isi-entrance

Q1 Permutations & Arrangements Distribution of Objects into Bins/Groups View

There are 8 balls numbered $1,2 , \ldots , 8$ and 8 boxes numbered $1,2 , \ldots , 8$. The number of ways one can put these balls in the boxes so that each box gets one ball and exactly 4 balls go in their corresponding numbered boxes is
(a) $3 \times {}^{8}\mathrm{C}_{4}$
(b) $6 \times {}^{8}\mathrm{C}_{4}$
(c) $9 \times {}^{8}\mathrm{C}_{4}$
(d) $12 \times {}^{8}C_{4}$

Q2 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

Let $a$ and $\beta$ be two positive real numbers. For every integer $n > 0$, define $a_n = \int_{\beta}^{n} \frac{a}{u(u^{a}+2+u^{-a})} du$. Then $\lim_{n \to \infty} a_n$ is equal to
(a) $1/(1+\beta^{a})$
(b) $\beta^{a}/(1+\beta^{-a})$
(c) $\beta^{a}/(1+\beta^{a})$
(d) $\beta^{-a}/(1+\beta^{a})$

Q3 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View

Let $f : \mathbb{R} \rightarrow \mathbb{R}^{2}$ be a function given by $f(x) = (x^{m}, x^{n})$, where $x \in \mathbb{R}$ and $m, n$ are fixed positive integers. Suppose that $f$ is one-one. Then
(a) Both $n$ and $m$ must be odd
(b) At least one of $m$ and $n$ must be odd
(c) Exactly one of $m$ and $n$ must be odd
(d) Neither $m$ nor $n$ can be odd.

Q4 Differentiating Transcendental Functions Limit involving transcendental functions View

$\lim_{x \to 2} \left[\frac{e^{x^{2}} - e^{2x}}{(x-2)e^{2x}}\right]$ equals
(a) 0
(b) 1
(c) 2
(d) 3

Q5 Circles Inscribed/Circumscribed Circle Computations View

A circle is inscribed in a triangle with sides $8, 15, 17$ cms. The radius of the circle in cms is
(a) 3
(b) $22/7$
(c) 4
(d) None of the above.

Q6 Trig Proofs Extremal Value of Trigonometric Expression View

Let $\alpha$, $\beta$ and $\gamma$ be the angles of an acute angled triangle. Then the quantity $\tan\alpha\tan\beta\tan\gamma$
(a) Can have any real value
(b) Is $\leq 3\sqrt{3}$
(c) Is $\geq 3\sqrt{3}$
(d) None of the above.

Q7 Differentiating Transcendental Functions Piecewise function analysis with transcendental components View

Let $f(x) = |x|\sin x + |x - \pi|\cos x$ for $x \in \mathbb{R}$. Then
(a) $f$ is differentiable at $x = 0$ and $x = \pi$
(b) $f$ is not differentiable at $x = 0$ and $x = \pi$
(c) $f$ is differentiable at $x = 0$ but not differentiable at $x = \pi$
(d) $f$ is not differentiable at $x = 0$ but differentiable at $x = \pi$

Q8 Straight Lines & Coordinate Geometry Perspective, Projection, and Applied Geometry View

Consider a rectangular cardboard box of height 3, breadth 4 and length 10 units. There is a lizard in one corner $A$ of the box and an insect in the corner $B$ which is farthest from $A$. The length of the shortest path between the lizard and the insect along the surface of the box is
(a) $\sqrt{5^{2} + 10^{2}}$
(b) $\sqrt{7^{2} + 10^{2}}$
(c) $4 + \sqrt{3^{2} + 10^{2}}$
(d) $3 + \sqrt{4^{2} + 10^{2}}$

Q9 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View

Recall that, for any non-zero complex number $w$ which does not lie on the negative real axis, $\arg(w)$ denotes the unique real number $\theta$ in $(-\pi, \pi)$ such that $w = |w|(\cos\theta + i\sin\theta)$. Let $z$ be any complex number such that its real and imaginary parts are both non-zero. Further, suppose that $z$ satisfies the relations $\arg(z) > \arg(z+1)$ and $\arg(z) > \arg(z+i)$. Then $\cos(\arg(z))$ can take
(a) Any value in the set $(-1/2, 0) \cup (0, 1/2)$ but none from outside
(b) Any value in the interval $(-1, 0)$ but none from outside
(c) Any value in the interval $(0, 1)$ but none from outside
(d) Any value in the set $(-1, 0) \cup (0, 1)$ but none from outside.

