8666 + Mcqs in Mathematics Page 45 McqOptions

2201.	\[\frac{1}{1!(n-1)\,!}+\frac{1}{3!(n-3)!}+\frac{1}{5!(n-5)!}+....=\] [AMU 2005]
A.	\[\frac{{{2}^{n}}}{n!}\]; for all even values of n
B.	\[\frac{{{2}^{n-1}}}{n!}\]; for all values of n i.e., all even odd values
C.	0
D.	None of these
Answer» C. 0

Discussion

2202.	\[\frac{{{C}_{0}}}{1}+\frac{{{C}_{1}}}{2}+\frac{{{C}_{2}}}{3}+....+\frac{{{C}_{n}}}{n+1}=\] [RPET 1996]
A.	\[\frac{{{2}^{n}}}{n+1}\]
B.	\[\frac{{{2}^{n}}-1}{n+1}\]
C.	\[\frac{{{2}^{n+1}}-1}{n+1}\]
D.	None of these
Answer» D. None of these

Discussion

2203.	\[{{C}_{1}}+2{{C}_{2}}+3{{C}_{3}}+4{{C}_{4}}+....+n{{C}_{n}}=\] [RPET 1995; MP PET 2002; Orissa JEE 2005]
A.	\[{{2}^{n}}\]
B.	\[n.\,\,{{2}^{n}}\]
C.	\[n.\,\,{{2}^{n-1}}\]
D.	\[n.\,\,{{2}^{n+1}}\]
Answer» D. \[n.\,\,{{2}^{n+1}}\]

Discussion

2204.	If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+..........+{{C}_{n}}{{x}^{n}}\], then \[\frac{{{C}_{1}}}{{{C}_{0}}}+\frac{2{{C}_{2}}}{{{C}_{1}}}+\frac{3{{C}_{3}}}{{{C}_{2}}}+....+\frac{n{{C}_{n}}}{{{C}_{n-1}}}=\] [BIT Ranchi 1986; RPET 1996, 97]
A.	\[\frac{n(n-1)}{2}\]
B.	\[\frac{n(n+2)}{2}\]
C.	\[\frac{n(n+1)}{2}\]
D.	\[\frac{(n-1)(n-2)}{2}\]
Answer» D. \[\frac{(n-1)(n-2)}{2}\]

Discussion

2205.	What is the sum of the coefficients of \[{{({{x}^{2}}-x-1)}^{99}}\] [Orissa JEE 2005]
A.	1
B.	0
C.	?1
D.	None of these
Answer» D. None of these

Discussion

2206.	The value of \[\sum\limits_{n=1}^{\infty }{\frac{^{n}{{C}_{0}}+...{{+}^{n}}{{C}_{n}}}{^{n}{{P}_{n}}}}\] is [Kerala (Engg.) 2005]
A.	\[{{e}^{2}}\]
B.	e
C.	\[{{e}^{2}}-1\]
D.	\[e-1\]
E.	\[{{e}^{2}}+1\]
Answer» D. \[e-1\]

Discussion

2207.	In the expansion of \[{{(1+x)}^{5}}\], the sum of the coefficient of the terms is [RPET 1992, 97; Kurukshetra CEE 2000]
A.	80
B.	16
C.	32
D.	64
Answer» D. 64

Discussion

2208.	\[\sum\limits_{k=0}^{10}{^{20}{{C}_{k}}=}\] [Orissa JEE 2004]
A.	\[{{2}^{19}}+\frac{1}{2}{{\,}^{20}}{{C}_{10}}\]
B.	\[{{2}^{19}}\]
C.	\[^{20}{{C}_{10}}\]
D.	None of these
Answer» B. \[{{2}^{19}}\]

Discussion

2209.	If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+..........+{{C}_{n}}{{x}^{2}},\] then \[C_{0}^{2}+C_{1}^{2}+C_{2}^{2}+C_{3}^{2}+......+C_{n}^{2}\] = [MP PET 1985; Karnataka CET 1995; MNR 1999]
A.	\[\frac{n!}{n!n!}\]
B.	\[\frac{(2n)!}{n!n!}\]
C.	\[\frac{(2n)!}{n!}\]
D.	None of these
Answer» C. \[\frac{(2n)!}{n!}\]

Discussion

2210.	The sum of the coefficients in the expansion of \[{{(1+x-3{{x}^{2}})}^{3148}}\] is [Karnataka CET 2003]
A.	7
B.	8
C.	? 1
D.	1
Answer» E.

