Phrases 2025 Previous Year Question Paper and Solution with PDF

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Previous Year Questions

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Previous Year Questions
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Phrases Previous Year Questions (PYQs)

Phrases 2025 PYQ

Qus : 1

The length of the projection of

$\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}$ on

$\vec{b} = -2\hat{i} + \hat{j} + 2\hat{k}$ , is equal to:

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Solution

Qus : 2

$8^{x-1}=(1/4)^{x}$ , then the value of

$\frac{1}{log_{x+1}4-log_{x+1}5}+\frac{1}{log_{1-x}4-log_{1-x}5}$ is

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Solution

Qus : 3

Consider the matrix

$B=\begin{pmatrix}{-1} & {-1} & {2} \\ {0} & {-1} & {-1} \\ {0} & {0} & {-1}\end{pmatrix}$ . The sum of all the entries of the matrix

$B^{19}$ is

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Solution

Qus : 4

The curve

$y=\frac{x}{1+x\tan x}$ attains maxima

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Solution

Qus : 5

The scores of students in a national level examination are normally distributed with a mean of 500 and a standard deviation of 100. If the value of the cumulative distribution of the standard normal random variable at 0.5 is 0.691, then the probability that a randomly selected student scored between 450 and 500 is

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Solution

Probability Between 450 and 500 (Normal Distribution)

The exam scores are normally distributed with a mean $\mu = 500$ and standard deviation $\sigma = 100$ .
Given: $P(Z \leq 0.5) = 0.691$
We need to find: $P(450 \leq X \leq 500)$

Step 1: Convert scores to Z-scores

Use the formula: $Z = \frac{X - \mu}{\sigma}$

For $X = 500$ : $Z = \frac{500 - 500}{100} = 0$ For $X = 450$ : $Z = \frac{450 - 500}{100} = -0.5$

Step 2: Find the probability using cumulative values

We calculate: $P(450 \leq X \leq 500) = P(-0.5 \leq Z \leq 0) = P(Z \leq 0) - P(Z \leq -0.5)$

From symmetry: $P(Z \leq -0.5) = 1 - P(Z \leq 0.5) = 1 - 0.691 = 0.309$ and $P(Z \leq 0) = 0.5$

Step 3: Final Calculation

$P(-0.5 \leq Z \leq 0) = 0.5 - 0.309 = \boxed{0.191}$

✅ Final Answer:

The probability that a randomly selected student scored between 450 and 500 is: 0.191

Qus : 6

Number of permutations of the letters of the word BANGLORE such that the string ANGLE appears together in all permutations, is

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Solution

Qus : 7

Let A and B be two square matrices of same order satisfying

$A^2+5A+5I =0$ and

$B^2+3B+I=0$ repectively. Where I is the identity matrix. Then the inverse of the matrix

$C= BA+2B+2A+4I$ is

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Solution

Qus : 8

The captains of five cricket teams, including India and Australia, are lined up randomly next to one other for a group photo. What is the probability that the captains of India and Australia will stand next to each other?

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Solution

Qus : 9

The value of

$\frac{d}{dx}\int ^{2\sin x}_{\sin {x}^2}{e}^{{t}^2}dt$ at

$x=\pi$

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Solution

Qus : 10

There are two coins, say blue and red. For blue coin, probability of getting head is 0.99 and for red coin, it is 0.01. One coin is chosen randomly and is tossed. The probability of getting head is

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Solution

Qus : 11

The number of all even integers between 99 and 999 which are not multiple of 3 and 5 is

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Solution

Qus : 12

Let A = {1,2,3, ... , 20}. Let

$R\subseteq A\times A$ such that R = {(x,y): y = 2x - 7}. Then the number of elements in R, is equal to

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Solution

Qus : 13

$\vec{a}, \vec{b}$ and

$\vec{c}$ are three vectors such that

$\vec{a} \times \vec{b}=\vec{c}$ ,

$\vec{a}.\vec{c} = 2$ and

$\vec{b}.\vec{c} = 1$ . If

$|\vec{b}| = 1$ , then the value of

$|\vec{a}|$ is

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Solution

Qus : 14

If x, y and z are three cube roots of 27, then the determinant of the matrix

$\begin{bmatrix}{x} & {y} & {z} \\ {y} & {z} & {x} \\ {z} & {x} & {y}\end{bmatrix}$ is

