NIMCET Mathematics Previous Year Question Paper and Solution with PDF

Solution

Qus : 4NIMCET PYQ

If $\vec{a}=\hat{i}-\hat{k}$, $\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$, then $\begin{bmatrix}{\vec{a}} & {\vec{b}} & {\vec{c}}\end{bmatrix}$ depends on

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Solution

Qus : 5NIMCET PYQ

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

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Solution

According to the given information, the figure should be as follows.

Let the height of tower = h

Qus : 6NIMCET PYQ

If $\vec{a}, \vec{b}$ are unit vectors such that $2\vec{a}+\vec{b} =3$ then which of the following statement is true?

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Solution

Qus : 7NIMCET PYQ

The area enclosed between the curves y² = x and y = |x| is

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Solution

Solving y² = x and y = x,

we get, y = 0, x = 0, y = 1, x = 1 Therefore,

Qus : 8NIMCET PYQ

$\int f(x)\mathrm{d}x=g(x)$, then $\int {x}^5f({x}^3)\mathrm{d}x$

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Solution

Qus : 9NIMCET PYQ

Test the continuity of the function at x = 2

$f(x)= \begin{cases} \frac{5}{2}-x & \text{ if } x<2 \\ 1 & \text{ if } x=2 \\ x-\frac{3}{2}& \text{ if } x>2 \end{cases}$

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Solution

LHL ≠ f(2)

Qus : 10NIMCET PYQ

$\lim _{{x}\rightarrow1}\frac{{x}^4-1}{x-1}=\lim _{{x}\rightarrow k}\frac{{x}^3-{k}^2}{{x}^2-{k}^2}=$, then find k

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Solution

Qus : 11NIMCET PYQ

The value of

2tan^-1[cosec(tan^-1x) - tan(cot^-1x)]

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Solution

Qus : 12NIMCET PYQ

The graph of function $f(x)=\log _e({x}^3+\sqrt[]{{x}^6+1})$ is symmetric about:

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Solution

Qus : 13NIMCET PYQ

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

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Solution

Qus : 42NIMCET PYQ

In a beauty contest, half the number of experts voted Mr. A and two thirds voted for Mr. B 10 voted for both and 6 did not for either. How may experts were there in all.

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Solution

Let the total number of experts be

N

E

is the set of experts who voted for miss

A

F

is the set of experts who voted for miss

B

.
Since

6

did not vote for either, n

(E \cup F) = N - 6

n (E) = \frac{N}{2}, n (F) = \frac{2}{3} N

and

n (E \cap F) = 10

.
So,

N - 6 = \frac{N}{2} + \frac{2}{3} N - 10

Solving the above equation gives

\frac{N}{6} = 4 \Rightarrow N = 24

Qus : 43NIMCET PYQ

A circle touches the x–axis and also touches the circle with centre (0, 3) and radius 2. The locus of the centre of the circle is

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Solution

Qus : 44NIMCET PYQ

The value of non-zero scalars α and β such that for all vectors

and

such that

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Solution

Qus : 45NIMCET PYQ

A force of 78 grams acts at the point (2,3,5). The direction ratios of the line of action being 2,2,1 . The magnitude of its moment about the line joining the origin to the point (12,3,4) is

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Solution

Qus : 46NIMCET PYQ

Number of real solutions of the equation

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Solution

Qus : 47NIMCET PYQ

The sum of infinite terms of a decreasing GP is equal to the greatest value of the function

in the interval [-2,3] and the difference between the first two terms is f'(0). Then the common ratio of GP is

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Solution

Qus : 48NIMCET PYQ

Number of onto (surjective) functions from A to B if n(A)=6 and n(B)=3, is

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Solution

Qus : 49NIMCET PYQ

If $|z|<\sqrt{3}-1$, then $|z^{2}+2z cos \alpha|$ is

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Solution

Qus : 50NIMCET PYQ

A computer producing factory has only two plants T₁ and T₂ produces 20% and plant T₂ produces 80% of the total computers produced. 7% of the computers produced in the factory turn out to be defective. It is known that P (computer turns out to be defective given that it is produced in plant T₁ 10P(computer turns out to be defective given that it is produced in plant T₂ ). A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant T₂ is

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Solution

Qus : 51NIMCET PYQ

If A > 0, B > 0 and A + B = $\frac{\pi}{6}$ , then the minimum value of $tanA + tanB$

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Solution

On differentiating

x= tanA + tan(π/6-A)

we get :

dx/dA = sec²A-sec²(π/6-A)

now putting

dx/dA=0

we get

cos²(A) = cos²(π/6-A) so 0≤A≤π/6

therefore

A=π/6-A from here we get A = π/12 = B

so minimum value of that function is

2tanπ/12 which is equal to 2(2-√3)

Qus : 52NIMCET PYQ

Inverse of the function $f(x)=\frac{10^x-10^{-x}}{10^{x}+10^{-x}}$ is

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Solution

Let $f(x)=y$, then

⇒ $\frac{10^x-10^-x}{10^x+10^-x}=y$

⇒ $\frac{10^{2x}-1}{10^{2x}+1}=y$

⇒ $10^{2x}=\frac{1+y}{1-y}$ By Componendo Dividendo Rule

⇒ $x=\frac{1}{2}\log _{10}\Bigg{(}\frac{1+y}{1-y}\Bigg{)}$

⇒ ${f}^{-1}(y)=\frac{1}{2}\log _{10}\Bigg{(}\frac{1+y}{1-y}\Bigg{)}$

⇒ ${f}^{-1}(x)=\frac{1}{2}\log _{10}\Bigg{(}\frac{1+x}{1-x}\Bigg{)}$

Qus : 53NIMCET PYQ

The tangent at the point (2, -2) to the curve $x^2 y^2-2x=4(1-y)$ does not passes through the point

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Solution

Qus : 54NIMCET PYQ

The integral $\int \sqrt{1+2 cot x(cosec x+cotx)} dx$ , $(0<x<\frac{\pi}{2})$ (where C is a constant of integration) is equal to

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Solution

Qus : 55NIMCET PYQ

If all the words, with or without meaning, are written using the letters of the word QUEEN add are arranged as in English Dictionary, then the position of the word QUEEN is

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Solution

Letters of the word QUEEN are E,E,N,Q,U

Words beginning with E (4!) = 24

Words beginning with N (4!/2!)=12

Words beginning with QE (3!) = 6

Words beginning with QN (3!/2!)= 3

Total words = 24+12+6+9=45

QUEEN is the next word and has rank 46th.

Qus : 56NIMCET PYQ

The curve satisfying the differential equation

and passing through the point (1,1) also passes through the point __________

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Solution

Qus : 57NIMCET PYQ

is equal to

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Solution

Qus : 58NIMCET PYQ

If S and S' are foci of the ellipse

, B is the end of the minor axis and BSS' is an equilateral triangle, then the eccentricity of the ellipse is

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Solution

Qus : 59NIMCET PYQ

The equation of the circle passing through the point (4,6) and whose diameters are along x + 2y - 5 =0 and 3x - y - 1=0 is

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Solution

Qus : 60NIMCET PYQ

In a parallelogram ABCD, P is the midpoint of AD. Also, BP and AC intersect at Q. Then AQ : QC =

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Solution

Qus : 61NIMCET PYQ

The median AD of ΔABC is bisected at E and BE is extended to meet the side AC in F. The AF : FC =

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Solution

Qus : 62NIMCET PYQ

Let $X_i, i = 1,2,.. , n$ be n observations and $w_i = px_i +k, i = 1,2, ,n$ where p and k are constants. If the mean of $x_i 's$ is 48 and the standard deviation is 12, whereas the mean of $w_i 's$ is 55 and the standard deviation is 15, then the value of p and k should be

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Solution

Qus : 69NIMCET PYQ

Not Available right now

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Qus : 73NIMCET PYQ

Let U and V be two events of a sample space S and P(A) denote the probability of an event A. Which of the following statements is true?

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Qus : 90NIMCET PYQ

Let $\vec{a}, \vec{b}, \vec{c} $ be distinct non-negative numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k}$ , $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ lie in a plane, then c is

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Solution

$\vec{a}=a\hat{i}+a\hat{j}+c\hat{k}\, ,\, \vec{b}=\hat{i}+\hat{k}\, \&\, \vec{c}=c\hat{i}+c\hat{j}+b\hat{k}$ are coplanar.

$\Rightarrow\begin{vmatrix}{a} & {a} & {c} \\ {1} & {0} & {1} \\ {c} & {c} & {b}\end{vmatrix}=0$

$\Rightarrow-ac-ab+ac+{c}^2=0$

Solution

Qus : 105NIMCET PYQ

A particle is at rest at the origin. It moves along the x −axis with an acceleration $x-x^2$ , where x is the distance of the particle at time t. The particle next comes to rest after it has covered a distance

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Solution

Qus : 116NIMCET PYQ

Which of the following is TRUE?

A. If $f$ is continuous on $[a,b]$, then $\int ^b_axf(x)\mathrm{d}x=x\int ^b_af(x)\mathrm{d}x$

B. $\int ^3_0{e}^{{x}^2}dx=\int ^5_0e^{{x}^2}dx+{\int ^5_3e}^{{x}^2}dx$

C. If $f$ is continuous on $[a,b]$, then $\frac{d}{\mathrm{d}x}\Bigg{(}\int ^b_af(x)dx\Bigg{)}=f(x)$

D. Both (a) and (b)

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Solution

Qus : 117NIMCET PYQ

The solution of the equation ${4\cos }^2x+6{\sin }^2x=5$ are

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Solution

Qus : 118NIMCET PYQ

Qus : 132NIMCET PYQ

If $\sin x=\sin y$ and $\cos x=\cos y$, then the value of x-y is

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Solution

Qus : 133NIMCET PYQ

Which of the following is NOT true?

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Solution

Qus : 144NIMCET PYQ

Among the given numbers below, the smallest number which will be divided by 9, 10, 15 and 20, leaves the remainders 4, 5, 10, and 15, respectively

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Solution

Qus : 145NIMCET PYQ

For $a\in R$ (the set of al real numbers), $a \ne 1$, $\lim _{{n}\rightarrow\infty}\frac{({1}^a+{2}^a+{\ldots+{n}^a})}{{(n+1)}^{a-1}\lbrack(na+1)(na+b)\ldots(na+n)\rbrack}=\frac{1}{60}$ . Then one of the value of $a$ is

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Solution

Qus : 154NIMCET PYQ

Given a set A with median $m_1 = 2$ and set B with median $m_2 = 4$

What can we say about the median of the combined set?

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Solution

Qus : 155NIMCET PYQ

If ${\Bigg{(}\frac{x}{a}\Bigg{)}}^2+{\Bigg{(}\frac{y}{b}\Bigg{)}}^2=1$, $(a{\gt}b)$ and ${x}^2-{y}^2={c}^2$ cut at right angles, then

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Solution

If $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{c^2}+\frac{y^2}{d^2}=1$ are orthogonal.

Then

$a^2-b^2=c^2-d^2$

Similarly

If $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $x^2-y^2=c^2$ are orthogonal.

It means

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{c^2}+\frac{y^2}{-c^2}=1$ are orthogonal

Then

$a^2-b^2=c^2-(-c^2)$

$a^2-b^2=2c^2$

Qus : 156NIMCET PYQ

Qus : 192NIMCET PYQ

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Solution

Qus : 193NIMCET PYQ

The function

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Solution

Qus : 194NIMCET PYQ

Two person A and B agree to meet 20 april 2018 between 6pm to 7pm with understanding that they will wait no longer than 20 minutes for the other. What is the probability that they meet?

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Solution

Qus : 195NIMCET PYQ

Three numbers a,b and c are chosen at random (without replacement) from among the numbers 1, 2, 3, ..., 99. The probability that

is divisible by 3 is,

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Solution

Qus : 196NIMCET PYQ

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

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Solution

The total number of ways in which numbers can be choosed =25 x 25=625 The number of ways in which either players can choose same numbers = 25

Probability that they win a prize = 25/625 = 1/25

Thus, the probability that they will not win a prize in a single trial = 1 - 1/25 = 24/25

Solution

Qus : 203NIMCET PYQ

$\lim_{x\to \infty} (\frac{x+7}{x+2})^{x+5}$ equal to

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Solution

Qus : 204NIMCET PYQ

If the mean of the squares of first n natural numbers be 11, then n is equal to?

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Solution

Qus : 205NIMCET PYQ

A and B are independent witness in a case. The chance that A speaks truth is x and B speaks truth is y. If A and B agree on certain statement, the probability that the statement is true is

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Solution

P(A speaks truth) = x

P(B speaks truth) = y

Since, both A and B agree on certain statement.

Hence, Total Probability =P(A)P(B)+P(A')P(B')

=xy + (1-x)(1-y)

If statement is true then it means both A and B speaks truth.

∴ Required Probability =

Qus : 206NIMCET PYQ

The number of common tangents to the circle $x^2+y^2=4$ and $x^2+y^2-6x-8y=24$ is

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Solution

Qus : 207NIMCET PYQ

Qus : 213NIMCET PYQ

Let

be defined by

. Find

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Solution

Qus : 214NIMCET PYQ

A man is known to speak the truth 2 out of 3 times. He threw a dice cube with 1 to 6 on its faces and reports that it is 1. Then the probability that it is actually 1 is

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Solution

Qus : 215NIMCET PYQ

For what value of p, the polynomial $x^4-3x^3+2px^2-6$ is exactly divisible by $(x-1)$

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Solution

Qus : 216NIMCET PYQ

The slope of two-lines

differ by

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Solution

Qus : 217NIMCET PYQ

Let A and B be two events such that

and

where

stands for the complement of event A. Then the events A and B are

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Solution

Qus : 218NIMCET PYQ

If $F(\theta)=\begin{bmatrix}{\cos \theta} & {-\sin \theta} & {0} \\ {\sin \theta} & {\cos \theta} & {0} \\ {0} & {0} & {1}\end{bmatrix}$ , then $F(\theta)F(\alpha)$ is equal to

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Solution

Qus : 219NIMCET PYQ

If the radius of the circle changes at the rate of

, at what rate does the circle's area change when the radius is 10m?

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Solution

Qus : 220NIMCET PYQ

The mean and variance of a random variable X having binomial distribution are 4 and 2 respectively. The P(X = 1) is

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Solution

np = 4

npq = 2

q = 1/2, p = 1/2, n = 8

p(X = 1) = 8C1 (1/2)(1/2)7

Qus : 221NIMCET PYQ

If $\frac{n!}{2!(n-2)!}$ and $\frac{n!}{4!(n-4)!}$ are in the ratio 2:1, then the value of n is

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Solution

Qus : 228NIMCET PYQ

In an acute-angled ΔABC the least value of secA+secB+secC is

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Solution

Qus : 229NIMCET PYQ

Find a matrix X such that 2A + B + X = 0, whose A =

and B =

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Solution

Qus : 230NIMCET PYQ

The general value of $\theta$, satisfying the equation $\sin \theta=\frac{-1}{2},\, \tan \theta=\frac{1}{\sqrt[]{3}}$

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Solution

Qus : 231NIMCET PYQ

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Solution

Qus : 232NIMCET PYQ

If in a triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in HP, then sin A, sin B, sin C are in

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Solution

If altitudes of a triangle are in HP then its side will be in AP because sides are inverse proportion to height as area is constant. a, b, c are sides of triangle.

a, b, c are in A.P.

sin A, sin B, sin C are in A.P.

Qus : 233NIMCET PYQ

The area of the triangle formed by the vertices whose position vectors are $3\widehat{i}+\widehat{j}$ , $5\widehat{i}+2\widehat{j}+\widehat{k}$ , $\widehat{i}-2\widehat{j}+3\widehat{k}$ is

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Solution

Qus : 234NIMCET PYQ

then the value of

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Solution

Qus : 235NIMCET PYQ

α, β are the roots of the an equation $x^2- 2x cosθ + 1 = 0$, then the equation having roots αn and βn is

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Solution

Qus : 236NIMCET PYQ

The standard deviation of 20 numbers is 30. If each of the numbers is increased by 4, then the new standard deviation will be

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Solution

Qus : 237NIMCET PYQ

The circles whose equations are

and

will touch one another externally, if

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Solution

Qus : 238NIMCET PYQ

The equation (x-a)³+(x-b)³+(x-c)³ = 0 has

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Solution

Let f(x) = (x – a)³ + (x – b)³ + (x – c)³.

Then f'(x) = 3{(x – a)² + (x – b)² + (x –c)²}

clearly , f'(x) > 0 for all x.

so, f'(x) = 0 has no real roots.

Hence, f(x) = 0 has two imaginary and one real root

Qus : 239NIMCET PYQ

The function $f(x)=\frac{x}{1+x\tan x}$ , $0\leq x\leq\frac{\pi}{2}$ is maximum when

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Solution

Qus : 240NIMCET PYQ

The locus of the orthocentre of the triangle formed by the lines (1+p)x-py+p(1+p)=0, (1+p)(x-q)+q(1+ q)=0 and y=0 where p≠q is

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Solution

Straight Line

Qus : 241NIMCET PYQ

Three positive number whose sum is 21 are in arithmetic progression. If 2, 2, 14 are added to them respectively then resulting numbers are in geometric progression. Then which of the following is not among the three numbers?

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Solution

Let the three terms in A.P. be a – d, a, a + d.

given that a – d + a + a + d = 21

a = 7

then the three term in A.P. are 7 – d, 7, 7 + d

According to given condition 9 – d, 9, 21 + d are in G.P.

(9)² = (9 – d) (21 + d)

81 = 189 + 9d – 21d – d²

81 = 189 – 12d – d²

d² + 12d – 108 = 0

d(d + 18) – 6 (d + 18) = 0

(d – 6) (d + 18) = 0

We get, d = 6, –18

Putting d = 6 in the term 7 – d, 7, 7 + d we get 1, 7, 13.

Qus : 242NIMCET PYQ

If $f\colon R\rightarrow R$ is defined by $f(x)=\begin{cases}{\frac{x+2}{{x}^2+3x+2}} & {,\, if\, x\, \in R-\{-1,-2\}} \\ {-1} & {,if\, x=-2} \\ {0} & {,if\, x=-1}\end{cases}$ , then f(x) is continuous on the set

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Solution

Qus : 243NIMCET PYQ

Equation of the common tangents with a positive slope to the circle

and

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Solution

Qus : 244NIMCET PYQ

, then

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Solution

Qus : 245NIMCET PYQ

A polygon has 44 diagonals, the number of sides are

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Solution

Qus : 246NIMCET PYQ

The area enclosed between the curves

and

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Solution

Qus : 247NIMCET PYQ

The value of A that satisfies the equation asinA + bcosA = c is equal to

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Solution

Qus : 248NIMCET PYQ

The probability of occurrence of two events E and F are 0.25 and 0.50, respectively. the probability of their simultaneous occurrence is 0.14. the probability that neither E nor F occur is

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Solution

Qus : 249NIMCET PYQ

Equation of the line perpendicular to x-2y=1 and passing through (1,1) is

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Solution

Qus : 250NIMCET PYQ

If tan x = - 3/4 and 3π/2 < x < 2π, then the value of sin2x is

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Solution

Qus : 251NIMCET PYQ

If $y={\tan }^{-1}\lgroup{\frac{3x-{x}^3}{1-3{x}^2}}\rgroup\, ,\, \frac{-1}{\sqrt[]{3}}{\lt}x{\lt}\frac{1}{\sqrt[]{3}}$ then $\frac{dy}{dx}$ is

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Solution

Incorrect question

Qus : 252NIMCET PYQ

and

, then f(A)=

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Solution

Qus : 253NIMCET PYQ

Find the principal value of

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Solution

Qus : 254NIMCET PYQ

The value of $\tan 9{^{\circ}}-\tan 27{^{\circ}}-\tan 63{^{\circ}}+\tan 81{^{\circ}}$ is equal to

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Solution

Qus : 255NIMCET PYQ

9 balls are to be placed in 9 boxes and 5 of the balls cannot fit into 3 small boxes. The number of ways of arranging one ball in each of the boxes is

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Solution

First off all select 5 boxes out 6 boxes in which 5 big ball can fit then arrange these ball in these 5 boxes and then put remaining 4 ball in any remaining box.

So Ans is [(6C5)5!](4!) = 6!4! = 17280

Qus : 256NIMCET PYQ

If cosθ = 4/5 and cosϕ = 12/13, θ and ϕ both in the fourth quadrant, the value of cos( θ + ϕ )is

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4
Go to Discussion

Solution

Qus : 257NIMCET PYQ

In a triangle, if the sum of two sides is x and their product is y such that (x+z)(x-z)=y, where z is the third side of the triangle , then triangle is

1
2
3
4
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Solution

Qus : 258NIMCET PYQ

Which of the following function is the inverse of itself?

1
2
3
4
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Solution

Qus : 259NIMCET PYQ

The value of sin36^o is

1
2
3
4
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Solution

Qus : 260NIMCET PYQ

If $H_1,H_2,\ldots,H_n$ are n harmonic means between a and b $(b\ne a)$;,then $\frac{{{H}}_n+a}{{{H}}_n-a}+\frac{{{H}}_n+b}{{{H}}_n-b}$

1
2
3
4
Go to Discussion

Qus : 264NIMCET PYQ

In a survey where 100 students reported which subject they like, 32 students in total liked Mathematics, 38 students liked Business and 30 students liked Literature. Moreover, 7 students liked both Mathematics and Literature, 10 students liked both Mathematics and Business. 8 students like both Business and Literature, 5 students liked all three subjects. Then the number of people who liked exactly one subject is

1
2
3
4
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Solution

Qus : 265NIMCET PYQ

If non-zero numbers a, b, c are in A.P., then the straight line ax + by + c = 0 always passes through a fixed point, then the point is

1
2
3
4
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Solution

Since a, b and c are in A. P. 2b = a + c

a –2b + c =0

The line passes through (1, –2).

Qus : 266NIMCET PYQ

If $32\, \tan ^8\theta=2\cos ^2\alpha-3\cos \alpha$ and $3\, \cos \, 2\theta=1$, then the general value of $\alpha$ =

1
2
3
4
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Solution

Qus : 267NIMCET PYQ

If A and B are two events and

, the A and B are two events which are

1
2
3
4
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Solution

Qus : 268NIMCET PYQ

If the lines x + (a – 1)y + 1 = 0 and 2x + a²y – 1 = 0 are perpendicular, then the condition satisfies by a is

1
2
3
4
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Solution

Qus : 269NIMCET PYQ

In a triangle ABC, $a\cos ^2\frac{C}{2}+\, c\, \, {\cos }^2\frac{A}{2}=\frac{3b}{2}$ then the sides of the triangle are in

1
2
3
4
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Solution

Qus : 270NIMCET PYQ

are positive real numbers whose product is a fixed number c, then the minimum of

1
2
3
4
Go to Discussion

Qus : 275NIMCET PYQ

If x² + 3xy + 2y² – x – 4y – 6 = 0 represents a pair of straight lines, their point of intersection is

1
2
3
4
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Solution

Qus : 276NIMCET PYQ

The four geometric means between 2 and 64 are

1
2
3
4
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Solution

Qus : 277NIMCET PYQ

The coefficient of

in the expansion of

1
2
3
4
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Solution

Qus : 278NIMCET PYQ

The equation of the tangent line to the curve y = 2x sin x at the point (π/2, π), is

1
2
3
4
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Solution

Qus : 279NIMCET PYQ

The lines $px+qy=1$ and $qx+py=1$ are respectively the sides AB, AC of the triangle ABC and the base BC is bisected at $(p,q)$. Equation of the median of the triangle through the vertex A is

1
2
3
4
Go to Discussion

Solution

Qus : 280NIMCET PYQ

Let

and

be the roots f the equation

and

are the roots of the equation

, then the value of r,

1
2
3
4
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Solution

Qus : 281NIMCET PYQ

If the graph of y = (x – 2)² – 3 is shifted by 5 units up along y-axis and 2 units to the right along the x-axis, then the equation of the resultant graph is

1
2
3
4
Go to Discussion

Solution

When y= f (x) is shifted by k units to the right along x

– axis, it become y= f (x - k )

Hence, new equation of

graph is y = (x - 4)² + 2

Qus : 282NIMCET PYQ

If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k},\hat{i}+\hat{j}+c\hat{k}$ , $(a,b,c\ne1)$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

1
2
3
4
Go to Discussion

Solution

Qus : 283NIMCET PYQ

How many natural numbers smaller than

can be formed using the digits 1 and 2 only?

1
2
3
4
Go to Discussion

Solution

Qus : 284NIMCET PYQ

The direction cosines of the vector a = (- 2i + j – 5k) are

1
2
3
4
Go to Discussion

Solution

Qus : 285NIMCET PYQ

Let $\vec{a}=\hat{i}+\hat{j}$ and $\vec{b}=2\hat{i}-\hat{k}$, the point of intersection of the lines $\vec{r}\times\vec{a}=\vec{b}\times\vec{a}$ and $\vec{r}\times\vec{b}=\vec{a}\times\vec{b}$ is

1
2
3
4
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Solution

Qus : 286NIMCET PYQ

The equation of the hyperbola with centre at the region, length of the transverse axis is 6 and one focus (0, 4) is

1
2
3
4
Go to Discussion

Solution

Qus : 287NIMCET PYQ

If the system of equations $3x-y+4z=3$ , $x+2y-3z=-2$ , $6x+5y+λz=-3 $ has atleast one solution, then $λ=$

1
2
3
4
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Solution

Qus : 288NIMCET PYQ

and

, then the value of

1
2
3
4
Go to Discussion

Qus : 295NIMCET PYQ

If n is an integer between 0 to 21, then find a value of n for which the value of $n!(21-n)!$ is minimum

1
2
3
4
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Solution

Qus : 296NIMCET PYQ

is continuous for

1
2
3
4
Go to Discussion

Solution

Qus : 297NIMCET PYQ

Let

and

be three vector such that |

| = 2, |

| = 3, |

| = 5 and

= 0. The value of

1
2
3
4
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Solution

Qus : 298NIMCET PYQ

Suppose $A_1,A_2,\ldots,A_{30}$ are 30 sets each with five elements and $B_1,B_2,B_3,\ldots,B_n$ are n sets (each with three elements) such that $\bigcup ^{30}_{i=1}{{A}}_i={{\bigcup }}^n_{j=1}{{B}}_i=S\, $ and each element of S belongs to exactly ten of the $A_i$'s and exactly 9 of the $B^{\prime}_j$'s. Then $n=$

1
2
3
4
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Solution

Qus : 299NIMCET PYQ

and

are unit vectors, then

does not exceeds

1
2
3
4
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Solution

Qus : 300NIMCET PYQ

= (i + 2j - 3k) and

=(3i -j + 2k), then the angle between (

) and (

)

1
2
3
4
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Solution

Qus : 301NIMCET PYQ

The eccentric angle of the extremities of latus-rectum of the ellipse $\frac{{x}^2}{{a}^2}^{}+\frac{{y}^2}{{b}^2}^{}=1$ are given by

1
2
3
4
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Solution

Qus : 302NIMCET PYQ

The vector

lies in the plane of the vector

and

and bisects the angle between

and

. Then which of the following gives possible values of

and

1
2
3
4
Go to Discussion

Solution

Qus : 303NIMCET PYQ

The number of elements in the power set P(S) of the set S = [2, (1, 4)] is

1
2
3
4
Go to Discussion

Solution

Qus : 304NIMCET PYQ

If α≠β and $\alpha^2=5\alpha-3,\beta^2=5\beta-3$, then the equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is

1
2
3
4
Go to Discussion

Qus : 311NIMCET PYQ

Let A and B two sets containing four and two elements respectively. The number of subsets of the A × B, each having at least three elements is

1
2
3
4
Go to Discussion

Solution

n(A) = 4

n(B) = 2

Then the number of subsets in A*B is 2⁸= 256

Qus : 312NIMCET PYQ

If a number x is selected at random from natural numbers 1,2,…,100, then the probability for $x+\frac{100}{x}{\gt}29$ is

