Mathematics Practice Questions
Exam Duration: 120 Mins Total Questions : 100
If f,\(g:R:\rightarrow R\), such that \(f:x\rightarrow sin\quad x\)and \(g:x\rightarrow \)x2,then gof equals
- (a)
sin x2
- (b)
x2 six x
- (c)
sin2 x
- (d)
none of these
If arg(z)<0, then arg.(-z)-arg (z), equals
- (a)
\(\pi \)
- (b)
\(-\pi \)
- (c)
\(-\cfrac { \pi }{ 2 } \)
- (d)
\(\cfrac { \pi }{ 2 } \)
If \(\left| \begin{matrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{matrix} \right| =0\) then
- (a)
a,b,c are in A.P.
- (b)
a,b,v are in G.P.
- (c)
\(\frac { 1 }{ a } ,\frac { 1 }{ b } ,\frac { 1 }{ c } \) are in A.P.
- (d)
\(\frac { 1 }{ a } ,\frac { 1 }{ b } ,\frac { 1 }{ c } \) are in G.P.
For a square matrix A, it is given that A.At=I; then A is a
- (a)
symmetric matrix
- (b)
skew-symmetric matrix
- (c)
diagonal matrix
- (d)
orthogonal matrix
The equation \(\sqrt { x+1 } \)- \(\sqrt { x-1 } \) = \(\sqrt { 4x-1 } \), has
- (a)
no solution
- (b)
one solution
- (c)
two solutions
- (d)
more than two solutions
If x=2+22/3+21/3 then the value of x3-6x2+6x is
- (a)
3
- (b)
2
- (c)
1
- (d)
None of these
The sides of an euilateral triangle, a square and a regular hexagon circumscribed in a circle are in
- (a)
A.P
- (b)
G.P
- (c)
H.P
- (d)
None of these
The equation of the common tangent touching the circle (x-3)2 + y2 =9 and the parabola y2 =4x above the x-axis,is
- (a)
\(\sqrt3y=3x+1\)
- (b)
\(\sqrt3y=-(x+3)\)
- (c)
\(\sqrt3.y=x+3\)
- (d)
\(\sqrt3y=-(3x+1)\)
If \(\vec { a } \times \vec { b } \quad =\quad \vec { b } \times \vec { c } \quad \neq \quad 0\) and \(\vec { a } +\vec { c } \neq 0\), then
- (a)
\(\vec { a } +\vec { c } \) is perpendicular to \(\vec { b } \)
- (b)
\(\vec { a } +\vec { c } \) is parallel to \(\vec { b } \)
- (c)
\(\vec { a } +\vec { c } =\vec { b } \)
- (d)
NONE OF THESE
Let \(\vec { a } \quad =\quad \hat { i } -\hat { k } ,\quad \vec { b } \quad =\quad \hat { xi } +\hat { j } +(1-x)\hat { k } \quad and\quad \vec { c } \quad =\quad y\hat { i } +x\hat { j } +(1+x-y)\hat { k } \). Then \([\vec { a } \vec { b } \vec { c } ]\) depends on
- (a)
only x
- (b)
only y
- (c)
neither x nor y
- (d)
both x and y
The mean of variate x is \(\overline{x}\). Then mean of the variate \(x+10\over k\), is
- (a)
\(\overline{x}\over k\)
- (b)
\(\overline{x}+10\over k\)
- (c)
\({\overline{x} \over k} +10\)
- (d)
\(k\ \overline{x}+10\)
Which of the following statements is false. If M and N are any two events, the probability that exactly one of them recurs is
- (a)
\(P(M)+P(N)-2P(M\cap N)\)
- (b)
\(P(M)+P(N)-P(M\cap N)\)
- (c)
\(P({ M }^{ C })+P({ N }^{ C })-2P({ M }^{ C }\cap { N }^{ C })\)
- (d)
\(P({ M }\cap { N }^{ C })+P({ M }^{ C }\cap N)\)
\(\underset { x->0 }{ lim } \frac { \sqrt { \frac { 1-cos2x }{ 2 } } }{ x } \)is:
- (a)
1
- (b)
-1
- (c)
0
- (d)
None of these
The chord joining the points where x = p and x = q on the curve y = ax2 + bx + c is parallel to the tangent at the point on the curve whose abscissa is
