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lim_x→a [(√a + 2x – √3x) / (√3a + x – 2√x] =

1) 2/3 2) 2/√3 3) 3√3/2 4) 2/3√3 Solution: (4) 2/3√3 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2... View Article

The function f (x) = [ln (1 + ax) – ln (1 – bx)] / x is not defined at x = 0. The value which should be assigned to f at x = 0, so that it is continuous at x = 0, is

1) a – b 2) a + b 3) ln a – ln b 4) none of these Solution: (2)... View Article

lim_x→0 [π^x – 1] / [√1 + x – 1]

1) does not exist 2) is equal to loge (π2) 3) is equal to 1 4) lies between 10 and... View Article

lim_x→0 [x2^x – x] / [1 – cos x] =

1) 2 log2 2) log 2 3) 1/2 log 2 4) 1/2 Solution: (1) 2 log2 \(\begin{array}{l}\lim _{x \rightarrow 0}... View Article

The value of lim_n→∞ [(1 + 2 + …. n] / (3n² + 5)] is

1) 1/3 2) 1/5 3) 1/6 4) 6 Solution: (3) ⅙ \(\begin{array}{l}\begin{array}{l} \lim _{n \rightarrow \infty} \frac{1+2+\cdots+n}{3{n^{2}}+5} \\ \Rightarrow \lim... View Article

If g(x) = (x² + 2x + 3) f(x), f(0) = 5 and lim_x→0 [f (x) – 5] / x = 4, then g’ (0) =

1) 22 2) 18 3) 20 4) 25 Solution: (1) 22 \(\begin{array}{l}g(x)=\left(x^{2}+2 x+3\right) f(x), f(0)=5\\ \lim _{x \rightarrow 0}\left(\frac{f(x)-5}{x}\right)=4\\ \lim... View Article

If L = lim_x→0 [a – √a² – x² – (x² / 4)] / x⁴, a > 0. If L is finite, then

1) a = 2 2) a = 1 3) a = 1/3 4) None of these Solution: (1) a =... View Article

If [x] denotes the greatest integer less than or equal to x for any real number x. Then, lim_n→∞ [n √2] / n =

1) 0 2) 2 3) √2 4) 1 Solution: (3) √2 \(\begin{array}{l}\begin{array}{l} \lim _{n \rightarrow \infty}\left[\frac{n \sqrt{2}}{n}\right] \\ x-1 \leq[x]<x... View Article

lim_x→0 [log (1 + 3x²) / {x (e^5x – 1)}] =

1) 3/5 2) 5/3 3) -3/5 4) -5/3 5) 1 Solution: (1) ⅗ \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{\log \left(1+3... View Article

The value of lim_x→3 [(x⁵ – 3⁵) / (x⁸ – 3⁸)] =

1) 5/8 2) 5/64 3) 5/216 4) 1/27 5) None of these Solution: (3) 5/216 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 3}... View Article

For a real number y, let [y] denote the greatest integer less than or equal to y. Then the function f (x) = [tan (π (x – π))] / [1 + [x]²]

1) discontinuous at some x 2) continuous at all x, but the derivative f’ (x) does not exist for some... View Article

lim_x→1 [(x^m – 1) / (xⁿ – 1)] =

Limits, Continuity, and Differentiability 1) n/m 2) m/n 3) 2m/n 4) 2n/m Solution: (2) m/n \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 1}... View Article

Let [t] denote the greatest integer less than or equal to t. Then the value of [latex]\int_{1}^{2}\left | 2x-[3x] \right |dx[/latex] is

Solution: Solution: To find \(\begin{array}{l}\int_{1}^{2}\left | 2x-[3x] \right |dx\end{array} \) Put 3x = t = \(\begin{array}{l}\frac{1}{3}\int_{3}^{6}\left | \frac{2t}{3}-[t] \right |dt\end{array}... View Article

Let the position vectors of points…..

Question:Let the position vectors of points ‘A’ and ‘B’ be \(\begin{array}{l}\hat{i}+\hat{j}+\hat{k}\end{array} \) and \(\begin{array}{l}2\hat{i}+\hat{j}+3\hat{k}\end{array} \) respectively. A point ‘P’ divides... View Article

Let a, b, c ∈ R be all non-zero and satisfy

Let a, b, c ∈ R be all non-zero and satisfy a3+b3+c3 = 2. If the matrix A = \(\begin{array}{l}\begin{bmatrix}... View Article

[latex]\lim_{x\to0}\left ( \tan (\frac{\pi }{4}+x) \right )^{\frac{1}{x}}[/latex] is equal to:

a) e b) e2 c) 2 d) 1 Solution: \(\begin{array}{l}\lim_{x\to0}\left ( \tan (\frac{\pi }{4}+x) \right )^{\frac{1}{x}}\end{array} \) Answer: (b)

For some θ∈(0, π/2) , if the eccentricity of the hyperbola, x²-y²sec² θ

For some θ∈(0, π/2) , if the eccentricity of the hyperbola, x2-y2sec2 θ = 10 is √5 times the eccentricity... View Article

If a curve y = f(x), passing through the point (1,2) is the solution of the differential equation, 2x²dy = (2xy+y²)dx, then f(1/2) is equal to

a) -1/(1+loge2) b) 1+loge 2 c) 1/(1+loge2) d) 1/(1-loge2) Solution: 2(dy/dx) = 2(y/x)+(y/x)2 HDE Since y = vx 2(v+x(dv/dx)) =... View Article

Let n>2 be an integer. Suppose that there are n metro stations in a city located along a circular path

Let n>2 be an integer. Suppose that there are n metro stations in a city located along a circular path.... View Article

The wavefront of distant source of unknown shape is approximately

(a) spherical (b) cylindrical (c) elliptical (d) plane Solution: Answer: (d) plane From a distant source, the shape of the... View Article

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