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:

Questions 02: In a class of 50 students, it was found that 30 students read “Hitavad”, 35 students read “Hindustan” and 10 read neither. How many students read both: “Hitavad” and “Hindustan” newspapers?

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

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

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:

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?

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?

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

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?

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

Questions 11: A, B, C are three sets of values of \(x\):

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:

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:

Questions 15: The number of values of \(k\) for which the linear equations

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?

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

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 \({x^2} + {y^2} = 4\).

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?

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

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

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:

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 :

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

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:

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

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

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

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

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

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

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

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

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

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

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:

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

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

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

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

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

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

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

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

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

Questions 47: Test  the  continuity  of  the  function  at  \(x = 2\) 

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

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

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: