(1+y²)dy/dx = y², which passes through the point (0,1), is

a) y2 = 1+y loge(1+e-x)/2b) y2 +1 = y ((loge(1+e-x)/2 )+2)c) y2 +1 = y ((loge(1+ex)/2 )+2)d) y2 = 1+y loge(1+ex)/2Solution: ⇒ (-1/y)+y =... View Article

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is

a) 1/6b) 1/5c) 1/4d) 1/7Solution:a = 3(25/2)(2a+24d) = (15/2)(2(a+25d)+14d)50a+600d = 15[2a+50d+14d]20a+600d = 960d60 = 360dd = 1/6Answer: a... View Article

The function, f(x) = (3x-7)x^2/3, x ∈ R is increasing for all x lying in:

a) (-∞, -14/15) U (0,∞)b) (-∞, 14/15)c) (-∞,0)U(14/15, ∞)d) (-∞,0)U(3/7, ∞)Solution:f’(x) =... View Article

The proposition p -> ∼(p˄q) is equivalent to

a) (∼p) ˅(∼ q)b) (∼ p) ˄qc) qd) (∼ p) ˅qSolution:∼(p˄ ∼q) → ∼ p˅qp → (p∼˅q)⇒ ∼ p˅(∼p˅q)⇒ ∼ p˅qAnswer: d... View Article

Consider the two sets: A = {m ∈ R : both the roots of x²-(m+1)x+m+4 = 0 are real} and B = [-3,5). Which of the following is not true ?

a) A-B = (-∞,-3) ∪ (5,∞)b) A ∩ B = {-3}c) B-A = (-3,5)d) A U B = RSolution:D ≥ 0(m+1)2 -4(m+4) ≥ 0m2-2m-15 ≥ 0(m-5)(m+3) ≥ 0m... View Article

[latex]\int_{-\pi }^{\pi }\left | \pi -\left | x \right | \right |dx[/latex] is equal to:

a) π2b) π2/2c) √2π2d) 2π2Solution: = 2[πx-x2/2]0π = 2[π2/2]= π2 Answer: a... View Article

A 5μF capacitor is charged fully by a 220V supply. It is then disconnected from the supply and is connected in series to another uncharged 2.5μ F capacitor. If the energy change during the charge redistribution is (X/100) J then value of X to the nearest integer is _________.

*NTA Answer is 36 Heat = Ui - Uf = 40,333.33 x 10-6J = 40.3 × 10-3 =(X/100) ⇒ x = 4.03 ≈ 4 Answer (4)... View Article