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# Board Paper of Class 12-Science Term-I 2021 Math Delhi(Set 4) - Solutions

General Instructions:
(i) This question paper comprises 50 questions out of which 40 questions are to be attempted as per instructions. All questions carry equal marks.
(ii) The question paper consists three Sections – Section A, B and C.
(iii) Section – A contains 20 questions. Attempt any 16 questions from Q.No. 1 to 20.
(iv) Section – B also contains 20 questions. Attempt any 16 questions from Q.No. 21 to 40.
(v) Section – C contains 10 questions including one Case Study. Attempt any 8 from Q.No. 41 to 50.
(vi) There is only one correct option for every Multiple Choice Question (MCQ). Marks will not be awarded for answering more than one option.
(vii) There is no negative marking.

• Question 1
Differential of  w.r.t x is
(a)

(b)

(c)

(d)  VIEW SOLUTION

• Question 2
The number of all possible matrices of order 2 × 3 with each entry 1 or 2 is
(a) 16
(b) 6
(c) 64
(d) 24 VIEW SOLUTION

• Question 3
A function f : R$\to$R is defined as f(x) = ${x}^{3}+1$. Then the function has
(a) no minimum value
(b) no maximum value
(c) both maximum and minimum values
(d) neither maximum value nor minimum value VIEW SOLUTION

• Question 4
If sin y = x cos (a + y), then $\frac{dx}{dy}$ is
(a)

(b)

(c)

(d)  VIEW SOLUTION

• Question 5
The points on the curve $\frac{{x}^{2}}{9}+\frac{{y}^{2}}{25}=1,$ where tangent is parallel to x-axis are
(a)
(b)
(c)
(d)  VIEW SOLUTION

• Question 6
Three points P(2x, x + 3), Q(0, x) and R(x + 3, x + 6) are collinear, then x is equal to
(a) 0
(b) 2
(c) 3
(d) 1 VIEW SOLUTION

• Question 7
The principal value of ${\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)+{\mathrm{sin}}^{-1}\left(-\frac{1}{\sqrt{2}}\right)$ is

(a) $\frac{\mathrm{\pi }}{12}$

(b) $\frac{\mathrm{\pi }}{3}$

(c) $\mathrm{\pi }$

(d) $\frac{\mathrm{\pi }}{6}$ VIEW SOLUTION

• Question 8
If (x2 + y2)2 = xy, then $\frac{dy}{dx}$ is

(a) $\frac{y+4x\left({x}^{2}+{y}^{2}\right)}{4y\left({x}^{2}+{y}^{2}\right)-x}$

(b) $\frac{y-4x\left({x}^{2}+{y}^{2}\right)}{x+4\left({x}^{2}+{y}^{2}\right)}$

(c) $\frac{y-4x\left({x}^{2}+{y}^{2}\right)}{4y\left({x}^{2}+{y}^{2}\right)-x}$

(d) $\frac{4y\left({x}^{2}+{y}^{2}\right)-x}{y-4x\left({x}^{2}+{y}^{2}\right)}$ VIEW SOLUTION

• Question 9
If a matrix A is both symmetric and skew symmetric, then A is necessarily a
(a) Diagonal matrix
(b) Zero square matrix
(c) Square matrix
(d) Identity matrix VIEW SOLUTION

• Question 10
Let set X = {1, 2, 3} and a relation R is defined in X as : R = {(1, 3), (2, 2), (3, 2)}, then minimum ordered pairs which should be added in relation R to make it reflexive and symmetric are
(a) {(1, 1), (2, 3), (1, 2)}
(b) {(3, 3), (3, 1), (1, 2)}
(c) {(1, 1), (3, 3), (3, 1), (2, 3)}
(d) {(1, 1), (3, 3), (3, 1), (1, 2)}
VIEW SOLUTION

• Question 11
A Linear Programming Problem is as follows :
Minimise                               z = 2x + y
subject to the constraints       x ≥ 3, x ≤ 9, y ≥ 0
xy ≥ 0, x + y ≤ 14
The feasible region has
(a) 5 corner points including (0, 0) and (9, 5)
(b) 5 corner points including (7, 7) and (3, 3)
(c) 5 corner points including (14, 0) and (9, 0)
(d) 5 corner points including (3, 6) and (9, 5)  VIEW SOLUTION

• Question 12
The function is continuous at = 0 for the value of k, as
(a) 3
(b) 5
(c) 2
(d) 8 VIEW SOLUTION

• Question 13
If Cij denotes the cofactor of element pij of the matrix $\mathrm{P}=\left[\begin{array}{ccc}1& -1& 2\\ 0& 2& -3\\ 3& 2& 4\end{array}\right]$, then the value of C31 ∙ C23 is
(a) 5
(b) 24
(c) –24
(d) –5 VIEW SOLUTION

• Question 14
The function y = x2eis decreasing in the interval
(a) (0, 2)
(b) (2, ∞)
(c) (–∞, 0)
(d) (–∞, 0) ∪ (2, ∞) VIEW SOLUTION

• Question 15
If R = {(x, y); x, y ∊ Z, x+ y2 ≤ 4} is a relation in set Z, then domain of R is
(a) {0, 1, 2}
(b) {–2, –1, 0, 1, 2}
(c) {0, –1, –2}
(d) {–1, 0, 1} VIEW SOLUTION

• Question 16
The system of linear equations
5x + ky = 5
3x + 3y = 5
will be consistent if
(a) k $\ne$ –3
(b) k = –5
(c) k = 5
(d) k$\ne$ 5  VIEW SOLUTION

• Question 17
The equation of the tangent to the curve y (1 + x2) = 2 – x, where it crosses the x-axis is
(a) x – 5y = 2
(b) 5x – y = 2
(c) + 5y = 2
(d) 5+ = 2  VIEW SOLUTION

• Question 18
If $\left[\begin{array}{cc}3c+6& a-d\\ a+d& 2-3b\end{array}\right]=\left[\begin{array}{cc}12& 2\\ -8& -4\end{array}\right]$ are equal, then value of abcd is
(a) 4
(b) 16
(c) –4
(d) –16 VIEW SOLUTION

• Question 19
The principal value of ${\mathrm{tan}}^{-1}\left(\mathrm{tan}\frac{9\mathrm{\pi }}{8}\right)$ is
(a) $\frac{\mathrm{\pi }}{8}$

(b) $\frac{3\mathrm{\pi }}{8}$

(c) $-\frac{\mathrm{\pi }}{8}$

(d) $-\frac{3\mathrm{\pi }}{8}$
VIEW SOLUTION

• Question 20
For two matrices P = $\left[\begin{array}{cc}3& 4\\ -1& 2\\ 0& 1\end{array}\right]$ and QT$\left[\begin{array}{ccc}-1& 2& 1\\ 1& 2& 3\end{array}\right]$ P – Q is
(a) $\left[\begin{array}{cc}2& 3\\ -3& 0\\ 0& -3\end{array}\right]$

(b) $\left[\begin{array}{cc}4& 3\\ -3& 0\\ -1& -2\end{array}\right]$

(c) $\left[\begin{array}{cc}4& 3\\ 0& -3\\ -1& -2\end{array}\right]$

(d) $\left[\begin{array}{cc}2& 3\\ 0& -3\\ 0& -3\end{array}\right]$ VIEW SOLUTION

• Question 21
The function f(x) = 2x3 – 15x2 + 36x + 6 is increasing in the interval
(a) (–∞, 2) ∪ (3, ∞)
(b) (–∞, 2)
(c) (–∞, 2] ∪ [3, ∞)
(d) [3, ∞) VIEW SOLUTION

• Question 22
If x = 2 cosθ – cos 2θ and y = 2 sinθ – 2θ, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is

(a)

(b)

(c)

(d) VIEW SOLUTION

• Question 23
What is the domain of the function cos–1 (2x – 3)?
(a) [–1, 1]
(b) (1, 2)
(c) (–1, 1)
(d) [1, 2] VIEW SOLUTION

• Question 24
A matrix A = [aij]3 × 3 is defined by

${a}_{\mathit{i}\mathit{j}}=\left\{\begin{array}{ll}2i+3j,& ij\end{array}\right\$

The number of elements in A which are more than 5, is
(a) 3
(b) 4
(c) 5
(d) 6 VIEW SOLUTION

