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CBSE Class 12 Mathematics 2023 question paper (65/2)

Maximum marks 80 · Time 3 hours · 3 sets

Try each question first, then open its model answer. Where the paper offers a choice, both questions are shown with OR between them.

Section A

1 mark each

  1. Q.11 mark
    If A=[0100]A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, then A2023A^{2023} is equal to

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  2. Q.21 mark
    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. Q.31 mark
    If [1212313a1]\begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & a & 1 \end{bmatrix} is non-singular matrix and a∈Aa \in A, then the set A is

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  4. Q.41 mark
    If ∣A∣=∣kA∣|A| = |kA|, where A is a square matrix of order 2, then sum of all possible values of k is

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  5. Q.51 mark
    If ddx[f(x)]=ax+b\frac{d}{dx}[f(x)] = ax + b and f(0)=0f(0) = 0, then f(x)f(x) is equal to

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  6. Q.61 mark
    Degree of the differential equation sin⁡x+cos⁡(dydx)=y2\sin x + \cos\left(\frac{dy}{dx}\right) = y^2 is

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  7. Q.71 mark
    The integrating factor of the differential equation (1−y2)dxdy+yx=ay(1 - y^2)\frac{dx}{dy} + yx = ay, (−1<y<1)(-1 < y < 1) is

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  8. Q.81 mark
    Unit vector along PQ→\overrightarrow{PQ}, where coordinates of P and Q respectively are (2, 1, −1-1) and (4, 4, −7-7), is

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  9. Q.91 mark
    Position vector of the mid-point of line segment AB is 3i^+2j^−3k^3\hat{i} + 2\hat{j} - 3\hat{k}. If position vector of the point A is 2i^+3j^−4k^2\hat{i} + 3\hat{j} - 4\hat{k}, then position vector of the point B is

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  10. Q.101 mark
    Projection of vector 2i^+3j^2\hat{i} + 3\hat{j} on the vector 3i^−2j^3\hat{i} - 2\hat{j} is

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  11. Q.111 mark
    Equation of a line passing through point (1, 1, 1) and parallel to z-axis is

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  12. Q.121 mark
    If the sum of numbers obtained on throwing a pair of dice is 9, then the probability that number obtained on one of the dice is 4, is :

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  13. Q.131 mark
    Anti-derivative of tan⁡x−1tan⁡x+1\frac{\tan x - 1}{\tan x + 1} with respect to xx is

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  14. Q.141 mark
    If (a, b), (c, d) and (e, f) are the vertices of ΔABC\Delta ABC and Δ\Delta denotes the area of ΔABC\Delta ABC, then ∣acebdf111∣2\begin{vmatrix} a & c & e \\ b & d & f \\ 1 & 1 & 1 \end{vmatrix}^2 is equal to

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  15. Q.151 mark
    The function f(x)=x∣x∣f(x) = x|x| is

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  16. Q.161 mark
    If tan⁡(x+yx−y)=k\tan\left(\frac{x + y}{x - y}\right) = k, then dydx\frac{dy}{dx} is equal to

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  17. Q.171 mark
    The objective function Z=ax+byZ = ax + by of an LPP has maximum value 42 at (4, 6) and minimum value 19 at (3, 2). Which of the following is true ?

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  18. Q.181 mark
    The corner points of the feasible region of a linear programming problem are (0, 4), (8, 0) and (203,43)\left(\frac{20}{3}, \frac{4}{3}\right). If Z=30x+24yZ = 30x + 24y is the objective function, then (maximum value of Z −- minimum value of Z) is equal to

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  19. In the following questions 19 & 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices :
    Q.191 mark
    Assertion (A) : Maximum value of (cos⁡−1x)2(\cos^{-1} x)^2 is π2\pi^2. Reason (R) : Range of the principal value branch of cos⁡−1x\cos^{-1} x is [−π2,π2]\left[\frac{-\pi}{2}, \frac{\pi}{2}\right].

