SEE (Grade Increment ) 2080 (2024)

General

Time : 3 hours    ♠♠♠  Full Marks: 75

Candidates are required to give their answer according to the given instructions.

Attempt all the questions.

Group A [10 \( \times \) 1 = 10]

Question 1.

If \( f(x) = x \) then what type of function is \( f(x) \) ? Write it.

Question 2.

If \( a, G \) and \( b \) are in a geometric series, write \( G \) in terms of \( a \) and \( b \).

Question 3.

Show the interval notation (\(-\)2,3) in the number line.

Question 4.

Define non-singular matrix.

Question 5.

Write the formula to find the angle between two straight lines having slopes \( m_1 \) and \( m_2 \) respectively.

Question 6.

In which condition a circle will be formed when a plane intersects a right circular cone? Write it.

Question 7.

Express \( \cos2A \) in terms of \( \cos A \) and \( \sin A \).

Question 8.

Write \(\cos C + \cos D \) in the form of product of cosine.

Question 9.

If the scalar product of two vectors \( \vec{a} \) and \( \vec{b} \) is zero, then write the relation between \( \vec{a} \) and \( \vec{b} \).

Question 10.

\(P' \) is the inversion point of a point \( P \) in an inversion circle having centre \( O \) and radius \( r \) then what is the product of \( OP \) and \( OP' \) ? Write it.

Group B  [ 8 \( \times \) 2 = 16 ]

Question 11.

If \( f(x) = 3x - 2 \), then find \( f^{-1} (x) \) .

Question 12.

Find the vertex of parabola formed from the quadratic equation \( y = x^2 - 2x + 1 \).

Question 13.

If the matrix \( A = \begin{bmatrix} 2 & 1 \\ x & 4 \end{bmatrix} \) and \( |A| = 10 \), find the value of \( x \).

Question 14.

Prove that the lines \( 2x + 3y = 8 \) and \( 4x + 6y = 7 \) are parallel to each other.

Question 15.

Prove that: \( \frac{1+\cos A}{\sin A} = \cot \frac{A}{2} \)

Question 16

Solve: \( 4\sin \theta - \sqrt{8} = 0 \) \( ( 0^\circ \le \theta \le 90^\circ ) \)

Question 17.

The position vectors of the points A and B are \( 3\vec{i} + 6\vec{j} \) and \( 5\vec{i} - 2\vec{j} \) respectively. If the point D divides line segment AB in the ratio

of 1: 1 then find the position vector of the point D.

Question 18.

If the third quartile of a continuous series is 25 and quartile deviation is \( \frac{13}{2} \), find the coefficient of quartile deviation.

 Group  C \([ 11 \times 3 = 33 ] \) 

Question 19.

Solve : \( x^3 - 6x^2 + 11x - 6 = 0 \)

Question 20.

The first term of an arithmetic series is 24 and its ninth term is 40. Find the sum of the first 15 terms of the series.

Question 21.

Function \( f(x) = \begin{cases} 3x - 1, & x \le 3 \\ 2x + 2, & x > 3 \end{cases} \) are given.

(i) For \( x = 2.9999\), find the value of \( f(x) \).

(ii) For \( x = 3.0001 \), find the value of \( f(x) \).

(iii) Is the function \( f(x) \) continuous at \( x = 3 \)? Give reason.

Question 22.

Solve using Cramer's rule :

\( 3x + y = 5, 4x - 3y = 11 \)

Question 23.

Find the angle between the pair of lines represented by the equation \[ 3x^2 + 7xy + 2y^2 = 0 \]

Question 24.

Prove that: \( \frac{\cos 2\theta }{1 - \sin 2\theta } = \frac{1 + \tan \theta }{ 1 - \tan \theta } \)

Question 25.

If \( \alpha + \beta + \gamma = \pi^c \) then prove that:

\( \cos 2\alpha + \cos 2\beta - \cos2\gamma = 1 - 4\sin\alpha. \sin\beta. \cos\gamma \)

Question 26.

A electric pole and a house are standing on the same plane ground. The angle of depression of the top of the electric pole from the roof of the house is 30\(^\circ \) and the angle of elevation of the roof of the house from the foot of the pole is 45\(^\circ \). If the height of the house is 30 m, find the height of the pole.

Question 27.

Find a 2 \( \times \) 2 transformation matrix which transforms a unit square into the quadrilateral \( \begin{bmatrix} 0 & 3 & 4 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix} \)

question 28.

Find the mean deviation from the median on the basis of given data.

Weight in kg	30 - 40	40 - 50	50 - 60	60 - 70	70 - 80
No. of students	3	2	5	3	1

Question 29.

Find the standard deviation from the given continuous series.

Class Interval	20 - 40	40 - 60	60 - 80	80 - 100	100 - 120
Frequency	13	12	8	9	8

Group D [ 4 \( \times \) 4 = 16 ]

Question 30.
Find the maximum value of the objective function \( F = 2x + 4y \) under the constraints \( x - y \le 4, x + y \le 6, x \ge 0, y \ge 0 \).
Question 31.
Find the radius of the circle \( x^2 + y^2 - 6x - 2y - 6 = 0 \). Also find the equation of the other circle having same radius and centre (6, 8).
Question 32.
Prove by vector method that the line segment joining the mid-points of any two sides of a triangle is half of the third side.
Question 33.
If \( R_1 \) represents the reflection on the X-axis and \( R_2 \) represents the reflection on the Y-axis, then which single transformation does combined transformation \( R_1oR_2 \) represent? Write it. Using this single transformation find the image of \( \Delta\)PQR, where P(1, 5), Q (5, 1) and R (8, 3). Also present \(\Delta \)PQR and the image on the same graph paper. 

 ♠♠♠ All The Best ♠♠♠

Subscribe My YouTube Channel Ambik Sir