Islamic University of Technology ইসলামিক ইউনিভার্সিটি অব টেকনোলজি ভর্তি পরীক্ষা ২০২৪-২০২৫

(13) .......

(14) .......

Neither the manager nor the assistants _______ available to answer questions at the moment.

Not only _______ the results, but he also provided a detailed analysis.

The old man along with his sons _______ working since morning.

Her $\underline{magnanimous}$ gesture was appreciated by everyone.

He displayed a $\underline{recalcitrant}$ attitude during the disciplinary hearing.

After reading his old letters an air of $\underline{melancholy}$ surrounded him.

It created a $\underline{tranquil}$ scene.

Choose the opposite of the italicized word.His $\underline{belligerent}$ tone escalated the argument.

Choose the opposite of the italicized word.His $\underline{apathetic}$ response showed he had no interest in the project.

Choose the opposite of the italicized word.It was an $\underline{arduous}$ journey.

Choose the opposite of the italicized word.His $\underline{frugal}$ lifestyle deceived him in the long run.

(12) .......

(15) .......

Let, $A = \begin{bmatrix} \alpha & 2 & 7 \\ -2 & \beta & -8 \\ -7 & 8 & \gamma \end{bmatrix}$ be a skew symmetric matrix, then what are the values of $\alpha, \beta, \gamma$?

Find the area bounded by the curves $f(x) = 2 - x^2$ and g(x)=x g(x) = x :

$\int x \sin(x^2) \cos(x^2) dx$ = ?

If one diagonal of a square is along the line 8x−15y=0 8x - 15y = 0 and one of its vertex is at (1, 2), then find the equation of sides of the square passing through this vertex.

Evaluate: $\lim_{x \to 1} (\log_5 5x^2)^{\frac{\log_x 5}{2}}$

$\int (\frac{4x+1}{x}) (4x+\log 3x)^2 dx$ = ?

A water tank has the shape of an upside-down cone with radius $2\text{m}$ and height $5\text{m}$ . The water is running out of the tank through a small hole at the bottom at a speed proportional to the square root of the depth of the water in the tank. Suppose that the water is running out at a rate of $3 \text{ m}^3/\text{min}$ when the depth of the water in the tank is $4\text{m}$ . Find the rate at which the water level is changing at this moment.

Consider the following figure. A is the town hall on Scott Street and D is a Post Office on Keach Avenue. Diagonal Road intersects Scott Street at B and Keach Avenue at C. Find the equation of the Peacock Street.

Consider the equation of a circle given by $(x-5)^2 + (y-1)^2 = 25$ . Which of the following two straight lines are the tangent lines of the above circle? (i $) y - 4 = \frac{4}{3} (x-1$) (ii $) y - 5 = \frac{3}{4} (x-8$)

For what values of x and y, matrix $A = \begin{bmatrix} 1 & 1 & 1 \\ 4 & x & 4 \\ 0 & y & 5 \end{bmatrix}$ will be singular (inverse does not exist):

How many three-digit numbers (first digit is non-zero) are there which contain the digit 2 at most once?

If $\vec{A} = 2\hat{i} - 3\hat{j} - 4\hat{k}$ and $\vec{B} = \hat{i} + \hat{j} - 5\hat{k}$ then the projection of $\vec{B}$ on $\vec{A}$ is:

Which one is the unit vector perpendicular to the plane formed by the following vectors: $\vec{A} = 2\hat{i} - 3\hat{j} - \hat{k}$ $\vec{B} = \hat{i} + 4\hat{j} - 2\hat{k}$

The area of the triangle whose vertices are (6,5),(−9,−4) (6, 5), (-9, -4) and (−5,0) (-5, 0) is:

The equation of the circle that touches y-axis at (0,4) (0, 4) and whose center lies on the line 5x−7y−2=0 5x - 7y - 2 = 0 is:

How many arrangements can be made out of the letters of the word “COURAGE” all of which have a consonant at the beginning?

A committee of 6 members is to be formed from a group consisting 7 boys and 4 girls. In how many ways, the committee can be formed so as to include at least 2 girls?

What is the slope of tangent of the curve $x^2 + y^2 - 2x - 3 = 0$ at the point $(-1, 1)$?

The velocity of the current in the Kirton Khola River is $5 \text{ m/s}$ . A boat, which has a velocity of $13 \text{ m/s}$ relative to the water, takes 2 minutes and 10 seconds to cross the river in a direction perpendicular to the current. What is the width of the river?

Imagine that you are riding a bicycle at a speed of $10 \text{m/s}$ along a straight road. At the same time, raindrops are falling vertically to the ground with a velocity of $4 \text{ m/s}$ . What is the velocity of the raindrops relative to you?

An isosceles triangle is inscribed in the parabola $y^2 = 36x$ such that one of its vertices is at the vertex of the parabola and the other two vertices lie at the ends of the latus rectum. What is the area of the triangle?

Find the angle between the complex number $\frac{(1+i)^3}{1-i^3}$ and $-i$

If p, q, r are real numbers and $p^2 + 4q^2 + 9r^2 - 2pq - 6qr - 3rp = 0$ then p, q, r in

There are 5 red balls and 9 blue balls in a box. Four balls are randomly picked from the box without replacement and the colors of the balls are observed. What is the probability that the sequence of colors of the picked balls are alternate?

If one root of the quadratic equation $4x^2 - 8kx - 9 = 0$ is negative to the other, then the resulting equation is:

A $68 \text{ kg}$ person lies in a hammock, as shown in figure given below. The rope at the person's head makes an angle $\phi$ of $40^\circ$ with the horizontal, while the rope at the person's feet makes an angle $\theta$ of $20^\circ$ . Find the tension in the two ropes.

A ball is thrown horizontally with speed v from the floor at the top of some stairs as shown in the figure given below. The width and height of each step are both equal to $l$ . What should v be so that the ball barely clears the corner of the step that is 4 steps down? Assume $l = 25 \text{ cm}$ and $g = 10 \text{ ms}^{-2}$ .

A subway train is traveling at $80 \text{ km h}^{-1}$ when it approaches a slower train $50 \text{ m}$ ahead traveling in the same direction at $25 \text{ km h}^{-1}$ . If the faster train begins decelerating at $2.1 \text{ ms}^{-2}$ while the slower train continues at constant speed, how soon and at what relative speed will they collide?

A student is staring idly out her dormitory window when she sees a water balloon fall past. If the balloon takes $0.22 \text{ s}$ to cross the $130 \text{ cm}$ high window, from what height above the top of the window was it dropped?

$\sqrt{8 + 4\sqrt{5}i}$ = ?

If one root of the equation $3x^2 - kx + 4 = 0$ is thrice the other then what are the values of k?

The middle term in the expansion of $(x+2)^6$ is-

The eccentricity of the ellipse $3x^2 + 5y^2 = 15$ is-

A weight of $5 \text{ N}$ is supported by two forces as shown below. The value of P is-

A body is projected with a velocity of $100 \text{ m/sec}$ at an angle of 30 30 degree with the horizon $(g = 9.8 \text{ m/sec}^2)$ . The maximum height of the projectile is-