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BCA Semester 1 – Mathematics I – Unit 5: Analytical Geometry

Comprehensive questions and detailed answers for Unit 5: Analytical Geometry. Perfect for exam preparation and concept clarity.

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Find a unit vector perpendicular to the plane containing points $P(1,−1,0), \space Q(2,1,−1), \space and \space R(−1,1,2).$

MediumTHEORY5 marks2019(TU FOHSS Final)

In how many ways can the letters of the word "Sunday" be arranged?How many arrangements begin with S?How many begin with S and do not end with y?

MediumTHEORY5 marks2019(TU FOHSS Final)

a) Define Conic section. Find the coordinates of vertices, eccentricity, and foci of the ellipse:

\quad \quad \quad 9x^{2}+4y^{2}-18x-16y-11=0

b) If $T:\mathbb{R}^{2}\to\mathbb{R}^{3}$ is defined by $T(x_{1},x_{2})=(x_{1}+x_{2},\,x_{2},\,x_{1})$ , find the matrix associated with $T$ .

MediumTHEORY10 marks2019(TU FOHSS Final)

Find the focus, vertex, equation of axis, equation of directrix, and length of latus rectum of the ellipse:

\quad \quad 4x2+9y2=36

MediumTHEORY5 marks2020(TU FOHSS Final)

If θ is the angle between two unit vectors $\vec a \space and \space \vec b$ , show that:

|\vec a−\vec b|=sin⁡\frac{\theta}{2}

MediumTHEORY5 marks2020(TU FOHSS Final)

\textbf{Prove by vector method: } $cos⁡(A+B) = cos⁡A⋅cos⁡B−sin⁡A⋅sin⁡B.

MediumTHEORY10 marks2020(TU FOHSS Final)

Find the equation of the circle passing through the points (1, 2), (3, 1), and (-3, -1).

MediumTHEORY10 marks2020(TU FOHSS Final)

Find the equation of the ellipse whose latus rectum is $3$ and eccentricity is $\frac{1}{\sqrt2}$ .

MediumNumerical5 marks2021(TU FOHSS Final)

Prove by vector method:

\cos(A - B) = \cos A \cos B + \sin A \sin B.

MediumNumerical5 marks2021(TU FOHSS Final)

a) Find the angle between vectors

\mathbf{u} = 4\mathbf{i} - 2\mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k}.

b) Find the Maclaurin series of $f(x) = \cos x.$

HardNumerical10 marks2021(TU FOHSS Final)

Find the eccentricity and foci of the ellipse: $25x^2 + 4y^2 = 100.$

MediumNumerical5 marks2022(TU FOHSS Final)

Find the equation of a parabola having vertex $(+, 2)$ and directrix $x = 4$ .

MediumNumerical5 marks2023(TU FOHSS Final)

a) Define a parabola with different parts using a figure and derive the standard equation of parabola:

y^2 = 4ax

b) In how many ways can the letters of the word “ARRANGE” be arranged so that all the vowels are always together?

HardNumerical10 marks2023(TU FOHSS Final)

Find the equation of the ellipse whose latus rectum is $5$ and the eccentricity is $\frac{1}{\sqrt2}$ .

MediumNumerical5 marks2024(TU FOHSS Final)

If $\vec a = \sqrt3 {i} + \hat{j}, \quad \vec {a} \times \vec b = (1, 2, 2),$ find the angle between $\vec{a}$ and $\vec{b}$ .

MediumNumerical5 marks2024(TU FOHSS Final)

a) Find the equation of a hyperbola in standard form having focus $(-2, 0)$ and directrix $x = -\frac{1}{2}$ .
b) In an examination paper on mathematics, $20$ questions are set. In how many different ways can you choose $18$ questions to answer?

HardNumerical10 marks2024(TU FOHSS Final)

Showing 16 questions

Sample Questions

Find a unit vector perpendicular to the plane containing points $P(1,−1,0), \space Q(2,1,−1), \space and \space R(−1,1,2).$

Marks: 5Chapter: Unit 5: Analytical Geometry

In how many ways can the letters of the word "Sunday" be arranged?How many arrangements begin with S?How many begin with S and do not end with y?

Marks: 5Chapter: Unit 5: Analytical Geometry

a) Define Conic section. Find the coordinates of vertices, eccentricity, and foci of the ellipse: \[ \quad \quad \quad 9x^{2}+4y^{2}-18x-16y-11=0 \] b) If $T:\mathbb{R}^{2}\to\mathbb{R}^{3}$ is defi

Marks: 10Chapter: Unit 5: Analytical Geometry

Find the focus, vertex, equation of axis, equation of directrix, and length of latus rectum of the ellipse: \[ \quad \quad 4x2+9y2=36\]

Marks: 5Chapter: Unit 5: Analytical Geometry

If θ is the angle between two unit vectors $\vec a \space and \space \vec b$, show that: \[|\vec a−\vec b|=sin⁡\frac{\theta}{2}\]

Marks: 5Chapter: Unit 5: Analytical Geometry

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BCA Semester 1 – Mathematics I – Unit 5: Analytical Geometry

Sample Questions

Find a unit vector perpendicular to the plane containing points \(P(1,−1,0), \space Q(2,1,−1), \space and \space R(−1,1,2).\)

In how many ways can the letters of the word "Sunday" be arranged?How many arrangements begin with S?How many begin with S and do not end with y?

a) Define Conic section. Find the coordinates of vertices, eccentricity, and foci of the ellipse: \[ \quad \quad \quad 9x^{2}+4y^{2}-18x-16y-11=0 \] b) If \(T:\mathbb{R}^{2}\to\mathbb{R}^{3}\) is defi

Find the focus, vertex, equation of axis, equation of directrix, and length of latus rectum of the ellipse: \[ \quad \quad 4x2+9y2=36\]

If θ is the angle between two unit vectors \(\vec a \space and \space \vec b\), show that: \[|\vec a−\vec b|=sin⁡\frac{\theta}{2}\]