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CBSE Class 12 Mathematics 2023 question paper (65/4)

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 x[12]+y[25]=[49]x\begin{bmatrix} 1 \\ 2 \end{bmatrix} + y\begin{bmatrix} 2 \\ 5 \end{bmatrix} = \begin{bmatrix} 4 \\ 9 \end{bmatrix}, then :

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  2. Q.21 mark
    The product [ab−ba][a−bba]\begin{bmatrix} a & b \\ -b & a \end{bmatrix}\begin{bmatrix} a & -b \\ b & a \end{bmatrix} is equal to :

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  3. Q.31 mark
    If A is a square matrix and A2=AA^2 = A, then (I+A)2−3A(I + A)^2 - 3A is equal to :

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  4. Q.41 mark
    If a matrix A=[123]A = [1 \quad 2 \quad 3], then the matrix AA' (where A' is the transpose of A) is :

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  5. Q.51 mark
    The value of ∣x+yy+zz+xzxy111∣\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix} is

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  6. Q.61 mark
    The function f(x)=∣x∣f(x) = |x| is

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  7. Q.71 mark
    If y=sin⁡2(x3)y = \sin^2 (x^3), then dydx\dfrac{dy}{dx} is equal to :

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  8. Q.81 mark
    ∫e5log⁡x dx\displaystyle\int e^{5 \log x}\, dx is equal to :

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  9. Q.91 mark
    If ∫0a3x2 dx=8\displaystyle\int_0^a 3x^2\, dx = 8, then the value of 'a' is :

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  10. Q.101 mark
    The integrating factor for solving the differential equation xdydx−y=2x2x\dfrac{dy}{dx} - y = 2x^2 is :

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  11. Q.111 mark
    The order and degree (if defined) of the differential equation, (d2ydx2)2+(dydx)3=xsin⁡(dydx)\left(\dfrac{d^2y}{dx^2}\right)^2 + \left(\dfrac{dy}{dx}\right)^3 = x \sin\left(\dfrac{dy}{dx}\right) respectively are :

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  12. Q.121 mark
    A unit vector along the vector 4i^−3k^4\hat{i} - 3\hat{k} is :

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  13. Q.131 mark
    If θ\theta is the angle between two vectors a⃗\vec{a} and b⃗\vec{b}, then a⃗⋅b⃗≥0\vec{a} \cdot \vec{b} \ge 0 only when :

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  14. Q.141 mark
    Distance of the point (p, q, r) from y-axis is :

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  15. Q.151 mark
    The solution set of the inequation 3x+5y<73x + 5y < 7 is :

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  16. Q.161 mark
    Which of the following points satisfies both the inequations 2x+y≤102x + y \le 10 and x+2y≥8x + 2y \ge 8 ?

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  17. Q.171 mark
    If the direction cosines of a line are (1a,1a,1a)\left(\dfrac{1}{a}, \dfrac{1}{a}, \dfrac{1}{a}\right), then :

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  18. Q.181 mark
    The probability that A speaks the truth is 45\dfrac{4}{5} and that of B speaking the truth is 34\dfrac{3}{4}. The probability that they contradict each other in stating the same fact is :

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  19. Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (a), (b), (c) and (d) as given below.
    Q.191 mark
    Assertion (A) : All trigonometric functions have their inverses over their respective domains. Reason (R) : The inverse of tan⁡−1x\tan^{-1} x exists for some x∈Rx \in \mathbb{R}.

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  20. Q.201 mark
    Assertion (A) : The lines r⃗=a⃗1+λb⃗1\vec{r} = \vec{a}_1 + \lambda \vec{b}_1 and r⃗=a⃗2+μb⃗2\vec{r} = \vec{a}_2 + \mu \vec{b}_2 are perpendicular, when b⃗1⋅b⃗2=0\vec{b}_1 \cdot \vec{b}_2 = 0. Reason (R) : The angle θ\theta between the lines r⃗=a⃗1+λb⃗1\vec{r} = \vec{a}_1 + \lambda \vec{b}_1 and r⃗=a⃗2+μb⃗2\vec{r} = \vec{a}_2 + \mu \vec{b}_2 is given by cos⁡θ=b⃗1⋅b⃗2∣b⃗1∣∣b⃗2∣\cos\theta = \dfrac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1||\vec{b}_2|}.

