2024

AISSCE(Board Exam)

Full Paper Analysis & Solutions

Mathematics

Multiple Choice Questions

1 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2024
#551
The position vectors of points P and Q are \(\vec{p}\) and \(\vec{q}\) respectively. The point R divides line segment PQ in the ratio 3:1 and S is the mid-point of line segment PR. The position vector of S is:
(A)
\(\frac{\vec{p}+3\vec{q}}{4}\)
(B)
\(\frac{\vec{p}+3\vec{q}}{8}\)
(C)
\(\frac{5\vec{p}+3\vec{q}}{4}\)
(D)
\(\frac{5\vec{p}+3\vec{q}}{8}\)
MEDIUM
2 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2024
#552
Let \(\vec{a}\) be any vector such that \(|\vec{a}|=a\) The value of \(|\vec{a}\times\hat{i}|^{2}+|\vec{a}\times\hat{j}|^{2}+|\vec{a}\times\hat{k}|^{2}\) is:
(A)
\(a^{2}\)
(B)
\(2a^{2}\)
(C)
\(3a^{2}\)
(D)
0
MEDIUM
3 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2024
#553
The vectors \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\), \(\vec{b}=\hat{i}-3\hat{j}-5\hat{k}\) and \(\vec{c}=-3\hat{i}+4\hat{j}+4\hat{k}\) represents the sides of
(A)
an equilateral triangle
(B)
an obtuse-angled triangle
(C)
an isosceles triangle
(D)
a right-angled triangle
MEDIUM
4 APPLY MCQ_SINGLE 1 Mark(s) 2024
#554
If \(\vec{a}\) and \(\vec{b}\) are two vectors such that \(|\vec{a}|=1,|\vec{b}|=2~and\vec{a}\cdot\vec{b}=\sqrt{3}\) then the angle between \(2\vec{a}\) and \(-\vec{b}\) is:
(A)
\(\frac{\pi}{6}\)
(B)
\(\frac{\pi}{3}\)
(C)
\(\frac{5\pi}{6}\)
(D)
\(\frac{11\pi}{6}\)
MEDIUM
5 APPLY MCQ_SINGLE 1 Mark(s) 2024
#555
If \(|\vec{a}|= 2\) and \(-3\le k\le2\), then \(|\vec{k}\vec{a}|\in\):
(A)
[-6, 4]
(B)
[0, 4]
(C)
[4, 6]
(D)
[0, 6]
MEDIUM
6 APPLY MCQ_SINGLE 1 Mark(s) 2024
#556
If \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}-\hat{k}\), then \(\vec{a}\) and \(\vec{b}\):
(A)
collinear vectors which are not parallel
(B)
parallel vectors
(C)
perpendicular vectors
(D)
unit vectors
MEDIUM
7 APPLY MCQ_SINGLE 1 Mark(s) 2024
#557
The unit vector perpendicular to both vectors \(\hat{i}+\hat{k}\) and \(\hat{i}-\hat{k}\) is:
(A)
\(2\hat{j}\)
(B)
\(\hat{j}\)
(C)
\(\frac{\hat{i}-\hat{k}}{\sqrt{2}}\)
(D)
\(\frac{\hat{i}+\hat{k}}{\sqrt{2}}\)
MEDIUM
8 APPLY MCQ_SINGLE 1 Mark(s) 2024
#558
For any two vectors \(\vec{a}\) and \(\vec{b}\), which of the following statements is always true?
(A)
\(\vec{a}.\vec{b}\ge
(B)
\vec{a}
(C)
(D)
\vec{b}
MEDIUM
9 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#559
The vector with terminal point \(A(2,-3,5)\) and initial point \(B(3, 4, 7)\) is:
(A)
\(\hat{i}-\hat{j}+2\hat{k}\)
(B)
\(\hat{i}+\hat{j}+2\hat{k}\)
(C)
\(-\hat{i}-\hat{j}-2\hat{k}\)
(D)
\(-\hat{i}+\hat{j}-2\hat{k}\)
EASY
10 APPLY MCQ_SINGLE 1 Mark(s) 2024
#560
Let \(\theta\) be the angle between two unit vectors \(\hat{a}\) and \(\hat{b}\) such that \(\sin\theta=\frac{3}{5}\). Then, \(\hat{a} \cdot \hat{b}\) is equal to:
(A)
\(\pm\frac{3}{5}\)
(B)
\(\pm\frac{3}{4}\)
(C)
\(\pm\frac{4}{5}\)
(D)
\(\pm\frac{4}{3}\)
MEDIUM
11 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#571
A function \(f:\mathbb{R}\rightarrow\mathbb{R}\) defined as \(f(x)=x^{2}-4x+5\) is:
(A)
injective but not surjective.
(B)
surjective but not injective.
(C)
both injective and surjective.
(D)
neither injective nor surjective.
