If the term from x in the expansion of [√x – (k / x2)]10 is 405, then the value of k is
1) ± 1 2) ± 3 3) ± 4 4) ± 2 Solution: (2) ± 3 \(\begin{array}{l}\begin{array}{l} \left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10} \text {... View Article
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If (1 + x)15 = a0 + a1x +…+ a15 x15, then ∑r=115 r (ar / ar – 1) is
1) 110 2) 115 3) 120 4) 135 Solution: (3) 120 \(\begin{array}{l}\begin{array}{l} \begin{array}{c} \text { We know }(1+x)^{15}=15_{c_{0}}+15_{c_{1}} x+15_{c_{2}} x^{2}+\cdots+15... View Article
If x = then √x – (1 / √x) =
1) 0.2 2) 4.8 3) 1.02 4) 5.2 Solution: (4) 5.2 \(\begin{array}{l}x=\frac{[729+6(2)(243)+15(4)(81)+20(8)(27)+15(16)(9)+6(32)(3)+64]}{1+4(4)+6(16)+4(64)+256} \\ =\frac{\left[3^{6}+6_{C_{1}} 3^{5} 2+6_{C_{2}} 2^{2} 3^{4}+6_{C_{3}} 2^{3} 3^{3}+6_{C_{4}}... View Article
If |x| < 1 / 2, then the coefficient of xr in the expansion of [1 + 2x] / [1 – 2x]2 is
1) r2r 2) (2r – 1)2r 3) r22r+1 4) (2r + 1)2r Solution: (4) (2r + 1)2r \(\begin{array}{l}\begin{array}{l} \begin{array}{l} \text... View Article
The coefficient of x2 in the binomial expansion of [(1 / 3) x1/2 + (x-¼)]10 is
1) 70/243 2) 60/423 3) 50/13 4) None of these Solution: (1) 70/243 \(\begin{array}{l}\begin{array}{l} \left(\frac{1}{3} x^{\frac{1}{2}}+x^{\frac{-1}{4}}\right)^{10} \text { is }... View Article
If in the expansion of (1 + x)21, the coefficients of xr and xr+1 are equal, then r is equal to
1) 9 2) 10 3) 11 4) 12 Solution: (2) 10 Given the coefficients of xr and xr+1are equal, 21Cr... View Article
For all integers n ≥ 1, which of the following is divisible by 9?
1) 8n + 1 2) 4n – 3n – 1 3) 32n + 3n + 1 4) 10n + 1... View Article
The digit at the unit place in the number 192005 + 112005 – 92005 is
1) 2 2) 1 3) 0 4) 8 5) 9 Solution: (2) 1 \(\begin{array}{l}\begin{array}{l} \text { Given } 19^{2005}+11^{2005}-9^{20005} \\... View Article
Coefficient of x19 in the polynomial (x – 1) (x – 2)…(x – 20) is equal to
1) 210 2) –210 3) 20! 4) None of these Solution: (2) –210 \(\begin{array}{l}\begin{array}{l} \text { Coefficient of } x^{19}... View Article
The coefficient of x100 in the expression of ∑j=0200 (1 + x)j is
1) 200C100 2) 201C102 3) 200C101 4) 201C100 Solution: (2) 201C102 \(\begin{array}{l}\text \ Rewrite \ the \ given \ expansion... View Article
If the second term, in the expression [latex][sqrt[13]{a}+frac{a}{sqrt{a^{-1}}}]^{n}[/latex] is 14a5/2, then the value of nC3 / nC2 is
1) 4 2) 3 3) 12 4) 6 Solution: (1) 4 \(\begin{array}{l}\begin{array}{l} \text { Given } T_{1+1}=14 a^{\frac{5}{2}} \\ \text... View Article
If the expression in powers of x of the function 1 / [(1 – ax) (1 – bx)] is a0 + a1x + a2x2 + a3x3 + ….. then the coefficient of xn in the expansion is
1) [an – bn] / (b – a) 2) [an+1 – bn+1] / (a – b) 3) [bn+1 – an+1]... View Article
The expression {x + (x3 – 1)1/2}5 + {x – (x3 – 1)1/2}5 is a polynomial of degree
1) 5 2) 6 3) 7 4) 8 Solution: (3) 7 \(\begin{array}{l}(x+a)^{n}+(x-a)^{n}=2\left[n_{C_{0}} x^{n} a^{0}+n_{C_{2}} x^{n-2} a^{2}+\cdots\right]\\ \left\{x+\left(x^{3}-1\right)^{\frac{1}{2}}\right\}^{5}+\left\{x-\left(x^{3}-1\right)^{\frac{1}{2}}\right\}^{5}\\ =2\left\{5 c_{0} x^{5}+5_{c_{2}}... View Article
For 2 ≤ r ≤ n, nCr + 2 nCr+1 + nCr+2 =
1) n+1Cr-1 2) 2 n+1Cr+1 3) 2 n+2Cr 4) n+2Cr+2 Solution: (4) n+2Cr+2 nCr + nCr+1 + nCr+1 + nCr+2... View Article
In the expansion of (2 – 3x3)20, if the ratio of 10th term to 11th term is 45/22, then x is equal to
1) – (2/3) 2) –3/2 3) (2 / 3)-1/3 4) (3 / 2)-1/3 Solution: (1) – (2/3) \(\begin{array}{l}\begin{array}{l} \text {... View Article
The coefficient of x5 in the expansion of (1 + x2)5 (1 + x)4 is
1) 30 2) 60 3) 40 4) None of the these Solution: (2) 60 \(\begin{array}{l}\text \ Rewrite \ the \... View Article
The larger of 9950 + 10050 and 10150 is
1) 9950 + 10050 2) Both are equal 3) 10150 4) None of these Solution: (3) 10150 \(\begin{array}{l}101^{50}=(100+1)^{50}=100^{50}+50.100^{49}+50.49 / 2.1100^{48}-\ldots... View Article
In the expansion of (1 + x + x2 + x3)6, the coefficient of x14 is
1) 130 2) 120 3) 128 4) 125 5) 115 Solution: (2) 120 \(\begin{array}{l}\begin{array}{l} \left(1+x+x^{2}+x^{3}\right)^{6}=(1+x)^{6}\left(1+x^{2}\right)^{6} \\ =\left(6_{c_{0}}+6_{c_{1}} x+6_{c_{2}} x^{2}+6_{c_{3}} x^{3}... View Article
The ninth term of the expansion (3x – 1 / 2x)8 is
1) 1 / (512 x9) 2) (- 1 / 512 x9) 3) (- 1 / 256x8) 4) (1 / 256... View Article
The coefficient of t24 in the expansion of (1 + t2)12 (1 + t12) (1 + t24) is
1) 12C6 + 2 2) 12C5 3) 12C6 4) 12C7 Solution: (1) 12C6 + 2 \(\begin{array}{l}\begin{array}{c} \left(1+t^{2}\right)^{12}\left(1+t^{12}\right)\left(1+t^{24}\right)=\left(12_{c_{0}}+12_{c_{1}} t^{2}+\right. 12_{c_{2}} t^{4}+12_{c_{3}}... View Article
