PGDBA 2023 Question Paper | PGDBA QA

PGDBA Question Paper | PGDBA Previous Year Paper

PGDBA 2023 Question Paper QA

The coefficient of \(x^2 y^6 z^4\) in the expansion of \((y z+x y)^6\left(1-\frac{z}{x}\right)^6\) is
1. 27720
2. 15
3. 33
4. 276
Choice B
15
PGDBA 2023 Question Paper QA

The sums of the first \(n\) terms of two arithmetic progressions are in the ratio of \((7 n+1)\) : \((4 n+27)\). The ratio of their \(11^{\text {th }}\) terms is
1. \(4: 3\)
2. \(5: 4\)
3. \(78: 71\)
4. \(148: 111\)
Choice D
\(148: 111\)
PGDBA 2023 Question Paper QA

Let \(P(x)\), and \(Q(x)\) be two distinct polynomials with degree at most 2 . Let \(a_0, \ldots, a_{n-1}\) be distinct elements of \(R\). Consider the following set \[ X=\left\{i \in\{0,1, \ldots,(n-1)\}: P\left(a_i\right) \neq Q\left(a_i\right)\right\} . \] Which of the following is always correct?
1. The number of distinct elements in the set \(X\) is at least \((n-2)\)
2. The number of distinct elements in the set \(X\) is less than \((n-2)\)
3. The number of distinct elements in the set \(X\) is greater than \((n-2)\)
4. The number of distinct elements in the set \(X\) is equal to \((n-2)\)
Choice A
The number of distinct elements in the set \(X\) is at least \((n-2)\)
PGDBA 2023 Question Paper QA

A government office assigns a distinct license plate for each vehicle registered under it. Each licence plate contains two letters of the English alphabet followed by four digits. In order to avoid confusion no licence plate is allowed to contain both the letter \(O\) and the number 0 . What is the maximum number of vehicles that can be registered by this office?
1. \( 10685236 \)
2. \( 6584611 \)
3. \( 6760000 \)
4. \( 4432536 \)
Choice B
\( 6584611 \)
PGDBA 2023 Question Paper QA

The domain of the function \(f(x)=\sqrt{\log _{10}\left(\frac{3 x-x^2}{2}\right)}\) is
1. \((1,2)\)
2. \([1,2]\)
3. \((0,3)\)
4. \((0,1] \cup[2, \infty)\)
Choice B
\([1,2]\)
PGDBA 2023 Question Paper QA

Suppose \(\sin \theta=\frac{3}{5}\), where \(\theta\) is an acute angle. Then the value of \(\left(500 \sin ^4 \frac{\theta}{2}+400 \sin ^2 \frac{\theta}{2}\right)\) is
1. \( 65 \)
2. \( 5 \)
3. \( 45 \)
4. \( 40 \)
Choice C
\( 45 \)
PGDBA 2023 Question Paper QA

Let \[ f(x)=\left\{\begin{array}{ll} x^\alpha \sin \frac{1}{x} & \text { if } x>0 \\ 2 x^3+x^2-2 x+\beta & \text { if } x \leq 0 \end{array}\right. \] be a continuous function. Then,
1. \(\alpha\lt1\) and \(\beta\gt0\)
2. \(\alpha \geq 1\) and \(\beta=0\)
3. \(-1\lt\alpha\lt1\) and \(\beta=0\)
4. \(\alpha\gt1\) and \(\beta\lt0\)
Choice B
\(\alpha \geq 1\) and \(\beta=0\)
PGDBA 2023 Question Paper QA

The value of the integral \(\int_0^{\pi / 2} \frac{\sin x}{\sin x+\cos x} d x\) is
1. \(\frac{\pi}{4}\)
2. \(\frac{\pi}{2}\)
3. \(\pi\)
4. \(\pi+2\)
Choice A
\(\frac{\pi}{4}\)
PGDBA 2023 Question Paper QA

The number of functions \(f:\{1,2,3,4,5\} \rightarrow\{1,2,3,4,5\}\) such that \(f(6-i)=f(i)\) for \(i=1,2,3,4,5\) is
1. \( 100 \)
2. \( 125 \)
3. \( 25 \)
4. \( 115 \)
Choice B
\( 125 \)
PGDBA 2023 Question Paper QA

