## Nimcet 2020 Solution

NIMCET 2020 PAPER AND SOLUTIONS

Questions 01: If $$\left( {\begin{array}{*{20}{c}}{15}\\8\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{15}\\7\end{array}} \right) = \left( {\begin{array}{*{20}{c}}n\\r\end{array}} \right)$$, then the values of $$n$$ and $$r$$ are:

(A) 16 and 7
(B) 16 and 8
(C) 16 and 9
(D) 30 and 15

We know the formula that $$\left( {\begin{array}{*{20}{c}}m\\k\end{array}} \right) + \left( {\begin{array}{*{20}{c}}m\\{k + 1}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{m + 1}\\{k + 1}\end{array}} \right)$$

Here $$m$$ = 15 and $$k$$ = 7, hence the desired sum is $$\left( {\begin{array}{*{20}{c}}{16}\\8\end{array}} \right)$$

Thus $$n$$ = 16 and $$r$$ = 8.

(A) 25
(B) 20
(C) 15
(D) 30

Suppose number of students reading Hidavat and Hindustan are $$n(A)$$ and $$n(B)$$, then

$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

$$\Rightarrow 50 - 10 = 30 + 35 - n(A \cap B)$$

$$\Rightarrow n(A \cap B) = 25$$

Questions 03: If $$A = \{ {4^x} - x - 1:x \in \mathbb{N}\}$$ and $$B = \{ 9(x - 1):x \in \mathbb{N}\}$$, where $$\mathbb{N}$$ is the set of natural numbers, then

(A) A ⊂ B
(B) A ⊆ B
(C) A ⊃ B
(D) A ⊇ B

Given that A = $${4^x} - 3x - 1$$ and B = 9(x – 1)

By putting x = 1, 2, 3, 4… we see that,

A = {0, 9, 54, …} and B = {0, 9, 18, 27, …}

The set A is a proper subset of B or $$A \subset B$$

Alternately we can expand A as

$${(1 + 3)^x} - 3x - 1$$

$$= 1 + 3x + {\,^x}{{\rm{C}}_2} \cdot {3^2} + .... - 3x - 1$$

= a multiple of 9.

Here A and B are both multiple of 9, but B contains each multiple of 9 while A does not. Thus A is a subset of B.

Questions 04: If $$A= \{x, y, z\}$$, then the number of subsets in power set of $$A$$ is:

(A) 6 (B) 8 (C) 7 (D) 9

A has 3 elements, hence number of subsets = $$2^3= 8$$.

Questions 05: How many words can be formed starting with letter D taking all letters from word DELHI so that the letters are not repeated:

(A) 4 (B) 12 (C) 24 (D) 120

In DELHI all the letters are different. The first letter is fixed, so the remaining 4 letters can be permuted in 4! ways.

Number of such words = 4! = 24.

Questions 06: Naresh has 10 friends, and he wants to invite 6 of them to a party. How many times will 3 particular friends never attend the party?

(A) 8 (B) 7 (C) 720 (D) 35

Out of the 10 friends, 3 are not attending the party, so Naresh has to select 6 friends out of 7 friends.

Number of ways = $$^7{{\rm{C}}_6} = 7$$

Questions 07: There is a young boy’s birthday party in which 3 friends have attended. The mother has arranged 10 games where a prize is awarded for a winning game. The prizes are identical. If each of the 4 children receives at least one prize, then how many distributions of prizes are possible?

(A) 80 (B) 84 (C) 70 (D) 72

Total number of prizes are 10 and there are 10 recipients, number of ways is equal to the number of solutions of the equation $$x + y + z + w = 10$$, where $$x,\;y,\;z,\,\,w \ge 1$$.

Hence the number of solutions = $$^{10 - 1}{{\rm{C}}_{4 - 1}} = {\,^9}{{\rm{C}}_3} = 84$$

Questions 08: Three cities A, B, C are equidistant from each other. A motorist travels from A to B at 30km/hour, from B to C at 40km/hour and from C to A at 50km/hour. Then the average speed is

(A) 39km/hour
(B) 40km/hour
(C) 38.3km/hour
(D) 37.6km/hour

Suppose the distance AB = BC = CA = $$x$$, then total journey = $$3x$$ and total time taken in the journey = $$\frac{x}{{30}} + \frac{x}{{40}} + \frac{x}{{50}} = \frac{{47x}}{{600}}$$.

