PYQ Vault

Day 278: CBSE Class 12 Maths

5 questions (8 parts). Try each one first, then open its answer.

  1. Question 1

    Integrals. Asked once: 2025

    22
    Evaluate : ∫0πsin⁡2pxsin⁡x  dx\displaystyle\int_0^{\pi} \dfrac{\sin 2px}{\sin x}\;dx, p∈Np \in N.

  2. Question 2

    Matrices. Asked once: 2025

    27 (b)
    A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on day I and sells 40 Chemistry, 45 Physics and 50 Maths books on day II. If the selling price for each such subject book is ₹ 150 (Chemistry), ₹ 175 (Physics) and ₹ 180 (Maths), then find his total sale in two days, using matrix method. If cost price of all the books together is ₹ 35,000, what profit did he earn after the sale of two days ?

  3. Question 3

    Relations and Functions. Asked once: 2026

    21 (b)
    Check whether f:Z×Z→Z×Zf : Z \times Z \to Z \times Z (where Z is the set of integers) defined as f(x,y)=(2y,3x)f(x, y) = (2y, 3x) is injective or not.

  4. Question 4 (case study, 4 parts)

    Vector Algebra. Asked once: 2025

    Three friends A, B and C move out from the same location O at the same time in three different directions to reach their destinations. They move out on straight paths and decide that A and B after reaching their destinations will meet up with C at his predecided destination, following straight paths from A to C and B to C in such a way that OA→=a⃗\overrightarrow{OA} = \vec{a}, OB→=b⃗\overrightarrow{OB} = \vec{b} and OC→=5a⃗−2b⃗\overrightarrow{OC} = 5\vec{a} - 2\vec{b} respectively. Based upon the above information, answer the following questions :
    36 (i)
    Complete the given figure to explain their entire movement plan along the respective vectors.
    36 (ii)
    Find vectors AC→\overrightarrow{AC} and BC→\overrightarrow{BC}.
    36 (iii) (a)
    If a⃗⋅b⃗=1\vec{a} \cdot \vec{b} = 1, distance of O to A is 1 km and that from O to B is 2 km, then find the angle between OA→\overrightarrow{OA} and OB→\overrightarrow{OB}. Also, find ∣a⃗×b⃗∣|\vec{a} \times \vec{b}|.
    36 (iii) (b)
    If a⃗=2i^−j^+4k^\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k} and b⃗=j^−k^\vec{b} = \hat{j} - \hat{k}, then find a unit vector perpendicular to (a⃗+b⃗)(\vec{a} + \vec{b}) and (a⃗−b⃗)(\vec{a} - \vec{b}).

  5. Question 5

    Application of Derivatives. Asked once: 2025

    22
    The radius of a cylinder is decreasing at a rate of 2 cm/s and the altitude is increasing at the rate of 3 cm/s. Find the rate of change of volume of this cylinder when its radius is 4 cm and altitude is 6 cm.