Discussion

2211.	The sum of coefficients in the expansion of \[{{(1+x+{{x}^{2}})}^{n}}\] is [EAMCET 2002]
A.	2
B.	\[{{3}^{n}}\]
C.	\[{{4}^{n}}\]
D.	\[{{2}^{n}}\]
Answer» C. \[{{4}^{n}}\]

Discussion

2212.	The sum of coefficients in \[{{(1+x-3{{x}^{2}})}^{2134}}\]is [Kurukshetra CEE 2001]
A.	? 1
B.	1
C.	0
D.	\[{{2}^{2134}}\]
Answer» C. 0

Discussion

2213.	\[{{n}^{n}}{{\left( \frac{n+1}{2} \right)}^{2n}}\] is [AMU 2001]
A.	Less than \[{{\left( \frac{n+1}{2} \right)}^{3}}\]
B.	Greater than \[{{\left( \frac{n+1}{2} \right)}^{3}}\]
C.	Less than \[{{(n!)}^{3}}\]
D.	Greater than \[{{(n!)}^{3}}\,\]
Answer» E.

Discussion

2214.	In the expansion of \[{{(1+x)}^{50}},\] the sum of the coefficient of odd powers of x is [UPSEAT 2001; Pb. CET 2004]
A.	0
B.	\[{{2}^{49}}\]
C.	\[{{2}^{50}}\]
D.	\[{{2}^{51}}\]
Answer» C. \[{{2}^{50}}\]

Discussion

2215.	The sum of coefficients in the expansion of \[{{(x+2y+3z)}^{8}}\] is [RPET 2000]
A.	\[{{3}^{8}}\]
B.	\[{{5}^{8}}\]
C.	\[{{6}^{8}}\]
D.	None of these
Answer» D. None of these

Discussion

2216.	If \[{{C}_{0}},{{C}_{1}},{{C}_{2}},.......,{{C}_{n}}\] are the binomial coefficients, then \[2.{{C}_{1}}+{{2}^{3}}.{{C}_{3}}+{{2}^{5}}.{{C}_{5}}+....\]equals [AMU 1999]
A.	\[\frac{{{3}^{n}}+{{(-1)}^{n}}}{2}\]
B.	\[\frac{{{3}^{n}}-{{(-1)}^{n}}}{2}\]
C.	\[\frac{{{3}^{n}}+1}{2}\]
D.	\[\frac{{{3}^{n}}-1}{2}\]
Answer» C. \[\frac{{{3}^{n}}+1}{2}\]

Discussion

2217.	If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+...+{{C}_{n}}{{x}^{n}}\], then the value of \[{{C}_{0}}+{{C}_{2}}+{{C}_{4}}+{{C}_{6}}+.....\] is [RPET 1997]
A.	\[{{2}^{n-1}}\]
B.	\[{{2}^{n-1}}\]
C.	\[{{2}^{n}}\]
D.	\[{{2}^{n-1}}-1\]
Answer» B. \[{{2}^{n-1}}\]

Discussion

2218.	\[^{n}{{C}_{0}}-\frac{1}{2}{{\,}^{n}}{{C}_{1}}+\frac{1}{3}{{\,}^{n}}{{C}_{2}}-......+{{(-1)} ^{n}}\frac{^{n}{{C}_{n}}}{n+1}=\]
A.	n
B.	1/n
C.	\[\frac{1}{n+1}\]
D.	\[\frac{1}{n-1}\]
Answer» D. \[\frac{1}{n-1}\]