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Section B

2 marks each

  1. Q.21 (a)2 marks
    Find the domain of y=sin⁡−1(x2−4)y = \sin^{-1}(x^2 - 4).

  2. OR

    Q.21 (b)2 marks
    Evaluate : cos⁡−1[cos⁡(−7π3)]\cos^{-1}\left[\cos\left(-\dfrac{7\pi}{3}\right)\right]
  3. Q.222 marks
    If (x2+y2)2=xy(x^2 + y^2)^2 = xy, then find dydx\dfrac{dy}{dx}.
  4. Q.232 marks
    Find the maximum and minimum values of the function given by f(x)=5+sin⁡2xf(x) = 5 + \sin 2x.
  5. Q.242 marks
    If the projection of the vector i^+j^+k^\hat{i} + \hat{j} + \hat{k} on the vector pi^+j^−2k^p\hat{i} + \hat{j} - 2\hat{k} is 13\dfrac{1}{3}, then find the value(s) of p.
  6. Q.25 (a)2 marks
    Find the vector equation of the line passing through the point (2, 1, 3) and perpendicular to both the lines x−11=y−22=z−33\dfrac{x-1}{1} = \dfrac{y-2}{2} = \dfrac{z-3}{3}; x−3=y2=z5\dfrac{x}{-3} = \dfrac{y}{2} = \dfrac{z}{5}.

  7. OR

    Q.25 (b)2 marks
    The equations of a line are 5x−3=15y+7=3−10z5x - 3 = 15y + 7 = 3 - 10z. Write the direction cosines of the line and find the coordinates of a point through which it passes.

Section C

3 marks each

  1. Q.263 marks
    Find : ∫x2+x+1(x+1)2(x+2) dx\displaystyle\int \dfrac{x^2 + x + 1}{(x+1)^2 (x+2)}\, dx
  2. Q.27 (a)3 marks
    Evaluate : ∫π/4π/2e2x(1−sin⁡2x1−cos⁡2x)dx\displaystyle\int_{\pi/4}^{\pi/2} e^{2x}\left(\dfrac{1 - \sin 2x}{1 - \cos 2x}\right) dx

  3. OR

    Q.27 (b)3 marks
    Evaluate : ∫−22x21+5x dx\displaystyle\int_{-2}^{2} \dfrac{x^2}{1 + 5^x}\, dx
  4. Q.28 (a)3 marks
    Find : ∫ex5−4ex−e2x dx\displaystyle\int \dfrac{e^x}{\sqrt{5 - 4e^x - e^{2x}}}\, dx

  5. OR

    Q.28 (b)3 marks
    Evaluate : ∫0π/2sin⁡x cos⁡5x dx\displaystyle\int_0^{\pi/2} \sqrt{\sin x}\, \cos^5 x\, dx
  6. Q.29 (a)3 marks
    Find the particular solution of the differential equation dydx=x+yx\dfrac{dy}{dx} = \dfrac{x + y}{x}, y(1)=0y(1) = 0.

  7. OR

    Q.29 (b)3 marks
    Find the general solution of the differential equation extan⁡y dx+(1−ex)sec⁡2y dy=0e^x \tan y\, dx + (1 - e^x)\sec^2 y\, dy = 0.
  8. Q.303 marks
    Solve the following linear programming problem graphically : Minimise : z=−3x+4yz = -3x + 4y subject to the constraints x+2y≤8x + 2y \le 8, 3x+2y≤123x + 2y \le 12, x,y≥0x, y \ge 0.
  9. Q.313 marks
    From a lot of 30 bulbs which include 6 defective bulbs, a sample of 2 bulbs is drawn at random one by one with replacement. Find the probability distribution of the number of defective bulbs and hence find the mean number of defective bulbs.