MEDIUM
12 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#572
Let \(R_{+}\) denote the set of all non-negative real numbers. Then the function \(f:R_{+}\rightarrow R_{+}\) defined as \(f(x)=x^{2}+1\) is :
(A)
one-one but not onto
(B)
onto but not one-one
(C)
both one-one and onto
(D)
neither one-one nor onto
EASY
13 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#573
Let \(f:R_{+}\rightarrow[-5,\infty)\) be defined as \(f(x)=9x^{2}+6x-5\), where \(R_{+}\) is the set of all non-negative real numbers. Then, f is:
(A)
one-one
(B)
onto
(C)
bijective
(D)
neither one-one nor onto
MEDIUM
14 APPLY MCQ_SINGLE 1 Mark(s) 2024
#574
A function \(f:R_{+}\rightarrow R\) (where \(R_{+}\) is the set of all non-negative real numbers) defined by \(f(x)=4x+3\) is:
(A)
one-one but not onto
(B)
onto but not one-one
(C)
both one-one and onto
(D)
neither one-one nor onto
MEDIUM
15 REMEMBER MCQ_SINGLE 1 Mark(s) 2024
#590
For what value of k, the function given below is continuous at \(x=0\) ? \(f(x)=\begin{cases}\frac{\sqrt{4+x}-2}{x},&x\ne0\\ k,&x=0\end{cases}\)
(A)
0
(B)
\(\frac{1}{4}\)
(C)
1
(D)
4
MEDIUM
16 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#591
A function \(f(x)=|1-x+|x||\) is:
(A)
discontinuous at \(x=1\) only
(B)
discontinuous at \(x=0\) only
(C)
discontinuous at \(x=0,1\)
(D)
continuous everywhere
MEDIUM
17 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2024
#600
Derivative of \(e^{\sin^{2}x}\) with respect to cos x is:
(A)
\(sin~x~e^{sin^{2}x}\)
(B)
\(cos~x~e^{sin^{2}x}\)
(C)
\(-2~cos~x~e^{sin^{2}x}\)
(D)
\(-2~sin^{2}x~cos~x~e^{sin^{2}x}\)
MEDIUM
18 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#601
If \(xe^{y}=1\), then the value of \(\frac{dy}{dx}\) at \(x=1\) is :
(A)
-1
(B)
1
(C)
-e
(D)
\(-\frac{1}{e}\)
EASY
19 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2024
#602
Derivative of \(e^{2x}\) with respect to \(e^{x}\), is:
(A)
\(e^{x}\)
(B)
\(2e^{x}\)
(C)
\(2e^{2x}\)
(D)
\(2e^{3x}\)
MEDIUM
20 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2024
#603
The derivative of \(2^{x}\) w.r.t. \(3^{x}\) is:
(A)
\((\frac{3}{2})^{x} \frac{\log~2}{\log~3}\)
(B)
\((\frac{2}{3})^{x}\frac{\log~3}{\log~2}\)
(C)
\((\frac{2}{3})^{x}\frac{\log~2}{\log~3}\)
(D)
\((\frac{3}{2})^{x}\frac{\log~3}{\log~2}\)
MEDIUM
21 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#604
The derivative of \(\tan^{-1}(x^{2})\) w.r.t. x is :
(A)
\(\frac{x}{1+x^{4}}\)
(B)
\(\frac{2x}{1+x^{4}}\)
(C)
\(-\frac{2x}{1+x^{4}}\)
(D)
\(\frac{1}{1+x^{4}}\)
EASY
22 APPLY MCQ_SINGLE 1 Mark(s) 2024
#605
The derivative of \(\sin(x^{2})\) w.r.t. x, at \(x=\sqrt{\pi}\) is :
(A)
1
(B)
-1
(C)
\(-2\sqrt{\pi}\)
(D)
\(2\sqrt{\pi}\)
MEDIUM
23 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2024
#607
Given a curve \(y=7x-x^{3}\) and x increases at the rate of 2 units per second. The rate at which the slope of the curve is changing, when \(x=5\) is:
(A)
\(-60~units/sec\)
(B)
\(60~units/sec\)
(C)
\(-70~units/sec\)
(D)
\(-140~units/sec\)
HARD
24 APPLY MCQ_SINGLE 1 Mark(s) 2024
#608
The function \(f(x)=\frac{x}{2}+\frac{2}{x}\) has a local minima at x equal to:
(A)
2
(B)
1
(C)
0
(D)
-2
MEDIUM
25 APPLY MCQ_SINGLE 1 Mark(s) 2024
#609
The function \(f(x)=kx-\sin~x\) is strictly increasing for
(A)
\(k\gt1\)
(B)
\(k\lt1\)
(C)
\(k\gt-1\)
(D)
\(k\lt-1\)
MEDIUM
26 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2024
#610
If the sides of a square are decreasing at the rate of \(1.5~cm/s\) the rate of decrease of its perimeter is:
(A)
\(1.5~cm/s\)
(B)
\(6~cm/s\)
(C)
\(3~cm/s\)
(D)
\(2.25~cm/s\)
MEDIUM
27 APPLY MCQ_SINGLE 1 Mark(s) 2024
#611
The function \(f(x)=x^{3}-3x^{2}+12x-18\) is:
(A)
strictly decreasing on R
(B)
strictly increasing on R
(C)
neither strictly increasing nor strictly decreasing on R
(D)
strictly decreasing on \((-\infty, 0)\)
MEDIUM
28 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#612
Let \(f(x)\) be a continuous function on [a, b] and differentiable on (a, b). Then, this function \(f(x)\) is strictly increasing in (a, b) if
(A)
\(f^{\prime}(x)\lt;0\), \(\forall x\in(a,b)\)
(B)
\(f^{\prime}(x)\gt;0\), \(\forall x\in(a,b)\)
(C)
\(f^{\prime}(x)=0\), \(\forall x\in(a,b)\)
(D)
\(f(x)\gt;0\), \(\forall x\in(a,b)\)
EASY
29 APPLY MCQ_SINGLE 1 Mark(s) 2024
#621
\(\int\frac{1}{x(\log~x)^{2}}dx\) is equal to:
(A)
\(2~log(log~x)+c\)
(B)
\(-\frac{1}{log~x}+c\)
(C)
\(\frac{(log~x)^{3}}{3}+c\)
(D)
\(\frac{3}{(log~x)^{3}}+c\)
MEDIUM
30 APPLY MCQ_SINGLE 1 Mark(s) 2024
#629
The value of \(\int_{-1}^{1}x|x|dx\) is:
(A)
\(\frac{1}{6}\)
(B)
\(\frac{1}{3}\)
(C)
\(-\frac{1}{6}\)
(D)
0
MEDIUM
31 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#630
The value of \(\int_{0}^{3}\frac{dx}{\sqrt{9-x^{2}}}\) is:
(A)
\(\frac{\pi}{6}\)
(B)
\(\frac{\pi}{4}\)
(C)
\(\frac{\pi}{2}\)
(D)
\(\frac{\pi}{18}\)
MEDIUM
32 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#631
\(\int_{-a}^{a}f(x)dx=0,\) if :
(A)
\(f(-x)=f(x)\)
(B)
\(f(-x)=-f(x)\)
(C)
\(f(a-x)=f(x)\)
(D)
\(f(a-x)=-f(x)\)
MEDIUM
33 APPLY MCQ_SINGLE 1 Mark(s) 2024
#632
\(\int_{0}^{\pi/2}\frac{\sin~x-\cos~x}{1+\sin~x~\cos~x}dx\) is equal to:
(A)
\(\pi\)
(B)
Zero (0)
(C)
\(\int_{0}^{\pi/2}\frac{2~\sin~x}{1+\sin~x~\cos~x}dx\)
(D)
\(\frac{\pi^{2}}{4}\)
MEDIUM
34 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#633
\(\int_{a}^{b}f(x)dx\) is equal to:
(A)
\(\int_{a}^{b}f(a-x)dx\)
(B)
\(\int_{a}^{b}f(a+b-x)dx\)
(C)
\(\int_{a}^{b}f(x-(a+b))dx\)
(D)
\(\int_{a}^{b}f((a-x)+(b-x))dx\)
MEDIUM
35 APPLY MCQ_SINGLE 1 Mark(s) 2024
#639
Area of the region bounded by curve \(y^{2}=4x\) and the X-axis between \(x=0\) and \(x=1\) is:
(A)
\(\frac{2}{3}\)
(B)
\(\frac{8}{3}\)
(C)
3
(D)
\(\frac{4}{3}\)
MEDIUM
36 ANALYZE MCQ_SINGLE 1 Mark(s) 2024
#643
The order of the differential equation \(\frac{d^{4}y}{dx^{4}}-sin(\frac{d^{2}y}{dx^{2}})=5\) is:
(A)
4
(B)
3
(C)
2
(D)
not defined
MEDIUM
37 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#644
The integrating factor of the differential equation \((x+2y^{2})\frac{dy}{dx}=y(y>0)\) is:
(A)
\(\frac{1}{x}\)
(B)
x
(C)
y
(D)
\(\frac{1}{y}\)
MEDIUM
38 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#645
The general solution of the differential equation \(x~dy+y~dx=0\) is:
(A)
\(xy=c\)
(B)
\(x+y=c\)
(C)
\(x^{2}+y^{2}=c^{2}\)
(D)
\(log~y=log~x+c\)
EASY
39 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#646
\(x~\log~x\frac{dy}{dx}+y=2~\log~x\) is an example of a :
(A)
variable separable differential equation.
(B)
homogeneous differential equation.
(C)
first order linear differential equation.
(D)
differential equation whose degree is not defined.
MEDIUM
40 APPLY MCQ_SINGLE 1 Mark(s) 2024
#647
The degree of the differential equation \((y^{\prime\prime})^{2}+(y^{\prime})^{3}=x~\sin(y^{\prime})\) is:
(A)
1
(B)
2
(C)
3
(D)
not defined
MEDIUM
41 APPLY MCQ_SINGLE 1 Mark(s) 2024
#648
The differential equation \(\frac{dy}{dx}=F(x,y)\) will not be a homogeneous differential equation, if \(F(x,y)\) is :
(A)
\(\cos~x-\sin(\frac{y}{x})\)
(B)
\(\frac{y}{x}\)
(C)
\(\frac{x^{2}+y^{2}}{xy}\)
(D)
\(\cos^{2}(\frac{x}{y})\)
MEDIUM
42 APPLY MCQ_SINGLE 1 Mark(s) 2024
#649
The order and degree of the differential equation \([1+(\frac{dy}{dx})^{2}]^{3}=\frac{d^{2}y}{dx^{2}}\) respectively are:
(A)
1, 2
(B)
2, 3
(C)
2, 1
(D)
2, 6
MEDIUM
43 APPLY MCQ_SINGLE 1 Mark(s) 2024
#650
The integrating factor of the differential equation \((1-x^{2})\frac{dy}{dx}+xy=ax,\) \(-1 \lt x\lt 1\) is:
(A)
\(\frac{1}{x^{2}-1}\)
(B)
\(\frac{1}{\sqrt{x^{2}-1}}\)
(C)
\(\frac{1}{1-x^{2}}\)
(D)
\(\frac{1}{\sqrt{1-x^{2}}}\)
MEDIUM
44 APPLY MCQ_SINGLE 1 Mark(s) 2024
#660
The Cartesian equation of the line passing through the point (1, -3, 2) and parallel to the line: \(\vec{r}=(2+\lambda)\hat{i}+\lambda\hat{j}+(2\lambda-1)\hat{k}\) is:
(A)
\(\frac{x-1}{2}=\frac{y+3}{0}=\frac{z-2}{-1}\)
(B)
\(\frac{x+1}{1}=\frac{y-3}{1}=\frac{z+2}{2}\)
(C)
\(\frac{x+1}{2}=\frac{y-3}{0}=\frac{z+2}{-1}\)
(D)
\(\frac{x-1}{1}=\frac{y+3}{1}=\frac{z-2}{2}\)
MEDIUM
45 APPLY MCQ_SINGLE 1 Mark(s) 2024
#661
The angle which the line \(\frac{x}{1}=\frac{y}{-1}=\frac{z}{0}\) makes with the positive direction of Y-axis is:
(A)
\(\frac{5\pi}{6}\)
(B)
\(\frac{3\pi}{4}\)
(C)
\(\frac{5\pi}{4}\)
(D)
\(\frac{7\pi}{4}\)
MEDIUM
46 APPLY MCQ_SINGLE 1 Mark(s) 2024
#662
The lines \(\frac{1-x}{2}=\frac{y-1}{3}=\frac{z}{1}\) and \(\frac{2x-3}{2p}=\frac{y}{-1}=\frac{z-4}{7}\) are perpendicular to each other for p equal to:
(A)
\(-\frac{1}{2}\)
(B)
\(\frac{1}{2}\)
(C)
2
(D)
3
MEDIUM
47 APPLY MCQ_SINGLE 1 Mark(s) 2024
#663
The vector equation of a line passing through the point (1, -1, 0) and parallel to Y-axis is :
(A)
\(\vec{r}=\hat{i}-\hat{j}+\lambda(\hat{i}-\hat{j})\)
(B)
\(\vec{r}=\hat{i}-\hat{j}+\lambda\hat{j}\)
(C)
\(\vec{r}=\hat{i}-\hat{j}+\lambda\hat{k}\)
(D)
\(\vec{r}=\lambda\hat{j}\)
MEDIUM
48 APPLY MCQ_SINGLE 1 Mark(s) 2024
#664
If a line makes an angle of \(\frac{\pi}{4}\) with the positive directions of both x-axis and z-axis, then the angle which it makes with the positive direction of y-axis is:
(A)
0
(B)
\(\frac{\pi}{4}\)
(C)
\(\frac{\pi}{2}\)
(D)
\(\pi\)
MEDIUM
49 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#665
If \(\alpha\), \(\beta\) and \(\gamma\) are the angles which a line makes with positive directions of x, y and z axes respectively, then which of the following is not true?
