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Published on: 22/05/2021
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Questions + Answers key
Take MCQ Maths Test1.
If a die is thrown and a card is selected at random from a deck of 52 playing cards, then the probability of getting an even number on the die and a spade card is
\(\frac{1}{2}\)
\(\frac{1}{4}\)
\(\frac{1}{8}\)
\(\frac{3}{4}\)
2.
If two events are independent, then
they must be mutually exclusive
the sum of their probabilities must be equal to 1
Both (a) and (b) are correct
None ofthe above is correct
3.
The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15,15), (0, 20). Let Z = px + qy where \(q>0\) .Then, the condition on p and q, so that the maximum of Z occurs at both the
points (15,15) and (0, 20) is
p = q
p = 2q
q = 2p
q = 3p
4.
If the plane \(2 x-3 y+6 z-11=0\) makes an angle \(\sin ^{-1} \alpha\) with X-axis, then the value of \(\alpha\) is
\(\frac{\sqrt{3}}{2}\)
\(\frac{\sqrt{2}}{3}\)
\(\frac{2}{7}\)
\(\frac{3}{7}\)
5.
The locus represented by xy + yz = 0 is
a pair of perpendicular lines
a pair of parallel lines
a pair of parallel planes
a pair of perpendicular planes
6.
The distance of the plane \(\vec{r}\left(\frac{2}{7} \hat{i}+\frac{3}{7} \hat{j}-\frac{6}{7} \hat{k}\right)=1\) from the origin is
1
7
\(\frac{1}{7}\)
None of these
7.
The value of \(\lambda\) for which the vectors \(\vec{a}=2 \hat{i}+\lambda \hat{j}+\hat{k} \text { and } \vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}\) are orthogonal os
0
1
\(\frac{3}{2}\)
\(\frac{-5}{2}\)
8.
The vector having initial and terminal points as \((2,5,0) \text { and }(-3,7,4)\) respectively is
\(-\hat{i}+12 \hat{j}+4 \hat{k}\)
\(5 \hat{i}+2 \hat{j}-4 \hat{k}\)
\(-5 \hat{i}+2 \hat{j}+4 \hat{k}\)
\(\hat{i}+\hat{j}+\hat{k}\)
9.
The magnitude of the vector \(6 \hat{i}+2 \hat{j}+3 \hat{k}\) is
5
7
12
1
10.
\(\int_{-\pi / 4}^{\pi / 4} \frac{d x}{1+\cos 2 x}\) is equal to
1
2
3
4
11.
\(\int \frac{x+\sin x}{1+\cos x} d x\) is equal to
\(\log |1+\cos x|+C\)
\(\log |x+\sin x|+C\)
\(x-\tan \frac{x}{2}+C\)
\(x \cdot \tan \frac{x}{2}+C\)
12.
The tangent to the curve \(y=e^{2 x}\) at the point (0,1) meets X-axis at
(0,1
\(\left(-\frac{1}{2}, 0\right)\)
(2,0)
(0,2)
13.
The curve \(y=x^{1 / 5}\) has at (0, 0)
a vertical tangent (parallel to Y-axis)
a horizontal tangent (parallel to X-axis)
an oblique tangent
no tangent
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
\(\frac{1}{10} \mathrm{rad} / \mathrm{s}\)
\(\frac{1}{20} \mathrm{rad} / \mathrm{s}\)
\(20 \mathrm{rad} / \mathrm{s}\)
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
\(\lambda=2\)
\(\lambda \neq 2\)
\(\lambda \neq-2\)
None of these
16.
Iff \(f(x)=\left|\begin{array}{ccc} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{array}\right|\) then
f(a) = 0
f(b) = 0
f(0) = 0
f(1) = 0
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
I
A
0
None ofthese
18.