Section D

5 marks each

  1. Q.325 marks
    Find the inverse of the matrix A=[1−1202−33−24]A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}. Using the inverse, A−1A^{-1}, solve the system of linear equations x−y+2z=1x - y + 2z = 1; 2y−3z=12y - 3z = 1; 3x−2y+4z=33x - 2y + 4z = 3.
  2. Q.335 marks
    Using integration, find the area of the region bounded by the parabola y2=4axy^2 = 4ax and its latus rectum.
  3. Q.34 (a)5 marks
    If N denotes the set of all natural numbers and R is the relation on N×NN \times N defined by (a, b) R (c, d), if ad(b+c)=bc(a+d)ad(b + c) = bc(a + d). Show that R is an equivalence relation.

  4. OR

    Q.34 (b)5 marks
    Let f:R−{−43}→Rf : \mathbb{R} - \left\{-\dfrac{4}{3}\right\} \to \mathbb{R} be a function defined as f(x)=4x3x+4f(x) = \dfrac{4x}{3x + 4}. Show that f is a one-one function. Also, check whether f is an onto function or not.
  5. Q.35 (a)5 marks
    Show that the following lines do not intersect each other : x−13=y+12=z−15\dfrac{x-1}{3} = \dfrac{y+1}{2} = \dfrac{z-1}{5}; x+24=y−13=z+1−2\dfrac{x+2}{4} = \dfrac{y-1}{3} = \dfrac{z+1}{-2}

  6. OR

    Q.35 (b)5 marks
    Find the angle between the lines 2x=3y=−z2x = 3y = -z and 6x=−y=−4z6x = -y = -4z.

Section E

  1. Let f(x) be a real valued function. Then its Left Hand Derivative (L.H.D.) : Lf′(a)=lim⁡h→0f(a−h)−f(a)−hLf'(a) = \lim_{h \to 0} \dfrac{f(a-h) - f(a)}{-h} Right Hand Derivative (R.H.D.) : Rf′(a)=lim⁡h→0f(a+h)−f(a)hRf'(a) = \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h} Also, a function f(x) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and both are equal. For the function f(x)={∣x−3∣,x≥1x24−3x2+134,x<1f(x) = \begin{cases} |x - 3|, & x \ge 1 \\ \dfrac{x^2}{4} - \dfrac{3x}{2} + \dfrac{13}{4}, & x < 1 \end{cases} answer the following questions :
    Q.36 (i)1 mark
    What is R.H.D. of f(x) at x = 1 ?
  2. Q.36 (ii)1 mark
    What is L.H.D. of f(x) at x = 1 ?
  3. Q.36 (iii) (a)2 marks
    Check if the function f(x) is differentiable at x = 1.

  4. OR

    Q.36 (iii) (b)2 marks
    Find f′(2)f'(2) and f′(−1)f'(-1).
  5. A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is 0⋅650{\cdot}65. The probability that many are not present and still the work gets completed on time is 0⋅350{\cdot}35. The probability that work will be completed on time when all workers are present is 0⋅800{\cdot}80. Let : E1E_1 : represent the event when many workers were not present for the job; E2E_2 : represent the event when all workers were present; and E : represent completing the construction work on time. Based on the above information, answer the following questions :
    Q.37 (i)1 mark
    What is the probability that all the workers are present for the job ?
  6. Q.37 (ii)1 mark
    What is the probability that construction will be completed on time ?
  7. Q.37 (iii) (a)2 marks
    What is the probability that many workers are not present given that the construction work is completed on time ?

  8. OR

    Q.37 (iii) (b)2 marks
    What is the probability that all workers were present given that the construction job was completed on time ?
  9. Sooraj's father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing for the other three sides as shown in the figure. He has 200 metres of fencing wire. Based on the above information, answer the following questions :
    Q.38 (i)2 marks
    Let 'x' metres denote the length of the side of the garden perpendicular to the brick wall and 'y' metres denote the length of the side parallel to the brick wall. Determine the relation representing the total length of fencing wire and also write A(x), the area of the garden.
  10. Q.38 (ii)2 marks
    Determine the maximum value of A(x).