- (a)
\(\frac{1}{2}\) (p + q)
- (b)
\(\frac{1}{2}\) (p - q)
- (c)
\(\frac{pq}{c}\)
- (d)
NONE OF THESE
The length of the tangent at the point \(t=\frac { \pi }{ 2 } \) on the curve \(x=\left( t+\sin { t } \right) ,y=\left( 1-\cos { t } \right) \) is
- (a)
1
- (b)
\(\sqrt { 2 } \)
- (c)
\(\sqrt { 3 } \)
- (d)
NONE OF THESE
The value of the integral \(\int _{ 1 }^{ 3 }{ \sqrt { 3+{ x }^{ 3 } } } dx\) lies in the interval
- (a)
(1,3)
- (b)
(2,30)
- (c)
\((4,2\sqrt { 30 } )\)
- (d)
none of these
The solution of the equation \(\frac { dy }{ dx } =\left( 4x+y+1 \right) ^{ 2 }\), is
- (a)
4x + y + 1 = 2 tan (x + c)
- (b)
4x + y + 1 = 2 tan (2x + c)
- (c)
4x + y + 1 = tan (2x + c)
- (d)
NONE OF THESE
If the resultant of two forces each of a unit magnitude,acting at a point ,is of unit magnitude, then angle between these forces, is
- (a)
\({ 45 }^{ \circ }\)
- (b)
\({ 60 }^{ \circ }\)
- (c)
\({ 90 }^{ \circ }\)
- (d)
\({ 120 }^{ \circ }\)
A man can exert a force of 100N and pulls on a rope fastened to the top of a post, the length of the rope beiong twice the length of the post.The horizontal force applied to the middle of the post that will keep it fromfailing, is
- (a)
100N
- (b)
200N
- (c)
\(100\sqrt { 3}\)N
- (d)
200N
If \(f:R\rightarrow R\) is given by f(x)=(3-x3)1/3, then fof(x) is equal to
- (a)
X1/3
- (b)
X3
- (c)
X
- (d)
3-X2
Let A={-1,0,1,2}, B={-4,-2,0,2} and \(f,g:A\rightarrow B\) be the function defined by \(f(x)={ x }^{ 2 }-x,\quad x\epsilon A\) and \(g(x)=2\left| x-\frac { 1 }{ 2 } \right| -1,\quad x\epsilon A\) . Then,
- (a)
f>g
- (b)
g>f
- (c)
f=g
- (d)
2f=g
If P(n): 49n+16n+\(\lambda \) is divisible by 64 for n ϵ N, then the least negative integral value of \(\lambda \) is
- (a)
-1
- (b)
-2
- (c)
-3
- (d)
-4
The solution of the differential equation \({ X }^{ 2 }\frac { dy }{ dx } ={ X }^{ 2 }+Xy+{ y }^{ 2 }\) is
- (a)
\(\tan ^{ -1 }{ \left( \frac { y }{ x } \right) } =2\log { x } +C\)
- (b)
\(\tan ^{ -1 }{ \left( \frac { y }{ x } \right) } =3\log { x } +C\)
- (c)
\(\tan ^{ -1 }{ \left( \frac { y }{ x } \right) } =\log { x } +C\)
- (d)
\(\tan ^{ -1 }{ \left( \frac { y }{ x } \right) } =4\log { x } +C\)
If p is the length of perpendicular from the origin on the line \(\frac { X }{ a } +\frac { Y }{ b } =1\) and a2 ,p2 and b2 are in AP, then a4+b4 is equal to
- (a)
1
- (b)
2
- (c)
3
- (d)
0
The angle between the vectors and \(a=2\hat { i } +2\hat { j } -\hat { k } \\ b=6\hat { i } -3\hat { j } +2\hat { k } \) is
- (a)
\({ cos }^{ -1 }\frac { 3 }{ 11 } \)
- (b)
\({ cos }^{ -1 }\frac { 2 }{ 11 } \)
- (c)
\({ cos }^{ -1 }\frac { 4 }{ 21 } \)
- (d)
\({ cos }^{ -1 }\frac { 3 }{ 22 } \)
The angle \({ tan }^{ -1 }\left( tan\frac { 3\pi }{ 4 } \right) \) is
- (a)
\(\frac { \pi }{ 2 } \)
- (b)
\(-\frac { \pi }{ 3 } \)
- (c)
\(-\frac { \pi }{ 4 } \)
- (d)
\(\frac { \pi }{ 4 } \)
Find the mean deviation about the mean of the following data.
Size | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 |
---|---|---|---|---|---|---|---|---|
Frequency | 3 | 3 | 4 | 14 | 7 | 4 | 3 | 4 |
- (a)
2.95
- (b)
3.24
- (c)
4
- (d)
2
Which of the following is wrong form of If p, then q' ?
- (a)
p is sufficient condition for q
- (b)
p only if q
- (c)
q is necessary condition for p
- (d)
\(\sim q\rightarrow p\)
\(\int{x^2+cos^2X\over x^2+1}.cosec^2x\ dx\) is equal to
- (a)
\(cot\ x+cot^{-1}x+C\)
- (b)
\(-e^{in\ tan^{-1}}x-cot\ x+C\)
- (c)
\(-cot\ x+cot^{-1}x+C\)
- (d)
\(-tan^{-1}x-{cosec\ x\over sec\ x}+C\)
If x, Y and z are positive integers, then value of expression ( X + Y) (Y + Z) (Z + X) is
- (a)
= 8 xyz
- (b)
> 8 xyz
- (c)
< 8 xyz
- (d)
= 4 xyz
The value of the expression \(2(1+\omega)(1+\omega^2)+3(2\omega+1)(2\omega^2+1)+4(3\omega+1)(3\omega^2+1)+..+(n+1)(n\omega+1)(n\omega^2+1)\) is (\(\omega\) is the cube root of unity)
- (a)
\(\frac{n^2(n+1)^2}{4}\)
- (b)
\(\left(\frac{n(n+1)}{2}\right)^2+n\)
- (c)
\(\left(\frac{n(n+1)}{2}\right)^2-n\)
- (d)
none of these
If a, b, c are sides of a triangle and \(\left| \begin{matrix} { a }^{ 2 } & { b }^{ 2 } & { c }^{ 2 } \\ { \left( a+1 \right) }^{ 2 } & { \left( b+1 \right) }^{ 2 } & { \left( c+1 \right) }^{ 2 } \\ { \left( a-1 \right) }^{ 2 } & { \left( b-1 \right) }^{ 2 } & { \left( c-1 \right) }^{ 2 } \end{matrix} \right| =0\) then
- (a)
\(\Delta ABC\) is an equilateral triangle
- (b)
\(\Delta ABC\) is a right-angled isosceles triangle
- (c)
\(\Delta ABC\) is an isosceles triangle
- (d)
none of the above
If \({ D }_{ k }=\left| \begin{matrix} { 2 }^{ k-1 } & \frac { 1 }{ k\left( k+1 \right) } & \sin { k\theta } \\ x & y & z \\ { 2 }^{ n }-1 & \frac { n }{ n+1 } & \frac { \sin { \left( \frac { n+1 }{ 2 } \right) \theta } \sin { \left( \frac { n\theta }{ 2 } \right) } }{ \sin { \left( \frac { \theta }{ 2 } \right) } } \end{matrix} \right| \) then \(\sum _{ k=1 }^{ n }{ { D }_{ k } } \) is equal to
- (a)
0
- (b)
independent of n
- (c)
independent of \(\theta\)
- (d)
independent of x, y and z
Let \(\Delta \neq 0\) and \({ \Delta }^{ c }\) denotes the determinant of cofactors, then \({ \Delta }^{ c }={ \Delta }^{ n-1 }\) where n (>0) is the order of \(\Delta \)
On the basis of above information, answer the question :
If a, b, c are the roots of the equation x3 - 3x2 + 3x + 7 = 0, then the value of \(\left| \begin{matrix} 2bc-{ a }^{ 2 } & { c }^{ 2 } & { b }^{ 2 } \\ { c }^{ 2 } & 2ac-{ b }^{ 2 } & { a }^{ 2 } \\ { b }^{ 2 } & { a }^{ 2 } & 2ab-{ c }^{ 2 } \end{matrix} \right| \) is
- (a)
9
- (b)
27
- (c)
81
- (d)
0
The remainder obtained, when 1! + 2! + 3! + ... + 175! is divided by 15 is