Discussion

2219.	If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+....+{{C}_{n}}{{x}^{n}}\], then \[{{C}_{0}}{{C}_{2}}+{{C}_{1}}{{C}_{3}}+{{C}_{2}}{{C}_{4}}+{{C}_{n-2}}{{C}_{n}}\]equals [RPET 1996]
A.	\[\frac{(2n)!}{(n+1)!(n+2)!}\]
B.	\[\frac{(2n)!}{(n-2)!(n+2)!}\]
C.	\[\frac{(2n)!}{(n)!(n+2)!}\]
D.	\[\frac{(2n)!}{(n-1)!(n+2)!}\]
Answer» C. \[\frac{(2n)!}{(n)!(n+2)!}\]

Discussion

2220.	\[2{{C}_{0}}+\frac{{{2}^{2}}}{2}{{C}_{1}}+\frac{{{2}^{3}}}{3}{{C}_{2}}+....+\frac{{{2}^{11}}}{11}{{C}_{10}}\] [MP PET 1999; EAMCET 1992]
A.	\[\frac{{{3}^{11}}-1}{11}\]
B.	\[\frac{{{2}^{11}}-1}{11}\]
C.	\[\frac{{{11}^{3}}-1}{11}\]
D.	\[\frac{{{11}^{2}}-1}{11}\]
Answer» B. \[\frac{{{2}^{11}}-1}{11}\]

Discussion

2221.	The value of \[^{15}C_{0}^{2}{{-}^{15}}C_{1}^{2}{{+}^{15}}C_{2}^{2}-....{{-}^{15}}C_{15}^{2}\]is [MP PET 1996]
A.	15
B.	? 15
C.	0
D.	51
Answer» D. 51

Discussion

2222.	The sum of the last eight coefficients in the expansion of \[{{(1+x)}^{15}}\] is
A.	\[{{2}^{16}}\]
B.	\[{{2}^{15}}\]
C.	\[{{2}^{14}}\]
D.	None of these
Answer» D. None of these

Discussion

2223.	The value of \[^{4n}{{C}_{0}}{{+}^{4n}}{{C}_{4}}{{+}^{4n}}{{C}_{8}}+....{{+}^{4n}}{{C}_{4n}}\]is
A.	\[{{2}^{4n-2}}+{{(-1)}^{n}}{{2}^{2n-1}}\]
B.	\[{{2}^{4n-2}}+{{2}^{2n-1}}\]
C.	\[{{2}^{2n-1}}+{{(-1)}^{n}}\,{{2}^{4n-2}}\]
D.	None of these
Answer» B. \[{{2}^{4n-2}}+{{2}^{2n-1}}\]

Discussion

2224.	If \[x+y=1\], then \[\sum\limits_{r=0}^{n}{{{r}^{2}}{{\,}^{n}}{{C}_{r}}{{x}^{r}}{{y}^{n-r}}}\]equals
A.	nxy
B.	\[nx(x+yn)\]
C.	\[nx(nx+y)\]
D.	None of these
Answer» D. None of these

Discussion

2225.	If the sum of the coefficients in the expansion of \[{{({{\alpha }^{2}}{{x}^{2}}-2\alpha \text{ }x+1)}^{51}}\]vanishes, then the value of \[\alpha \] is [IIT 1991; Pb. CET 1988]
A.	2
B.	?1
C.	1
D.	? 2
Answer» D. ? 2

Discussion

2226.	Coefficients of \[{{x}^{r}}[0\le r\le (n-1)]\] in the expansion of \[{{(x+3)}^{n-1}}+{{(x+3)}^{n-2}}(x+2)\]\[+{{(x+3)}^{n-3}}{{(x+2)}^{2}}+...+{{(x+2)}^{n-1}}\]
A.	\[^{n}{{C}_{r}}({{3}^{r}}-{{2}^{n}})\]
B.	\[^{n}{{C}_{r}}({{3}^{n-r}}-{{2}^{n-r}})\]
C.	\[^{n}{{C}_{r}}({{3}^{r}}+{{2}^{n-r}})\]
D.	None of these
Answer» C. \[^{n}{{C}_{r}}({{3}^{r}}+{{2}^{n-r}})\]

