CUET UG Mathematics Flashcards

24 sample cards from the Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What are the order and degree of a differential equation?

   The order is the highest order derivative present; the degree is the power of the highest order derivative when the equation is a polynomial in derivatives (free of radicals/fractions in derivatives).

2. State the general method to solve a first-order linear differential equation dy/dx + Py = Q.

   Find the integrating factor I.F. = e^(∫P dx); the solution is y·(I.F.) = ∫ Q·(I.F.) dx + C.

3. How do you solve a variable separable differential equation?

   Rewrite as f(y) dy = g(x) dx so each variable is on its own side, then integrate both sides: ∫f(y) dy = ∫g(x) dx + C.

4. What is the formula for the dot (scalar) product of two vectors a and b, and what does it equal geometrically?

   a·b = |a||b|cos θ, where θ is the angle between them; it is a scalar and equals zero when the vectors are perpendicular.

5. What is the cross (vector) product a × b and its magnitude?

   a × b is a vector perpendicular to both a and b with magnitude |a||b|sin θ; its magnitude equals the area of the parallelogram formed by a and b.

6. How do you find the projection of vector a on vector b?

   Projection of a on b = (a·b)/|b|.

7. What are the direction cosines of a line, and what relation do they satisfy?

   Direction cosines l, m, n are the cosines of the angles the line makes with the x, y, z axes; they satisfy l² + m² + n² = 1.

8. Write the vector equation of a line passing through point a with direction vector b.

   r = a + λb, where λ is a scalar parameter.

9. What is the equation of a plane in normal form, and what does each term mean?

   r·n̂ = d, where n̂ is the unit normal vector to the plane and d is the perpendicular distance of the plane from the origin.

10. How is the angle between two planes found?

    It equals the angle between their normal vectors: cos θ = |n1·n2|/(|n1||n2|).

11. What is the shortest distance between two skew lines r = a1 + λb1 and r = a2 + μb2?

    Shortest distance = |(a2 - a1)·(b1 × b2)| / |b1 × b2|.

12. State the formula for conditional probability P(A|B).

    P(A|B) = P(A ∩ B)/P(B), provided P(B) ≠ 0.

13. State Bayes' Theorem.

    P(Eᵢ|A) = [P(Eᵢ)·P(A|Eᵢ)] / Σ[P(Eⱼ)·P(A|Eⱼ)], where the Eⱼ form a partition of the sample space.

14. When are two events A and B said to be independent in probability terms?

    A and B are independent if P(A ∩ B) = P(A)·P(B), equivalently P(A|B) = P(A).

15. State the multiplication theorem of probability for two events A and B.

    P(A ∩ B) = P(A)·P(B|A) = P(B)·P(A|B).

16. In linear programming, what is the feasible region and where does the optimal solution occur?

    The feasible region is the set of all points satisfying every constraint (including non-negativity). The optimal value of the objective function occurs at a corner (vertex) point of the feasible region (Corner Point Theorem).

17. What is the difference between a bounded and an unbounded feasible region in LPP regarding optimal solutions?

    A bounded feasible region always has both a maximum and a minimum at corner points. An unbounded region may have no maximum (or no minimum); existence must be verified using an open half-plane test.

18. In optimisation problems, what is the general procedure to find the absolute maximum/minimum of a continuous function on a closed interval [a,b]?

    Find critical points where f'(x)=0 or is undefined within [a,b], evaluate f at these points and at the endpoints a and b, then compare values — the largest is the absolute maximum and the smallest the absolute minimum.

19. In numerical applications, what does the LCM and HCF relationship for two numbers a and b state?

    HCF(a,b) × LCM(a,b) = a × b (the product of two numbers equals the product of their HCF and LCM).

20. What is the rule of allegation (alligation) used for in numerical applications?

    It finds the ratio in which two ingredients at different prices/concentrations must be mixed to produce a mixture of a desired mean value: (Quantity cheaper)/(Quantity dearer) = (Dearer price - Mean)/(Mean - Cheaper price).

21. What is an index number, and how is the simple aggregative price index computed?

    An index number measures the relative change in a variable (e.g. price) over time relative to a base period. Simple aggregative price index = (Σ current year prices / Σ base year prices) × 100.

22. In time-based data, what are the four components of a time series?

    Secular trend (long-term direction), seasonal variation (regular short-term periodic), cyclical variation (recurring over years), and irregular/random variation.

23. In financial mathematics, what is the formula for the future value of an ordinary annuity with payment R, rate i per period, for n periods?

    FV = R·[((1 + i)ⁿ - 1)/i], where each payment is made at the end of each period.

24. In inferential statistics, what is the difference between a population parameter and a sample statistic, and what is a null hypothesis?

    A parameter is a numerical characteristic of the whole population (e.g. μ); a statistic is computed from a sample (e.g. x̄) and estimates the parameter. The null hypothesis (H₀) is the default assumption of no effect/no difference that a test seeks to reject or fail to reject.

What this deck covers

Mathematics flashcards FAQ