Hence the average speed = $$\frac{{{\rm{Total}}\;\,{\rm{Journey}}}}{{{\rm{Total}}\;\,{\rm{time}}}}$$

$$= \frac{{3x}}{{\frac{{47x}}{{600}}}} = \frac{{1800}}{{47}} = 38.3$$

Questions 09: A problem in Mathematics is given to 3 students A, B and C. If the probability of A solving the problem is $$\frac{1}{2}$$ and B not solving it is $$\frac{1}{4}$$. The whole probability of the problem being solved is $$\frac{{63}}{{64}}$$, then what is the probability of solving it by C?

(A) $$\frac{1}{8}$$
(B) $$\frac{1}{{64}}$$
(C) $$\frac{7}{8}$$
(D) $$\frac{1}{2}$$

Probabilities that B will solve the problem = $$1 - \frac{1}{4} = \frac{3}{4}$$. Let probability that C will solve the problem = $$p$$, then probability that none of them is able to solve the problem = $$\frac{1}{2} \times \frac{1}{4} \times (1 - p) = \frac{{1 - p}}{8}$$.

From the given information,

$$\frac{{1 - p}}{8} = \frac{1}{{64}} \Rightarrow p = \frac{7}{8}$$

Questions 10: A and B play a game where each is asked to select a number from 1 to 25. If the two numbers match, both win a prize. The probability that they will not win a prize in a single trial is

(A) $$\frac{1}{{25}}$$
(B) $$\frac{{24}}{{25}}$$
(C) $$\frac{2}{{25}}$$
(D) $$\frac{3}{{25}}$$

A can select any number from the first 25 numbers with probability $$\frac{{25}}{{25}}$$. Now B has to select a number different from A, this can be done with probability $$\frac{{24}}{{25}}$$.

Required probability = $$\frac{{24}}{{25}}$$.

Questions 11: A, B, C are three sets of values of $$x$$:

A: 2, 3, 7, 1, 3, 2, 3
B: 7, 5, 9, 12, 5, 3, 8
C: 4, 4, 11, 7, 2, 3, 4
Select the correct statement among the following:
(A) Mean of A is equal to Mode of C.
(B) Mean of C is equal to Median of B.
(C) Median of B is equal to Mode of A.
(D) Mean, Median and Mode of A are same.

Questions 12: Standard deviation for the following distribution is:

Questions 13: If A = $$\left[ {\begin{array}{*{20}{c}}{\cos \alpha }&{\sin \alpha }\\{ - \sin \alpha }&{\cos \alpha }\end{array}} \right]$$, then for any value of $$n$$, the value of $$A_n$$ is:

(A) $$\left[ {\begin{array}{*{20}{c}}{\sin n\alpha }&{\cos n\alpha }\\{ - \cos n\alpha }&{\sin n\alpha }\end{array}} \right]$$
(B) $$\left[ {\begin{array}{*{20}{c}}{\cos n\alpha }&{\sin n\alpha }\\{\sin n\alpha }&{\cos n\alpha }\end{array}} \right]$$
(C) $$\left[ {\begin{array}{*{20}{c}}{\cos n\alpha }&{\sin n\alpha }\\{\sin n\alpha }&{ - \cos n\alpha }\end{array}} \right]$$
(D) $$\left[ {\begin{array}{*{20}{c}}{\cos n\alpha }&{\sin n\alpha }\\{ - \sin n\alpha }&{\cos n\alpha }\end{array}} \right]$$

Questions 14: Roots of equation $$a{x^2}-2bx + c = 0$$ are $$n$$ and $$m$$, then the value of $$\frac{b}{{a{n^2} + c}} + \frac{b}{{a{m^2} + c}}$$ is:

(A) $$\frac{c}{a}$$
(B) $$\frac{b}{a}$$
(C) $$\frac{a}{c}$$
(D) $$\frac{a}{b}$$

Questions 15: The number of values of $$k$$ for which the linear equations

$$4x + ky + z = {\rm{ }}0,\;\;kx + 4y + z = {\rm{ }}0,{\rm{ }}2x + 2y + z = {\rm{ }}0$$, possess a non-zero solution is
(A) 2
(B) 1
(C) 0
(D) 3

Questions 16: Let $$A = ({a_{ij}})$$and $$B = ({b_{ij}})$$ be two square matrices of order $$n$$ and $$\det (A)$$ denotes the determinant of A. Then, which of the following is not correct?