On using elementary row operation \(R_{1} \rightarrow R_{1}-3 R_{2}\) in the following matrix equation \(\left[\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right]=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right],\) we have
\(\left[\begin{array}{cc}-5 & -7 \\ 3 & 3\end{array}\right]=\left[\begin{array}{cc}1-7 \\ 0 & 3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]\)
\(\left[\begin{array}{cc}-5 & -7 \\ 3 & 3\end{array}\right]=\left[\begin{array}{cc}1 & 2 \\ 0 & 3\end{array}\right]\left[\begin{array}{cc}-1 & -3 \\ 1 & 1\end{array}\right]\)
\(\left[\begin{array}{cc}-5 & -7 \\ 3 & 3\end{array}\right]=\left[\begin{array}{cc}1 & 2 \\ 1 & -7\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]\)
\(\left[\begin{array}{rr}4 & 2 \\ -5 & -7\end{array}\right]=\left[\begin{array}{cc}1 & 2 \\ -3 & -3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]\)
19.
The matrix \(\left[\begin{array}{ccc}0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0\end{array}\right]\) is a
diagonal matrix
symmetric matrix
skew-symmetric matrix
scalar matrix
20.
If A and B are square matrices of the sameorder, then (A + B) (A - B) is equal to
A2-B2
A2 - BA - AB - B2
A2 - B2 + BA - AB
A2 - BA + B2 + AB
21.
Let \(f: R \rightarrow R\) be the functions defined by \(f(x)=x^{3}+5\). Then, \(f^{-1}(x)\) is
\((x+5)^{1 / 3}\)
\((x-5)^{1 / 3}\)
\((5-x)^{1 / 3}\)
5-x
22.
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
720
120
0
None of these
1.
(c)
\(\frac{1}{8}\)
2.
(d)
None ofthe above is correct
3.
(d)
q = 3p
4.
(c)
\(\frac{2}{7}\)
5.
(d)
a pair of perpendicular planes
6.
(a)
1
7.
(d)
\(\frac{-5}{2}\)
8.
9.
(b)
7
10.
(a)
1
11.
12.
The equation of curve is y = e2x
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 = e2X 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 \(\left(\frac{-1}{2}, 0\right)\)
13.
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 = x1/5 has a vertical tangent at (0, 0), which is parallel to Y-axis.
14.
(b)
\(\frac{1}{20} \mathrm{rad} / \mathrm{s}\)
15.
\(A^{-1} \text {exist iff }|A| \neq 0\)
16.
Clearly,
\( f(a) =\left|\begin{array}{ccc} 0 & 0 & a-b \\ 2 a & 0 & a-c \\ a+b & a+c & 0 \end{array}\right| \)
\(=[(a-b)\{2 a \cdot(a+c)\}] \neq 0 \)
\( \therefore \ f(b) =\left|\begin{array}{ccc} 0 & b-a & 0 \\ b+a & 0 & b-c \\ 2 b & b+c & 0 \end{array}\right|\)
\( =-(b-a)[2 b(b-c)] \)
\( =-2 b(b-a)(b-c) \neq 0 \)
17.
(a)
I
18.
Given, \(\left[\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right]=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]\)
On applying \(R_{1} \rightarrow R_{1}-3 R_{2},\) we get
\(\left[\begin{array}{cc} 4-9 & 2-9 \\ 3 & 3 \end{array}\right]=\left[\begin{array}{cc} 1-0 & 2-9 \\ 0 & 3 \end{array}\right]\left[\begin{array}{cc} 2 & 0 \\ 1 & 1 \end{array}\right]\)
\(\Rightarrow\)\(\left[\begin{array}{rr} -5 & -7 \\ 3 & 3 \end{array}\right]=\left[\begin{array}{rr} 1 & -7 \\ 0 & 3 \end{array}\right] \cdot\left[\begin{array}{ll} 2 & 0 \\ 1 & 1 \end{array}\right]\)
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.
(A + B) (A - B) = A(A - B) + B(A - B)
= A2 - AB + BA - B2
21.
Let \(y=r(x)=x^{2}+5\)
\(\Rightarrow\) Then, \(x^{3}=y-5 \rightarrow x=(y-5)^{\prime \prime}\)
\(\Rightarrow f^{-1}(y)=(y-5)^{3} \quad\left[\because y=f(x) \rightarrow x=f^{-1}\langle y\}\right.\)
or \(f^{-1}(x)=\{x-5)^{10}\)
22.
One-one onto mapping is possible only ifn(A) = n(B)
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