- (a)
5
- (b)
0
- (c)
3
- (d)
8
If n is even positive integer, then the condition that the greatest term in the expansion of (1 + x)n may have the greatest coefficient also is
- (a)
\((\frac{n}{n+2})\)
- (b)
\((\frac{n+1}{n})\)
- (c)
\((\frac{n}{n+4})\)
- (d)
none of these
If \(\alpha\),\(\beta\),\(\gamma \),\(\delta \) are in AP and \(\int _{ 0 }^{ 2 }{ f(x)dx=-4 } \), where \(f(x)=\left| \begin{matrix} x+\alpha & x+\beta & x+\alpha +\gamma \\ x+\beta & x+\gamma & x-1 \\ x+\gamma & x+\delta & x-\beta +\delta \end{matrix} \right| \) then the common difference d is
- (a)
1
- (b)
-1
- (c)
2
- (d)
-2
If A, G and H are respectively arithmetic, geometric and harmonic means between a and b both being unequal and positive. then \(A=\frac { a+b }{ 2 } \Rightarrow a+b=2A,G=\sqrt { ab } \Rightarrow ab={ G }^{ 2 }and\quad H=\frac { 2ab }{ a+b } \Rightarrow { G }^{ 2 }=AH\)
From above discussion we can say that a. b are the roots of the equation x2-- 2A x + G2 = 0
Now, quadratic equation x2--Px + Q=0 and quadratic equation a ( b - c )x2 + b ( c - a )x + c ( a - b ) = 0 have a root common and satisfy the relation b = \(\frac { 2ac }{ (a+c) } \),where a, b, c are real numbers.
The ratio of the AM, GM and HM of the roots of the given quadratic equation is
- (a)
1 ; 2 : 3
- (b)
1 : 1 : 2
- (c)
2 : 2: 3
- (d)
1 : 1 : 1
The sum of the squares of three distinct real numbers which are in strictly increasing GP is S2. If their sum is \(\alpha\) S.
If \(\alpha\)2 = 2, then the value of [r] is (where [.] denotes the greatest integer function and r is common ratio of GP)
- (a)
0
- (b)
1
- (c)
2
- (d)
3
If in \(\triangle ABC\), tan A + tan B + tan C = 6 and tan A tan B = 2, then sin2A : sin2B : sin2 C is
- (a)
8:9:5
- (b)
8:5:9
- (c)
5:9:8
- (d)
5:8:5
Let \(\vec { a } (x)=\sin { x } \hat { i } +(\cos { x) } \hat { j } \) and \(\vec { b } (x)=(\cos { 2x)\hat { i } +(sin2x)\hat { j } } \) be two variable vectors \((x\epsilon R)\) then \(\vec { a } (x)\quad \vec { b } (x)\) are
- (a)
collinear for unique Value of x
- (b)
perpendicular for infinntely many values of x
- (c)
Zero vectors for unique values of x
- (d)
none of the above
The acute angle between two lines whose direction cosines are given by the relation between l + m + n = 0 and l2 + m2 - n2 = 0 is
- (a)
\(\pi / 2\)
- (b)
\(\pi /\ 3\)
- (c)
\(\pi / 4\)
- (d)
none of these
If \(\alpha, \beta, \gamma\) are the angles which a line makes with the coordinate axes, then
- (a)
sin2 \(\alpha\) = cos2 \(\beta\) + cos2 \(\gamma\)
- (b)
cos2 \(\alpha\) = cos2 \(\beta\) + cos2 \(\gamma\)
- (c)
cos2 \(\alpha\) = cos2 \(\beta\) + cos2 \(\gamma\) = 1
- (d)
