Time: 3 Hrs.
Max. Marks: 80
General instructions:
- Section A:
Q. 1 contains Eight multiple choice type ofquestions, each carrying Two marks.
Q. 2 contains Four very short answer type questions, each carrying One mark. - Section B:
Contains Twelve short answer type questions, each carryingTwo marks. (Attempt any Eight) - Section C:
Contains Twelve short answer type questions, each carrying Three marks. (Attempt any Eight) - Section D:
Contain Eight long answer type questions, each carrying Four marks.(Attempt any Five). - Use of log table is allowed. Use of calculator is not allowed.
- Figures to the right indicate full marks.
- For each multiple choice type of question, it is mandatory to write the correct answer along with its alphabet, e.g. (a)……/ (b)……./ (c)………etc. No marks shall be given, if ONLY the correct answer or the alphabet of correct answer is written. Only the first attempt will be considered for evaluation.
- Start answer to each section on a new page.
Section-A
Q.1. Select and write the correct answer for the following multiple choice type of questions: [16 Marks] [2M each]
i. If A = {1, 2, 3, 4, 5} then which of the following is not true?
(a) \exists\;x\;\in A such that x + 3= 8
(b) \exists\;x\;\in A such that x+2<9
(c) \forall\;x\;\in\;A,\;x\;+6\;\geq9
(d) \exists\;x\;\in\;A such that x + 6 < 10
ii. In \triangle ABC, \left(a+b\right)\cdot\cos\;C+\left(b+c\right)\cdot\cos\;A\;+\left(c+a\right)\cdot\cos\;B\; is equal to _______.
(a) a – b + c
(b) a + b – c
(c) a + b + c
(d) a – b – c
iii. If \left|\overline a\right|=5,\;\left|\overline b\right|=13 and \left|\overline a\times\overline b\right|=25 then \left|\overline a\cdot\overline b\right| is equal to _______.
(a) 30
(b) 60
(c) 40
(d) 45
iv. The vector equation of the line passing through the point having position vector 4\widehat i-\widehat j+2\widehat k and parallel to vector -2\widehat i-\widehat j+\widehat k is given by______.
(a) \left(4\widehat i-\widehat j-2\widehat k\right)+\lambda\left(-2\widehat i-\widehat j+\widehat k\right)
(b) \left(4\widehat i-\widehat j+2\widehat k\right)+\lambda\left(2\widehat i-\widehat j+\widehat k\right)
(c) \left(4\widehat i-\widehat j+2\widehat k\right)+\lambda\left(-2\widehat i-\widehat j-\widehat k\right)
(d) \left(4\widehat i-\widehat j+2\widehat k\right)+\lambda\left(-2\widehat i-\widehat j+\widehat k\right)
v. Let f(1) = 3, f'\left(1\right)=-\frac13,g\left(1\right)=-4 and g'\left(1\right)=-\frac83. The derivation of \sqrt{\left[f\left(x\right)\right]^2+\left[g\left(x\right)\right]^2} w. r. t. x at x = 1 is ______.
(a) -\frac{29}{25}
(b) \frac73
(c) \frac{31}{15}
(d) \frac{29}{15}
vi. If the mean and variance of a binomial distribution are 18 and 12 respectively, then n is equal to ___.
(a)36
(b) 54
(c) 18
(d) 27
vii. The value of \int x^x\left(1+\log\;x\right) dx is equal to ______.
(a) \frac12\left(1+\log\;x\right)^2+c
(b) x^{2x}+c
(c) x^x\cdot\log\;x+c
(d) x^x+c
viii. The area bounded by the line y = x, X-axis and the lines x = -1 and x = 4 is equal to _____.
(in square units)
(a) \frac2{17}
(b) 8
(c) \frac{17}2
(d) \frac12
Q.2. Answer the following questions: [4 Marks] [1M each]
i. Write the negation of the statement: ‘\exists\;n\in N such that n+8>11‘
ii. Write unit vector in the opposite direction to \overline u=8\widehat i+3\widehat j-\widehat k
iii. Write the order of the differential equation \sqrt{1+\left(\frac{dy}{dx}\right)^2}=\left(\frac{d^2y}{dx^2}\right)^\frac32
iv. Write the condition for the function f(x), to be strictly increasing, for all x ∈ R.
Section-B
Attempt any EIGHT of the following questions: [16 Marks] [2M each]
Q.3. Using truth table, prove that the statement patterns p ⟷ q and (p ∧ q) ∨ (∼p ∧ ∼q) are logically equivalent.
Q.4. Find the adjoint of the matrix \begin{bmatrix}2&-2\\4&3\end{bmatrix}
Q.5. Find the general solution of \tan^2\theta=1.
Q.6. Find the co-ordinates of the points of intersection of the lines represented by x^2-y^2-2x+1=0.