(A)
\(cos^{2}\alpha+cos^{2}\beta+cos^{2}\gamma=1\)
(B)
\(sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma=2\)
(C)
\(cos~2\alpha+cos~2\beta+cos~2\gamma=-1\)
(D)
\(cos~\alpha+cos~\beta+cos~\gamma=1\)
MEDIUM
50 APPLY MCQ_SINGLE 1 Mark(s) 2024
#666
If a line makes an angle of \(30^{\circ}\) with the positive direction of x-axis, \(120^{\circ}\) with the positive direction of y-axis, then the angle which it makes with the positive direction of z-axis is:
(A)
\(90^{\circ}\)
(B)
\(120^{\circ}\)
(C)
\(60^{\circ}\)
(D)
\(0^{\circ}\)
MEDIUM
51 APPLY MCQ_SINGLE 1 Mark(s) 2024
#667
Direction ratios of a vector parallel to line \(\frac{x-1}{2}=-y=\frac{2z+1}{6}\) are:
(A)
\(2,-1,6\)
(B)
\(2, 1, 6\)
(C)
\(2, 1, 3\)
(D)
\(2,-1, 3\)
MEDIUM
52 APPLY MCQ_SINGLE 1 Mark(s) 2024
#668
The coordinates of the foot of the perpendicular drawn from the point \((0, 1, 2)\) on the x-axis are given by:
(A)
\((1,0,0)\)
(B)
\((2,0,0)\)
(C)
\((\sqrt{5},0,0)\)
(D)
\((0,0,0)\)
MEDIUM
53 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#669
The distance of point \(P(a,b,c)\) from y-axis is :
(A)
b
(B)
\(b^{2}\)
(C)
\(\sqrt{a^{2}+c^{2}}\)
(D)
\(a^{2}+c^{2}\)
EASY
54 APPLY MCQ_SINGLE 1 Mark(s) 2024
#670
If the direction cosines of a line are \(\sqrt{3}k, \sqrt{3}k\), \(\sqrt{3}k,\) then the value of k is:
(A)
\(\pm1\)
(B)
\(\pm\sqrt{3}\)
(C)
\(\pm3\)
(D)
\(\pm\frac{1}{3}\)
MEDIUM
55 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2024
#674
Of the following, which group of constraints represents the feasible region given below ?
(A)
\(x+2y\le76\), \(2x+y\ge104\), \(x, y\ge0\)
(B)
\(x+2y\le76\), \(2x+y\le104,\) \(x, y\ge0\)
(C)
\(x+2y\ge76\), \(2x+y\le104\), \(x, y\ge0\)
(D)
\(x+2y\ge76\), \(2x+y\ge104,\) \(x, y\ge0\)
MEDIUM
56 REMEMBER MCQ_SINGLE 1 Mark(s) 2024
#675
The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called :
(A)
feasible solutions
(B)
constraints
(C)
optimal solutions
(D)
infeasible solutions
EASY
57 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#676
The common region determined by all the constraints of a linear programming problem is called :
(A)
an unbounded region
(B)
an optimal region
(C)
a bounded region
(D)
a feasible region
EASY
58 APPLY MCQ_SINGLE 1 Mark(s) 2024
#677
The number of corner points of the feasible region determined by constraints \(x\ge0, y\ge0, x+y\ge4\) is:
(A)
0
(B)
1
(C)
2
(D)
3
MEDIUM
59 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#678
A linear programming problem deals with the optimization of a/an:
(A)
logarithmic function
(B)
linear function
(C)
quadratic function
(D)
exponential function
EASY
60 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#686
If A and B are events such that \(P(A/B)=P(B/A)\ne0,\) then :
(A)
\(A\subset B\), but \(A\ne B\)
(B)
\(A=B\)
(C)
\(A\cap B=\phi\)
(D)
\(P(A)=P(B)\)
MEDIUM
61 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#687
Let E and F be two events such that \(P(E)=0\cdot1\), \(P(F)=0\cdot3,\) \(P(E\cup F)=0\cdot4\) then \(P(F|E)\) is:
(A)
0.6
(B)
0.4
(C)
0.5
(D)
0
MEDIUM
62 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024
#688
Let E be an event of a sample space S of an experiment, then \(P(S|E)=\)
(A)
\(P(S\cap E)\)
(B)
\(P(E)\)
(C)
1
(D)
0
EASY
63 APPLY MCQ_SINGLE 1 Mark(s) 2024
#689
If \(P(A|B)=P(A^{\prime}|B)\), then which of the following statements is true?
(A)
\(P(A)=P(A^{\prime})\)
(B)
\(P(A)=2~P(B)\)
(C)
\(P(A\cap B)=\frac{1}{2}P(B)\)
(D)
\(P(A\cap B)=2~P(B)\)
MEDIUM
64 APPLY MCQ_SINGLE 1 Mark(s) 2024
#729
If A is a square matrix of order 3 such that the value of \(|adj\cdot A|=8,\) then the value of \(|A^{T}|\) is:
(A)
\(\sqrt{2}\)
(B)
\(-\sqrt{2}\)
(C)
8
(D)
\(2\sqrt{2}\)
MEDIUM
65 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2024
#732
\(If~A=[\begin{matrix}2&1\\ -4&-2\end{matrix}].\) then the value of \(I-A+A^{2}-A^{3}+...is\):
(A)
\([\begin{matrix}-1&-1\\ 4&3\end{matrix}]\)
(B)
\([\begin{matrix}3&1\\ -4&-1\end{matrix}]\)
(C)
\([\begin{matrix}0&0\\ 0&0\end{matrix}]\)
(D)
\([\begin{matrix}1&0\\ 0&1\end{matrix}]\)
MEDIUM

VSA

1 APPLY VSA 2 Mark(s) 2024
#1244
Evaluate : $\sec^{2}(\tan^{-1}\frac{1}{2})+cosec ^{2}(\cot^{-1}\frac{1}{3})$
MEDIUM
2 UNDERSTAND VSA 2 Mark(s) 2024
#1245
If $x=e^{x/y}$, prove that $\frac{dy}{dx}=\frac{log~x-1}{(log~x)^{2}}$
MEDIUM
3 REMEMBER VSA 2 Mark(s) 2024
#1246
Check the differentiability of $f(x)=\begin{cases}x^{2}+1,&0\le x<1\\ 3-x,&1\le x\le2\end{cases}$ at $x=1.$
MEDIUM
4 UNDERSTAND VSA 2 Mark(s) 2024
#1247
Evaluate: $\int_{0}^{\pi/2}sin~2x~cos~3x~dx$
MEDIUM
5 UNDERSTAND VSA 2 Mark(s) 2024
#1248
Given $\frac{d}{dx}F(x)=\frac{1}{\sqrt{2x-x^{2}}}$ and $F(1)=0$, find $F(x)$.