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Solution

Qus : 15

Let

$A=\{{5}^n-4n-1\colon n\in N\}$ and

$B=\{{}16(n-1)\colon n\in N\}$ be sets. Then

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Solution

Qus : 16

Let

$\vec{a}$ ,

$\vec{b}$ and

$\vec{c}$ be unit vectors such that the angle between them is

${\cos }^{-1}\Bigg{\{}\frac{1}{4}\Bigg{\}}$ . If

$\vec{b}=2\vec{c}+\lambda \vec{a}$ , where

$\lambda$ > 0 and

$\vec{b}=4$ , then

$\lambda$ is equal to

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Solution

Qus : 17

A tower subtends angles

$\alpha, 2\alpha$ and

$3\alpha$ respectively at points A, B and C which are lying on a

horizontal line through the foot of the tower. Then

$\frac{AB}{BC}$ is equal to

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Solution

Qus : 18

$\vec{a}$ and

$\vec{b}$ are twp vectors such that |

$\vec{a}$ |=3, |

$\vec{b}$ |=4 and |

$\vec{a}+\vec{b}$ |=1, then the value of

$|\vec{a}-\vec{b}|$ is

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Solution

Qus : 19

$\vec{a}=\hat{i}+\hat{j}+\hat{k}$ ,

$\vec{b}=2\hat{i}-\hat{j}+3\hat{k}$ and

$\vec{c}=\hat{i}-2\hat{j}+\hat{k}$ , then a vector of magnitude

$\sqrt{22}$ which is parallel to

$2\vec{a}-\vec{b}+3\vec{c}$ is

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Solution

Qus : 20

Consider the sample space

$\Omega={\{(x,y):x,y\in{\{1,2,3,4\}\}}}$ where each outcome is equally likely. Let A = {x ≥ 2} and B = {y > x} be two events. Then which of the following is NOT true?

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Solution

Qus : 21

Let the line

$\frac{x}{4}+\frac{y}{2}=1$ meets the x-axis and y-axis at A and B, respectively. M is the midpoint of side AB, and M' is the image of the point M across the line x + y = 1. Let the point P lie on the line x + y = 1 such that the

$\Delta$ ABP is an isosceles triangle with AP = BP. Then the distance between M' and P is

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Solution

Qus : 22

Which one of the following is NOT a correct statement?

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Solution

Qus : 23

An equilateral triangle is inscribed in the parabola

$y^2 = x$ . One vertex of the triangle is at the vertex of the parabola. The centroid of triangle is

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Solution

Qus : 24

The angles of depression of the top and bottom of an 8m tall building from the top of a multi storied building are 30° and 45°, respectively. What is the height of the multistoried building and the distance between the two buildings?

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Solution

Qus : 25

The number of accidents per week in a town follows Poisson distribution with mean 2. If the probability that there are three accidents in two weeks time is

$ke^{-6}$ , then the value of k is

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Solution

Qus : 26

$B=sin^2 y+cos^4 y$ , then for all real y

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Solution

Qus : 27

Let

$F_1, F_2$ be foci of hyperbola

$\frac{{x}^2}{{a}^2}-\frac{{y}^2}{{b}^2}=1$ , a>0, b>0, and let O be the origin. Let M be an arbitrary point on curve C and above X-axis and H be a point on

$MF_1$ such that

$MF_2\perp{{F}}_1{{F}}_2$ ,

$MF_1\perp{{O}}{{H}}$ ,

$|OH|=\lambda |OF_2|$ with

$\lambda \in(2/5, 3/5)$ , then the range of the eccentricity

$e$ is

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Solution

Qus : 28

A circle with its center in the first quadrant touches both the coordinate axes and the line x -y - 2 = 0. Then the area of the circle is

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Solution

Qus : 29

$\alpha$ and

$\beta$ are the two roots of the quadratic equation

$x^2 + ax + b = 0, (ab \ne 0)$ then the quadratic roots whose roots

$\frac{1}{\alpha^3+\alpha}$ and

$\frac{1}{\beta^3+\beta}$ is

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Solution

Qus : 30

The maximum value of

$sinx+sin(x+1)$ is

$k cos \frac{1}{2}$ . Then the value of k is

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Solution

Qus : 31

Let

$\mathbb{R}\rightarrow\mathbb{R}$ be any function defined as

$f(x)=\begin{cases}{{x}^{\alpha}\sin \frac{1}{{x}^{\beta}}} & {,x\ne0} \\ {0} & {,x=0}\end{cases}$ ,

$\alpha , \beta \in \mathbb{R}$ . Which of the following is true? (

$\mathbb{R}$ denotes the set of all real numbers)

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Solution

Qus : 32

The number of 3-digit integers that are multiple of 6 which can be formed by using the digits 1,2,3,4,5,6 without repetition is

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Solution

Qus : 33

The circle

$x^2 + y^2+ \alpha x+ \beta y+ \gamma=0$ is the image of the circle

$x^2 + y^2- 6x- 10y+ 30=0$ across the line 3x + y = 2. The value of

$[\alpha+ \beta+ \gamma]$ is (where [.] represents the floor function.)