(A) If A is a diagonal matrix, then $$\det (A) = {a_{11}}{a_{22}}......{a_{nn}}$$
(B) $$\det (AB) = \det (A) \cdot \det (B)$$
(C) $$\det (cA) = c\det (A)$$
(D) $$\det (A) = \det ({A^T})$$where $${A^T}$$ denotes the transpose of the matrix A.

Questions 17: The tangent to the ellipse $$x^2 + 16y^2 = 16$$ and making 60° angle with positive $$x$$ axis is:

(A) x – √3 y  + 7 = 0
(B) √3 x – y + 8 = 0
(C) √3 x – y + 7 = 0
(D) x + √3y + 7 = 0

The equation of the ellipse is $$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{1} = 1$$ and the slope of the tangent = $$\tan 60 = \sqrt 3$$.

If the equation of the tangent is $$y = mx + c$$, then $${c^2} = {a^2}{m^2} + {b^2}$$

$$\Rightarrow {c^2} = 16 \times 3 + 1 = 49$$

Or $$c = \pm 7$$

Equation of the tangent is $$y = \sqrt 3 x \pm 7$$

Questions 18: Find the number of point(s) of intersection of the ellipse $$\frac{{{x^2}}}{4} + \frac{{{{(y - 1)}^2}}}{9} = 1$$ and the circle $${x^2} + {y^2} = 4$$.

(A) 4
(B) 3
(C) 2
(D) 1

Questions 19: An arithmetic progression has 3 as its first term. Also, the sum of the first 8 terms is twice the sum of the first 5 terms. Then what is the common difference?

(A) 3/4
(B) 1/2
(C) 1/4
(D) 4/3

Questions 20: If $$a + b + c = 0$$, then the value of $$\frac{{{a^2}}}{{bc}} + \frac{{{b^2}}}{{ac}} + \frac{{{c^2}}}{{ac}}$$ is:

(A) 1 (B) 0 (C) 3 (D) – 1

Questions 21: Find $$\mathop {\lim }\limits_{x \to 0} \,\,\left( {{x^2}{e^{\sin \frac{1}{x}}}} \right)$$

(A)  1
(B) l
imit does not exist
(C) i
nfinity
(D) None of these

Questions 22:  If $$f(x) = \left\{ {\;\begin{array}{*{20}{c}}{{x^2}}&{x \le 0}\\{2\sin x}&{x > 0}\end{array}} \right.$$, then $$x = 0$$ is:

(A) Point of minima
(B) Point of maxima
(C) Point of discontinuity
(D) None of these

The graph of the function is given here. The point at $$x = 0$$ is the point of minima.

Questions 23: If   $$g(x) = \left\{ {\begin{array}{*{20}{c}}{\frac{{{x^2} - 2x}}{{2x}}}&{x \ne 0}\\k&{x = 0}\end{array}} \right.$$  is a continuous function  at  $$x = 0$$,  then  the value of $$k$$ is :

(A) 2
(B)
1/2
(C)
1
(D)
None of these

The value of the function$$g(x)$$ when $$x \ne 0$$ is $$\frac{x}{2} - \frac{1}{2}$$.

When $$x = 0 + h$$ or $$x = 0 - h$$, then $$g(x) = - \frac{1}{2}$$.

As the function is continuous, so $$k = - \frac{1}{2}$$.

Questions 24: Find the interval(s) on which the graph $$y = 2{x^3}{e^x}$$ is  increasing:

(A) (– 3, 0) and (0, ∞)
(B)
(– 3/2, 0) and (0, ∞)
(C)
(– 3, ∞) only
(D)
None of these

Questions 25: If $$\int {{{\sec }^2}x\,\,{\rm{cose}}{{\rm{c}}^4}x} \,dx$$ = $$- \frac{1}{3}{\cot ^3}x + k\tan x - 2\cot x + c$$, then the value of $$k$$ is:

(A) 1
(B) 2
(C) 3
(D) 4

Questions 26: Evaluate $$\int {{e^x}} \left( {\frac{{1 + \sin x\cos x}}{{{{\cos }^2}x}}} \right)dx$$

(A) $${e^x}\cos x + c$$
(B) $${e^x}\sec x\tan x + c$$
(C)  $${e^x}\tan x + c$$
(D) $${e^x}{\cos ^2}x - 1 + c$$