Discussion

2227.	The sum of the coefficients of even power of x in the expansion of \[{{(1+x+{{x}^{2}}+{{x}^{3}})}^{5}}\]is [EAMCET 1988]
A.	256
B.	128
C.	512
D.	64
Answer» D. 64

Discussion

2228.	If n is an integer greater than 1, then \[a{{-}^{n}}{{C}_{1}}(a-1){{+}^{n}}{{C}_{2}}(a-2)+....+{{(-1)}^{n}}(a-n)=\] [IIT 1972]
A.	\[a\]
B.	0
C.	\[{{a}^{2}}\]
D.	\[{{2}^{n}}\]
Answer» C. \[{{a}^{2}}\]

Discussion

2229.	If \[{{(1+x-2{{x}^{2}})}^{6}}=1+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+....+{{a}_{12}}{{x}^{12}}\], then the expression \[{{a}_{2}}+{{a}_{4}}+{{a}_{6}}+....+{{a}_{12}}\]has the value [RPET 1986, 99; UPSEAT 2003]
A.	32
B.	63
C.	64
D.	None of these
Answer» E.

Discussion

2230.	If the sum of the coefficients in the expansion of \[{{(x-2y+3z)}^{n}}\] is 128 then the greatest coefficient in the expansion of \[{{(1+x)}^{n}}\]is
A.	35
B.	20
C.	10
D.	None of these
Answer» B. 20

Discussion

2231.	The sum of all the coefficients in the binomial expansion of \[{{({{x}^{2}}+x-3)}^{319}}\] is [Bihar CEE 1994]
A.	1
B.	2
C.	? 1
D.	0
Answer» D. 0

Discussion

2232.	In the expansion of \[{{(1+x)}^{n}}\] the sum of coefficients of odd powers of x is [MP PET 1986, 93, 2003]
A.	\[{{2}^{n}}+1\]
B.	\[{{2}^{n}}-1\]
C.	\[{{2}^{n}}\]
D.	\[{{2}^{n-1}}\]
Answer» E.

Discussion

2233.	The value of \[\frac{{{C}_{1}}}{2}+\frac{{{C}_{3}}}{4}+\frac{{{C}_{5}}}{6}+.....\]is equal to [Karnataka CET 2000]
A.	\[\frac{{{2}^{n}}-1}{n+1}\]
B.	\[n{{.2}^{n}}\]
C.	\[\frac{{{2}^{n}}}{n}\]
D.	\[\frac{{{2}^{n}}+1}{n+1}\]
Answer» B. \[n{{.2}^{n}}\]

Discussion

2234.	If \[{{(1+x)}^{15}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+......+{{C}_{15}}{{x}^{15}},\] then \[{{C}_{2}}+2{{C}_{3}}+3{{C}_{4}}+....+14{{C}_{15}}=\] [IIT 1966]
A.	\[{{14.2}^{14}}\]
B.	\[{{13.2}^{14}}+1\]
C.	\[{{13.2}^{14}}-1\]
D.	None of these
Answer» C. \[{{13.2}^{14}}-1\]

Discussion

2235.	If a and d are two complex numbers, then the sum to \[(n+1)\] terms of the following series \[a{{C}_{0}}-(a+d){{C}_{1}}+(a+2d){{C}_{2}}-........\] is
A.	\[\frac{a}{{{2}^{n}}}\]
B.	\[na\]
C.	0
D.	None of these
Answer» D. None of these

Discussion

2236.	\[^{10}{{C}_{1}}{{+}^{10}}{{C}_{3}}{{+}^{10}}{{C}_{5}}{{+}^{10}}{{C}_{7}}{{+}^{10}}{{C}_{9}}=\] [MP PET 1982]
A.	\[{{2}^{9}}\]
B.	\[{{2}^{10}}\]
C.	\[{{2}^{10}}-1\]
D.	None of these
Answer» B. \[{{2}^{10}}\]