• Question 25
If a function f defined by

is continuous at $x=\frac{\mathrm{\pi }}{2},$ then the value of k is
(a) 2
(b) 3
(c) 6
(d) –6 VIEW SOLUTION

• Question 26
For the matrix $\mathrm{X}=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right]$, (X2 – X) is
(a) 2 I
(b) 3 I
(c) I
(a) 5 I VIEW SOLUTION

• Question 27
Let X = {x2 : xN} and the function f : N → X is defined by f(x) = x2, xN. Then this function is
(a) injective only
(b) not bijective
(c) surjective only
(d) bijective VIEW SOLUTION

• Question 28
The corner points of the feasible region for a Linear Programming problem are P(0, 5), Q(1, 5), R(4, 2) and S(12, 0). The minimum value of the objective function Z = 2x + 5y is at the point
(a) P
(b) Q
(c) R
(d) S VIEW SOLUTION

• Question 29

The equation of the normal to the curve ay2 = x3 at the point (am2, am3) is
(a) 2y – 3mx + am3 = 0
(b) 2x + 3my – 3am4am2 = 0
(c) 2x + 3my + 3am4 – 2am2 = 0
(d) 2x + 3my – 3am4 – 2am2 = 0

VIEW SOLUTION

• Question 30
If A is a square matrix of order 3 and |A| = –5, then |adj A| is
(a) 125
(b) –25
(c) 25
(d) ±25 VIEW SOLUTION

• Question 31
The simplest form of  tan–1 $\left[\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right]$ is
(a) $\frac{\mathrm{\pi }}{4}-\frac{x}{2}$

(b) $\frac{\mathrm{\pi }}{4}+\frac{x}{2}$

(c) $\frac{\mathrm{\pi }}{4}-\frac{1}{2}{\mathrm{cos}}^{-1}x$

(d) ​$\frac{\mathrm{\pi }}{4}+\frac{1}{2}{\mathrm{cos}}^{-1}x$ VIEW SOLUTION

• Question 32
If for the matric A =$\left[\begin{array}{cc}\mathrm{\alpha }& -2\\ -2& \mathrm{\alpha }\end{array}\right]$$\left|{\mathrm{A}}^{3}\right|=125,$ then the value of $\mathrm{\alpha }$ is
(a) $±$ 3
(b) –3
(c) $±$ 1
(d) 1 VIEW SOLUTION

• Question 33
If y = sin(m sin–1 x), then which one of the following equations is true?
(a) $\left(1-{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}+{m}^{2}y=0$

(b) $\left(1-{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}+{m}^{2}y=0$

(c) $\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}-{m}^{2}y=0$

(d) ​$\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}-{m}^{2}x=0$ VIEW SOLUTION

• Question 34
The principal value of $\left[{\mathrm{tan}}^{-1}\sqrt{3}-{\mathrm{cot}}^{-1}\left(-\sqrt{3}\right)\right]$ is
(a) $\mathrm{\pi }$
(b) $-\frac{\mathrm{\pi }}{2}$
(c) 0
(d) $2\sqrt{3}$ VIEW SOLUTION

• Question 35
The maximum value of ${\left(\frac{1}{x}\right)}^{x}$ is
(a) ${e}^{\frac{1}{e}}$

(b) e

(c) ${\left(\frac{1}{e}\right)}^{\frac{1}{e}}$

(d) ee VIEW SOLUTION

• Question 36
Let matrix X = [xij] is given by . Then the matrix Y = [mij], where mij = Minor of xij, is
(a)
(b)
(c)
(d)  VIEW SOLUTION

• Question 37
A function f: RR defined by f(x) = 2 + x2 is
(a) not one-one
(b) one-one
(c) not onto
(d) neither one-one nor onto VIEW SOLUTION

• Question 38
A Linear Programming Problem is as follows:
Maximise / Minimise objective function Z = 2x – y +5
Subject to the constraints
3x + 4y ≤ 60
x + 3y ≤ 30
x ≥ 0, ≥ 0
If the corner points of the feasible region are A (0, 10), B(12, 6), C(20, 0) and O(0, 0), then which of the following is true?
(a) Maximum value of Z is 40
(b) Minimum value of Z is – 5
(c) Difference of maximum and minimum values of Z is 35
(d) At two corner points, value of Z are equal VIEW SOLUTION