Questions 27: If $${I_n} = \int\limits_0^a {({x^2} - {y^2})\,dx}$$, where $$n$$ is  a  positive  integer,  then  the  relation  between $${I_n}$$ and $${I_{n - 1}}$$ is

(A) $${I_n} = \frac{{2n{a^2}}}{{2n + 1}}{I_{n - 1}}$$
(B)
$${I_n} = \frac{{2n^2{a^2}}}{{2n - 1}}{I_{n - 1}}$$
(C) $${I_n} = \frac{{2n{a^2}}}{{2n - 1}}{I_{n - 1}}$$
(D) $${I_n} = \frac{{2n^2{a^2}}}{{2n+ 1}}{I_{n - 1}}$$

Questions 28: The value of $$\int\limits_{ - 2}^2 {(a{x^5} + b{x^3} + c)} \,dx$$ depends on the

(A) Value of $$b$$
(B) Value of $$c$$
(C) Value of $$a$$
(D) Value of $$a$$ and $$b$$

Questions 29: Find the area bounded by the line $$y = 3 - x$$, the parabola $$y = {x^2} - 9$$ and $$x \ge - 1$$, $$y \ge 0$$.

(A) 7/2
(B) 11/2
(C) 9/2
(D) None of these

Questions 30: If $$\vec a,\vec b,\vec c$$   are three non-coplanar vectors, then $$(\vec a + \vec b + \vec c).\left[ {\left( {\vec a + \vec b} \right) \times \left( {\vec a + \vec c} \right)} \right]$$

(A) 0
(B)  $$\left[ {\vec a\;\vec b\;\vec c} \right]$$
(C) $$2\left[ {\vec a\;\vec b\;\vec c} \right]$$
(D) $$\left[ {\vec a\;\vec b\;\vec c} \right]$$

Questions 31: Two forces $${F_1}$$ and $${F_2}$$ are used to pull a car, which met an accident. The angle between the two forces is  $$\theta$$. Find the values of  $$\theta$$ for which the resultant force is equal to:

(A) $$\theta = 0$$
(B) $$\theta = 45$$
(C) $$\theta = 90$$
(D) $$\theta = 135$$

Questions 32:  If $$\vec a,\vec b,\vec c,\vec d\;$$ are four vectors such that $$\vec a + \vec b + \vec c\;\;$$Is collinear with  $$\vec d$$ and  $$\vec b + \vec c + \;\vec d\;$$  is collinear with $$\vec a,$$ then $$\vec a + \vec b + \vec c + \vec d\;$$ is

(A) $$\vec 0$$
(B) collinear with $$\vec a + \vec d\;$$
(C)
collinear with $$\vec a - \vec d$$
(D)
collinear with $$\vec b - \vec c$$

Questions 33: Forces of magnitude 5, 3, 1 units act in the directions 6i + 2j + 3k, 3i – 2j + 6k, 2i – 3j – 6k respectively on a particle which is displaced from the point (2, – 1, – 3) to (5, – 1, 1). The total work done by the force is

(A) 21 units
(B) 5 units
(C)
33 units
(D)
105 units

Questions 34:  The position vectors of points A and B are  $$\vec a$$ and $$\vec b$$ . Then the position vector of point P dividing AB in the ratio $$m:n$$ is

(A) $$\frac{{n\vec a + m\vec b}}{{m + n}}$$
(B) $$\frac{{n\vec a + m\vec b}}{{m - n}}$$
(C) $$\frac{{n\vec a - m\vec b}}{{m + n}}$$
(D) None of these

Questions 35: If $$\vec a,\vec b,\vec c\;$$ are three non-zero vectors with no two of which are $$\vec a + 2\vec b\;\;$$is collinear with  $$\vec c$$ and  $$\vec b + \overrightarrow {3c} \;$$  is collinear with $$\vec a,$$ then $$\left| {\vec a + 2\vec b + 6\vec c} \right|$$ will be equal to

(A) 0
(B) 9
(C) 1
(D) None of above

Questions 36: Vertices of the vectors i – 2j + 2k, 2i + j – k and 3i – j + 2k form a triangle. This triangle is

(A) Equilateral triangle
(B) Right angle triangle
(C) Two sides are equal in length
(D)
None of the above