If \(p, q, r\) are strictly positive real numbers, and \(p x+q y+r z=0, q x+r y+p z=0\) and \(r x+p y+q z=0\), then there is a real number \(\lambda \neq 1\) such that \(x: y: z\) is the same as
1. \(1: \lambda:-\lambda^2\)
2. \(1: \lambda^2: \lambda\)
3. \(1: \lambda: \lambda^2\)
4. \(1:-\lambda: \lambda^2\)
Choice A
\(1: \lambda:-\lambda^2\)
PGDBA 2023 Question Paper QA

The area enclosed by the curves \(y=\cos ^{-1} x\) and \(y=\sin ^{-1} x\) over the range \(0 \leq x \leq \frac{1}{\sqrt{2}}\) is
1. \(\sqrt{2}-1\)
2. \(1-\frac{1}{\sqrt{2}}\)
3. \(\frac{\pi}{4}\)
4. \(\frac{\pi}{4 \sqrt{2}}\)
Choice A
\(\sqrt{2}-1\)
PGDBA 2023 Question Paper QA

A man standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is \(60^{\circ}\). When he goes 48 meters away from the bank along the line joining the person and the tree, he finds the angle to the tree to be \(30^{\circ}\). Then the height of the tree is
1. \(24 \sqrt{3}\) meters
2. \(\sqrt{3}\) meters
3. \(4 \sqrt{3}\) meters
4. 26 meters
Choice A
\(24 \sqrt{3}\) meters
PGDBA 2023 Question Paper QA

In a \((8 \times 8)\) chessboard, numbers are placed on each of the 64 squares such that the number on each square is the average of its neighboring squares (that is, the squares with which it shares a side). Also it is known that the sum of all the numbers is 640 . Which of the following is true:
1. There exists a way of placing the numbers in the chessboard such that the average of the numbers on the four corner squares is strictly greater than 10
2. There exists a way of placing the numbers in the chessboard such that the product of all the numbers must be strictly less than \(10^{64}\)
3. There exists a way of placing the numbers in the chessboard such that the average of the numbers on the four corner squares is strictly less than 10
4. The number of ways the chessboard can be filled subject to the given conditions is less than 10
Choice D
The number of ways the chessboard can be filled subject to the given conditions is less than 10
PGDBA 2023 Question Paper QA

Consider the following figure, where each square is a house. Some pairs of houses are connected by a narrow lane, as indicated in the diagram by connecting lines.

Now, the owners decide to paint these houses (each with a single color) keeping in mind that two connected houses can't be painted with the same color. What will be the minimum number of different colors needed to paint all these houses?
1. \( 2 \)
2. \( 3 \)
3. \( 4 \)
4. at least \( 5 \)
Choice B
\( 3 \)
PGDBA 2023 Question Paper QA

Consider the function \[ f(x)=\left\{\begin{array}{ll} \frac{1-\cos (x)}{x^2} & \text { for } x \neq 0 \\ 1 & \text { if } x=0 \end{array}\right. \] Then, which of the following statements is correct?
1. \(f\) has a discontinuity and it is not removable
2. \(f\) has a discontinuity and it is removable
3. \(f\) is continuous everywhere and is differentiable
4. \(f\) is continuous everywhere but is not differentiable
Choice B
\(f\) has a discontinuity and it is removable
PGDBA 2023 Question Paper QA

If \(w, x, y, z\) are positive real numbers then the least value of \[ (w+2 x+3 y+4 z)\left(\frac{1}{w}+\frac{1}{2 x}+\frac{1}{3 y}+\frac{1}{4 z}\right) \] is
1. \( 16 \)
2. \( \frac { 125 } { 6 } \)
3. \( 20 \)
4. \( 9 \)
Choice A
\( 16 \)
PGDBA 2023 Question Paper QA

The equation of the straight line of slope \(\frac{5}{2}\), which touches the parabola \(y^2=5 x\), is
1. \(5 x+2 y+1=0\)
2. \(5 x-2 y+1=0\)
3. \(5 x+2 y-1=0\)
4. \(5 x-2 y-1=0\)
Choice B
\(5 x-2 y+1=0\)
PGDBA 2023 Question Paper QA