MEDIUM
6 REMEMBER VSA 2 Mark(s) 2024
#1249
Find the position vector of point C which divides the line segment joining points A and B having position vectors $\hat{i}+2\hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively in the ratio $4:1$ externally. Further, find $|\vec{AB}|:|\vec{BC}|$ .
MEDIUM
7 REMEMBER VSA 2 Mark(s) 2024
#1250
Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors. Prove that $|\vec{a}\times\vec{b}|\le|\vec{a}||\vec{b}|$ . State the condition under which equality holds, i.e., $|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|$
EASY
8 COMPETENCY UNDERSTAND VSA 2 Mark(s) 2024
#1268
If $f(x)=|tan~2x|$, then find the value of $f^{\prime}(x)$ at $x=\frac{\pi}{3}$
MEDIUM
9 UNDERSTAND VSA 2 Mark(s) 2024
#1269
If $y=cosec(cot^{-1}x)$, then prove that $\sqrt{1+x^{2}}\frac{dy}{dx}-x=0$ .
MEDIUM
10 REMEMBER VSA 2 Mark(s) 2024
#1270
If M and m denote the local maximum and local minimum values of the function $f(x)=x+\frac{1}{x}(x\ne0)$ respectively, find the value of $(M-m)$
MEDIUM
11 UNDERSTAND VSA 2 Mark(s) 2024
#1271
Find: $\int\frac{e^{4x}-1}{e^{4x}+1}dx$
MEDIUM
12 ANALYZE VSA 2 Mark(s) 2024
#1272
Show that $f(x)=e^{x}-e^{-x}+x-tan^{-1}x$ is strictly increasing in its domain.
MEDIUM
13 UNDERSTAND VSA 2 Mark(s) 2024
#1288
Check whether the function $f(x)=x^{2}|x|$ is differentiable at $x=0$ or not.
MEDIUM
14 UNDERSTAND VSA 2 Mark(s) 2024
#1289
If $y=\sqrt{tan\sqrt{x}}$ , prove that $\sqrt{x}\frac{dy}{dx}=\frac{1+y^{4}}{4y}$
HARD
15 UNDERSTAND VSA 2 Mark(s) 2024
#1290
Show that the function $f(x)=4x^{3}-18x^{2}+27x-7$ has neither maxima nor minima.
MEDIUM
16 UNDERSTAND VSA 2 Mark(s) 2024
#1291
Find: $\int x\sqrt{1+2x}dx$
MEDIUM
17 APPLY VSA 2 Mark(s) 2024
#1292
Evaluate: $\int_{0}^{\frac{\pi^{2}}{4}}\frac{sin\sqrt{x}}{\sqrt{x}}dx$
MEDIUM
18 UNDERSTAND VSA 2 Mark(s) 2024
#1293
If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $(\vec{a}+\vec{b})\perp\vec{a}$ and $(2\vec{a}+\vec{b})\perp\vec{b}$ , then prove that $|\vec{b}|=\sqrt{2}|\vec{a}|$.
MEDIUM
19 COMPETENCY UNDERSTAND VSA 2 Mark(s) 2024
#1294
In the given figure, ABCD is a parallelogram. If $\vec{AB}=2\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{DB}=3\hat{i}-6\hat{j}+2\hat{k}$ , then find $\vec{AD}$ and hence find the area of parallelogram ABCD.
MEDIUM
20 UNDERSTAND VSA 2 Mark(s) 2024
#1312
If $y=cos^{3}(sec^{2}2t)$, find $\frac{dy}{dt}$ .
MEDIUM
21 UNDERSTAND VSA 2 Mark(s) 2024
#1313
If $x^{y}=e^{x-y},$ prove that $\frac{dy}{dx}=\frac{log~x}{(1+log~x)^{2}}.$
MEDIUM
22 APPLY VSA 2 Mark(s) 2024
#1314
Find the interval in which the function $f(x)=x^{4}-4x^{3}+10$ is strictly decreasing.
MEDIUM
23 COMPETENCY UNDERSTAND VSA 2 Mark(s) 2024
#1315
The volume of a cube is increasing at the rate of $6~cm^{3}/s.$ How fast is the surface area of cube increasing, when the length of an edge is 8 cm?
MEDIUM
24 APPLY VSA 2 Mark(s) 2024
#1316
Find: $\int\frac{1}{x(x^{2}-1)}dx.$
MEDIUM
25 UNDERSTAND VSA 2 Mark(s) 2024
#1334
Verify whether the function f defined by $f(x)=\begin{cases}x~sin(\frac{1}{x}),x\ne0\\ 0&,x=0\end{cases}$ is continuous at $x=0$ or not.
MEDIUM
26 APPLY VSA 2 Mark(s) 2024
#1335
Check for differentiability of the function f defined by $f(x)=|x-5|$, at the point $x=5$.
MEDIUM
27 COMPETENCY UNDERSTAND VSA 2 Mark(s) 2024
#1336
The area of the circle is increasing at a uniform rate of $2~cm^{2}/sec$. How fast is the circumference of the circle increasing when the radius $r=5$ cm?
MEDIUM
28 UNDERSTAND VSA 2 Mark(s) 2024
#1337
Find: $\int cos^{3}x~e^{log~sin~x}dx$
MEDIUM
29 UNDERSTAND VSA 2 Mark(s) 2024
#1338
Find: $\int\frac{1}{5+4x-x^{2}}dx$
MEDIUM
30 APPLY VSA 2 Mark(s) 2024
#1339
Find the vector equation of the line passing through the point (2, 3, -5) and making equal angles with the co-ordinate axes.
MEDIUM

SA

1 UNDERSTAND SA 3 Mark(s) 2024
#1251
If $x~cos(p+y)+cos~p~sin(p+y)=0$ prove that $cos~p\frac{dy}{dx}=-cos^{2}(p+y),$ where p is a constant.