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Solution

Qus : 34

Let

$\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$ ,

$\vec{b}=\hat{i}+2\hat{j}-\hat{k}$ and

$\vec{c}=3\hat{i}+\hat{j}+\lambda \hat{k}$ be the co-terminal edges

of a parallelopiped whose volume is 5 units. Then the value of

$\lambda$ is

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Solution

Qus : 35

The area enclosed between the curve y = sin x, y = cosx,

$0\leq x\leq\frac{\pi}{2}$ is

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Solution

Qus : 36

Given the equation

$x+y =1$ ,

$x^2+y^2 =2$ ,

$x^5 +y^5 =A$ . Let N be the number of solution pairs (x,y) to this system of equations. Then AN is equal to

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Solution

Qus : 37

Number of three digit numbers that can be formed using 0, 1, 2, 3 and 5 where these digits are allowed to repeat any number of times, is equal to

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Solution

Qus : 38

Let

$g:\mathbb{R}\rightarrow \mathbb{R}$ and

$h:\mathbb{R}\rightarrow \mathbb{R}$ , be two functions such that

$h(x) = sgn(g(x))$ . Then select which of the following is not true?(

$\mathbb{R}$ denotes the set of all real numbers, sgn stands for signum function)

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Solution

Qus : 39

An airplane, when 4000m high from the ground, passes vertically above another airplane at an instant when the angles of elevation of the two airplanes from the same point on the ground are 60° and 30°, respectively. Find the vertical distance between the two airplanes.

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Solution

Qus : 40

Suppose

$t_1, t_2, ...t_55$ are in AP such that

$\sum ^{18}_{l=0}{{t}}_{3l+1}=1197$ and

${{t}}_7+{{3}}t_{22}=174$ . If

$\sum ^9_{l=1}{{{t}}_l}^2=947b$ , then the value of

$b$ is

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Solution

Qus : 41

Let E and F be two events such that P(E) > 0 and P(F) > 0. Which one of the following is NOT equivalent to the condition that

$P(E) =P(E|F)$ ?

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Solution

Qus : 42

What is the general solution of the equation

$tan \theta + cot \theta = 2$ ?

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Solution

Qus : 43

$cos^2(10°)cos(20°)cos(40°)cos(50°) cos(70°) = \alpha+\frac{\sqrt{3}}{16} cos(10°)$ , then

$3\alpha^{-1}$ is equal to

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Solution

Qus : 44

The slope of the normal line to the curve

$x = t^2 + 3t - 8$ and

$y = 2t^2 - 2t - 5$ at the point (2,-1) is

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Solution

Qus : 45

A tower subtends an angle of 30° at a point on the same level as the foot of the tower. At a second point h meters above the first, the depression of the foot of the tower is 60°. What is the horizontal distance of the tower from the point?

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Solution

Qus : 46

There are 40 female and 20 male students in a class. If the average heights of female and male students are 5.15 feet and 5.66 feet, respectively, then the average height (in feet) of all the students in the class equals

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Solution

Qus : 47

Let

$x$ be a positive real number such that

$x^{(8log_5(x)-24)}=5^{-4}$ . Then the product of all possible values of x is =

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Solution

Qus : 48

The obtuse angle between lines 2y = x + 1 and y = 3x + 2 is

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Solution

Qus : 49

What is the value of

$\lim _{{x}\rightarrow\infty}-(x+1)\Bigg{(}{e}^{\frac{1}{x+1}}-1\Bigg{)}$ ?

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4
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Solution

Qus : 50

The value of

$\int ^{\frac{\pi}{2}}_0\frac{(1+2\cos x)}{({2+\cos x)}^2}dx$

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4
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Solution

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Phrases Previous Year Questions (PYQs)

Phrases 2025 PYQ

Solution

Solution

Solution

Solution

Solution

Probability Between 450 and 500 (Normal Distribution)

Step 1: Convert scores to Z-scores

Step 2: Find the probability using cumulative values

Step 3: Final Calculation

✅ Final Answer:

Solution

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