Questions 37: If the volume of a parallelepiped whose adjacent edges are $$\vec a = 2i + 3j + 4k,$$$$\vec b = i + \alpha j + 2k,$$ is 15, then $$\alpha$$ is:

(A) 1
(B) 5/2
(C)
9/2
(D)
0

Questions 38: Solve the equation $${\sin ^2}x - \sin x - 2 = 0$$ for $$x$$ on the interval $$0 \le x \le 2\pi$$

(A) $$x = - \frac{\pi }{2}$$ only
(B) $$x = \frac{\pi }{4}\;\;{\rm{and}}\;\frac{{2\pi }}{7}$$
(C) $$x = \frac{{2\pi }}{3}\;\;{\rm{and}}\;\frac{{2\pi }}{5}$$
(D) None of these

Questions 39: If $$\frac{{\tan x}}{2} = \frac{{\tan y}}{3} = \frac{{\tan z}}{5}\;$$and $$x + y + z = \pi$$, then the value of $${\tan ^2}x + {\tan ^2}y + {\tan ^2}z$$

(A) 38/3
(B) 38
(C) 114
(D) None of these

Questions 40:  Find the value of sin 12°sin 48°sin 54°:

(A) 1/8
(B) 1/6
(C) 1/2
(D) 1/4

Questions 41: If cos x = tan y, cot y = tan z and cot z = tan x, then sin x is:

(A)  $$\frac{{\sqrt 5 + 1}}{2}$$
(B)  $$\frac{{\sqrt 5 - 1}}{2}$$
(C)  $$\frac{{\sqrt 5 + 1}}{4}$$
(D) $$\frac{{\sqrt 5 - 1}}{4}$$

Questions 42: The  value  of  $$\tan \left( {45 + \frac{A}{2}} \right)$$  is

(A) cot θ – sec θ
(B) cot θ + sec θ
(C) tan θ – sec θ
(D) tan θ + sec θ

Questions 43: The  value  of  sin 10° sin 50° sin 70°  is  :

(A) 1/4
(B) 1/2
(C) 3/4
(D) 1/8

Questions 44: The expression  $$\frac{{\tan A}}{{1 - \cot A}} + \frac{{\cot A}}{{1 - \tan A}}$$ can  be written  as

(A) sin A cos A + 1
(B) sec A cosec A + 1
(C) tan A + cot A
(D) sec A + cosec A

Questions 45: Angle  of  elevation  of  the  top  of  the  tower  from  3  points  (collinear)  A,  B  and  C  on  a road  leading  to  the  foot  of  the  tower  are  30°,  45°  and  60°,  respectively.  The  ratio  of AB  and  BC  is

(A) $$\sqrt 3 :1$$
(B) $$\sqrt 3 :2$$
(C) $$1:2$$
(D) $$2:\sqrt 3$$

Questions 46: The  area  enclosed between  the  curves $${y^2} = x$$ and $$y = \left| x \right|$$ is

(A) 2/3  sq.  unit
(B) 1   sq. unit
(C) 1/6    sq.  unit
(D) 1/3  sq.  Unit

Questions 47: Test  the  continuity  of  the  function  at  $$x = 2$$

$$f(x) = \left\{ {\begin{array}{*{20}{c}}{\frac{5}{2} - x,}&{x < 2}\\{1,}&{x = 2}\\{x - \frac{3}{2},}&{x > 2}\end{array}} \right.$$
(A) Continuous  at  $$x = 2$$
(B) Discontinuous  at  $$x = 2$$
(C) Semi continuous  at  $$x = 2$$
(D) None of the above

Questions 48: The value  of $$2{\tan ^{ - 1}}\left[ {{\rm{cosec}}({{\tan }^{ - 1}}x) - \tan ({{\cot }^{ - 1}}x)} \right]$$ is:

(A) tanx
(B) cotx
(C) tan⁻¹x
(D) cosec⁻¹x

Questions 49: If $$3\sin x + 4\cos x = 5$$, then $$6\tan \frac{x}{2} - 9{\tan ^2}\frac{x}{2}$$ is:

(A) 1
(B) 3
(C) 4
(D) 6

Questions 50: If  A  is  a  subset  of  B  and  B  is  a  subset  of  C, then  cardinality of  $${\rm{A}} \cup {\rm{B}} \cup {\rm{C}}$$  is  equal  to: