CBSE 12th Standard Maths Subject Ncert Exemplar MCQ Questions 2021 Part - II | QB365

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CBSE 12th Standard Maths Subject Ncert Exemplar MCQ Questions 2021 Part - II

Published on: 22/05/2021

QB365 Provides the updated NCERT Examplar Questions for Class 12 Maths, and also provide the detail solution for each and every ncert examplar questions , QB365 will give all kind of study materials will help to get more marks

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Questions:

Multiple Choice Question

21 x 1 = 21

The coordinates of the foot of the perpendicular drawn from the point (2. 5. 7) on the X-axis are given by

(a)

(2.0.0)

(b)

(0,5,0)

(c)

(0,0,7)

(d)

(0,5,7)

If \(|\vec{a}|=10,|\vec{b}|=2\) and \(\vec{a} \cdot \vec{b}=12\) 12, then the value of \(|\vec{a} \times \vec{b}|\) is

(a)

(b)

(c)

(d)

The angle between two vectors \(\vec{a} \text { and } \vec{b}\) with magnitude, \(\sqrt{3} \text { and } 4\) respectively and \(\vec{a} \cdot \vec{b}=2 \sqrt{3}\) is

(a)

\(\frac{\pi}{6}\)

(b)

\(\frac{\pi}{3}\)

(c)

\(\frac{\pi}{2}\)

(d)

\(\frac{5 \pi}{2}\)

The position vector of the point which divides the join of points with position vectors \(\vec{a}+\vec{b}\) and \(2 \vec{a}-\vec{b}\) in the ratio 1: 2 is

(a)

\(\frac{3 \vec{a}+2 \vec{b}}{3}\)

(b)

\(\vec{a}\)

(c)

\(\frac{5 \vec{a}-\vec{b}}{3}\)

(d)

\(\frac{4 \vec{a}+\vec{b}}{3}\)

The magnitude of the vector \(6 \hat{i}+2 \hat{j}+3 \hat{k}\) is

(a)

(b)

(c)

(d)

\(\int_{a+c}^{b+c} f(x) d x\) is equal to

(a)

\(\int_{a}^{b} f(x-c) d x\)

(b)

\(\int_{a}^{b} f(x+c) d x\)

(c)

\(\int_{a}^{b} f(x) d x\)

(d)

\(\int_{a-c}^{b-c} f(x) d x\)

\(\int_{0}^{\pi / 2} \cos x e^{\sin x} d x\) is equal to

(a)

e + 1

(b)

e -1

(c)

(d)

-e

\(\int \frac{x+\sin x}{1+\cos x} d x\) is equal to

(a)

\(\log |1+\cos x|+C\)

(b)

\(\log |x+\sin x|+C\)

(c)

\(x-\tan \frac{x}{2}+C\)

(d)

\(x \cdot \tan \frac{x}{2}+C\)

The tangent to the curve \(y=e^{2 x}\) at the point (0,1) meets X-axis at

(a)

(0,1

(b)

\(\left(-\frac{1}{2}, 0\right)\)

(c)

(2,0)

(d)

(0,2)

10.

The curve \(y=x^{1 / 5}\) has at (0, 0)

(a)

a vertical tangent (parallel to Y-axis)

(b)

a horizontal tangent (parallel to X-axis)

(c)

an oblique tangent

(d)

no tangent

11.

Which of the following functions is decreasing on \(\left(0, \frac{\pi}{2}\right)\)?

(a)

sin2x

(b)

tanx

(c)

cosx

(d)

cos3x

12.

The function \(f(x)=\tan x-x\)

(a)

always increases

(b)

always decreases

(c)

never increases

(d)

sometimes increases and sometimes decreases

13.

If \(y=x(x-3)^{2}\) decreases for the values of x given by

(a)

\(1<x<3\)

(b)

\(x<0\)

(c)

\(x>0\)

(d)

\(0<x<\frac{3}{2}\)

14.

A ladder, 5 m long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/s, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 m from the wall is

(a)

\(\frac{1}{10} \mathrm{rad} / \mathrm{s}\)

(b)

\(\frac{1}{20} \mathrm{rad} / \mathrm{s}\)

(c)

\(20 \mathrm{rad} / \mathrm{s}\)

(d)

10 rad/s

15.

If \(A=\left|\begin{array}{llr} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{array}\right|\) then A^-I exists, if

(a)

\(\lambda=2\)

(b)

\(\lambda \neq 2\)

(c)

\(\lambda \neq-2\)

(d)

None of these

16.

For any two matrices A and B, we have

(a)

AB=BA

(b)

AB ≠BA

(c)

AB = 0

(d)

None of these

17.

If matrix \(A=\left[a_{i j}\right]_{2 \times 2},\ where \ a_{i j}=\left\{\begin{array}{l}1, \text { if } i \neq j \\ 0, \text { if } i=j\end{array}\right.\) Then \(A^{2}\) is equal to

(a)

(b)

(c)

(d)

None ofthese

18.