MEDIUM
2 REMEMBER SA 3 Mark(s) 2024
#1252
Find the value of a and b so that function f defined as : $$ f(x) = \begin{cases} \frac{x-2}{|x-2|} + a, & \text{if } x < 2 \\ a+b, & \text{if } x = 2 \\ \frac{x-2}{|x-2|} + b, & \text{if } x > 2 \end{cases} $$ is a continuous function.
MEDIUM
3 APPLY SA 3 Mark(s) 2024
#1253
Find the intervals in which the function $f(x)=\frac{log~x}{x}$ is strictly increasing or strictly decreasing.
MEDIUM
4 REMEMBER SA 3 Mark(s) 2024
#1254
Find the absolute maximum and absolute minimum values of the function f given by $f(x)=\frac{x}{2}+\frac{2}{x}$ , on the interval [1, 2].
EASY
5 UNDERSTAND SA 3 Mark(s) 2024
#1255
Find: $\int\frac{x^{2}+1}{(x^{2}+2)(x^{2}+4)}dx$
MEDIUM
6 UNDERSTAND SA 3 Mark(s) 2024
#1256
Find: $\int\frac{2+sin~2x}{1+cos~2x}e^{x}dx$
MEDIUM
7 UNDERSTAND SA 3 Mark(s) 2024
#1257
Evaluate: $\int_{0}^{\pi/4}\frac{1}{sin~x+cos~x}dx$
MEDIUM
8 REMEMBER SA 3 Mark(s) 2024
#1258
Solve the following linear programming problem graphically: Maximise $z=4x+3y.$ subject to the constraints $x+y\le800$, $2x+y\le1000$, $x\le400$, $x,y\ge0$.
MEDIUM
9 COMPETENCY UNDERSTAND SA 3 Mark(s) 2024
#1259
The chances of P, Q and R getting selected as CEO of a company are in the ratio 4: 1: 2 respectively. The probabilities for the company to increase its profits from the previous year under the new CEO, P, Q or R are 0-3, 0-8 and 0.5 respectively. If the company increased the profits from the previous year, find the probability that it is due to the appointment of R as CEO.
HARD
10 REMEMBER SA 3 Mark(s) 2024
#1273
If $x=e^{cos~3t}$ and $y=e^{sin~3t}$ , prove that $\frac{dy}{dx}=-\frac{y~log~x}{x~log~y}$
MEDIUM
11 UNDERSTAND SA 3 Mark(s) 2024
#1274
Show that: $\frac{d}{dx}(|x|)=\frac{x}{|x|},x\ne0$
MEDIUM
12 UNDERSTAND SA 3 Mark(s) 2024
#1275
Evaluate : $\int_{-2}^{2}\sqrt{\frac{2-x}{2+x}}dx$
HARD
13 APPLY SA 3 Mark(s) 2024
#1276
Find: $\int\frac{1}{x[(log~x)^{2}-3~log~x-4]}dx$
MEDIUM
14 UNDERSTAND SA 3 Mark(s) 2024
#1277
Find the particular solution of the differential equation given by $2xy+y^{2}-2x^{2}\frac{dy}{dx}=0$ $y=2$, when $x=1.$
MEDIUM
15 UNDERSTAND SA 3 Mark(s) 2024
#1278
Find the general solution of the differential equation : $y~dx=(x+2y^{2})~dy$
MEDIUM
16 COMPETENCY UNDERSTAND SA 3 Mark(s) 2024
#1279
The position vectors of vertices of $\Delta$ ABC are $A(2\hat{i}-\hat{j}+\hat{k}),$ $B(\hat{i}-3\hat{j}-5\hat{k})$ and $C(3\hat{i}-4\hat{j}-4\hat{k})$ Find all the angles of $\Delta$ AВС.
HARD
17 UNDERSTAND SA 3 Mark(s) 2024
#1280
A pair of dice is thrown simultaneously. If X denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of X.
MEDIUM
18 UNDERSTAND SA 3 Mark(s) 2024
#1281
Find: $\int x^{2}\cdot sin^{-1}(x^{3/2})dx$
MEDIUM
19 REMEMBER SA 3 Mark(s) 2024
#1295
A relation R on set $A=\{1,2,3,4,5\}$ is defined as $R=\{(x,y):|x^{2}-y^{2}|<8\}$. Check whether the relation R is reflexive, symmetric and transitive.
EASY
20 REMEMBER SA 3 Mark(s) 2024
#1296
A function f is defined from $R\rightarrow R$ as $f(x)=ax+b$, such that $f(1)=1$ and $f(2)=3$ Find function $f(x)$. Hence, check whether function $f(x)$ is one-one and onto or not.
MEDIUM
21 REMEMBER SA 3 Mark(s) 2024
#1297
If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y),$ prove that $\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}} .$
HARD
22 APPLY SA 3 Mark(s) 2024
#1298
If $y=(tan~x)^{x}$, then find $\frac{dy}{dx}$ .