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  20. Q.201 mark
    Assertion (A) : If a line makes angles α,β,γ\alpha, \beta, \gamma with positive direction of the coordinate axes, then sin⁡2α+sin⁡2β+sin⁡2γ=2\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2. Reason (R) : The sum of squares of the direction cosines of a line is 1.

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Section B

2 marks each

  1. Q.21 (a)2 marks
    Evaluate sin⁡−1(sin⁡3π4)+cos⁡−1(cos⁡π)+tan⁡−1(1)\sin^{-1}\left(\sin \frac{3\pi}{4}\right) + \cos^{-1}(\cos \pi) + \tan^{-1}(1).

  2. OR

    Q.21 (b)2 marks
    Draw the graph of cos⁡−1x\cos^{-1} x, where x∈[−1,0]x \in [-1, 0]. Also, write its range.
  3. Q.222 marks
    A particle moves along the curve 3y=ax3+13y = ax^3 + 1 such that at a point with xx-coordinate 1, y-coordinate is changing twice as fast at xx-coordinate. Find the value of a.
  4. Q.232 marks
    If a⃗\vec{a}, b⃗\vec{b}, c⃗\vec{c} are three non-zero unequal vectors such that a⃗⋅b⃗=a⃗⋅c⃗\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}, then find the angle between a⃗\vec{a} and b⃗−c⃗\vec{b} - \vec{c}.
  5. Q.242 marks
    Find the coordinates of points on line x1=y−12=z+12\frac{x}{1} = \frac{y - 1}{2} = \frac{z + 1}{2} which are at a distance of 11\sqrt{11} units from origin.
  6. Q.25 (a)2 marks
    If y=ax+by = \sqrt{ax + b}, prove that y(d2ydx2)+(dydx)2=0y\left(\frac{d^2y}{dx^2}\right) + \left(\frac{dy}{dx}\right)^2 = 0.

  7. OR

    Q.25 (b)2 marks
    If f(x)={ax+b;0<x≤12x2−x;1<x<2f(x) = \begin{cases} ax + b & ; \quad 0 < x \le 1 \\ 2x^2 - x & ; \quad 1 < x < 2 \end{cases} is a differentiable function in (0, 2), then find the values of a and b.

Section C

3 marks each

  1. Q.26 (a)3 marks
    Evaluate ∫0π/4log⁡(1+tan⁡x) dx\int\limits_{0}^{\pi/4} \log (1 + \tan x) \, dx.

  2. OR

    Q.26 (b)3 marks
    Find ∫dxsin⁡3xcos⁡(x−α)\int \frac{dx}{\sqrt{\sin^3 x \cos (x - \alpha)}}.
  3. Q.273 marks
    Find ∫ecot⁡−1x(1−x+x21+x2)dx\int e^{\cot^{-1} x}\left(\frac{1 - x + x^2}{1 + x^2}\right) dx.
  4. Q.283 marks
    Evaluate ∫log⁡2log⁡31(ex+e−x)(ex−e−x) dx\int\limits_{\log \sqrt{2}}^{\log \sqrt{3}} \frac{1}{(e^x + e^{-x})(e^x - e^{-x})} \, dx
  5. Q.29 (a)3 marks
    Find the general solution of the differential equation : (xy−x2) dy=y2 dx(xy - x^2) \, dy = y^2 \, dx.

  6. OR

    Q.29 (b)3 marks
    Find the general solution of the differential equation : (x2+1)dydx+2xy=x2+4(x^2 + 1)\frac{dy}{dx} + 2xy = \sqrt{x^2 + 4}
  7. Q.30 (a)3 marks
    Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of the random variable.