Discussion

2237.	The area of a parallelogram formed by the lines \[ax\pm by\pm c=0\], is [IIT 1973]
A.	\[\frac{{{c}^{2}}}{ab}\]
B.	\[\frac{2{{c}^{2}}}{ab}\]
C.	\[\frac{{{c}^{2}}}{2ab}\]
D.	None of these
Answer» C. \[\frac{{{c}^{2}}}{2ab}\]

Discussion

2238.	The triangle formed by the lines \[x+y-4=0,\,\] \[3x+y=4,\] \[x+3y=4\] is [RPET 2002; IIT 1983; MNR 1992; UPSEAT 2001]
A.	Isosceles
B.	Equilateral
C.	Right?angled
D.	None of these
Answer» B. Equilateral

Discussion

2239.	Two lines are drawn through (3, 4), each of which makes angle of 45o with the line \[x-y=2\], then area of the triangle formed by these lines is [RPET 2000]
A.	9
B.	9/2
C.	2
D.	2/9
Answer» C. 2

Discussion

2240.	The area of the triangle bounded by the straight line \[ax+by+c=0,\,\,\,\,(a,b,c\ne 0)\] and the coordinate axes is [AMU 2000]
A.	\[\frac{1}{2}\frac{{{a}^{2}}}{\|bc\|}\]
B.	\[\frac{1}{2}\frac{{{c}^{2}}}{\|ab\|}\]
C.	\[\frac{1}{2}\frac{{{b}^{2}}}{\|ac\|}\]
D.	0
Answer» C. \[\frac{1}{2}\frac{{{b}^{2}}}{\|ac\|}\]

Discussion

2241.	Area of the parallelogram whose sides are \[x\cos \alpha +y\sin \alpha =p\] \[x\cos \alpha +y\sin \alpha =q,\,\,\] \[x\cos \beta +y\sin \beta =r\] and \[x\cos \beta +y\sin \beta =s\] is
A.	\[\pm (p-q)(r-s)\,\text{cosec}(\alpha -\beta )\]
B.	\[(p+q)(r-s)\,\text{cosec }(\alpha +\beta )\]
C.	\[(p+q)(r+s)\,\text{cosec }(\alpha -\beta )\]
D.	None of these
Answer» B. \[(p+q)(r-s)\,\text{cosec }(\alpha +\beta )\]

Discussion

2242.	The equation to the sides of a triangle are \[x-3y=0\], \[4x+3y=5\] and \[3x+y=0\]. The line \[3x-4y=0\]passes through [EAMCET 1994]
A.	The incentre
B.	The centroid
C.	The circumcentre
D.	The orthocentre of the triangle
Answer» E.

Discussion

2243.	If A is (2, 5), B is (4, -11) and C lies on \[9x+7y+4=0\], then the locus of the centroid of the \[\Delta ABC\] is a straight line parallel to the straight line is [MP PET 1986]
A.	\[7x-9y+4=0\]
B.	\[9x-7y-4=0\]
C.	\[9x+7y+4=0\]
D.	\[7+9y+4=0\]
Answer» D. \[7+9y+4=0\]

Discussion

2244.	A straight line through the point (1, 1) meets the x-axis at 'A' and the y-axis at 'B'. The locus of the mid-point of AB is [UPSEAT 2004]
A.	\[2xy+x+y=0\]
B.	\[x+y-2xy=0\]
C.	\[x+y+2=0\]
D.	\[x+y-2=0\]
Answer» C. \[x+y+2=0\]

Discussion

2245.	The point moves such that the area of the triangle formed by it with the points (1, 5) and (3, ?7) is 21sq. unit. The locus of the point is [Kerala (Engg.) 2002]
A.	\[6x+y-32=0\]
B.	\[6x-y+32=0\]
C.	\[x+6y-32=0\]
D.	\[6x-y-32=0\]
Answer» B. \[6x-y+32=0\]

Discussion

2246.	If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is [IIT 1992, Karnataka CET 1999; DCE 2000,01]
A.	Square
B.	Circle
C.	Straight line
D.	Two intersecting lines
Answer» B. Circle

Discussion

2247.	The locus of a point so that sum of its distance from two given perpendicular lines is equal to 2 unit in first quadrant, is [Bihar CEE 1994]
A.	\[x+y+2=0\]
B.	\[x+y=2\]
C.	\[x-y=2\]
D.	None of these
Answer» C. \[x-y=2\]

Discussion

2248.	Locus of the points which are at equal distance from \[3x+4y-11=0\]and \[12x+5y+2=0\]and which is near the origin is [MNR 1987]
A.	\[21x-77y+153=0\]
B.	\[99x+77y-133=0\]
C.	\[7x-11y=19\]
D.	None of these
Answer» C. \[7x-11y=19\]

Discussion

2249.	A point moves so that square of its distance from the point (3, -2) is numerically equal to its distance from the line \[5x-12y=13\]. The equation of the locus of the point is [Roorkee 1974]
A.	\[13{{x}^{2}}+13{{y}^{2}}-83x+64y+182=0\]
B.	\[{{x}^{2}}+{{y}^{2}}-11x+16y+26=0\]
C.	\[{{x}^{2}}+{{y}^{2}}-11x+16y=0\]
D.	None of these
Answer» B. \[{{x}^{2}}+{{y}^{2}}-11x+16y+26=0\]

Discussion

2250.	The triangle formed by \[{{x}^{2}}-9{{y}^{2}}=0\]and \[x=4\]is [Orissa JEE 2004]
A.	Isosceles
B.	Equilateral
C.	Right angled
D.	None of these
Answer» B. Equilateral

Discussion

Explore topic-wise MCQs in Joint Entrance Exam - Main (JEE Main).

\[\frac{1}{1!(n-1)\,!}+\frac{1}{3!(n-3)!}+\frac{1}{5!(n-5)!}+....=\] [AMU 2005]

\[\frac{{{C}_{0}}}{1}+\frac{{{C}_{1}}}{2}+\frac{{{C}_{2}}}{3}+....+\frac{{{C}_{n}}}{n+1}=\] [RPET 1996]

\[{{C}_{1}}+2{{C}_{2}}+3{{C}_{3}}+4{{C}_{4}}+....+n{{C}_{n}}=\] [RPET 1995; MP PET 2002; Orissa JEE 2005]

If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+..........+{{C}_{n}}{{x}^{n}}\], then \[\frac{{{C}_{1}}}{{{C}_{0}}}+\frac{2{{C}_{2}}}{{{C}_{1}}}+\frac{3{{C}_{3}}}{{{C}_{2}}}+....+\frac{n{{C}_{n}}}{{{C}_{n-1}}}=\] [BIT Ranchi 1986; RPET 1996, 97]

What is the sum of the coefficients of \[{{({{x}^{2}}-x-1)}^{99}}\] [Orissa JEE 2005]

The value of \[\sum\limits_{n=1}^{\infty }{\frac{^{n}{{C}_{0}}+...{{+}^{n}}{{C}_{n}}}{^{n}{{P}_{n}}}}\] is [Kerala (Engg.) 2005]

In the expansion of \[{{(1+x)}^{5}}\], the sum of the coefficient of the terms is [RPET 1992, 97; Kurukshetra CEE 2000]

\[\sum\limits_{k=0}^{10}{^{20}{{C}_{k}}=}\] [Orissa JEE 2004]

If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+..........+{{C}_{n}}{{x}^{2}},\] then \[C_{0}^{2}+C_{1}^{2}+C_{2}^{2}+C_{3}^{2}+......+C_{n}^{2}\] = [MP PET 1985; Karnataka CET 1995; MNR 1999]