sin2 \(\alpha\) = sin2 \(\beta\) = 1 + cos2 \(\gamma\)
A number is chosen at random from among the first 30 natural numbers. The probability of the number chosen being a prime, is
- (a)
1/3
- (b)
3/10
- (c)
1/30
- (d)
11/30
1, 3, 5, 7, or 9 is
- (a)
\(\left( \frac { 2 }{ 5 } \right) ^{ n }\)
- (b)
\(\left( \frac { 1 }{ 2 } \right) ^{ n }\)
- (c)
\(\frac { { 2 }^{ n }-1 }{ { 5 }^{ n } } \)
- (d)
\(\frac { { 5 }^{ n }-{ 4 }^{ n } }{ { 10 }^{ n } } \)
If \(\overline { E } \) and \(\overline { F } \) are the complementary events of the events E and F respectively, then
- (a)
P (E/F) + P (\(\overline { E } \)/F) = 1
- (b)
P (E/F) + P (E/\(\overline { F } \)) = 1
- (c)
P (\(\overline { E } \)/F) + P (E/\(\overline { F } \)) = 1
- (d)
P (E/\(\overline { F } \)) + P (\(\overline { E } \)/\(\overline { F } \)) = 1
Let f(x)=\(\begin{cases} 1+x & 0\le x\le 2 \\ 3-x & 2<x\le 3 \end{cases}\), then fof(x)
- (a)
\(=\begin{cases} 2+x & 0\le x\le 1 \\ 2-x & 1<x\le 2 \end{cases}\)
- (b)
\(=\begin{cases} 2+x & 0\le x\le 2 \\ 4-x & 2<x\le 3 \end{cases}\)
- (c)
\(=\begin{cases} 2+x & 0\le x\le 2 \\ 2-x & 2<x\le 3 \end{cases}\)
- (d)
none of these
if \(f(x)=\left( \frac { x-1 }{ x+1 } \right) ,\) then which of the following statement(s) is/are correct
- (a)
\(f\left( \frac { 1 }{ x } \right) =f(x)\)
- (b)
\(f\left( \frac { 1 }{ x } \right) =-f(x)\)
- (c)
\(f\left( -\frac { 1 }{ x } \right) =\frac { 1 }{ f(x) } \)
- (d)
\(f\left( -\frac { 1 }{ x } \right) =-\frac { 1 }{ f(x) } \)
The integer n for which \(\lim _{ x\rightarrow 0 }{ \frac { (\cos { x-1)( } \cos { x-{ e }^{ x }) } }{ { x }^{ n } } } \) is a finite non-zero number is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
If [x] denotes the greatest integer ≤ x, then \(\lim _{ x\rightarrow \infty }{ \frac { 1 }{ { n }^{ 3 } } \{ [{ 1 }^{ 2 }x]+[{ 2 }^{ 2 }x]+[{ 3 }^{ 2 }x] } +.....+[{ n }^{ 2 }x]\} \) equals
- (a)
x/2
- (b)
x/3
- (c)
x/6
- (d)
0
If length of perpendiculars from origin on tangent and normal at 't' are p and PI' respectively, then the value of 9p2 + p1 is equal to
- (a)
9a2
- (b)
9a2sin2(t/2)
- (c)
\(9a^{ 2 }cos^{ 2 }\left( \frac { 3t }{ 2 } \right) \)
- (d)
a2
The value of \(\lim _{ n\rightarrow \infty }{ \left( \frac { { 1 }^{ k }+{ 2 }^{ k }+...+{ n }^{ k } }{ { n }^{ k-1 } } \right) } \) is
- (a)
\(\frac { 1 }{ k+2 } \)
- (b)
\(\frac { 1 }{ k+1 } \)
- (c)
\(\frac { 1 }{ k+3 } \)
- (d)
0