  8. OR

    Q.30 (b)3 marks
    A and B throw a die alternately till one of them gets a '6' and wins the game. Find their respective probabilities of wining, if A starts the game first.
  9. Q.313 marks
    Solve the following linear programming problem graphically : Minimize : Z=5x+10yZ = 5x + 10y subject to constraints : x+2y≤120x + 2y \le 120, x+y≥60x + y \ge 60, x−2y≥0x - 2y \ge 0, x≥0x \ge 0, y≥0y \ge 0

Section D

5 marks each

  1. Q.32 (a)5 marks
    If A=[−3−2−4212213]A = \begin{bmatrix} -3 & -2 & -4 \\ 2 & 1 & 2 \\ 2 & 1 & 3 \end{bmatrix}, B=[120−2−1−20−11]B = \begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix}, then find AB and use it to solve the following system of equations : x−2y=3x - 2y = 3 2x−y−z=22x - y - z = 2 −2y+z=3-2y + z = 3

  2. OR

    Q.32 (b)5 marks
    If f(α)=[cos⁡α−sin⁡α0sin⁡αcos⁡α0001]f(\alpha) = \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}, prove that f(α)⋅f(−β)=f(α−β)f(\alpha) \cdot f(-\beta) = f(\alpha - \beta)
  3. Q.33 (a)5 marks
    Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4, 2, −6-6), Q(5, −3-3, 1), R(12, 4, 5) and S(11, 9, −2-2). Use these equations to find the point of intersection of diagonals.

  4. OR

    Q.33 (b)5 marks
    A line ll passes through point (−1-1, 3, −2-2) and is perpendicular to both the lines x1=y2=z3\frac{x}{1} = \frac{y}{2} = \frac{z}{3} and x+2−3=y−12=z+15\frac{x + 2}{-3} = \frac{y - 1}{2} = \frac{z + 1}{5}. Find the vector equation of the line ll. Hence, obtain its distance from origin.
  5. Q.345 marks
    Using integration, find the area of region bounded by line y=3xy = \sqrt{3}x, the curve y=4−x2y = \sqrt{4 - x^2} and y-axis in first quadrant.
  6. Q.355 marks
    A function f:[−4,4]→[0,4]f : [-4, 4] \to [0, 4] is given by f(x)=16−x2f(x) = \sqrt{16 - x^2}. Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a)=7f(a) = \sqrt{7}.

Section E

  1. Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area 75π75\pi cm2^2. Based on the above information, answer the following questions :
    Q.36 (i)1 mark
    If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r.
  2. Q.36 (ii)1 mark
    Find dVdr\frac{dV}{dr}.
  3. Q.36 (iii)(a)2 marks
    Find the radius of cylinder when its volume is maximum.

  4. OR

    Q.36 (iii)(b)2 marks
    For maximum volume, h > r. State true or false and justify.
  5. Recent studies suggest that roughly 12% of the world population is left handed. Depending upon the parents, the chances of having a left handed child are as follows : A : When both father and mother are left handed : Chances of left handed child is 24%. B : When father is right handed and mother is left handed : Chances of left handed child is 22%. C : When father is left handed and mother is right handed : Chances of left handed child is 17%. D : When both father and mother are right handed : Chances of left handed child is 9%. Assuming that P(A)=P(B)=P(C)=P(D)=14P(A) = P(B) = P(C) = P(D) = \frac{1}{4} and L denotes the event that child is left handed. Based on the above information, answer the following questions :
    Q.37 (i)1 mark
    Find P(L/C)P(L/C)
  6. Q.37 (ii)1 mark
    Find P(L‾/A)P(\overline{L}/A)
  7. Q.37 (iii)(a)2 marks
    Find P(A/L)P(A/L)

  8. OR

    Q.37 (iii)(b)2 marks
    Find the probability that a randomly selected child is left handed given that exactly one of the parents is left handed.
  9. The use of electric vehicles will curb air pollution in the long run. The use of electric vehicles is increasing every year and estimated electric vehicles in use at any time t is given by the function V : V(t)=15t3−52t2+25t−2V(t) = \frac{1}{5}t^3 - \frac{5}{2}t^2 + 25t - 2 where t represents the time and t = 1, 2, 3.... corresponds to year 2001, 2002, 2003, ....... respectively. Based on the above information, answer the following questions :
    Q.38 (i)2 marks
    Can the above function be used to estimate number of vehicles in the year 2000 ? Justify.
  10. Q.38 (ii)2 marks
    Prove that the function V(t) is an increasing function.