On using elementary column operations \(C_{2} \rightarrow C_{2}-2 C_{1}\) in the following matrix equation \(\left[\begin{array}{cc}1 & -3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right]\), we have

(a)

\(\left[\begin{array}{cc}1 & -5 \\ 0 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ -2 & 2\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ 2 & 0\end{array}\right]\)

(b)

\(\left[\begin{array}{cc}1 & -5 \\ 0 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ -0 & 2\end{array}\right]\)

(c)

\(\left[\begin{array}{cc}1 & -5 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}1 & -3 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & 1 \\ -2 & 4\end{array}\right]\)

(d)

\(\left[\begin{array}{cc}1 & -5 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & -5 \\ 2 & 0\end{array}\right]\)

19.

The matrix \(\left[\begin{array}{ccc}0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0\end{array}\right]\) is a

(a)

diagonal matrix

(b)

symmetric matrix

(c)

skew-symmetric matrix

(d)

scalar matrix

20.

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

(a)

720

(b)

120

(c)

(d)

None of these

21.

For the set A = {1, 2, 3}, define a relation R in the set A as follows
R = {(1, 1), (2,2), (3, 3), (1, 3)}
Then, the ordered pair to be added to R to make it the smallest equivalence relation is

(a)

(1, 3)

(b)

(3, 1)

(c)

(2, 1)

(d)

(1, 2)

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Answers Key:

Multiple Choice Question

21 x 1 = 21

(d)
(0,5,7)

(d)
16

(b)
\(\frac{\pi}{3}\)

(b)
7

(b)
\(\int_{a}^{b} f(x+c) d x\)

(b)
e -1

The equation of curve is y = e^2x
Since, it passes through the point (0, 1).
\(\begin{aligned} &\therefore \quad \cdot \frac{d y}{d x}=e^{2 x} \cdot 2=2 \cdot e^{2 x}\\ &\Rightarrow\left(\frac{d y}{d x}\right)_{(0,1)}=2 \cdot e^{2 \cdot 0}=2=\text { Slope of tangent to the curve } \end{aligned}\)
Equation of tangent is y -1 = 2(x - 0)
\(\Rightarrow \quad y=2 x+1\)
Since, tangent to curve y = e^2X at the point (0, 1) meets X-axis i.e., y = 0
\(\therefore \quad 0=2 x+1 \Rightarrow x=-\frac{1}{2}\)
So, the required point is \(\left(\frac{-1}{2}, 0\right)\)

10.

We have,\(y=x^{1 / 5}\)
\(\begin{array}{l} \Rightarrow \quad \frac{d y}{d x}=\frac{1}{5} x^{\frac{1}{5}-1}=\frac{1}{5} x^{-4 / 5} \\ \therefore \quad\left(\frac{d y}{d x}\right)_{(0,0)}=\frac{1}{5} \times(0)^{-4 / 5}=\infty \end{array}\)
S6, the curve Y = x^1/5 has a vertical tangent at (0, 0), which is parallel to Y-axis.

11.

In the interval \(\left(0, \frac{\pi}{2}\right), f(x)=\cos x\)
\(\Rightarrow \quad f^{\prime}(x)=-\sin x\)
which gives \(f^{\prime}(x)<0 \text { in }\left(0, \frac{\pi}{2}\right)\)
Hence, f(x) = cos x is decreasing in \(\left(0, \frac{\pi}{2}\right)\)

12.

We have, f(x) = tan x - x
\(\therefore \quad f^{\prime}(x)=\sec ^{2} x-1 \Rightarrow f^{\prime}(x) \geq 0, \forall x \in R\)
So, f(x) always increases

13.

We have,\(y=x(x-3)^{2}\)
\( \therefore \quad \frac{d y}{d x} =x \cdot 2(x-3) \cdot 1+(x-3)^{2} \cdot 1 \\ =2 x^{2}-6 x+x^{2}+9-6 x=3 x^{2}-12 x+9 \\ =3\left(x^{2}-3 x-x+3\right)=3(x-3)(x-1) \)
So, y = x(x - 3)² decreases for (1, 3).
[since, y' < 0 for all x E (1,3), hence y is decreasing on (1, 3)]

14.

(b)
\(\frac{1}{20} \mathrm{rad} / \mathrm{s}\)

15.

\(A^{-1} \text {exist iff }|A| \neq 0\)

16.

(d)
None of these

17.

(a)
I

18.

Given, \(\left[\begin{array}{rr}1 & -3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right]\)
On applying \(C_{2} \rightarrow C_{2}-2 C_{1},\)
we get \(\left[\begin{array}{rr}1 & -3-2 \\ 2 & 4-4\end{array}\right]=\left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1-6 \\ 2 & 4-4\end{array}\right]\)

\(\Rightarrow \left[\begin{array}{rr}1 & -5 \\ 2 & 0\end{array}\right]=\left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{rr}3 & -5 \\ 2 & 0\end{array}\right]\)

19.

\(\text { Let } A=\left[\begin{array}{ccc} 0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0 \end{array}\right] \text { . Then, } A^{\prime}=-A\)

20.

One-one onto mapping is possible only ifn(A) = n(B)

21.

Clearly R is reflexive and transitive. For R to be symmetric we should add (3, 1) in R.