MEDIUM
23 UNDERSTAND SA 3 Mark(s) 2024
#1299
Find: $\int\frac{x^{2}}{(x^{2}+4)(x^{2}+9)}dx$
MEDIUM
24 UNDERSTAND SA 3 Mark(s) 2024
#1300
Evaluate: $\int_{1}^{3}(|x-1|+|x-2|+|x-3|)dx$
MEDIUM
25 COMPETENCY UNDERSTAND SA 3 Mark(s) 2024
#1301
Find the particular solution of the differential equation given by $x^{2}\frac{dy}{dx}-xy=x^{2}cos^{2}(\frac{y}{2x})$ given that when $x=1$, $y=\frac{\pi}{2}$
HARD
26 COMPETENCY UNDERSTAND SA 3 Mark(s) 2024
#1302
Solve the following linear programming problem graphically: Maximise $z=500x+300y,$ subject to constraints $x+2y\le12$, $2x+y\le12$, $4x+5y\ge20$, $x\ge0$, $y\ge0$
MEDIUM
27 COMPETENCY UNDERSTAND SA 3 Mark(s) 2024
#1303
E and F are two independent events such that $P(\overline{E})=0\cdot6$ and $P(E\cup F)=0\cdot6$ Find $P(F)$ and $P(\overline{E}\cup\overline{F})$
MEDIUM
28 SA 3 Mark(s) 2024
#1317
Given that $y=(sin~x)^{x}\cdot x^{sin~x}+a^{x},$ find $\frac{dy}{dx}$
29 COMPETENCY APPLY SA 3 Mark(s) 2024
#1318
Evaluate $\int_{0}^{\frac{\pi}{4}}\frac{x~dx}{1+cos~2x+sin~2x}$
MEDIUM
30 UNDERSTAND SA 3 Mark(s) 2024
#1319
Find: $\int e^{x}[\frac{1}{(1+x^{2})^{\frac{3}{2}}}+\frac{x}{\sqrt{1+x^{2}}}]dx$
HARD
31 UNDERSTAND SA 3 Mark(s) 2024
#1320
Find: $\int\frac{3x+5}{\sqrt{x^{2}+2x+4}}dx$
MEDIUM
32 REMEMBER SA 3 Mark(s) 2024
#1321
Find the particular solution of the differential equation $\frac{dy}{dx}=y~cot~2x,$ given that $y(\frac{\pi}{4})=2.$
MEDIUM
33 COMPETENCY UNDERSTAND SA 3 Mark(s) 2024
#1322
Find the particular solution of the differential equation $(xe^{\frac{y}{x}}+y)dx=x~dy$, given that $y=1$ when $x=1$
HARD
34 REMEMBER SA 3 Mark(s) 2024
#1323
Solve the following linear programming problem graphically: Maximise $Z=2x+3y$ subject to the constraints: $x+y\le6$, $x\ge2$, $y\le3$, $x,y\ge0$
MEDIUM
35 COMPETENCY UNDERSTAND SA 3 Mark(s) 2024
#1324
A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.
MEDIUM
36 COMPETENCY UNDERSTAND SA 3 Mark(s) 2024
#1325
A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.
HARD
37 UNDERSTAND SA 3 Mark(s) 2024
#1340
Find $\frac{dy}{dx}$ , if $(cos~x)^{y}=(cos~y)^{x}.$
MEDIUM
38 REMEMBER SA 3 Mark(s) 2024
#1341
If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, prove that $\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$
HARD
39 REMEMBER SA 3 Mark(s) 2024
#1342
If $x=a~sin^{3}\theta$, $y=b~cos^{3}\theta$ then find $\frac{d^{2}y}{dx^{2}}$ at $\theta=\frac{\pi}{4}$
HARD
40 REMEMBER SA 3 Mark(s) 2024
#1343
Evaluate : $\int_{0}^{\pi}\frac{e^{cos~x}}{e^{cos~x}+e^{-cos~x}}d~x$
MEDIUM
41 UNDERSTAND SA 3 Mark(s) 2024
#1344
Find: $\int\frac{2x+1}{(x+1)^{2}(x-1)}dx$
HARD
42 REMEMBER SA 3 Mark(s) 2024
#1345
Find the particular solution of the differential equation $\frac{dy}{dx}-2xy=3x^{2}e^{x^{2}};y(0)=5.$
MEDIUM
43 UNDERSTAND SA 3 Mark(s) 2024
#1346
Solve the following differential equation $x^{2}dy+y(x+y)dx=0$
MEDIUM
44 COMPETENCY UNDERSTAND SA 3 Mark(s) 2024
#1347
Find a vector of magnitude 4 units perpendicular to each of the vectors $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\hat{k}$ and hence verify your answer.
MEDIUM
45 REMEMBER SA 3 Mark(s) 2024
#1348
The random variable X has the following probability distribution where a and b are some constants: $P(X)$ for X=1 is 0.2, X=2 is a, X=3 is a, X=4 is 0.2, X=5 is b. If the mean $E(X)=3$, then find values of a and b and hence determine $P(X\ge3)$
MEDIUM

LA

1 REMEMBER LA 5 Mark(s) 2024
#1260
A relation R on set $A=\{-4,-3,-2,-1,0,1,2,3,4\}$ be defined as $R=\{(x,y):x+y$ is an integer divisible by 2). Show that R is an equivalence relation. Also, write the equivalence class [2].
MEDIUM
2 COMPETENCY REMEMBER LA 5 Mark(s) 2024
#1261
It is given that function $f(x)=x^{4}-62x^{2}+ax+9$ attains local maximum value at $x=1$ Find the value of 'a', hence obtain all other points where the given function f(x) attains local maximum or local minimum values.
MEDIUM
3 COMPETENCY UNDERSTAND LA 5 Mark(s) 2024
#1262
The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that volume of cylinder so formed is maximum.
HARD
4 REMEMBER LA 5 Mark(s) 2024
#1263
Using integration, find the area of the region enclosed between the circle $x^{2}+y^{2}=16$ and the lines $x=-2$ and $x=2.$
MEDIUM
5 COMPETENCY REMEMBER LA 5 Mark(s) 2024
#1264
Find the equation of the line passing through the point of intersection of the lines $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and $\frac{x-1}{0}=\frac{y}{-3}=\frac{z-7}{2}$ and perpendicular to these given lines.
HARD
6 COMPETENCY REMEMBER LA 5 Mark(s) 2024
#1265
Two vertices of the parallelogram ABCD are given as $A(-1,2,1)$ and $B(1,-2,5)$. If the equation of the line passing through C and D is $\frac{x-4}{1}=\frac{y+7}{-2}=\frac{z-8}{2}$ then find the distance between sides AB and CD. Hence, find the area of parallelogram ABCD.
HARD
7 REMEMBER LA 5 Mark(s) 2024
#1282
Show that a function $f:R\rightarrow R$ defined by $f(x)=\frac{2x}{1+x^{2}}$ is neither one-one nor onto. Further, find set A so that the given function $f:R\rightarrow A$ becomes an onto function.
MEDIUM
8 UNDERSTAND LA 5 Mark(s) 2024
#1283
A relation R is defined on $N\times N$ (where N is the set of natural numbers) as: $(a, b)~R~(c,d)\Leftrightarrow a-c=b-d$ Show that R is an equivalence relation.