The sum of the coefficients in the expansion of \[{{(1+x-3{{x}^{2}})}^{3148}}\] is [Karnataka CET 2003]

The sum of coefficients in the expansion of \[{{(1+x+{{x}^{2}})}^{n}}\] is [EAMCET 2002]

The sum of coefficients in \[{{(1+x-3{{x}^{2}})}^{2134}}\]is [Kurukshetra CEE 2001]

\[{{n}^{n}}{{\left( \frac{n+1}{2} \right)}^{2n}}\] is [AMU 2001]

In the expansion of \[{{(1+x)}^{50}},\] the sum of the coefficient of odd powers of x is [UPSEAT 2001; Pb. CET 2004]

The sum of coefficients in the expansion of \[{{(x+2y+3z)}^{8}}\] is [RPET 2000]

If \[{{C}_{0}},{{C}_{1}},{{C}_{2}},.......,{{C}_{n}}\] are the binomial coefficients, then \[2.{{C}_{1}}+{{2}^{3}}.{{C}_{3}}+{{2}^{5}}.{{C}_{5}}+....\]equals [AMU 1999]

If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+...+{{C}_{n}}{{x}^{n}}\], then the value of \[{{C}_{0}}+{{C}_{2}}+{{C}_{4}}+{{C}_{6}}+.....\] is [RPET 1997]

\[^{n}{{C}_{0}}-\frac{1}{2}{{\,}^{n}}{{C}_{1}}+\frac{1}{3}{{\,}^{n}}{{C}_{2}}-......+{{(-1)} ^{n}}\frac{^{n}{{C}_{n}}}{n+1}=\]

If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+....+{{C}_{n}}{{x}^{n}}\], then \[{{C}_{0}}{{C}_{2}}+{{C}_{1}}{{C}_{3}}+{{C}_{2}}{{C}_{4}}+{{C}_{n-2}}{{C}_{n}}\]equals [RPET 1996]

\[2{{C}_{0}}+\frac{{{2}^{2}}}{2}{{C}_{1}}+\frac{{{2}^{3}}}{3}{{C}_{2}}+....+\frac{{{2}^{11}}}{11}{{C}_{10}}\] [MP PET 1999; EAMCET 1992]

The value of \[^{15}C_{0}^{2}{{-}^{15}}C_{1}^{2}{{+}^{15}}C_{2}^{2}-....{{-}^{15}}C_{15}^{2}\]is [MP PET 1996]

The sum of the last eight coefficients in the expansion of \[{{(1+x)}^{15}}\] is

The value of \[^{4n}{{C}_{0}}{{+}^{4n}}{{C}_{4}}{{+}^{4n}}{{C}_{8}}+....{{+}^{4n}}{{C}_{4n}}\]is

If \[x+y=1\], then \[\sum\limits_{r=0}^{n}{{{r}^{2}}{{\,}^{n}}{{C}_{r}}{{x}^{r}}{{y}^{n-r}}}\]equals

If the sum of the coefficients in the expansion of \[{{({{\alpha }^{2}}{{x}^{2}}-2\alpha \text{ }x+1)}^{51}}\]vanishes, then the value of \[\alpha \] is [IIT 1991; Pb. CET 1988]

Coefficients of \[{{x}^{r}}[0\le r\le (n-1)]\] in the expansion of \[{{(x+3)}^{n-1}}+{{(x+3)}^{n-2}}(x+2)\]\[+{{(x+3)}^{n-3}}{{(x+2)}^{2}}+...+{{(x+2)}^{n-1}}\]

The sum of the coefficients of even power of x in the expansion of \[{{(1+x+{{x}^{2}}+{{x}^{3}})}^{5}}\]is [EAMCET 1988]

If n is an integer greater than 1, then \[a{{-}^{n}}{{C}_{1}}(a-1){{+}^{n}}{{C}_{2}}(a-2)+....+{{(-1)}^{n}}(a-n)=\] [IIT 1972]