If we rotate the axes of the rectangular hyperbola x2 -y2 = a2 through an angle \(\pi/4\) in the clockwise direction, then the equation x2 - y2 = a2 reduces to xy \(=\frac { { a }^{ 2 } }{ 2 } ={ \left( \frac { a }{ \sqrt { 2 } } \right) }^{ 2 }={ c }^{ 2 }\) (say) Since x = ct, y = \(\frac {c}{t}\) satisfies.xy = c2. \(\therefore\) ( x, y ) = \((ct,\frac{c}{t})\)\((t\neq0)\) is called a "t' point on the rectangular hyperbola. If e1 and e2 are the eccentricities of the hyperbolas xy = 9 and x2- y2 = 25, then (e1,e2) lie on a circle C1 with centre origin then the (radius)2 of the director circle of C1 is
- (a)
2
- (b)
4
- (c)
8
- (d)
16
If ∝,β are the roots ax2+c=bx,then the equation (a+c y)2=b2y in y has the roots
- (a)
\({ \alpha }^{ -1 },\beta ^{ -1 }\)
- (b)
\({ \alpha }^{ 2 },\beta ^{ 2 }\)
- (c)
\({ \alpha \beta }^{ -1 },\alpha ^{ -1 }\beta \)
- (d)
\({ \alpha }^{ -2 },\beta ^{ -2 }\)
The number of ordered 4-tuple (x,y,z,w),(x,y,z,w ∈ [1,10]) which satisfies the inequality \({ 2 }^{ sin^{ 2 }x }3^{ sin^{ 2 }y }4^{ sin^{ 2 }z }5^{ cos^{ 2 }w }\ge 120\) is
- (a)
0
- (b)
144
- (c)
81
- (d)
infinite
Let consider quadratic equation ax2 + bx + c = 0 .... (i)
where \(a,b,c\epsilon R\) and \(a\neq 0\). If Eq. (i) has roots, \(\alpha ,\beta \)
\(\therefore \quad \alpha +\beta =-\frac { b }{ a } ,\alpha \beta =\frac { c }{ a } \) and Eq. (i) can be written as ax2 + bx + c = a(x - \(\alpha \))(x - \(\beta \)).
Also, if a1 , a2 , a3, a4 , .... are in AP, then \({ a }_{ 2 }-{ a }_{ 1 }={ a }_{ 3 }-{ a }_{ 2 }={ a }_{ 4 }-{ a }_{ 3 }=....\neq 0\) and if b1 , b2 , b3 , b4 , ... are in GP, then \(\frac { { b }_{ 2 } }{ { b }_{ 1 } } =\frac { { b }_{ 3 } }{ { b }_{ 2 } } =\frac { { b }_{ 4 } }{ { b }_{ 3 } } =...\neq 1\) Now, if c1 , c2 , c3 , c4 ,.... are in HP, then \(\frac { 1 }{ { c }_{ 2 } } -\frac { 1 }{ { c }_{ 1 } } =\frac { 1 }{ { c }_{ 3 } } -\frac { 1 }{ { c }_{ 2 } } =\frac { 1 }{ { c }_{ 4 } } -\frac { 1 }{ { c }_{ 3 } } =...\neq 0\)
On the basis of above information, answer the following questions:
Let p and q be roots of the equation x2 - 2x + A = 0 and let r and s be the roots of the equation x2 - 18x + B = 0. If p < q < r < s are in arithmetic progression. Then the values of A and B respectively are
- (a)
-5, 67
- (b)
-3, 77
- (c)
67, -5
- (d)
77, -3
The number of solutions of |[x] - 2x|= 4, where [x] denotes the greatest integer ≤ x, is
- (a)
infinite
- (b)
4
- (c)
3
- (d)
2
If f(x) = x + tan x and f is inverse of g' then g' (x) is equal to
- (a)
\(\frac { 1 }{ 1+(g(x)-x)^{ 2 } } \)
- (b)
\(\frac { 1 }{ 1-(g(x)-x)^{ 2 } } \)
- (c)
\(\frac { 1 }{ 2+(g(x)-x)^{ 2 } } \)
- (d)
\(\frac { 1 }{ 2-(g(x)-x)^{ 2 } } \)
Let xcos y + y cos x = 5. Then
- (a)
at x = 0, y = 0, y' = 0
- (b)
at x = 0, y = 1, y' = 0
- (c)
at x = y = 1, y' = - 1
- (d)
at x = 1, Y = 0, y' = 1