MEDIUM
9 COMPETENCY UNDERSTAND LA 5 Mark(s) 2024
#1284
Find the equation of the line which bisects the line segment joining points $A(2,3,4)$ and $B(4,5,8)$ and is perpendicular to the lines $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$
HARD
10 COMPETENCY APPLY LA 5 Mark(s) 2024
#1285
Solve the following system of equations, using matrices: $\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4$ $\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1$ , $\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$ where x, y, $z\ne0$
HARD
11 REMEMBER LA 5 Mark(s) 2024
#1286
If $A=\begin{bmatrix}1&cot~x\\ -cot~x&1\end{bmatrix}$ show that $A^{\prime}A^{-1}=\begin{bmatrix}-cos~2x&-sin~2x\\ sin~2x&-cos~2x\end{bmatrix}$
MEDIUM
12 REMEMBER LA 5 Mark(s) 2024
#1287
If $A_{1}$ denotes the area of region bounded by $y^{2}=4x,$ $x=1$ and x-axis in the first quadrant and $A_{2}$ denotes the area of region bounded by $y^{2}=4x,$ $x=4$, find $A_{1}:A_{2}$.
MEDIUM
13 COMPETENCY UNDERSTAND LA 5 Mark(s) 2024
#1304
If $A=[\begin{bmatrix}1&-2&0\\ 2&-1&-1\\ 0&-2&1\end{bmatrix}],$ find $A^{-1}$ and use it to solve the following system of equations: $x-2y=10$, $2x-y-z=8$, $-2y+z=7$
HARD
14 REMEMBER LA 5 Mark(s) 2024
#1305
If $A=[\begin{bmatrix}-1&a&2\\ 1&2&x\\ 3&1&1\end{bmatrix}]$ and $A^{-1}=[\begin{bmatrix}1&-1&1\\ -8&7&-5\\ b&y&3\end{bmatrix}],$ find the value of $(a+x)-(b+y)$.
HARD
15 REMEMBER LA 5 Mark(s) 2024
#1306
Evaluate: $\int_{0}^{\frac{\pi}{4}}\frac{sin~x+cos~x}{9+16~sin~2x}dx$
MEDIUM
16 APPLY LA 5 Mark(s) 2024
#1307
Evaluate: $\int_{0}^{\frac{\pi}{2}}sin~2x~tan^{-1}(sin~x)dx$
MEDIUM
17 REMEMBER LA 5 Mark(s) 2024
#1308
Using integration, find the area of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1,$ included between the lines $x=-2$ and $x=2$.
MEDIUM
18 COMPETENCY REMEMBER LA 5 Mark(s) 2024
#1309
The image of point $P(x,y,z)$ with respect to line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ is $P^{\prime}(1,0,7)$ Find the coordinates of point P.
HARD
19 COMPETENCY REMEMBER LA 5 Mark(s) 2024
#1326
Sketch the graph of $y=x|x|$ and hence find the area bounded by this curve, X-axis and the ordinates $x=-2$ and $x=2,$ using integration.
MEDIUM
20 UNDERSTAND LA 5 Mark(s) 2024
#1327
Using integration, find the area bounded by the ellipse $9x^{2}+25y^{2}=225$, the lines $x=-2,$ $x=2$, and the X-axis.
MEDIUM
21 UNDERSTAND LA 5 Mark(s) 2024
#1328
Let $A=R-\{5\}$ and $B=R-\{1\}$. Consider the function $f:A\rightarrow B$, defined by $f(x)=\frac{x-3}{x-5}$ Show that f is one-one and onto.
MEDIUM
22 UNDERSTAND LA 5 Mark(s) 2024
#1329
Check whether the relation S in the set of real numbers R defined by $S=\{(a,b)$: where $a-b+\sqrt{2}$ is an irrational number is reflexive, symmetric or transitive.
MEDIUM
23 COMPETENCY APPLY LA 5 Mark(s) 2024
#1330
If $A=[\begin{bmatrix}2&1&-3\\ 3&2&1\\ 1&2&-1\end{bmatrix}],$ find $A^{-1}$ and hence solve the following system of equations: $2x+y-3z=13$, $3x+2y+z=4$, $x+2y-z=8$
HARD
24 COMPETENCY REMEMBER LA 5 Mark(s) 2024
#1331
Find the distance between the line $\frac{x}{2}=\frac{2y-6}{4}=\frac{1-z}{-1}$ and another line parallel to it passing through the point (4, 0, -5).
MEDIUM
25 COMPETENCY REMEMBER LA 5 Mark(s) 2024
#1332
If the lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-7}$ are perpendicular to each other, find the value of k and hence write the vector equation of a line perpendicular to these two lines and passing through the point (3, -4, 7).
HARD
26 COMPETENCY REMEMBER LA 5 Mark(s) 2024
#1349
If $A=[\begin{bmatrix}1&2&-3\\ 2&0&-3\\ 1&2&0\end{bmatrix}],$ then find $A^{-1}$ and hence solve the following system of equations: $x+2y-3z=1$, $2x-3z=2$, $x+2y=3$
HARD
27 COMPETENCY REMEMBER LA 5 Mark(s) 2024
#1350
Find the product of the matrices $[\begin{bmatrix}1&2&-3\\ 2&3&2\\ 3&-3&-4\end{bmatrix}][\begin{bmatrix}-6&17&13\\ 14&5&-8\\ -15&9&-1\end{bmatrix}]$ and hence solve the system of linear equations: $x+2y-3z=-4$, $2x+3y+2z=2$, $3x-3y-4z=11$
HARD
28 UNDERSTAND LA 5 Mark(s) 2024
#1351
Find the area of the region bounded by the curve $4x^{2}+y^{2}=36$ using integration.
MEDIUM
29 COMPETENCY UNDERSTAND LA 5 Mark(s) 2024
#1352
Find the co-ordinates of the foot of the perpendicular drawn from the point (2, 3, -8) to the line $\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}$ Also, find the perpendicular distance of the given point from the line.
HARD
30 COMPETENCY REMEMBER LA 5 Mark(s) 2024
#1353
Find the shortest distance between the lines $L_{1}$ & $L_{2}$ given below :
$L_{1}$: The line passing through (2, -1, 1) and parallel to $\frac{x}{1}=\frac{y}{1}=\frac{z}{3}$ and
$L_{2}:\vec{r}=\hat{i}+(2\mu+1)\hat{j}-(\mu+2)\hat{k}$
HARD
31 COMPETENCY UNDERSTAND LA 5 Mark(s) 2024
#1354
Solve the following L.P.P. graphically: Maximise $Z=60x+40y$ Subject to $x+2y\le12$, $2x+y\le12$, $4x+5y\ge20$, $x,y\ge0$
MEDIUM