• Question 39
If x = –4 is a root of  then the sum of the other two roots is
(a) 4
(b) –3
(c) 2
(d) 5 VIEW SOLUTION

• Question 40
The absolute maximum value of the function $\mathrm{f}\left(x\right)=4x–\frac{1}{2}{x}^{2}$ in the interval $\left[–2,\frac{9}{2}\right]$ is
(a) 8
(b) 9
(c) 6
(d) 10 VIEW SOLUTION

• Question 41
In a sphere of radius r, a right circular cone of height h having maximum curved surface area is inscribed. The expression for the square of curved surface of cone is
(a) 2π2rh (2rh + h2)
(b) π2hr (2rh + h2)
(c) 2π2r(2rh2h3)
(d) 2π2r2 (2rhh2) VIEW SOLUTION

• Question 42
The corner points of the feasible region determined by a set of constrains (linear inequalities) are P(0, 5), Q(3, 5), R(5, 0) and S(4, 1) and the
objective function is Z = ax + 2by where a, b > 0. The condition on a and b such that the maximum Z occurs at Q and S is
(a) a – 5b = 0
(b) a – 3b = 0
(c) a – 2b = 0
(d) a – 8b = 0 VIEW SOLUTION

• Question 43
If curves y2 = 4x and xy = c cut at right angles, then the value of c is
(a) $4\sqrt{2}$
(b) 8
(c) $2\sqrt{2}$
(d) $-4\sqrt{2}$ VIEW SOLUTION

• Question 44
The inverse of the matrix $\mathrm{X}=\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$ is
(a) $24\left[\begin{array}{ccc}1/2& 0& 0\\ 0& 1/3& 0\\ 0& 0& 1/4\end{array}\right]$

(b) $\frac{1}{24}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

(c) $\frac{1}{24}\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$

(d) $\left[\begin{array}{ccc}1/2& 0& 0\\ 0& 1/3& 0\\ 0& 0& 1/4\end{array}\right]$ VIEW SOLUTION

• Question 45
For an L.P.P. the objective function is Z = 4x + 3y, and the feasible region determined by a set of constraints (linear inequations) is shown in the  graph. Which one of the following statements is true ?
(a) Maximum value of Z is at R.
(b) Maximum value of Z is at Q.
(c) Value of Z at R is less than the value at P.
(d) Value of Z at Q is less than the value at R. VIEW SOLUTION

• Question 46 In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹400 × (depth)2. Association will like to have minimum cost.
Based on this information, answer the any 4 of the following questions.

Let side of square plot is x m and its depth is h metres, then cost c for the pit is
(a)

(b)

(c) $\frac{250}{h}+{h}^{2}$

(d)  VIEW SOLUTION

• Question 47 In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹400 × (depth)2. Association will like to have minimum cost.
Based on this information, answer the any 4 of the following questions.

Value of h (in m) for which $\frac{dc}{dh}=0$ is
(a) 1.5
(b) 2
(c) 2.5
(d) 3 VIEW SOLUTION

• Question 48 In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹400 × (depth)2. Association will like to have minimum cost.
Based on this information, answer the any 4 of the following questions.

$\frac{{d}^{2}c}{d{h}^{2}}$ is given by

(a) $\frac{25000}{{h}^{3}}+800$

(b) $\frac{500}{{h}^{3}}+800$

(c) $\frac{100}{{h}^{3}}+800$

(b) $\frac{500}{{h}^{3}}+2$
VIEW SOLUTION

• Question 49 In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹400 × (depth)2. Association will like to have minimum cost.
Based on this information, answer the any 4 of the following questions.

Value of x (in m) for minimum cost is
(a) 5
(b) $10\sqrt{\frac{5}{3}}$
(c) $5\sqrt{5}$
(d) 10 VIEW SOLUTION

• Question 50 In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹400 × (depth)2. Association will like to have minimum cost.
Based on this information, answer the any 4 of the following questions.

Total minimum cost of digging the pit (in ₹) is
(a) 4,100
(b) 7,500
(c) 7,850
(d) 3,220 VIEW SOLUTION
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