If f(x) =\(\left| \begin{matrix} { x }^{ n } & sin\quad x & cos\quad x \\ n! & sin(n\pi /2) & cos(n\pi /2) \\ a & { a }^{ 2 } & { a }^{ 3 } \end{matrix} \right| \) then the value of \(\frac { { d }^{ n } }{ { dx }^{ n } } \) (f(x)) at x = 0 for n = 2m + 1 is
- (a)
-1
- (b)
0
- (c)
1
- (d)
independent of a
The range of the function f(x)=\(\frac{1+x^2}{x^2}\) is equal to
- (a)
[0,1]
- (b)
(0,1)
- (c)
(1,∞)
- (d)
[1,∞)
If\(\left[ \begin{matrix} x+y & 2x+z \\ x-y & 2z+w \end{matrix} \right] =\left[ \begin{matrix} 4 & 7 \\ 0 & 10 \end{matrix} \right] \) , then the values of x,y,z and w respectively are
- (a)
2,2,3,4
- (b)
2,3,1,2
- (c)
3,3,0,1
- (d)
None of these
Find the modulus \(\frac{1+i}{1-i}\)
- (a)
1
- (b)
2
- (c)
-1
- (d)
-2
If A =\(\left[ \begin{matrix} 1 & 2 \\ 0 & 2 \end{matrix} \right] \), then A4-24(A-I) =
- (a)
I-A
- (b)
2I-A
- (c)
I+A
- (d)
A-2I
If A =\(\left[ \begin{matrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{matrix} \right] \) then A3-7A2+10A =
- (a)
5I+A
- (b)
5I-A
- (c)
5I
- (d)
6I
If iz3+z2-z+i=0, then |z| equals
- (a)
2
- (b)
1
- (c)
0
- (d)
None of these
Length of intercept made by the circle x2+y2-16x+4y-36=0 on x-axis is
- (a)
20
- (b)
10
- (c)
5
- (d)
None of these
Find the term independent of x in the expansion of \(\left( \sqrt [ 3 ]{ x } +\frac { 1 }{ 2\sqrt [ 3 ]{ x } } \right) ,\quad x>0\)
- (a)
\(18_{ { C }_{ 9 } }\frac { 1 }{ { 2 }^{ 9 }\quad } \)
- (b)
\({ 18 }_{ { C }_{ 8 } }\frac { 1 }{ { 2 }^{ 8 } } \)
- (c)
\({ 18 }_{ { C }_{ 7 } }\frac { 1 }{ { 2 }^{ 8 } } \)
- (d)
None of these
\(\lim _{ x\rightarrow 0 }{ \frac { |x| }{ x } } \) is
- (a)
1
- (b)
-1
- (c)
0
- (d)
Does not exist
Which term of the G.P.2,1,\({1\over2},{1\over4},...\) is \({1\over128}?\)
- (a)
9th
- (b)
8th
- (c)
7th
- (d)
5th
If a focal chord of the parabola y2=ax is 2x-y-8=0, then equation of the directrix is
- (a)
x+4=0
- (b)
x-4=0
- (c)
y-4=0
- (d)
y+4=0
Let f(x) = \(=\frac { In(1+ax)-In(1-bx) }{ x } ,x\neq 0.\) If f(x) is continuous at x =0, then f(0) =
- (a)
a - b
- (b)
a + b
- (c)
b - a
- (d)
In a + In b
Find the sum to n terms of the series: 5+11+19+29+41...
- (a)
\({n(n+2)(n+4)\over3}\)
- (b)
\({n(n+1)(n+2)\over3}\)
- (c)
\({n(n+2)(n+3)\over3}\)
- (d)
None of these
If the average of the numbers 1, 2, 3, ...,98, 99, x is 100x, then the value of x is
- (a)
\(\frac { 51 }{ 100 } \)
- (b)
\(\frac { 50 }{ 99 } \)
- (c)
\(\frac { 50 }{ 101 } \)
- (d)
\(\frac { 51 }{ 99 } \)
The Fahrenheit temperature F and absolute temperature K satisfy a linear equation. Given that K - 273 when F = 32 and that K = 373 when F = 212 Find the value of F, When K = 0
- (a)
459.4
- (b)
-459.4
- (c)
287.3
- (d)
-287.3
Which of the following statement is a conjunction?