Let \(M=\left[\begin{array}{ccc}0 & 1 & -\alpha \\ -1 & 0 & 5 \\ \alpha & -5 & 0\end{array}\right]\) and \(y =\left(\begin{array}{l}y_1 \\ y_2 \\ y_3\end{array}\right)\), where \(\alpha\) is non-zero and \(y_1, y_2, y_3\) are real numbers. Let \(M x = y\) for some vector \(x\). Then the value of \(x ^T y\) is
1. \(\alpha\)
2. \(- \alpha\)
3. \( 0 \)
4. \(\alpha y_2\)
Choice C
\( 0 \)
PGDBA 2023 Question Paper QA

Let \(f\) be a function defined by \[ f(x)=\left\{\begin{array}{ll} x^2 \sin \frac{1}{x} & \text { for } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. \] Then, which of the following statements is correct?
1. \(f\) is differentiable at 0 , but \(f^{\prime}(x)\) is not continuous at 0
2. \(f\) is not differentiable at 0
3. \(f\) is differentiable at 0 , and \(f^{\prime}(x)\) is continuous at 0
4. \(f\) is differentiable at 0 , and \(f^{\prime}(x)\) is not continuous at 0 but \(\lim _{x \rightarrow 0} f^{\prime}(x)\) exists.
Choice B
\(f\) is not differentiable at 0
PGDBA 2023 Question Paper QA

The equation of the locus of point \(P\) which maintains the distance from two fixed points \(R=(0,2)\) and \(S=(0,-2)\) satisfying the equality \(|R P+S P|=6\) is
1. \(\frac{x^2}{5^2}+\frac{y^2}{9^2}=1\)
2. \(\frac{x^2}{5}+\frac{y^2}{9}=1\)
3. \(\frac{x^2}{5^2}-\frac{y^2}{9^2}=1\)
4. \(\frac{x^2}{5}-\frac{y^2}{9}=1\)
Choice B
\(\frac{x^2}{5}+\frac{y^2}{9}=1\)
PGDBA 2023 Question Paper QA

The value of \(\frac{2}{0 !+1 !+2 !}+\frac{3}{1 !+2 !+3 !}+\cdots+\frac{n}{(n-2) !+(n-1) !+n !}\) is
1. \(1+\frac{1}{n}\)
2. \(1-\frac{1}{n}\)
3. \(1-\frac{1}{n !}\)
4. \(1+\frac{1}{n !}\)
Choice C
\(1-\frac{1}{n !}\)
PGDBA 2023 Question Paper QA

Let \(PQR\) be a right-angled triangle with the right angle at \(P\) and the angles \(\theta_1\) and \(\theta_2\) at \(Q\) and \(R\), respectively. Let the length of \(PQ\) be \(\sqrt{2}\) and the length of RP be 3 . Then the value of \(\cot \left(\theta_1\right)+\cot \left(\theta_2\right)\) is
1. \(\frac{\sqrt{11}}{3}\)
2. \(\frac{11}{3 \sqrt{2}}\)
3. \(\frac{3 \sqrt{2}}{11}\)
4. \(33 \sqrt{2}\)
Choice B
\(\frac{11}{3 \sqrt{2}}\)
PGDBA 2023 Question Paper QA

If there are 10 red balls and 12 blue balls, and these are arranged by drawing one ball at a time at random, then what is the probability that the last ball in the order is of color red?
1. \(\frac{10}{22}\)
2. \(\frac{10 !}{22 !}\)
3. \(\frac{1}{2}\)
4. \(\frac{12}{22}\)
Choice A
\(\frac{10}{22}\)
PGDBA 2023 Question Paper QA

The value of \(\lim _{x \rightarrow 0}(1+2 x)^{(x+3) / x}\) is
1. \(e\)
2. \(e^6\)
3. \(e^4\)
4. \(1\)
Choice B
\(e^6\)
PGDBA 2023 Question Paper QA

If the line \(3 x+4 y-7=0\) divides the line segment joining the points \((2,1)\) and \((-2,1)\) in the ratio \(\lambda: 1\), then the value of \(\lambda\) is
1. \(\frac{1}{3}\)
2. \( 3 \)
3. \( 1 \)
4. \(\frac{1}{2}\)
Choice A
\(\frac{1}{3}\)