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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) 6 b) 2 c) 8 d) 4 Solution: f’(x) = k(x+1)x(x-1) f’(x) = k[ x3-x] Integrating both sides f(x)... 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/2 b) 2π/3 c) π/9 d) 0 Solution: cos α = \(\begin{array}{l}\frac{\vec{p}.\vec{q}}{\left | p \right |\left | q \right... 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 = \(\begin{array}{l}\begin{bmatrix} 2 & -1 &1 \\ -1 &0 &2... 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 \(\begin{array}{l}\int \frac{dy}{y}=\int \frac{2-x}{x^{2}(x-1)}dx\end{array} \), \(\begin{array}{l}\int \frac{dy}{y}=-\int \frac{x-2}{x^{2}(x-1)}dx\end{array}... 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... View Article

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

A = \(\begin{array}{l}\begin{bmatrix} x &1 \\ 1 & 0 \end{bmatrix}\end{array} \) A2 = \(\begin{array}{l}\begin{bmatrix} x &1 \\ 1 & 0... 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 ⇒ \(\begin{array}{l}\left (\frac{2}{5} \right... 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 = r for y = 2 r = |(2-α-2)/1| = |α| for x = 3 r = |(α-3)/1| =... 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 \(\begin{array}{l}\lim_{x\to0}\left \{ \frac{1}{x^{8}}\left ( 1-\cos \frac{x^{2}}{2} -\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2}\cos \frac{x^{2}}{4}\right ) \right \}= 2^{-k}\end{array} \) then the value of... 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) 0 b) 2 c) 1/2 d) 1 Solution: \(\begin{array}{l}\lim_{x\to0}\left | \frac{1-x+\left | x \right |}{\lambda -x+[x]} \right |= L\end{array}... 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 m b) intersect when l = 1 and m =... 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 = -p αβ = 2 (1/α)+(1/β) = -q 1/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) 23/16 b) 79/16 c) 23/6 d) 79/24 Solution: A= \(\begin{array}{l}\int_{\frac{1}{2}}^{1}(x^{2}+1)dx+\int_{1}^{2}(x+1)dx\end{array} \) = \(\begin{array}{l}\left ( \frac{x^{3}}{3}+x\right )_{\frac{1}{2}}^{1}+\left ( \frac{x^{2}}{2}+x \right... 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)/2 b) y2 +1 = y ((loge(1+e-x)/2 )+2) c) y2 +1 = y ((loge(1+ex)/2 )+2) d)... 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) 1/6 b) 1/5 c) 1/4 d) 1/7 Solution: a = 3 (25/2)(2a+24d) = (15/2)(2(a+25d)+14d) 50a+600d = 15[2a+50d+14d] 20a+600d =... 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) = (3x-7)(2/3x1/3)+(x2/3×3) = (6x-14+9x)/3x1/3... View Article

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

a) (∼p) ˅(∼ q) b) (∼ p) ˄q c) q d) (∼ p) ˅q Solution: ∼(p˄ ∼q) → ∼ p˅q... 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... View Article

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

a) π2 b) π2/2 c) √2π2 d) 2π2 Solution: \(\begin{array}{l}\int_{-\pi }^{\pi }\left | \pi -\left | x \right | \right... 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 \(\begin{array}{l}=\frac{1}{2}\frac{C_{1}C_{2}}{C_{1}+C_{2}}(V_{1}-V_{2})^{2}\end{array} \) \(\begin{array}{l}=\frac{1}{2}\frac{5\times 2.5}{7.5}(220-0)^{2}\mu J\end{array} \) \(\begin{array}{l}=\frac{5}{6}\times 220\times 220\times 10^{-6}J\end{array} \)... View Article

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