PYQ Vault

Day 280: CBSE Class 12 Maths

5 questions (14 parts). Try each one first, then open its answer.

  1. Question 1 (case study, 4 parts)

    Application of Derivatives. Asked once: 2024

    The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is θ\theta. On the basis of the above information, answer the following questions :
    36 (i)
    Express θ\theta in terms of height of the camera installed on the pole and x.
    36 (ii)
    Find dθdx\dfrac{d\theta}{dx}.
    36 (iii) (a)
    Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.
    36 (iii) (b)
    If the rate of change of angle of elevation with respect to time of another car at a distance of 50 m from the base of the pole is 3101\dfrac{3}{101} rad/s, then find the speed of the car.

  2. Question 2

    Matrices. Asked once: 2023

    2
    If [2054]=P+Q\begin{bmatrix} 2 & 0 \\ 5 & 4 \end{bmatrix} = P + Q, where P is a symmetric and Q is a skew symmetric matrix, then Q is equal to

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  3. Question 3 (case study, 4 parts)

    Relations and Functions. Asked once: 2025

    Let A be the set of 30 students of class XII in a school. Let f:A→Nf : A \to N, N is a set of natural numbers such that function f(x)=f(x) = Roll Number of student x. On the basis of the given information, answer the following :
    37 (i)
    Is f a bijective function ?
    37 (ii)
    Give reasons to support your answer to (i).
    37 (iii) (a)
    Let R be a relation defined by the teacher to plan the seating arrangement of students in pairs, where R={(x,y):x, yR = \{(x, y) : x,\ y are Roll Numbers of students such that y=3x}y = 3x\}. List the elements of R. Is the relation R reflexive, symmetric and transitive ? Justify your answer.
    37 (iii) (b)
    Let R be a relation defined by R={(x,y):x, yR = \{(x, y) : x,\ y are Roll Numbers of students such that y=x3}y = x^3\}. List the elements of R. Is R a function ? Justify your answer.

  4. Question 4

    Vector Algebra. Asked once: 2025

    12
    Let p⃗\vec{p} and q⃗\vec{q} be two unit vectors and α\alpha be the angle between them. Then (p⃗+q⃗)(\vec{p} + \vec{q}) will be a unit vector for what value of α\alpha ?

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  5. Question 5 (case study, 4 parts)

    Application of Derivatives. Asked once: 2024

    Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as F=V2500−V4+14F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14. On the basis of the above information, answer the following questions :
    36 (i)
    Find F, when V = 40 km/h.
    36 (ii)
    Find dFdV\dfrac{dF}{dV}.
    36 (iii) (a)
    Find the speed V for which fuel consumption F is minimum.
    36 (iii) (b)
    Find the quantity of fuel required to travel 600 km at the speed V at which dFdV=−0⋅01\dfrac{dF}{dV} = -0{\cdot}01.