1
2
3
4
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Solution

Qus : 313NIMCET PYQ

, then value of

1
2
3
4
Go to Discussion

Solution

Qus : 314NIMCET PYQ

The slope of the function

1
2
3
4
Go to Discussion

Solution

Qus : 315NIMCET PYQ

If X and Y are two sets, then X∩Y ' ∩ (X∪Y) ' is

1
2
3
4
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Solution

Qus : 316NIMCET PYQ

Given it has extremum values at x=1 and x=2
⇒f′(1)=0 and f′(2)=0
Given f(x) is a fourth degree polynomial
Let $f(x)=a{x}^4+b{x}^3+c{x}^2+dx+e$

Given

$\lim _{{x}\rightarrow0}[1+\frac{f(x)}{{x}^2}]=3$

$\lim _{{x}\rightarrow0}\lbrack1+\frac{a{x}^4+b{x}^3+c{x}^2+\mathrm{d}x+e}{{x}^2}\rbrack=3$

$\lim _{{x}\rightarrow0}\lbrack1+a{x}^2+bx+c+\frac{d}{x}+\frac{e}{{x}^2}\rbrack=3$

For limit to have finite value, value of 'd' and 'e' must be 0

⇒d=0 & e=0

Substituting x=0 in limit

⇒ c+1=3

⇒ c=2

$f^{\prime}(x)=4a{x}^3+3b{x}^2+2cx+d$

$x=1$ and $x=2$ are extreme values,

⇒$f^{\prime}(1)=0$ and $f^{\prime}(2)=0

⇒ $4a+3b+4=0$ and $32a+12b+8=0$

By solving these equations

we get, $a=\frac{1}{2}$ and $b=-2$

So,

$f(x)=\frac{x^{4}}{2}-2x^{3}+2x^{2}$

⇒$f(x)=x^{2}(\frac{x^{2}}{2}-2x+2)$

⇒$f(2)=2^{2}(2-4+2)$

⇒$f(2)=0$

Qus : 324NIMCET PYQ

If P(1,2), Q(4,6), R(5,7) and S(a,b) are the vertices of a parallelogram PQRS, then

1
2
3
4
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Solution

Qus : 325NIMCET PYQ

The maximum value of 4 sin²x + 3 cos²x + sin(x/2) + cos(x/2) is

1
2
3
4
Go to Discussion

Solution

Qus : 326NIMCET PYQ

If $a\, \cos \theta+b\, \sin \, \theta=2$ and $a\, \sin \, \theta-b\, \cos \, \theta=3$ , then ${a}^{2^{}}+{b}^2=$

1
2
3
4
Go to Discussion

Solution

Qus : 327NIMCET PYQ

The solution of (e^x + 1) y dy = (y + 1) e^xdx is

1
2
3
4
Go to Discussion

Qus : 332NIMCET PYQ

The critical point and nature for the function f(x, y) = x² –2x + 2y² + 4y – 2 is

1
2
3
4
Go to Discussion

Solution

Qus : 333NIMCET PYQ

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

1
2
3
4
Go to Discussion

Solution

P(x)=50,

P(A ∩ B)' = 10

So P(A ∪ B) = 50 - 10 = 40.

So P(A ∩ B) = P(A) + P(B) - P(A ∪ B)

= 30 + 35 - 40 = 25

Solution Contribution by Priyanka Soni

Qus : 334NIMCET PYQ

There are 50 questions in a paper. Find the number of ways in which a student can attempt one or more questions :

1
2
3
4
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Solution

Qus : 335NIMCET PYQ

If y = cos²x², find dx/dy

1
2
3
4
Go to Discussion

Solution

Let y = (cos x²)²

= 2 (cos x2) sin x2 .2x

= -4 (cos x2) sin x2

Qus : 336NIMCET PYQ

If A = {4^x - 3x - 1 : x ∈ N} and B = {9(x - 1) : x ∈ N}, where N is the set of natural numbers, then

1
2
3
4
Go to Discussion

Solution

A = {0,9,54...}

B = {0,9,18,27...}

So, A ⊂ B

Qus : 337NIMCET PYQ

Consider the following frequency distribution table.

Class Interval	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency	180	$f_1$	34	180	136	$f_2$	50

If the total frequency is 686 and the median is 42.6, then the value of $f_1$;and $f_2$ are

1
2
3
4
Go to Discussion

1
2
3
4
Go to Discussion

Solution

^(10-3)C₆= ⁷C₆ = 7

Qus : 344NIMCET PYQ

If the points

lie in the region corresponding to the acute angle between the lines and then

1
2
3
4
Go to Discussion

Solution

Qus : 345NIMCET PYQ

1
2
3
4
Go to Discussion

Solution

Function is the form of

therefore using by L'Hospital rule

Again apply L'Hospital Rule,

Putting x = 0, we get

Qus : 346NIMCET PYQ

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?