Q.7. A line makes angles of measure 45o and 60o with the positive directions of the Y and Z axes respectively. Find the angle made by the line with the positive direction of the X-axis.
Q.8. Find the vector equation of the plane passing through the point having position vector 2\widehat i+3\widehat j+4\widehat k and perpendicular to the vector 2\widehat i+\widehat j-2\widehat k.
Q.9. Divide the number 20 into two parts such that sum of their squares is minimum.
Q.10. Evaluate: \int x^9\cdot sec^2\left(x^{10}\right)dx
Q.11. Evaluate: \int\frac1{25-9x^2}dx
Q.12. Evaluate: \int\limits_{-\frac\pi4}^\frac\pi4\frac1{1-\sin x\;}dx
Q.13. Find the area of the region bounded by the parabola Y2 = 16x and its latus rectum.
Q.14. Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by:
\begin{array}{cc}f\left(x\right)=\frac15,&for\;0\leq x\leq5\=0,&otherwise\end{array}Find the probability that:
i. waiting time is between I to 3 minutes.
ii. waiting time is more than 4 minutes.
Section-C
Attempt any EIGHT of the following questions: [24 Marks] [3M each]
Q.15. Express the following switching circuit in the symbolic form of logic. Construct the switching table
and interpret it:
Q.16. Prove that: 2\;\tan^{-1}\left(\frac13\right)+\cos^{-1}\left(\frac35\right)=\frac\pi2\;
Q.17. In ∆ABC if a = 13, b = 14, c = 15 then find the values of (i) sec A (ii) \cos ec\frac A2
Q.18. A line passes through the points (6, —7, —1) and (2, —3, 1). Find the direction ratios and the direction
cosines of the line. Show that the line does not pass through the origin.
Q.19. Find the cartesian and vector equations of the line passing through A(1, 2, 3) and having direction ratios 2, 3, 7.
Q.20. Find the vector equation of the plane passing through points A(1, 1, 2), B(O, 2, 3) and C(4, 5, 6).
Q.21. Find the nth order derivative of log x.
Q.22. The displacement of a particle at time t is given by s = 2t3 – 5t2 + 4t – 3. Find the velocity and displacement at the time when the acceleration is 14 ft/sec2.
Q.23. Find the equations of tangent and normal to the curve y = 2x3 – x2 + 2 at point \left(\frac12,2\right).
Q.24. Three coins are tossed simultaneously, X is the number of heads. Find the expected value and
variance of X.
Q.25. Solve the differential equation: x\frac{dy}{dx}=x\cdot\tan\left(\frac yx\right)+y.
Q.26. Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the
probability that:
i. all the five cards are spades.
ii. none is spade.
Section-D
Attempt any FIVE of the following questions: [20 Marks] [4M each]
Q.27. Find the inverse of
\begin{bmatrix}\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\0&0&1\end{bmatrix}by elementary row transformations.
Q.28. Prove that homogeneous equation of degree two in x and y, ax2 + 2hxy + by2 = 0 represents a pair of lines passing through the origin if h^2-ab\geq0. Hence show that equation x2 + y2 = 0 does not represent a pair of lines.
Q.29. Let \overline a and \overline b be non-collinear vectors. If vector \overline r is coplanar with \overline a and \overline b then show that there exist unique scalars t1 and t2 such that \overline r=t_1\overline a+t_2\overline b. For \overline r=2\widehat i+7\widehat j+9\widehat k, \overline a=\widehat i+2\widehat j, \overline b=\widehat j+3\widehat k, find t1 and t2.
Q.30. Solve the linear programming problem graphically.
Maximize: z = 3x + 5y
Subject to: x +4y ≤ 24,
3x + y ≤ 21,
x + y ≤ 9,
x ≥ 0, y ≥ 0
Also find the maximum value of z.
Q.31. If x = f(t) and y g(t) are differentiable functions oft so that y is a function of x and if \frac{dx}{dt}\neq0.
then prove that \frac{dy}{dx}=\frac{\displaystyle\frac{dy}{dt}}{\displaystyle\frac{dx}{dt}}.
Hence find the derivative of 7x w. r. t. x7.
Q.32. Evaluate: \int e^{\sin-1_x}\left(\frac{x+\sqrt{1-x^2}}{\sqrt{1-x^2}}\right)dx.
Q.33. Prove that: \int\limits_a^bf\left(x\right)dx=\int\limits_a^bf\left(a+b-x\right)dx
Hence evaluate: \int\limits_0^3\frac{\sqrt x}{\sqrt x+\sqrt{3-x}}dx.
Q.34. If a body cools from 80oC to 50oC at room temperature of 250C in 30 minutes, find the temperature
of the body after 1 hour.