Q10 Sine and Cosine Rules Heights and distances / angle of elevation problem View

An aeroplane $P$ is moving in the air along a straight line path which passes through the points $P_1$ and $P_2$, and makes an angle $\alpha$ with the ground. Let $O$ be the position of an observer. When the plane is at the position $P_1$ its angle of elevation is $30^\circ$ and when it is at $P_2$ its angle of elevation is $60^\circ$ from the position of the observer. Moreover, the distances of the observer from the points $P_1$ and $P_2$ respectively are 100 metres and $500/3$ metres.
Then $\alpha$ is equal to
(a) $\tan^{-1}\{(2-\sqrt{3})/(2\sqrt{3}-1)\}$
(b) $\tan^{-1}\{(2\sqrt{3}-3)/(4-2\sqrt{3})\}$
(c) $\tan^{-1}\{(2\sqrt{3}-2)/(5-\sqrt{3})\}$
(d) $\tan^{-1}\{(6-\sqrt{3})/(6\sqrt{3}-1)\}$

Q11 Number Theory Divisibility and Divisor Analysis View

The sum of all even positive divisors of $1000$ is
(a) $2170$
(b) $2184$
(c) $2325$
(d) $2340$

Q12 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

The equation $x^{2} + (b/a)x + (c/a) = 0$ has two real roots $\alpha$ and $\beta$. If $a > 0$, then the area under the curve $f(x) = x^{2} + (b/a)x + (c/a)$ between $\alpha$ and $\beta$ is
(a) $(b^{2} - 4ac)/2a$
(b) $(b^{2} - 4ac)^{3/2}/6a^{3}$
(c) $-(b^{2} - 4ac)^{3/2}/6a^{3}$
(d) $-(b^{2} - 4ac)/2a$

Q13 Stationary points and optimisation Find absolute extrema on a closed interval or domain View

The minimum value of $x_1^{2} + x_2^{2} + x_3^{2} + x_4^{2}$ subject to $x_1 + x_2 + x_3 + x_4 = a$ and $x_1 - x_2 + x_3 - x_4 = b$ is
(a) $(a^{2} + b^{2})/4$
(b) $(a^{2} + b^{2})/2$
(c) $(a+b)^{2}/4$
(d) $(a+b)^{2}/2$

Q14 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View

The value of $\lim_{n \to \infty} \frac{\sum_{r=0}^{n} {}^{2n}C_{2r} \times 3^{r}}{\sum_{r=0}^{n-1} {}^{2n}C_{2r+1} \times 3^{r}}$ is
(a) 0
(b) 1
(c) $\sqrt{3}$
(d) $(\sqrt{3}-1)/(\sqrt{3}+1)$

Q15 Approximating the Poisson to the Normal distribution View

For any real number $x$, let $\tan^{-1}(x)$ denote the unique real number $\theta$ in $(-\pi/2, \pi/2)$ such that $\tan\theta = x$. Then $\lim_{n \to \infty} \sum_{m=1}^{n} \tan^{-1}\left\{\frac{1}{1+m+m^{2}}\right\}$
(a) Is equal to $\pi/2$
(b) Is equal to $\pi/4$
(c) Does not exist
(d) None of the above.

Q16 Number Theory Prime Number Properties and Identification View

Let $n$ be an integer. The number of primes which divide both $n^{2}-1$ and $(n+1)^{2}-1$ is
(a) At most one.
(b) Exactly one.
(c) Exactly two.
(d) None of the above.

Q17 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View

The value of $\lim_{n \to \infty} \sum_{r} \frac{6n}{9n^{2} - r^{2}}$ is
(a) 0
(b) $\log(3/2)$
(c) $\log(2/3)$
(d) $\log(2)$

Q18 Permutations & Arrangements Lattice Path / Grid Route Counting View

A person $X$ standing at a point $P$ on a flat plane starts walking. At each step, he walks exactly 1 foot in one of the directions North, South, East or West. Suppose that after 6 steps $X$ comes to the original position $P$. Then the number of distinct paths that $X$ can take is
(a) 196
(b) 256
(c) 344
(d) 400

Q19 Conic sections Locus and Trajectory Derivation View

Consider the branch of the rectangular hyperbola $xy = 1$ in the first quadrant. Let $P$ be a fixed point on this curve. The locus of the mid-point of the line segment joining $P$ and an arbitrary point $Q$ on the curve is part of
(a) A hyperbola
(b) A parabola
(c) An ellipse
(d) None of the above.

Q20 Number Theory Properties of Integer Sequences and Digit Analysis View

The digit at the unit place of $$(1! - 2! + 3! - \ldots + 25!)^{(1! - 2! + 3! - \ldots + 25!)}$$ is
(a) 0
(b) 1
(c) 5
(d) 9

Q21 Sine and Cosine Rules Multi-step composite figure problem View

Let $A_1, A_2, \ldots, A_n$ be the vertices of a regular polygon and $A_1A_2$, $A_2A_3$, $\ldots$, $A_{n-1}A_n$, $A_nA_1$ be its $n$ sides. If $\left(\frac{1}{A_1A_2}\right) - \left(\frac{1}{A_1A_4}\right) = \frac{1}{A_1A_3}$, then the value of $n$ is
(a) 5
(b) 6
(c) 7
(d) 8

isi-entrance

Papers (32)

2010 solved