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

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

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

(a) $\frac{\mathrm{cos\theta }+\cos 2\mathrm{\theta }}{\mathrm{sin\theta }-\sin 2\mathrm{\theta }}$

(b) $\frac{\mathrm{cos\theta }-\cos 2\mathrm{\theta }}{\sin 2\mathrm{\theta }-\mathrm{sin\theta }}$

(c) $\frac{\mathrm{cos\theta }-\cos 2\mathrm{\theta }}{\mathrm{sin\theta }-\sin 2\mathrm{\theta }}$

(d) $\frac{\cos 2\mathrm{\theta }-\mathrm{cos\theta }}{\sin 2\mathrm{\theta }+\mathrm{sin\theta }}$ VIEW SOLUTION

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

A matrix A = [a_ij]_{3 × 3} is defined by

$a_{\mathit{i}\mathit{j}}=\left\{\begin{array}{ll}2i+3j,& i<j\\ 5,& i=j\\ 3i-2j,& i>j\end{array}\right.$

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

If a function f defined by

$f\left(x\right)=\left\{\begin{array}{ll}\frac{k \cos x}{\mathrm{\pi }-2x},& \mathrm{if} x\ne \frac{\mathrm{\pi }}{2}\\ 3,& \mathrm{if} x=\frac{\mathrm{\pi }}{2}\end{array}\right.$
is continuous at $x=\frac{\mathrm{\pi }}{2},$ then the value of k is
(a) 2
(b) 3
(c) 6
(d) –6 VIEW SOLUTION

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

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

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

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

VIEW SOLUTION

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

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

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

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) $\pm$ 3
(b) –3
(c) $\pm$ 1
(d) 1 VIEW SOLUTION

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

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

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) e^e VIEW SOLUTION

Let matrix X = [x_ij] is given by $\mathrm{X} = \left[\begin{array}{ccc}1& –1& 2\\ 3& 4& –5\\ 2& –1& 3\end{array}\right]$. Then the matrix Y = [m_ij], where m_ij = Minor of x_ij, is
(a) $\mathrm{X} = \left[\begin{array}{ccc}7& –5& –3\\ 19& 1& –11\\ –11& 1& 7\end{array}\right]$
(b) $\mathrm{X} = \left[\begin{array}{ccc}7& –19& –11\\ 5& –1& –1\\ 3& 11& 7\end{array}\right]$
(c) $\mathrm{X} = \left[\begin{array}{ccc}7& 19& –11\\ –3& 11& 7\\ –5& –1& –1\end{array}\right]$
(d) $\mathrm{X} = \left[\begin{array}{ccc}7& 19& –11\\ –1& –1& 1\\ –3& –11& 7\end{array}\right]$ VIEW SOLUTION

A function f: R → R 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

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, y ≥ 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

If x = –4 is a root of ${\left|\begin{array}{ccc}x& 2& 3\\ 1& x& 1\\ 3& 2& x\end{array}\right|}_{ }=0,$ then the sum of the other two roots is
(a) 4
(b) –3
(c) 2
(d) 5 VIEW SOLUTION

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

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(2rh² – h³)
(d) 2π²r² (2rh – h²) VIEW SOLUTION

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

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

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

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

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) $\frac{50}{h}+400 h^2$

(b) $\frac{12500}{h}+400 h^2$

(c) $\frac{250}{h}+h^2$

(d) $\frac{250}{h}+400 h^2$ VIEW SOLUTION

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

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

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

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