- (a)
Ram and Shyam are friends
- (b)
Both Ram and Shyam are tall
- (c)
Both Ram and Shyam are enemies
- (d)
None of these
Evaluate cosec480+cosec960+cosec1920+cosec3840
- (a)
-1
- (b)
0
- (c)
2
- (d)
1
\(cos^{-1}(\frac{1}{2})+2sin^{-1}(\frac{1}{2})\) is equal to
- (a)
\(\frac{\pi}{4}\)
- (b)
\(\frac{\pi}{6}\)
- (c)
\(\frac{\pi}{3}\)
- (d)
\(\frac{2\pi}{3}\)
\(cos^{-1}\{\frac{1}{2}x^2+\sqrt{1-x^2}\sqrt{1-\frac{x^2}{4}}\}=cos^{-1}\frac{x}{2}-cos^{-1}x\) holds for
- (a)
|x|≤1
- (b)
x∈R
- (c)
0≤x≤1
- (d)
-1≤x≤0
The area of the quadrilateral ABCD, where A(0,4,1), B(2,3,-1), C(4,5,0), and D(2,6,2), is equal to
- (a)
9sq. units
- (b)
18sq.units
- (c)
27sq.units
- (d)
81sq.units
Let f(x)=1+x, g(x)=x2+x+1, then (f+g)(x) at x=0 is
- (a)
2
- (b)
5
- (c)
6
- (d)
9
Solution of the differential equation \(\frac { dx }{ dy } -\frac { x\log { x } }{ 1+\log { x } } =\frac { { e }^{ y } }{ 1+\log { x } } \), if y(1) = 0
- (a)
\({ x }^{ x }={ e }^{ y{ e }^{ y } }\)
- (b)
\({ e }^{ y }={ x }^{ { e }^{ y } }\)
- (c)
\({ x }^{ x }=y{ e }^{ y }\)
- (d)
none of these
The 4th term in the expansion of (x - 2y)12 is
- (a)
1760x3y9
- (b)
-1760x9y3
- (c)
1760x9y3
- (d)
None of these
Four candidates A, B, C and D have applied for the assignment to coach a school cricket team. If A is twice as likely to be selected as Band B and Care given about the same chance of being selected, while C is twice as likely to be selected as D, what is the probability that A will not be selected?
- (a)
\(\frac{4}{9}\)
- (b)
\(\frac{5}{9}\)
- (c)
\(\frac{9}{4}\)
- (d)
\(\frac{9}{5}\)
Let a, b, c, d, e be the observations with mean m and standard deviation s. The standard deviation of the observations a + k, b + k, c + k, d + k, e + k is
- (a)
s
- (b)
ks
- (c)
s + k
- (d)
\(\frac { s }{ k } \)
Three numbers are chosen from 1 to 20. Find the probability that they are not consecutive.
- (a)
\(\frac{186}{190}\)
- (b)
\(\frac{187}{190}\)
- (c)
\(\frac{188}{190}\)
- (d)
\(\frac{18}{^{20}C_{3}}\)
If siny=xsin(a+y), then dy/dx is
- (a)
sin(a+y)
- (b)
sin2(a+y)
- (c)
\(\cfrac { { sin }^{ 2 }(a+y) }{ sina } \)
- (d)
\(\cfrac { sin\quad (a+y) }{ sina } \)
Statement-I : If a, b , \(\theta\) are real numbers , then minimum and maximum values of asin\(\theta\) + bcos\(\theta\) are respectively - \(\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \) and \(\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \).
Statement II : the equation asin\(\theta\) + bcos\(\theta\) = c (a, b,c \(\epsilon \), R)has a solution only it c2 > a2 + b2
- (a)
If both statement-I and statement -II are trur and statement -II is the correct explantion of statement -I
- (b)
If both statement -I and statement II are true but statment-II is not the correct explanation of statement-I
- (c)
If statement -I is true but statement -II is false
- (d)
if statement -I is false and statement -II is true
If j(x) = x100 + x99 + ...+ x + 1, then f'(1) is equal to
- (a)
5050
- (b)
5049
- (c)
5051
- (d)
50051