If \[{{(1+x-2{{x}^{2}})}^{6}}=1+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+....+{{a}_{12}}{{x}^{12}}\], then the expression \[{{a}_{2}}+{{a}_{4}}+{{a}_{6}}+....+{{a}_{12}}\]has the value [RPET 1986, 99; UPSEAT 2003]

If the sum of the coefficients in the expansion of \[{{(x-2y+3z)}^{n}}\] is 128 then the greatest coefficient in the expansion of \[{{(1+x)}^{n}}\]is

The sum of all the coefficients in the binomial expansion of \[{{({{x}^{2}}+x-3)}^{319}}\] is [Bihar CEE 1994]

In the expansion of \[{{(1+x)}^{n}}\] the sum of coefficients of odd powers of x is [MP PET 1986, 93, 2003]

The value of \[\frac{{{C}_{1}}}{2}+\frac{{{C}_{3}}}{4}+\frac{{{C}_{5}}}{6}+.....\]is equal to [Karnataka CET 2000]

If \[{{(1+x)}^{15}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+......+{{C}_{15}}{{x}^{15}},\] then \[{{C}_{2}}+2{{C}_{3}}+3{{C}_{4}}+....+14{{C}_{15}}=\] [IIT 1966]

If a and d are two complex numbers, then the sum to \[(n+1)\] terms of the following series \[a{{C}_{0}}-(a+d){{C}_{1}}+(a+2d){{C}_{2}}-........\] is

\[^{10}{{C}_{1}}{{+}^{10}}{{C}_{3}}{{+}^{10}}{{C}_{5}}{{+}^{10}}{{C}_{7}}{{+}^{10}}{{C}_{9}}=\] [MP PET 1982]

The area of a parallelogram formed by the lines \[ax\pm by\pm c=0\], is [IIT 1973]

The triangle formed by the lines \[x+y-4=0,\,\] \[3x+y=4,\] \[x+3y=4\] is [RPET 2002; IIT 1983; MNR 1992; UPSEAT 2001]

Two lines are drawn through (3, 4), each of which makes angle of 45o with the line \[x-y=2\], then area of the triangle formed by these lines is [RPET 2000]

The area of the triangle bounded by the straight line \[ax+by+c=0,\,\,\,\,(a,b,c\ne 0)\] and the coordinate axes is [AMU 2000]

Area of the parallelogram whose sides are \[x\cos \alpha +y\sin \alpha =p\] \[x\cos \alpha +y\sin \alpha =q,\,\,\] \[x\cos \beta +y\sin \beta =r\] and \[x\cos \beta +y\sin \beta =s\] is

The equation to the sides of a triangle are \[x-3y=0\], \[4x+3y=5\] and \[3x+y=0\]. The line \[3x-4y=0\]passes through [EAMCET 1994]

If A is (2, 5), B is (4, -11) and C lies on \[9x+7y+4=0\], then the locus of the centroid of the \[\Delta ABC\] is a straight line parallel to the straight line is [MP PET 1986]

A straight line through the point (1, 1) meets the x-axis at 'A' and the y-axis at 'B'. The locus of the mid-point of AB is [UPSEAT 2004]

The point moves such that the area of the triangle formed by it with the points (1, 5) and (3, ?7) is 21sq. unit. The locus of the point is [Kerala (Engg.) 2002]

If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is [IIT 1992, Karnataka CET 1999; DCE 2000,01]

The locus of a point so that sum of its distance from two given perpendicular lines is equal to 2 unit in first quadrant, is [Bihar CEE 1994]

Locus of the points which are at equal distance from \[3x+4y-11=0\]and \[12x+5y+2=0\]and which is near the origin is [MNR 1987]

A point moves so that square of its distance from the point (3, -2) is numerically equal to its distance from the line \[5x-12y=13\]. The equation of the locus of the point is [Roorkee 1974]

The triangle formed by \[{{x}^{2}}-9{{y}^{2}}=0\]and \[x=4\]is [Orissa JEE 2004]