1
2
3
4
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Solution

Qus : 347NIMCET PYQ

A problem in Mathematics is given to 3 students A, B, and C. If the probability of A solving the problem is 1/2 and B not solving it is 1/4 . The whole probability of the problem being solved is 63/64 , then what is the probability of solving it by C?

1
2
3
4
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Solution

Qus : 348NIMCET PYQ

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

1
2
3
4
Go to Discussion

Solution

Probability of winning a prize = $\frac{1}{25}$
Thus, probability of not winning = $1-\frac{1}{25}=\frac{24}{25}$

Qus : 349NIMCET PYQ

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:

1
2
3
4
Go to Discussion

Solution

A: 2, 3, 7, 1, 3, 2, 3

Increasing Order : A: 1, 2, 2, 3, 3, 3, 7

Mode = 3 (occurs maximum number of times)

Median = 3 (the middle term)

Mean =$\frac{(1+2+2+3+3+3+7)}{7}$

$=\frac{21}{7} = 3$

Hence. Mean=Median=Mode

Qus : 350NIMCET PYQ

Standard deviation for the following distribution is

Size of item	6	7	8	9	10	11	12
Frequency	3	6	9	13	8	5	4

1
2
3
4
Go to Discussion

Solution

Total number of items in the distribution = Σ f_i = 3 + 6 + 9 + 13 + 8 + 5 + 4 = 48.

The Mean (x̅) of the given set = $\rm \dfrac{\sum f_i x_i}{\sum f_i}$.

⇒ x̅ = $\rm \dfrac{6\times3+7\times6+8\times9+9\times13+10\times8+11\times5+12\times4}{48}=\dfrac{432}{48}$ = 9.

Let's calculate the variance using the formula: $\rm \sigma^2 =\dfrac{\sum x_i^2}{n}-\bar x^2$.

$\rm \dfrac{\sum {x_i}^2}{n}=\dfrac{6^2\times3+7^2\times6+8^2\times9+9^2\times13+10^2\times8+11^2\times5+12^2\times4}{48}=\dfrac{4012}{48}$ = 83.58.

∴ σ² = 83.58 - 9² = 83.58 - 81 = 2.58.

And, Standard Deviation (σ) = $\rm \sqrt{\sigma^2}=\sqrt{Variance}=\sqrt{2.58}$ ≈ 1.607.

Qus : 351NIMCET PYQ

1
2
3
4
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Solution

Qus : 352NIMCET PYQ

Roots of equation are $ax^2-2bx+c=0$ are n and m , then the value of $\frac{b}{an^2+c}+\frac{b}{am^2+c}$ is

1
2
3
4
Go to Discussion

Solution

Qus : 353NIMCET PYQ

The number of values of k for which the linear equations

4x + ky + z = 0

kx + 4y + z = 0

2x + 2y + z = 0

posses a non-zero solution is

1
2
3
4
Go to Discussion

Solution

Since, equation has non-zero solution.
Δ = 0

Qus : 354NIMCET PYQ

Let A = (a_ij) and B = (b_ij) be two square matricesof order n and det(A) denotes the determinant of A. Then, which of the following is not correct.

1
2
3
4
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Solution

Qus : 355NIMCET PYQ

The tangent to an ellipse x² + 16y² = 16 and making angel 60° with X-axis is:

1
2
3
4
Go to Discussion

Solution

Qus : 356NIMCET PYQ

Find the number of point(s) of intersection of the ellipse

and the circle x² + y² = 4

1
2
3
4
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Solution

Qus : 357NIMCET PYQ

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?

1
2
3
4
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Solution

Qus : 358NIMCET PYQ

If a + b + c = 0, then the value of

1
2
3
4
Go to Discussion

Solution

Qus : 359NIMCET PYQ

Find

1
2
3
4
Go to Discussion

Solution

Qus : 360NIMCET PYQ

, then x = 0 is

1
2
3
4
Go to Discussion

Solution

Qus : 369NIMCET PYQ

Qus : 375NIMCET PYQ

Two forces F₁ and F₂ are used to pull a car, which met an accident. The angle between the two forces is θ . Find the values of θ for which the resultant force is equal to

1
2
3
4
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Solution

Qus : 376NIMCET PYQ

Consider the following frequency distribution table.

Class interval	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency	180	$f_1$	34	180	136	$f_2$	50

If the total frequency is 685 & median is 42.6 then the values of $f_1$ and $f_2$ are

1
2
3
4
Go to Discussion

Solution

Qus : 377NIMCET PYQ

are four vectors such that

is collinear with

and

is collinear with

then

1
2
3
4
Go to Discussion

Solution

Qus : 378NIMCET PYQ

The sum of infinite terms of decreasing GP is equal to the greatest value of the function $f(x) = x^3 + 3x – 9$ in the interval [–2, 3] and difference between the first two terms is f '(0). Then the common ratio of the GP is

1
2
3
4
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Solution

Qus : 379NIMCET PYQ

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

1
2
3
4
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